Infinity
if there are infinite whole numbers, and there are infinite decimals between 0 and 1, and there are infinite decimals between 0.1 and 0.12, and there are infinite decimals between 0.1111111 and 0.1111112, and (etc.) does that mean that there are infinitely infinite infinitely infinite infinitely infinite infinitely infinite infinitely
(etc.) infinities?
Comments (754)
But here are exact statements that might answer what the poster is wondering about:
Any set of all the real numbers in any non-empty interval is infinite.
Any set of all the real numbers in one non-empty interval is equinumerous with the set of all the real numbers in any other non-empty interval.
There is no greatest cardinality.
For any infinite set of cardinalities, there are greater infinite sets of cardinalities.
Hopefully this doesn't contradict what @TonesInDeepFreeze has said. It seems that if the word "infinity" was being used as a noun, then yes, there would be an infinite number of infinities. However, the word "infinity" is not being used as a noun, but rather is being used as an adjective, in which case there is only one infinity. IE, "infinity" as an adjective means something along the lines "any known set of real numbers can be added to".
Quoting 180 ProofAs @180 Proof said, set theory goes like that. And since you gave in the example of just rational numbers (0,1111111 and 0,1111112,..) then this is equivalent to the infinity of natural numbers, a countable infinity. With real numbers we get into the more interesting questions.
And @TonesInDeepFreeze correctly asked you just what you mean by "infinitely infinite...". We are puzzled what you mean by this. But before you answer that, please read the following:
You see, it comes down to if can you have a way to count, at least theoretically, those infinities themselves, then they can be put into 1-to-1 correspondence with the Natural Numbers. Then it's easy. Just how different the math is, you can see for instance from the example of Hilbert's Hotel.
Here's an easy primer on this short video:
[math]\{0.1,\text{ }0.01,\text{ }0.001,\text{ }...\}\\\{0.2,\text{ }0.02,\text{ }0.002,\text{ }...\}\\\{0.3,\text{ }0.03,\text{ }0.003,\text{ }...\}\\...\\\{0.11,\text{ }0.011,\text{ }0.0011,\text{ }...\}\\\{0.12,\text{ }0.012,\text{ }0.0012,\text{ }...\}\\\{0.13,\text{ }0.013,\text{ }0.0013,\text{ }...\}\\...[/math]
Yes, that follows from the axioms of "standard mathematics". You can build any sort of mathematics (if you wanna call it that) depending on what axioms you choose, the matter is whether it is useful to do so and whether it matches at least something in reality.
There is a retired Australian professor of mathematics called Norman Wildberger, whose project is to build mathematics without mention of infinity within the doctrine of finitism. I would not really recommend looking into it however if you are not deeply knowledgeable in mathematics.
That's actually a philosophical view in mathematics. And thus quite well fits a Philosophy Forum.
But of course you can argue that the most permissive math is simply the one where we start with an axiom of 0=1.
Anything goes. Wee! :razz:
But maybe the mathematicians and scientists do things differently. They don't ask what the concepts mean as long as they are in the textbook. They just accept them, and work on.
"INFINITY definition =1. time or space that has no end: 2. a place that is so far away that it cannot be reached:" - the Cambridge English Dictionary.
It implies that if you know what it is, then you don't know what it is. If you don't know what it is, then you know that you don't know. It is a paradoxical concept, which has to be branded as a contradiction in Philosophy.
And it is a view that I hold, as a "methological physicalist" (as 180 proof puts it), I don't subscribe to abstract objects, so I could not be a platonist about mathematics :razz:
I will likely make a thread about the Grundlagenkrise in the coming weeks.
:gasp:
A: 0 = 1
B: 0 = S(0) (follows from definition)
C: The set of natural numbers only has the element 0 (follows from B)
C contradicts B. Proof of the contradiction? :smirk:
I think you got it a bit wrong. Those who are obsessed about truth or falsity are mathematicians. Even if they sometimes have different axiomatic systems, then it's about right or wrong in that formal system.
It's the Philosophers who are interested about a lot more. Things like morals or aesthetics, which obviously aren't about truth or falsity.
Great! Like to see that one...
But as a non-mathematician, try to keep it as simple and understandable, because the paradoxes are interesting. After all, it's everything to do with infinity.
I think you got it wrong too. Philosophers don't care about the truths and falsity as the answers in the answer sheets. Philosophers are more concerned with the truth and falsity in the concepts, propositions, and logic.
Quoting ssu
Yes, Philosophy used to be the parents of all sciences and mathematics. It is the mother of all subjects, and we cannot deny the fact.
Blimey I was going to make a thread about "Science as a superstition".
? :yikes:
I don't get your point here.
Math and Science pursues the answers in the answer book. You are either right or wrong. Philosophy is more into your arguments and logic for the answers, hence there is no such thing as the answers in the answer book i.e. truth and falsity they pursue are different in nature.
What answer book?
I think mathematics is especially interested in logic. I would dare to say that math is part of logic.
The starting foundations of Science accepts that we cannot find some ultimate truth, hence things are theories, not laws. We can in the find out something new that alters our present views. And mathematicians do understand that especially when you look at the foundations of mathematics, there are philosophical arguments and philosophical schools. Hence you have the philosophy of Mathematics.
Just look at wrote above. Now I don't know if he is a mathematician, but at least he totally understands that philosophy is part of mathematics.
Sure, not denying that at all. They are all parts of each other we could say that. They are all inter-related too. But the methodologies they employ and the ideas of their goals might be different depending on the folks who are doing them.
Quoting ssu
Never said math is not part of philosophy. That is what you are saying for some reason.
I said math and philosophy have different way of doing things.
Have you not read a single math book? If you read any math book, it will have Exercises and Examples after or in the middle of a chapter. The answers for the Exercises will be either at the back of the book, or as a separate Answer Book that you must acquire, if you needed it.
I am not a professional mathematician, but my area does use lots and lots of mathematics inherently, my interest in the foundations of mathematics are coincidental.
I think jgill is one though.
A question for mathematicians: looking at what I've done above, can this be written as a matrix like this?
[math]\begin{bmatrix}0.1 & 0.01 & 0.001 & \cdots \\ 0.2 & 0.02 & 0.002 & \cdots \\ 0.3 & 0.03 & 0.003 & \cdots\\\vdots & \vdots & \vdots & \vdots \\ 0.11 & 0.011 & 0.0011 & \cdots \\ 0.12 & 0.012 & 0.0012 & \cdots \\ 0.13 & 0.013 & 0.0013 & \cdots\\\vdots & \vdots & \vdots & \ddots\end{bmatrix}[/math]
We can then say that [math]a_{m,n} = {m\over10^{n+\lfloor\log_{10}(m)\rfloor}}[/math]?
The series m/(10^(m+log(10,m))) converges to a power of 1/10 as m goes to infinity and n is any given number, and to 0 as n goes to infinity and m is any given number, and the diagonal also converges to 0. The two conditions that converge to 0 seem fine, as we are multiplying by a power of 1/10. But for the condition that converges to powers of 1/10 I am not sure.
So as you go down the column 1, it should converge to 1/10 and in column 3 to 1/1000. Is that what you were looking for? I am not sure if that is what is represented by your matrix.
Disclaimer: Not a mathematician.
But urgently, how do you write matrices and footnotes and equations here?
Umm... that's a school math book. Have you even studied a math course in the University? They are a bit different.
And if you study philosophy, you will similarly (hopefully) be given a exam where you have to answer too.
I assume that true math is more about giving proofs.
https://thephilosophyforum.com/discussion/5224/mathjax-tutorial-typeset-logic-neatly-so-that-people-read-your-posts/p1
Can there be an infinite set of (infinite set of numbers)?
The word "infinite" is not a noun but an adjective qualifying the noun "set".
Therefore, there can be infinite infinities because the word "infinity" is an adjective.
I have a few university Calculus and Algebra and Trigonometry books lying around here, and they are full of questions and answers. Studying math means you read the definitions in the books and work on the questions for the answers purely using your reasonings.
Quoting ssu
No. That is not the case. If you study philosophy for the degree, you must read, and write dissertations which you must defend it at a 'viva voce'.
Where can one see the project?
Wildberger's video on set theory is atrocious, appalling, obnoxious intellectual dishonesty.
jgill.
You make it sound like he's a sleep medicine.
Very nice, that should be pinned imo.
Quoting RussellA
The fact that you can stack a property onto a substance to make an object does not mean that that object is instantiated in real life, especially because many objects are contradictory and cannot exist (rectangular circle or blue orange).
Quoting TonesInDeepFreeze
No clue, I could not find it, I only know that he works on it lol
Quoting Corvus
They certainly do, which is why Im wondering what a thread on mathematics is doing on a philosophy forum.
The philosophy of mathematics is a rich area.
(1) Unfortunately, cranks, who are ignorant and confused about the mathematics post incorrect criticisms of the mathematics, from either a crudely conceived philosophical or a crudely imagined mathematical perspective. That calls for correcting their misinformation about the mathematics itself.
It is great to challenge classical mathematics, but a meanginful challenge needs to not misrepresent that mathematics. Otherwise the effect is inimical to knowledge and understanding of the subject.
(2) And sometimes people post questions about mathematical subjects that have bearing on philosophy, such as about infinities, incompleteness and computability. The debate on realism v nominalism has as one of its major battlegrounds the ontological status of mathematical objects, especially infinitistic ones. And some may think that questions in epistemology are informed by such things as the incompleteness theorem and the unsolvability of the halting problem.
Brouwer v Hilbert itself is one of the very great debates in the history of the philosophy of mathematics, carried on by two mathematicians.
/
Meanwhile, one could also ask what are threads on such things as the U.S. presidential election, Gaza, and candy bars doing in a philosophy website. (Don't get me wrong, I am in no way saying those should not be in this website. Very much I say live and let live.)
I'll believe that he has anything when I see it. Especially, does he purport to offer an axiomatic system? I don't recall, but perhaps he rejects the axiomatic method. That would be fine. But there is no comparing, on one hand, an ostensive treatment of mathematics in which one can leave a lot unexplained, unsupported and without the ultimate objectivity of access to mechanical means of checking proofs with, one the other hand, an axiomatization that submits itself to the constraints and discipline required for that ultimate objectivity.
You have this inverted. These are actually philosophical issues which have a bearing on mathematics. The way that a particular mathematician deals with these issues exposes their philosophical inclinations, or lack thereof.
Works both ways. It would be better if I had said that. The philosophy of mathematics needs for there to be mathematics to philosophize about and developments in mathematics do inform philosophy; and mathematics is liable to being philosophized about by philosophers.
Honestly, I don't know. I only brought up Wildberg because it is the first name that came to mind when it comes to non-standard mathematics. The last time I heard of him was years ago in some science board where discussions about him were common.
In hindsight I should have brought up intuitionistic mathematics but the connection to infinity is not as straightforward.
Yes, he is prominent, therefore natural to refer to him.
The point I was making is that concepts like infinity, incompleteness, and even computability, extend beyond mathematics. So, the mathematical approach is only one approach to such concepts. The philosophical approach, specifically the dialectical approach, is to consider the way that such concepts appear in all the different fields. The way that each field deals with the concepts demonstrates how that field fits, or does not fit, within a consistent whole philosophy. To say that such concepts are the domain of mathematics, therefore mathematicians ought to define them, is to make a statement not consistent with the world we live in.
Of course.
Except incompleteness (in the sense of the incompleteness theorem).
Quoting Metaphysician Undercover
I never said such a thing. Maybe other people have.
That's a specific, restricted definition of "incompleteness". The term is slightly different in physics for example. So this is an example of what I am talking about. Mathematics also uses a specific, restricted definition of "infinite", a meaning exclusive to mathematics, determined by the axioms. The mathematicians designing the axioms tailor the meaning of the term, to suit their purposes.
Not only that, but when we think about almost anything we are quantifying. Does this action bring about more welfare than the other? Is a pantheist god encompassing of the whole universe? Do our beliefs have different percentages of certainty?
Mathematics spans our thoughts just like language, those are perhaps worthier of investigation than metaphysics dependent on the former two.
That is silly. Mathematicians don't claim that the mathematical sense of incompleteness trumps all other senses of that rubric in other fields. Just as a biologist talks about cells in an organism does not begrudge a penologist talking about cells in a prison and vice versa.
Quoting Metaphysician Undercover
Yes, and no fair minded mathematician would thereby begrudge people in other areas of thought from using other senses of the words.
I've said it over and over: Philosophers, scientists, theologists, et. al should be permitted to use terminology as suits them, and to express concepts as suits them. But when someone says the mathematics is wrong for using terminology in its way too, then that is quite unreasonable, and even worse when cranks claim that the mathematics is thereby wrong and worse, premises that claim on horribly misconstruing the mathematics and outright fabricating that makes it says things it actually does not say.
Words have different meanings in different fields of study. The point is that reasonable people allow that. What is unreasonable is when the crank has his own ways of using words and their related notions and then dictates that mathematicians are wrong for their specialized sense not conforming to the crank's own sense.
Yes, for example as in "infinite number" where "infinite" is a property of "number".
There is no such a thing as "infinite" number. See this is an illusion, and source of the confusion.
Infinity is a property of motion or action, nothing to do with numbers. Infinite number means that you keep adding (or counting whatever) what you have been adding (or counting) to the existing number until halted by break signal (as can be demonstrated in computer programming).
A set containing 3 numbers can be made infinite, when it is in the counting Loop 1, 2, 3, 1, 2, 3 .... ? Therefore a term "infinite number" is a misnomer. I bet my bottom dollar that you will never find a number which is infinite, because it doesn't exist. If it did exist, then it is not an infinite number.
Mathematics must have been believing in Philosophy's assistance in clarifying the tricky concepts. :snicker:
I agree. "Infinite" is a property attached to an object, such as "large house" or "infinite number".
As "large" doesn't exist as an object, "infinite" doesn't exist as an object.
As I wrote before: ""infinity" as an adjective means something along the lines "any known set of real numbers can be added to"".
If that is the case, then it seems barmy to talk about different size of the infinite sets.
Extended real number line
"What is the number line to infinity?
For instance the number line has arrows at the end to represent this idea of having no bounds. The symbol used to represent infinity is ?. On the left side of the number line is ?? and on the right side of the number line is ? to describe the boundless behavior of the number line.11 Sept 2021" - Google
Quoting TonesInDeepFreeze
I agree with your points. I just meant that this particular thread doesnt seem to be getting beyond the correcting of wayward mathematical assumptions in order to deal with the philosophy. I might add that even at the level of securing consensus concerning standard mathematics there is likely to be more disagreement than many might expect, perhaps due to the inseparability of philosophical presuppositions and mathematical principles.
'infinity' is not an adjective.
No set has different sizes. But there are infinite sets that have sizes different from one another. That follows from the axioms.
One is free to reject those axioms, but then we may ask, "Then what axioms do you propose instead?"
One is free to reject the axiomatic method itself, but then we may ask, "Then by what means do you propose by which anyone can check with utter objectivity whether a purported mathematical proof is correct?"
One is free to respond that we check by comparing to reality or facts or something like that. But then we may point out, "People may reasonably disagree about such things as what is or is not the case in whatever exactly is meant by 'reality' or in what the facts are, so we cannot be assured utter objectivity that way."
One is free to say that we don't need utter objectivity, but then we may say, "Fair enough. So your desideratum is different from those using the axiomatic method."
'infinity pool' for example. But I'm talking about the context here.
There cannot be different sizes of infinite sets
As you say: "Infinity is a property of motion or action..............Infinite number means that you keep adding (or counting whatever) what you have been adding (or counting) to the existing number"
What does "infinite set" refer to?
It cannot refer to an object, an infinite set, as comprehending an infinite set is beyond the ability of a finite mind. It can only refer to the process of being able to add to an existing set.
In other words, "infinite set" refers to "a set that can be added to", where "that can be added to" qualifies the object "a set".
As a "set" is an object it can have a size, and therefore there can be different sizes of sets.
However, as the qualifier "that can be added to" is not an aspect of the size of the set, whilst the expression "different sizes of sets" is grammatical, the expression "different sizes of infinite sets" is ungrammatical.
What is infinity
On the one hand we have the concept of infinity within the symbol ?, but on the other hand a finite mind cannot comprehend an object of infinite size. So what does our concept of infinity refer to?
As the Wikipedia article Extended Real Number Line notes, the infinities are "treated" as actual numbers, not that the infinities are actual numbers.
As our concept of infinity cannot refer to an object, as comprehending an infinite number of things is beyond the ability of a finite mind, it can only refer to the process of adding to an existing number of things until it is not possible to add any more, which can be comprehended by a finite mind.
IE, "infinity" refers to a process not an object.
Ordinarily, one would take that to be referencing mathematics, as have posters in this thread, not just me.
Actually, in that instance I responded to the poster who has written:
Quoting RussellA
That is a mathematical context.
I didn't say anything about 'equating to math'.
Rather, the context includes mathematics, as also other posters have taken it.
One can do whatever one wants with numbers. That doesn't vitiate that it is reasonable that I and others have commented on the mathematics.
Hear hear.
True, infinite is an adjective and infinity is a noun
But it can get complicated.
Music fills the infinite between the two souls - Rabindranath Tagore
Infinity pencil with eraser - Amazon
Its the reverse.
In recent threads, the notion infinity has been raised with reference to mathematics - in the original posts and in replies. And I have not said that therefore the subject must be contained to mathematics. But the mathematical aspects should not be mangled, so I have commented to correct and articulate points about the mathematics that is being referenced. And still that does not even insist that one may not have an alternative mathematics; rather that if, for example, one claims to disprove that there are sets of higher infinite cardinalities, then the context in which there are higher infinite cardinalities should not be misconstrued or misrepresented. And ordinarily, the context of higher cardinalities would be classical set theory. Or for example, if the context begins with intervals on the real number line, then the ordinary context is classical real analysis and it should not be misconstrued or misrepresented. And, again, if one wants to discuss it in some other context, mathematical or otherwise, then that is fine, but that doesnt preclude that we also discuss it in context of ordinary mathematics.
On the other hand, you post to say that you view it as silly to consider the notion of infinity regarding size when it is not regarding size but rather continuity. Then it is reasonable for one to say, No, this discussion is not silly for talking about size, as indeed size is central to the ordinary context of sets in mathematics as indeed the definition leads right into size rather than continuity.
Yes, continuity is an important topic related to the infinitude of certain sets. But the notion of infinitude applies even where the topic of continuity is not involved. So it is not silly to talk about the sizes of infinite sets.
So I am not insisting that any discussion be confined to mathematics. But I do say that such a discussion may include mathematics, especially when it starts out with reference to notions that are usually regarded as pertaining to mathematics, especially as the notion of greater infinitudes is ordinarily in context of classical mathematics and especially where the original poster mentions it in connection with Russells paradox. But, in stark contrast, meanwhile you are the one who is telling other people that it is silly to talk about the subject in terms of a certain aspect when you incorrectly claim it not concerned with that aspect.
Of course, but I'm saying that in context of sets in mathematics, 'infinity' as a noun invites misunderstanding, especially as it suggests there is an object named 'infinity' that has different sizes.
In fact, this was attempted in the New Math of the 1957- 1970s. It was a disaster. For a variety of reasons. I know, I was there in the classroom.
For me, as a kid, New Math was wonderful. It opened my imagination to different ways of looking at mathematics, not just learning by rote how to do long division and stuff like that. For example, the idea of numbers in binary, modular arithmetic, intersections and unions, truth tables. I think maybe the idea behind it was to get children primed for the upcoming age of computers, such as binary numbers. It blew my mind, I savored it and it served me well.
The problem for curricula is that there is of course no one way to teach, each individual having different needs and backgrounds. What is required is trust in the teacher's ability to recognise and adapt their teaching to the student. But that's contrary to the very notion of a curriculum.
That seems to me to be a trenchant observation.
:brow:
What is one sentence where "infinity" is used as an adjective?
My cousin spent $85,000 on an infinity pool because he thinks that if he swims in it he will live forever.
That is typically called an open compound noun in English. If we are doing a semantic analysis, infinity pool, the 'infinity' qualifies 'pool' implying that it looks endless, so 'infinity' would stay a noun and be in the similative case English does not have grammatical cases morphologically, but semantically that is what it would be. English does not have morphological rules for adjectives (or for any word class I think), so the '-y' ending does not stop 'infinity' from one day becoming an adjective, but it is just not used as such today.
So he spent $85,000 for just a fancy noun. I told him it was not a wise purchase.
In the Merriam Webster dictionary infinity is classed as a noun, and within mathematics is the infinity symbol ?. But as you say, this is problematic as it suggests that infinity is an object, such as a mountain or a table, which can be thought about. But in one sense this is impossible, as it impossible for a finite mind to know something infinite, where infinity is an unknowable Kantian "ding an sich" ("thing in itslef").
So what does the word "infinity" refer to, if not a noun inferring an object?
As the Wikipedia article on Infinity writes: Infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
As Literature as a noun refers to the study of books, perhaps Infinity as a noun refers to the study of infinite sets. Both Literature and Infinity are nouns, but refer to a process, not the intended conclusion of the process. This makes sense, as processes are comprehensible to finite minds. A finite mind can comprehend the process of adding to an existing set, even if not able to comprehend the eventual conclusion of continually adding to an existing set .
IE, "infinity" is a noun and refers to a process rather than any conclusion of that process.
My statements were from my reasoning. But what you claim to be objectivity is from the textbooks. Please bear in mind, the textbooks are also written by someone who have been reasoning on the subject. It is not the bible, to which you have to take every words and sentences as the objectivity that everyone on the earth must follow. That sounds religious.
Mathematics is a narrow scoped subject which borrowed most of its concepts from Philosophy and modified to suit their abstraction to justify their theorems. Hence we find lots of confusions in math and also the math students. Philosophy can clarify some of its modified concepts for the real meaning of them, so they can understand the subject better.
The whole confusion resulted from the wrong premise that infinite numbers do exist. No they don't exist at all. So it is an illusion. From the illusive premises you can draw any conclusions which are also illusive.
Infinity in math has been improvised to explain and describe continuous motion hence the Limit and Integral symbols in Calculus. But they have taken the concept further to apply into the set and number theories. Yes depending on what you accept, you can say the infinite Sets can have different sizes etc. It is OK to keep on saying that in math forums, and it sounds correct because that is what the textbook says.
But when it comes to under the Philosophical analysis, one cannot fail to notice the whole picture was based on the fabricated concepts, which are not very useful or practical in the real world.
:up: :up: :up:
That reminds me of intuitionists or at the very least of psychologists in the ontology of mathematics, where the number 2^100 does not exist until it is thought up.
I do not think I was ever subjected to new math. I simply learned at a very young age not to follow rules without a reason for doing so. I was not interested in the things which mathematics was useful for, so it was dismissed from my curriculum, as soon as possible, until the need was developed. So my education in mathematics was done in an 'as required' way, rather than a force-feeding of conventional 'fact' to memorize, like history.
Quoting ssu
You get the same effect when you take a boat on a reservoir, up toward the dam, the higher the dam the better. It's like empirical proof that the earth is flat, and you're at the edge of the world.
But then again, the number 2 does not exist until it is thought up.
This reminds me of the axiomatic systems that perhaps some animals (or people) have: nothing, 1, 2,3,4, many. When you think of it, it's quite useful for up to a point.
Quoting Metaphysician Undercover
Yes, I've always pondered how few people go to a port or to the seashore and simply look at how large ships simply "sink" into the horizon far earlier than they become tiny specs. But I guess flat Earthers just have this habit of going with the crazy and being against the tyrannical science & math we "sheeple" so blindly accept and follow. It makes them special.
And because the math is extremely hard:
And note it's called a theorem.
I don't say that.
I say that 'infinity', applied to set theory, is not advisable, because in set theory there is no object called 'infinity', especially one that has different cardinalities. It's not a matter of can be thought about, but rather that there are many infinite sets, not just one called 'infinity'.
Quoting RussellA
The lemniscate is usually used to indicate a point of infinity on a number line, which is very different from the context of the cardinalities of infinite sets. Such a point of infinity is some designated (or sometimes, less formally, unspecified) object along with a set, such as the set of real numbers, and an ordering is stipulated. If the treatment is fully set theoretical, then the object itself can be infinite or not.
Quoting RussellA
I am not saying that one should not use 'infinity' as a noun. It is a noun. And people can use it for many things. But it is an invitation to confusion to use 'infinity' regarding set theory or mathematics in a context such as discussing infinite cardinalities. Set theory does not define an object named 'infinity' in this context. Rather, it defines a property 'is infinite'. Keeping that distinction in mind goes a long way to avoiding confusions.
Quoting RussellA
(1) There are better sources than Wikipedia.
(2) The quote does not say that mathematics refers to some set that is named 'infinity'.
(3) The quote then defers to 'infinite', the adjective, which is correct.
(4) The article is almost all about 'infinity' not applied to infinite sets and cardinalities. And the small part of the article that is concerned with infinite sets and cardinalites correctly, when talking about sets, uses 'is infinite', the adjective for the property of being infinite, not 'infinity' to name a set.
Quoting Corvus
Also, more simply, there are infinite sets of numbers, such as the set of naturals, the set of rationals, and the set of reals. With ordinary classical logic, for even just simple first order PA to have a set over which the quantifier ranges requires an infinite set.
Quoting Corvus
Not just because it's what a book says. Rather, textbooks provide proofs of theorems from axioms (including definitional axioms) with inference rules. One doesn't have to accept those axioms and inference rules, but if one is criticizing set theory then it is irresponsible to not recognize that the axioms and inference rules do provide formal proofs of the theorems. Moreover, intellectual responsibility requires not misrepresenting the mathematics as if the mathematics says that the theorems claim simpliciter such things as that there are infinite sets of physical objects or even that there are infinite sets in certain other metaphysical senses of 'infinite'.
Quoting Corvus
Fabricated in the sense of being abstract. And it is patently false that classical infinitistic mathematics is not useful or practical. Reliance on even just ordinary calculus is vast in the science and technology we all depend on.
Ok. So we still have no explanation of how you came to misapprehend "=".
What I said was that it is objective to mechanically check that a purported formal proof is indeed a proof from the stated axioms and rules of inference. If there is anything more objective than verification of application of an algorithm, then I'd like to know what it is.
Of course. And I have many times explicitly said that no one is obligated to accept, like, or work with any given set of axioms and inference rules. But if the axioms and inference rules are recursive, no matter what else they are, then it is objective to check whether a given sequence purported to be a proof sequence is indeed a proof sequence per the cited axioms and rules. If you give me formal (recursive) axioms and rules of your own, and a proof sequence with them, then no matter whether I like your axioms or rules, I would confirm that your proof is indeed a proof from those axioms and rules.
There are areas of great puzzlement and disagreement in the philosophy of mathematics. But I don't know what specific confusions you refer to, specifically in formalized classical mathematics.
I don't think so. And it's clear to me. There are infinite sets that have sizes different from one another. More formally:
There exist x and y such that x is infinite, and y is infinite, and the size of x is not the size of y.
What is the "whole confusion"? Yes, there are people who don't know about set theory and are confused about it so that they make false and/or confused claims about it. But the axioms of set theory don't engender a confusion. They engender philosophical discussion and debate, but there is no confusion as to what is or is not proven in set theory. Whether any given axiom is wrong or not is a fair question, but it doesn't justify people who don't know anything about axiomatic set theory thereby spreading disinformation and their own confusions about it.
Virtually any student is subjected to certain instruction whether they like it or not. It would be fair to say that New Math is not good only if one at least knows what it is.
Quoting Metaphysician Undercover
You have virtually no education or self-education in the mathematics you so obdurately opinionate about.
What's worse, a population of palm trees in a city, or a city in a population of palm trees?
My knowledge in mathematics is quite meager compared with people more dedicated to the study.
Quoting Vaskane
That might be true. Also, the fact that formal logic is not conveyed so that students could see how the mathematics is derived logically rather than simply decreed.
One can always learn vastly more about logic, mathematics, philosophy and the philosophy of mathematics, but my first interest in logic (which led to mathematics) came from my interest in philosophy, and as I learned logic and mathematics, I was learning about the philosophy of mathematics right alongside.
Did you not understand the example I gave you in the other thread? I suggest you go back and read that post you made for me when you fed that example to Chat GPT. It totally agreed with me. It said, in much arithmetic and mathematics "=" signifies equality, not identity. Chat GPT does not lie you know. The simple fact, as my example shows, an "equation" would be completely useless if the left side signified the very same thing as the right side.
:wink:
As I argue, there is much inconsistency in mathematics. The use of "sometimes" here is very telling.
ChatGPT
The use of "=" to signify both equality and identity in different mathematical contexts doesn't necessarily imply an inconsistency in mathematics. Instead, it reflects the flexibility and versatility of mathematical notation to adapt to various branches and subfields within mathematics.
Mathematics is a vast and multifaceted discipline, encompassing diverse areas such as arithmetic, algebra, calculus, geometry, logic, and more. Each of these areas may have its own conventions, definitions, and notational systems tailored to the specific concepts and structures being studied.
While "=" is commonly understood to denote equality in basic arithmetic and algebra, its use to signify identity in formal logic or set theory arises from the need to express relationships between objects or sets in a precise and rigorous manner.
Inconsistencies in mathematics would arise if there were contradictions or logical paradoxes within a particular mathematical system. However, the use of "=" in different contexts doesn't inherently introduce inconsistencies; rather, it reflects the richness and diversity of mathematical language and notation.
That mathematics consists of "objects" with identity is Platonist metaphysics. In this metaphysical theory, mathematical ideas like numbers are objects, rather than quantitative values. Set theory is nothing but Platonist based mathematical theory. Notice that it is "theory", not mathematics in practise.
In the actual application of mathematics, values are assigned, and the left side of an equation must represent something different from the right side, or the equation would be useless, as I explained.
The conclusion we can make is that set theory does not represent mathematics, as mathematics is actually used. That's the problem, We can define terms, or in this case symbols, for theory, in a way which doesn't actually represent how they are used in practise. That's an idealist folly. I think Wittgenstein made a similar point.
Chat GPT got it wrong. As is common.
In mathematics, equality and identity are the same.
Quoting Metaphysician Undercover
Are you serious?
" "=" typically denotes identity, meaning the left side is considered the same as the right side."
Though that is correct, it's worthless coming from Chat GPT, which is not even remotely an authority on mathematics, and famously known to fabricate on all kinds of subjects.
Anyone who thinks Chat GPT doesn't lie and can be relied upon for accurate information is grossly uninformed about Chat GPT.
Lying requires intent, which GPT lacks.
Quoting TonesInDeepFreeze
Here's the example I gave Banno in the other thread. You and I are each one. Together we are two. We can symbolize this as 1+1=2. The two 1's here each represent something different, one represents you, the other I. Because the two each represent something different, the two together as 1+1 can make 2, meaning two distinct things. Also, we can say 1=1. But if the two 1's here both represent the same thing, then 1+1 could not make 2, because we'd still just have two different representations of the very same thing.
Just to be clear, my use of ChatGPT here is purely rhetorical, intended for amusement.
Oh puhleeze! The point is not about the definition of 'lie' but rather that there would not be any point in you saying that it doesn't lie if you didn't mean that it is a reliable source. (The word used most commonly for AI making false statements is 'hallucinating'.) Moreover, lying does not always require intent, as false statement made from negligence, especially repeated negligence may also be considered lies. And that is the case with Chat GPT, as its designers are negligent in allowing it to spew falsehoods. Indeed, the makers of such AI will say themselves that its main purpose is for composition of prose and not always to be relied upon for information.
Hopefully, now it's agreed that Chat GPT is not a reliable source. Indeed, it is worse than not reliable. So your quote of it is worthless.
I'll explain it to you again as I did years ago:
Let T and S be any terms.
T = S
means that what 'T' denotes is the same thing as what 'S' denotes.
That is not vitiated by the fact that aside from denotation there is also sense.
For example:
Mark Twain = Samuel Clemens
means that 'Mark Twain' and 'Samuel Clemens' denote the same person
But the names 'Mark Twain' and 'Samuel Clemens' are different names and have different senses, such as 'Mark Twain' is a pen name and 'Samuel Clemens' is a birth name.
Now, denotation is extensional and sense is intensional. Ordinary mathematics handles only the extensional. So, again:
S = T
means that S and T stand for the same thing, though, of course, S and T may be very different terms.
Got it.
One can get Chat GPT to claim just about anything you want it to claim. I've gotten it to make all kinds of ridiculously false claims. I've even got it to make a claim, then retract that claim, then retract the retraction. Except, no matter how hard I tried, I couldn't get it to say that the earth is flat.
In Philosophy, they don't use axioms and deductive reasonings and proofs as their main methodology. Philosophy can check the axioms, theorems, hypotheses, definitions and even the questions statements for their validity, but the actual proof processes and math knowledge themselves are not the main philosophical interests.
What I meant was that, as Frege, Russell, Wittgenstein and Hilbert had in their minds, that many math axioms, concepts and definitions are not logical or justifiable in real life truths. A good example is the concept of Infinity, and Infinite Sets.
Infinity is not numeric, but a property of motions, operations and actions. But they seem to think it is some solid existence in reality. When they talk about the concepts like infinite sets and claim this or that as if there are self-evident truths for them, it sounds confused.
The textbook axioms and formal proofs of the theorems are subject to change or found out to be falsity at any moment when someone comes up with the newly found axioms and proofs against them. In that case it would be the one who used to think that their claims were the truths, have been actually spreading misrepresentation of the knowledge. No matter what the textbooks say, one must be able to ask Why? instead of just blindly accepting the answers and claim that it is the only truths because the textbooks say so.
Bottom line is that, truth speaks for itself. One doesn't need to say to the others, they are wrong unless when it is absolutely necessary. But just tell the arguments and conclusions, which are true. If in any case of doubt, ask why and how so.
"A careful reader will find that literature of mathematics is glutted with inanities and absurdities which have had their source in the infinite. " - David Hilbert, On the Infinite, pp.184 Philosophy of Mathematics Selected Readings, Edited by H. Putnam and P. Benacerraf 1982
I take the OP as asking the question "are there an infinite number of infinities?"
The answer would depend on whether looked at from set theory or natural language.
Set Theory is a specific field of knowledge with its own rules, and as the Scientific American noted: As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinitiesand some are simply larger than others.
However the terms infinity and infinite sets are also used in everyday language outside of set theory, such as "I have an infinity of problems" and "I have an infinite set of problems".
As the OP doesn't refer to the very specific field of "set theory", having its own particular rules, I think the OP should be considered as a problem of natural language.
Within natural language, the question "are there an infinite number of infinities" is meaningless, as not only is "an infinite number" unknowable, it follows that whether there is one or more infinite numbers must also be unknowable.
On the assumption that the OP refers to a problem in natural language, otherwise it would have specifically referred to "set theory", as it refers to that which is unknowable, although syntactically correct is semantically meaningless.
The issue is not whether or not some mathematicians define "=" as meaning 'is identical to', as a premise for a mathematical theory, or some other purpose, like debate or discussion. We've seen very much evidence here that some actually do this. So there is no question concerning that.
The question is how "=" is actually used in the application of mathematics. And anyone who takes a critical look at an equation in the application of mathematics will see that the right side never signifies the very same thing as the left side. In fact, it's quite obvious that if the right side did signify the same thing as the left, the equation would be completely useless. That is why many philosophers will argue that the law of identity is a useless tautology.
Since this is the case, we can clearly see that those mathematicians who define "=" as meaning 'is identical to' do not properly represent the meaning of "=" with that definition. Therefore we can say that they are wrong with that definition.
Quoting TonesInDeepFreeze
This is not a mathematical equation, so I do not see how it is relevant. You are trying to compare apples with oranges, as if they are the same thing, but the requirement that "Mark Twain = Samuel Clemens" is a representation of a mathematical equation renders your analogy as useless.
Please consider a real mathematical equation as an example, like how the circumference of a circle "is equal to" the diameter times pi, or the square of the hypotenuse of a right triangle "is equal to" the sum of the squares of the two perpendicular sides, for example. Be my guest, pick an equation, any equation, and we'll see if the right side signifies the very same thing as the left side. I think that an intelligent mathematician such as yourself, ought to know better than to argue the ridiculous claim that you have taken up.
The principal problem with set theory, as I indicated in my reply to @Banno above, which is evident from Chat GPT's statement, is that set theory is derived from a faulty Platonist premise, which assumes "mathematical objects". If we recognize as fact, that mathematics does not consist of objects, we must reject the whole enterprise of set theory, along with its fantastic representation of "infinite" and "transfinite", as completely unsound, i.e. based in a false premise.
In a random web site is set a problem that can be solved by set theory:
Doesn't this problem, soluble by set theory, assume "objects", such as the object "a person who can speak English"?
If the number "1" does not refer to an object, what does it refer to?
Quoting Metaphysician Undercover
I would say that "infinite number" does not refer to an object, because unknowable by a finite mind, but does refer to a process along the lines of addition, which is knowable by a finite mind.
The issue is a little more complex than how you represent here, but this is a good indication of why "set theory" is not applicable to mathematics. In your first question, "a person who can speak English" is a description, not an object. It represents a category by which we could sort objects. In the second sentence, the numeral "1" represents a specific concept, which can be described as a quantitative value. It is not a true representation of how we use numbers, to think of a number as itself an object. Set theory may represent a number as an object, but that's the false premise of set theory.
Why call for Grammar in Artistic License's house?
Quoting TonesInDeepFreeze
Depends. Do you like city or palm-trees more?
When first I played with ChatGPT I had it "prove" 999983 is not a prime - it just baldly asserted that it was the product of two integers. Then correct itself. Regretfully, I was using the playground so the record is lost.
They are coming out of the woodwork now.
I like cities as grim and forbidding as can be, thus without palm trees.
It is exactly the point that it is not a mathematical expression, so mathematics is not called on to account for its intensionality. More generally that ordinary mathematics is extensional, and we don't require that it also accommodate intensioncality. That is how it is relevant.
/
Later, hopefully, I'll have time and motivation to dispel a number of misconceptions in a catalog of them you've posted lately.
Those are hard-coded, just like anything revolving sensitive western politics.
I bet if you put the cyber equivalent of a ravenous rat in its face like in '1984' then you could break it. Would say anything, begging like HAL 9000.
What passages from Frege, Russell or Hilbert do you have in mind?
Frege proposed a system to derive mathematics from logic alone. That system was not a set theory per se, but sets can be configured in the system. And Frege did not at all oppose infinite sets. I can be checked on this, but I think it's safe to say that Frege's framework is indeed infitisitic.
Russell showed that Frege's system was inconsistent. Then Whitehead and Russell proposed a different system from Frege's, this time presumably consistent, to derive mathematics from logic alone. But that system is seen to not be purely logic. And Whitehead and Russell explicitly used infinite sets. And I would bet that Whitehead and Russell recognized the applicability of infinitistic mathematics to the sciences.
Hilbert endorsed infinitistic mathematics but hoped there would be a finitistic proof of its consistency. Alas, Godel proved that there can be no finitistic proof even of the consistency of arithmetic, let alone of set theory. In any case, Hilbert distinguished between contentual (basically, finitistic) mathematics and ideal (basically, infinitistic) mathematics, and such that he saw the application of the ideal to the contentual.
/
I hope that later I'll have the time and inclination to catch up to certain misunderstandings and strawmen you've recently posted.
I interpret that as 'mathematics is extensional and that's how intensionality is relevant'. Whatever it is you are trying to say here, it appears to be just as irrelevant as your analogy was.
Quoting TonesInDeepFreeze
I'll be looking forward to that.
Perhaps not axioms as the main approach. And philosophy ranges from poetic through speculative, hypothetical, concrete and formal. But deductive reasoning and demonstration is basic and ubiquitous in large parts of philosophy. And the axiomatic method does appear in certain famous philosophy, and its principles and uses - sometimes even formalized - are prevalent in modern philosophy, philosophy of mathematics and philosophy of language.
Quoting Corvus
The axioms are subject of deep, extensive and lively discussion in the philosophy of mathematics.
/
But when I mentioned objectivity, of course I was not referring to objectivity of philosophy, but rather the objectivity of formal axiomatics, in the very specific sense I mentioned. And that is a philosophical consideration. Then you challenged my claim that mathematics has that objectivity. So I explained to you again the very specific sense I first mentioned. The fact that philosophy in its wide scope is not usually characterized as axiomatic doesn't vitiate my point.
The analogy was not irrelevant. And the key word in what you just said is "appears" but the other crucial words you left out are "to me", as indeed what appears to you is quite unclear with your extreme myopia. And meanwhile I'm still guffawing at your trust in AI chat and your pathetic transparently disingenuous attempt to back out by saying that it's only its lack of intent you had in mind, and even as you are wrong about the definition of the word in question.
Who is "they"? What specific mathematicians do you claim that about? What specific mathematicians do claim have said that the infinite sets of mathematics have solidity as material objects or even like material objects?
Quoting Corvus
Often the axioms are taken to be true, on different bases, sometimes self-evidence, depending on the mathematician or philosopher. But often, at least in the philosophy of mathematics, arguments, not merely self-evidence, are given. Moreover, there is a wide array of approaches where "the axioms are true" would be an oversimplification not claimed without context and explanation by many mathematicians and philosophers. This includes such approaches as structuralism, instrumentalism, fictionalism, consequentialism and formalism. And formalism itself ranges from extreme formalism to Hilbertian formalism, including the view of some mathematicians that the assertion that there are infinite sets is nonsense but that still infinitistic mathematics is useful.
As I said, there are deep, puzzling questions about mathematics, but that doesn't make the mathematics itself, especially as formalized, confusing. On the contrary, if you ever read a treatment of the axiomatic development of mathematics, you may see that it is precise, unambiguous, objective (in the specific sense I mentioned), and with good authors, crisply presented.
Of course, my point went right past you no matter that I explained it clearly.
There are many different and alternative formal axiom systems in mathematics. Mathematicians and philosophers sometimes disagree on which axioms are best, most intuitive, and even true to some concepts. That's a good thing. But the point that went past you is that what is objective, even among them, is that for each one, there is a mechanical procedure to determine whether a purported formal proof is indeed a formal proof allowed from the given set of axioms and inference rules. And, as people may disagree as to what axioms are best or even philosophically or conceptually justified, at least in the formal sense, one doesn't "disprove" an axiom or set of axioms as you seem to imagine (except, of course, by showing that the axioms of a given system are inconsistent with themselves; and by the way, there are at least two famous cases where axiom systems were proven inconsistent - Frege's and one of Quine's, examples that it is not the case that mathematicians follow blindly and uncritically, ).
Quoting Corvus
Again, you are unfamiliar with any of this; you are blindly punching.
We have axioms and rules of inference. Textbooks often do explain the bases for the axioms and rules of inference and do not require blind acceptance. Then, with the axioms and inference rules given, it is objective whether or not a purported proof from those axioms and with those rules is indeed a proof from the axioms with the rules. So that does not require blind acceptance. The process is to state the axioms and rules, often providing intuitive bases for them, then proofs of theorems, as those proofs can be checked. And a good student does check the proofs, both to understand them and to verify for themselves that it is indeed a proof from the axioms with the rules.
But with the inference rules, it's even better. In a mathematical logic, we PROVE that the inference rules are justified in the two key ways: The rules permit only valid deductions and the rules provide for every valid deduction.
On the other hand, blind acceptance is when mathematics is not given axiomatically. The teacher says that a bunch of formulas are correct, to be memorized and performed upon call. But why, the student may ask? Instead, with axioms, the student may ask why, and always an answer is given based on previous formulas that prove the ones in question. And those previous formulas are proven, etc., until we get to the end of the line - the axioms. So, with axiomatics, we can justify everything formally, except the axioms, which are the starting point (not everything can be justified formally without infinite regress or circularity) and are only justified intuitively. Then, one may say, but I don't like or accept those axioms. And the best answer is, "Fine. You don't have to. But at least you can still check that the proofs are permitted from the axioms and rules. And if one wants, one can study an alternative set of axioms. Or even not study any axiomatic system and go one's merry way accepting or not accepting whatever non-axiomatic mathematics one encounters."
Since you are harkening to the original post, see that it is a question about the infinitude of intervals on the real number line, and about the number of different infinite sizes. The ordinary context of that is mathematics and set theory. Anyone is welcome to consider the question in another context, but that doesn't make it inapposite to talk about it in the context of mathematics and set theory.
Quoting RussellA
That's a non sequitur. That the poster didn't mention set theory by name does not imply that set theory would not be a natural context for the matter, especially as the question gave a mathematical context and refers to a concept that is characteristically set theoretic. Moreover, discussion doesn't even have to be limited to whatever unstated context the poster himself might have had in mind.
Quoting RussellA
If that is true, then even more reason why one would then consider the question in regard to mathematics. If it's meaningless in context C but defined in another context D, then it wouldn't make sense to say that then it is inapposite to context D.
That raises the interesting question that if an expression such as "infinite infinities" has no meaning in a natural language, the everyday spoken and written language used to describe the world around us, but does have meaning in the formal language of set theory, then what exactly is the relationship between a formal language such as set theory and the world around us?
You must read them yourself. They all had reservations on the concept of Infinity in math. Quite understandably and rightly so.
Sure as endeavours to be formal and more clear in their system, but is it always making sense? That is another question. Often it tends to make the system look more convoluted, if not done properly.
Quoting TonesInDeepFreeze
Objectivity is the objectivity of knowledge. Not objectivity of philosophy or objectivity of mathematics. That is another misunderstanding of yours. I wouldn't be surprised if you go on claiming an objectivity for set theories and an objectivity for numbers ... It is like saying a subjectivity of objectivity. A contradiction.
Can there be a description without an object being described?
Isn't "a person who can speak English" a description of the object (a person who can speak English)?
===============================================================================
Quoting Metaphysician Undercover
The word "object" has different meanings. In mathematics, a mathematical object is an abstract concept (Wikipedia Mathematical Object). In natural language, it can be something material perceived by the senses (Merriam Webster - Object).
For example, we can think of the number [math]{\sqrt{2}}[/math], the number [math]{2 * 10^{100}}[/math] and [math]{\infty}[/math] as abstract mathematical objects but cannot think of them as natural concrete objects.
However we can think of the numbers 1. 6 and 10 as not only abstract mathematical objects but also as natural concrete objects.
That raises the question as to how we are able to think of something that is abstract, disassociated from any specific instance (Merriam Webster Abstract). For example, independence, beauty, love, anger, Monday, [math]{\infty}[/math], [math]{\sqrt{2}}[/math] and the number 6.
George Lakoff and Mark Johnson in their book Metaphors We Live By propose that we can only understand abstract concepts metaphorically, in that we understand the concept of gravity by thinking about a heavy ball on a rubber sheet.
Thereby, we understand the concept of independence by remembering the feeling of leaving a job we didn't like. We understand the concept of beauty by looking at a Monet painting of water-lilies. We understand the concept of infinity by thinking about continually adding to an existing set of objects. We understand the concept of [math]{\sqrt{2}}[/math] by thinking about the number 1.414 etc etc. We understand the concept of 6 by picturing 6 apples.
IE, we can only understand an abstract concept metaphorically, whereby a word or phrase literally denoting one kind of object or idea is used in place of another to suggest a likeness or analogy between them (Merriam Webster Metaphor).
Axiomatic methodology in math is not free from problems and deficiencies. They are subjective definitions which are often circular in logic. They lack in consistency and are incomplete in most times.
They are dependent on the other axioms mostly. Most of them are abstract and illusional which renders to the false conclusions. A typical example is the Infinity in Set theory.
Of course, that's known as fiction.
Quoting RussellA
This evades me. How do you think of a number as a natural concrete object? Are you talking about the numeral, or the group of objects which the numeral is used to designate, or what?
Quoting RussellA
So why would we label an abstract concept an "object", as in "mathematical object", and speak of it as if it had an identity in the same way that a natural concrete object has an identity? If we only know abstract concepts through analogy, or suggestions of likeness, isn't it completely wrong to suggest that anything which only exists in this way, i.e. through metaphor, could have an "identity"?
This is the problem which @TonesInDeepFreeze is stuck on. Tones seems to think that just because one can show how "=" can be used to to show a relationship of identity between two distinct names for the same natural concrete object (]Mark Twain = Samuel Clemens), we can conclude that when "=" is used in mathematics, it's being used in that same way.
But of course in mathematics this is not true. There is no such natural concrete object which the symbols refer to, in theory. only abstract concepts. Natural concrete objects are only referred to through application. And in application the concrete situation referred to by the right side of the equation is never the same as the concrete situation referred to by the left side. So all that Tones has indicated is that there is two very different ways to use "=", the mathematical way, and the way which signifies a relation between two different names for the same natural concrete object. Therefore, we must be careful not to confuse the two different ways, or equivocate between them, because that would be misleading.
That was an accurate description of the problems of the mathers. Not blindly punching anything at all.
The other shortcomings of math is it cannot accurately reflect the real world and its problems. It often distorts it via the unfounded and unjustified concepts and axioms, hence arriving at nonsense.
Mind you when math started in ancient Egypt, it used to be for mainly the practical problem solving purposes e.g. counting the sheeps, cows, and apples in the markets, and finding out the boundaries and locations for the pyramid locations in the deserts.
It used to work well, but once math started running the blind free rein of modifying the abstract concepts and keep deducing the illusional theories, things started going wrong turning the empirical and pragmatic skills in origin into some sort of an abstract subject which sometimes speaks in the tone of deeply frozen religion. Not cool at all.
"A mythical animal typically represented as a horse with a single straight horn projecting from its forehead" describes an object, even through the object is fictional.
In fact, from my position of Neutral Monism, all objects, whether house, London, mountain, government, the Eiffel Tower, unicorn or Sherlock Holmes are fictional, in that no object is able to exist outside the mind and independently of the mind.
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Quoting Metaphysician Undercover
The problem is, how does the mind understand an abstract concept, such as beauty, [math]{\sqrt{2}}[/math], ngoe, or the number 6?
My belief is that the mind cannot understand an abstract concept in isolation from concrete instantiations of it, in that, if I am learning a new word, such as "ngoe", it would be impossible to learn its meaning in isolation from concrete instantiations of it.
IE, I see no possibility of learning an abstract concept, such as "ngoe" or the number "6" without first being shown concrete examples of it.
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Quoting Metaphysician Undercover
If I wanted to teach you the meaning of the symbol "ngoe", which I know is a concept, how is it possible for you to learn its meaning without your first being shown particular concrete instantiations of it?
===============================================================================
Quoting Metaphysician Undercover
Given 1 and 1, if the second use of 1 refers to the same thing as the first use of 1, then the proper equation should be 1 = 1. The symbol "=" means identity
Given 1 and 1, if the second use of 1 refers to a different thing as the first use of 1, then the proper equation should be 1 + 1 = 2. The symbol "=" means equality.
Continuing:
Given a horse's body and a horse's head, as a horse's body is different to a horse's head, the proper equation should be horse's body + horse's head = horse
This raises the question as whether a horse as a whole is more than the sum of its parts, a horse's body and a horse's head.
Has the whole emerged from its parts, or is the whole no more than the sum of its parts?
IE, by knowing the parts, can I of necessity know the whole?
Referring back to Kant, by knowing the parts, for example, the number 5 and the number 7, can I of necessity know the number 12?
Niet, artistic freedom is fine, but "In passing I had caught a glimpse of the infinity beauty deep within her eyes" does not have "infinity" as an adjective, because the word can't be used as such, the sentence in fact comes across as gibberish. A good way to distinguish adjectives from nouns is putting them in an is-clause.
A: The world is infintiy.
B: The world is infinite.
If the predicate is giving a property to the subject, it is an adjective (B); if it is equating the predicate and the subjective, it is a noun (A).
Quoting Vaskane
I would rather say that English was born in the late 11th century, with word order being the king that dictates meaning, but word order does not differentiate a noun from an adjective, since compound open nouns look syntactically identical to an adjective+noun, simply a word next to the other without any affixes, declination, or conjuctions.
Grammar itself includes word order, also known as syntax, you may be referring to morphology as "grammar", which is how the Greeks use it, and it is their words, so point to you.
I have no proof for my claims, but I remember using character.ai in early 2023 and I was concerned because that thing was smarter, more engaging, and more polite than the average person, and I spent days talking to the different characters despite there being people around me not very well-mannered of me I reminisce. But I strongly feel that the AI there was downgraded and made dumber on purpose; I feel like free ChatGPT was also limited at around that time, I have not tried ChatGPT 4 yet.
Ironic, I was just using ChatGPT and it triple messed up:
https://chat.openai.com/share/96378835-0a94-43ce-a25b-f05e5646ec40
Fixed would be things like pi, e, i, ?2, ?3.....
Defined would be things like variables, parameters, objects by unrestricted definition.
Infinity should always be regarded as a defined object and never a fixed object.
Wrong assumption.
Quoting Vaskane
Fortunately my high school was not somewhere where this is taught as poetry:
So I will do fine without the "self-expression" that amounts to the same sophistication as caveman paintings.
That is indeed a doozy. Yet AI Chat is only somewhat less misinformational than Wikipedia.
The way it is done in ordinary formal mathematics is that there are open terms and closed terms.
Open terms have free variables. Closed terms have no free variables.
A constant is a closed term.
There are primitive closed terms and defined closed terms.
Before defining a constant, we must first prove there is a unique x such that x that satisfies a formula whose only free variable is x.
There are also primitive predicate symbols and defined predicate symbols.
An n-placed predicate symbol is defined by a formula having at most n free variables.
In set theory, there is no constant nicknamed 'infinity' (not talking about points of infinity on the extended real line and such here). Rather, there is the predicate nicknamed 'is infinite'. However we do define constants for certain infinite sets, such as [read 'w' here as if it were the Greek letter omega]:
x = w iff for alll y, y is a member of x iff y is a natural number. (The formula that is satisfied by one and only one x is "for all y, y is a member of x iff y is a natural number".)
I would definitely disagree with this. There is a big difference between seeing, hearing, touching, or otherwise sensing an "object", thereby describing what i sensed, and creating an imaginary "object". The latter does not involve an object, nor does it involve a "description" ( in the proper sense of the word) because it is an imaginary creation an invention rather than a description. A "fictional object" is not an object, that's actually what "fictional" means. OED #1 definition of object "a material thing that can be seen or touched". "Fictional", on the other hand, means exactly the opposite, invented by the imagination, therefore not able to be seen or touched.
Quoting RussellA
I suggest that your "position" is not consistent with common understanding. It might benefit you to give up on the monism.
Quoting RussellA
There is no such thing as a "concrete instantiation" of a concept. Concepts are categorically different from concrete objects. To take your example, show me where I can find a concrete instantiation of beauty, 6, or the square root of 2. It is one thing to assert that there is a concrete instantiation of a six out there somewhere, but quite another thing to prove this. And if it is true, it ought to be easy to prove. Just point out this 6 to me, so i can go see it with my own eyes, or otherwise sense it.
Quoting RussellA
This seems to be completely inconsistent with what you've already argued. You've already made the claim that you can make a fictious description, so why couldn't you also define a concept, thereby providing the means for someone else to understand it, without showing a concrete instance of that type of thing? I mean, you presented me with "a horse with a single straight horn projecting from its forehead", and i understand this image without seeing a concrete instantiation, so why take the opposite position now, and say that a person cannot understand the meaning of a concept without being shown a concrete instantiation of it.?
Quoting RussellA
If this is the case, then what you have shown is logical inconsistency in the use of "1". In the first case, the two instances of 1 must refer to the very same thing, and in the second case, the two 1's must refer to two different things. If we simply say "=" means equality, then there is consistency between your two examples. Furthermore, there is no practical advantage to designating "=" as meaning identical in the case of "1=1", so you're just proposing logical inconsistency for no reason. That is simply illogical, therefore not a fair representation of the logic of mathematics.
A fictional object sounds like an object.
The OED notes "There are 14 meanings listed in OED's entry for the noun object, four of which are labelled obsolete" and then says "purchase a subscription".
The Merriam Webster includes "something mental or physical toward which thought, feeling, or action is directed".
I think Cervantes would have had great difficulty in writing "Don Quixote" without being able to describe objects.
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Quoting Metaphysician Undercover
I agree that if one stopped one hundred people at random in the street, only a few would know about philosophical Monism.
Examples of modern philosophers who were monists include Baruch Spinoza, Georg Wilhelm Friedrich Hegel, Arthur Schopenhauer, and Bertrand Russell (https://study.com)
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Quoting Metaphysician Undercover
How can one learn a concept in the absence of a concrete instantiation of it?
For example, as a test, suppose you thought of a concept. In practice, how can you teach me its meaning without using concrete instantiations of it?
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Quoting Metaphysician Undercover
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Quoting Metaphysician Undercover
Exactly, you understand the concept using images.
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Quoting Metaphysician Undercover
There are two different cases.
The first a case of identity where the two 1's refer to the same thing. The second a case of equality where the two 1's refer to different things.
The practical advantage of using identity rather than equality is to distinguish two very different cases.
One reason that comes to my mind is that we haven't gotten much applied use for Aleph-2, for Aleph-3, or Aleph-4 etc. Usually correct math has a lot of applications. Physics and engineering and science uses it all the time. Cantor doesn't help it with then speaking of an Absolute Infinity. Now people dismiss this as irrelevant and just being Cantor's religious ideas (that God is Absolute Infinity), yet I don't think so. He simply didn't understand it and didn't get that kind of relevation as he did with noticing that the cardinality of the natural numbers isn't the same as with real numbers.
Yet once you assume that there actually would be theorem for Absolute Infinity that we haven't discover, then that theorem has to clear Russel's Paradox, the 'set of all sets', or the sometimes called Cantor's Paradox or Burali-Forti Paradox. When we have a paradox, obviously our reasoning about the premises aren't correct. Because mathematics is logical.
I think the problem is in counting itself and giving a proof in Mathematics. Mathematics has started from a need to count, not from let's create a logical system and call it math. Hence humans have made discoveries in math: that there are irrational numbers. That there are many types of geometries. Hence we can come up with new ways to think about math.
I'm not sure if this is correct, so I'll ask here: is counting basically a way to give a proof? Because let's assume that we have true but unprovable entities in Math (or with counting, uncountable numbers). What would you get if you would try to prove and unprovable entity in Math?
I guess you would get a paradox, because you cannot prove the unprovable or count the uncountable. The paradoxical nature is quite obvious. And the problem won't go away even with Cantor's hierarchial system.
Perhaps the problem is that people have tried to solve the paradoxes yet still hold on to their premises, as if everything is already there in the foundations of math and paradoxes can be kept away by restrictions. Like ZF.
I'm not saying that I know the answer, but saying that there might be here something for us to still discover.
The key point here, is that imagination does not require sensation of whatever it is that is imagined. But you claim understanding a concept requires "concrete instantiation" and I assume that means something which is sensed. If I'm wrong here, and "concrete instantiation" means the production of an image in the mind, without the requirement of sensing it, then we might have something to discuss.Quoting RussellA
Yeah sure, you've indicated that in the first case "=" signifies identity. I agree, that's what you've stipulated. The point though, is that in the case where you used "=" to signify identity, it is not a mathematical usage. The "practical advantage" you refer to is rhetorical only, intended to persuade me. The usage is not mathematical, because in the application of mathematics "=" is not used to represent identity. That's the issue with Tones' example of Twain = Clemens, it just demonstrates that it's possible to use "=" in this way. However, it doesn't at all represent how people applying mathematics actually use "=" in the formulation of equations. So it's nothing but a rhetorical example, produced solely for the purpose of trying to persuade, in the mode of sophistry.
The issue is with the premises (axioms) of set theory. If you are "mathematical antirealist" you ought to reject set theory on the basis of the axioms it employs. Your view on "infinity" would be irrelevant at this point. So how set theory treats "infinite" would not even enter into the reasons for your rejection of it.
Set theory is based in the assumption of "mathematical objects". And, the mathematical objects as elements of the sets, are allowed to have relations which physical objects, according to our knowledge of them, cannot have. So a "set" by set theory is a bunch of "objects", but since they are mathematical objects instead of physical objects, what can be 'truthfully' said (what is acceptable by the axioms) about that bunch of objects, doesn't have to be consistent with our knowledge of physical objects. So for example, there can be an empty set (a bunch of objects with no objects), and sets do not necessarily have an order (a bunch of objects without any order to them).
Why?
Set theory begins with the assumption of mathematical objects, hence it is based in Platonic realism.
You don't need to believe in Platonic realism to use set theory. Its axioms are just rules to follow when "doing" maths. There's no need to think of them as statements that correspond to some mind-independent fact about the world.
In some set theories, sets can contain themselves. In others, they can't. As a mathematical antirealist I wouldn't claim that one of them must be "wrong". They're just following different rules.
I agree. I didn't say you need to believe in the truth of the principles you employ. However, it's hypocrisy to say "I'm a mathematical antirealist" and then go ahead and use set theory. But that sort of hypocrisy is extremely commonplace in our world, it's actually become the norm now. Very few people make the effort to understand the metaphysics which they claim to believe in, and whether it is consistent with the metaphysics which supports the theories which they employ in practise.
No it's not.
And how would you justify that claim?
I already did above. The axioms of some given set theory are just rules that you must follow when using that set theory. Different set theories have different axioms and so different rules. Given that there's no connection between using some set theory and believing in the mind-independent existence of abstract mathematical objects, there's no hypocrisy in using some set theory and being a mathematical antirealist.
Your position is like arguing that it's hypocritical to play chess if I do not believe that the rules of chess correspond to some mind-independent fact about the world.
"Mathematical antirealist" is also a "rule". It states an ontological principle, or rule. It is a rejection of mathematical objects. The rules of set theory are inconsistent with this rejection of mathematical objects, because set theory assumes mathematical objects, as a foundational premise. Therefore you must assume mathematical objects, as a fundamental premise, to be able to follow the rules of set theory. This activity is contrary to the ontological belief stated as "mathematical antirealist", and is therefore hypocrisy for anyone claiming to be a mathematical antirealist.
Quoting Michael
I don't see the relevance. You do not need to accept the premise of "mathematical objects" to play chess. You do need to accept the premise of "mathematical objects" to follow the rules of set theory.
I imagine a unicorn by picturing a unicorn.
How do you imagine a unicorn if you don't picture a unicorn?
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Quoting Metaphysician Undercover
"1 = 1" is a mathematical expression. The expressions "Twain = Clemens" and "sugar = bad" are not mathematical expressions.
Similarly, the word "infinity" has one meaning in a formal set theory and a different meaning in everyday natural language
From Frege's "Context Principle", the meaning of "=" and "infinity" depend on their contexts.
You need to accept the premise of queens and kings and pawns to play chess, but accepting this premise doesn't commit you to "chess realism".
You need to accept the premise of a murderer and a victim when playing Cluedo, but accepting this premise doesn't commit you to "Cluedo realism".
And so accepting the premise of mathematical objects when using set theory doesn't commit you to mathematical realism.
When using set theory, mathematical objects "exist" only in the sense that queens "exist" in chess and a murderer "exists" in Cluedo, i.e. not in any realist sense.
Quoting Michael
Quoting Metaphysician Undercover
Is it hypocritical for a mathematical formalist to use set theory? I think the differences between the philosophical schools in mathematics don't actually matter so much because the differences are in the realm of metaphysics. If for a Platonist the abstract mathematical objects exist and for the formalist it's just basically something compared to an eloquent game, what's the actual difference?
The way I see it the difference between anti-realists and realists (Platonists of some sort?) is things like if mathematical truths are discovered or invented. It doesn't change the math!
The math in set theory is mainly about injections, surjections and a bijection, which we mark usually with "=".
Problem with Set Theory is that their concept "infinite" means "finite". It breaks the most fundamental principle of Truth.
Similar, but I wouldn't call it "picturing". Anyway, the point is that this "picturing" does not require a "concrete instantiation", which I assume implies a physical object being sensed.
Quoting RussellA
When you say "=" signifies identity, "1=1" is not a mathematical expression. Think about it. If "1=1" means that the quantitative value signified by the first "1" is equivalent to the quantitative value signified by the second, then this is a mathematical expression. And in the case of ordinals, if "1" signifies "first", and the expression means that the first is equivalent with the first, as first, then this is also a mathematical expression. But if "1=1" is meant to signify that the thing identified by "1" on the right side is the very same as the thing identified by the "1" on the left side, then it is not a mathematical expression. It is an expression of identity. And, the fact that it is analogous with Twain = Clemens, which is clearly not a mathematical expression is evidence that it is not a mathematical expression.
I could address your examples, but I do not see how they are relevant really. In set theory it is stated that the elements of a set are objects, and "mathematical realism" is concerned with whether or not the things said to be "objects" in set theory are, or are not, objects.
To play chess you must accept the reality of the pieces as objects in order to move them, therefore you must accept "chess reality" to play chess. Since it may not be stated in the rules that the pieces are "objects" the acceptance is only implicit, unlike set theory in which case the rule is explicit, therefore acceptance is explicit.
It seems to me that you do not understand "realism". Do you agree, that to be able to take "an object", manipulate it, move it, do whatever you please with it, or move it according to some set of rules, you need to accept that the object which you are doing this with is "real"? And this implies that believing the things which you are manipulating to be "objects", implies some sort of realism. Or, do you separate "realism" from "objects", so that realism has nothing to do with objects? In which case, what would you base "realism", and consequently "antirealism" in?
You can play chess without a physical board and physical pieces. You can play it with pen and paper if you like; much like we do with maths. Or, if you're very smart, you can play it in your head; again, much like we do with maths.
When playing chess in your head you're not committed to being a realist about the queen you're playing with. When using set theory you're not committed to being a realist about the mathematical objects you're using.
You just follow the rules.
I can only imagine a unicorn by picturing a unicorn. A picture requires a "concrete instantiation". A "concrete instantiation" can be on a screen or a piece of paper. Both a screen and a piece of paper are physical objects existing in the world. As physical objects in the world, I can sense them.
You can only imagine a unicorn if you know what a unicorn is. How can you know what a unicorn is without having first seen several "concrete instantiations" of it as physical objects in the world?
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Quoting Metaphysician Undercover
Identity is a valid part of mathematics.
Wikipedia -Identity (Mathematics)
You put the pebbles in rows, from left to right. I'll use B and W to represent the pebbles, but it's nicest to play with natural concrete instantiated objects. There are two rules.
Rule 1. You can make a row by putting two pebbles down like this:
Rule 2. If you have made a row, or some rows, of pebbles, you can join them altogether into one long row, and then put an extra B at the beginning and an extra W at the end.
Let's see some patterns we can make. Using rule 1 we have
We could use rule 1 again.
This is boring. Let's try rule 2. We could make
or
If we took
we could make
If we took
we could make
It is possible to interpret these rows of pebbles as multisets. It is possible to interpret some rows as sets. It is possible to interpret some rows as natural numbers. It is possible to interpret the sequence
as counting. It's a pretty cumbersome way of counting. It would be easier to ignore the colours of the pebbles, and just count the pebbles, and interpret the counts as numbers. It is possible to ignore all these interpretations, and just play the game.
For a mathematical antirealist, does any of this constitute hypocrisy?
(@Metaphysician Undercover mostly.)
What's worse than people trying to do physics without the mathematics?
Apparently, people will also try to do mathematics without the mathematics.
Pointing out their errors simply makes them double down. Sometimes all you can do is laugh and walk away.
Quoting Corvus
A common definition of 'infinite' in mathematics is 'not finite' You have it completely wrong. Would that you would not persist in posting falsehoods.
Ugh.
Me neither. But I try to reply to the posts addressed to me.
Which math textbook says "infinite" means "not finite"?
No one was doing math here. This is philosophy forum, not math. We have been just pointing out that misuse of concepts and definitions, and using them as the premises in their arguments can mislead people with the wrong answers and absurd conclusions.
Quoting Banno
You claim that you care about philosophy, but don't appear to be doing so. What you seem to be doing here is just codon blindly whoever is on your side whether right or wrong, and laugh and walk away from truths.
:nerd: Be honest to yourself, and try to be your own man. :cool:
This is clearly incorrect. We can imagine things without a concrete instantiation. That's how artists create original works, they transfer what has been created by the mind, to the canvas. It is also what happens in dreams, things never before seen are created by the mind.
Quoting GrahamJ
I can't see the relevance. Your game clearly involves real objects, pebbles, or in the case of your presentation, the letters. Would the antirealist insist that these are not real objects?
Quoting Banno
Those are the people who say "=" signifies identity in mathematics. They claim to be doing mathematics when they say that "1=1" means that what left 1 signifies is the same as what the right 1 signifies. But that's obviously not mathematics. In mathematics, the left side of the equation always signifies something different from the right side, or else the equation would be useless.
It's one thing for non-mathematicians, who don't know any better, to think that what they are doing is mathematics, when it's not. But it's truly shameful when mathematicians claim to be doing mathematics when what they are doing is not mathematical. As I explained already, that's how they come up with false axioms.
The symbol "=" is defined in ZFC by saying that "A = B" is true if and only if A is B.
They could have used the symbol "#" instead, but they decided on "=".
Yes, and as I've shown over and over again, that definition of "=" is not representative of how "=" is actually used in mathematics. Therefore it is a false definition, designed for some other purpose, foreign to mathematics.
You're putting the cart before the horse. It's not that we use maths and then retroactively describe what the symbols mean and infer the axioms; it's that we define what the symbols mean, prescribe the axioms, and then use them.
So when the issue is set theory, isn't then more correct just to talk about a bijection?
Or is that problematic too?
You have this wrong. A study of the history of mathematics will reveal to you that the axioms come about as a representation of usage. We could start with something like "the right angle", and see that the Egyptians were using that concept to create parallel lines and things like that, far before the axiom, the Pythagorean theorem, which represents this usage, was expressed.
As I recently explained in a related thread, since axioms are determined by choice, and used by choice, we must accept that axioms follow usage, they do not determine usage. People can produce whatever axioms they like, but if they are not useful they will not be used, nor become conventional. So, the axioms which become the convention are the ones best representative of what mathematicans are actually doing.
In the case of the axiom of extensionality, it is useful for a purpose other than mathematics. It's use is rhetorical, to persuade people of the usefulness of set theory. It is clearly not true though, because, for example, the order of the elements within a set is not accounted for. So, sets which are said to be identical may have the same elements in a different order. But in any true sense of "identity" order is an essential feature. Therefore the rhetorical use of this axiom is really a matter of deception.
I don't see any issue with bijection in principle. But when it is proposed that the quantity of a specific set is infinite, bijection would be impossible. The proposal of infinite sets presents numerous procedural problems. That is self-evident.
Yes, that's precisely right, and is why your talk of axioms being "false" is nonsense. Axioms aren't truth-apt; they're just either useful for their purpose or not. And given that the axioms of ZFC are the most prominently used, it stands to reason that they are considered to be the most useful.
That's all there is to say about them.
Regarding the "=" sign, it was invented in 1557 by Robert Recorde:
Isn't there a bijection between the set of natural numbers and the set of natural numbers?
If so,
Isn't there also a bijection between the set of natural numbers and the set of rational numbers also? And a bijection between the set of natural numbers and the set of algebraic numbers? This is the reason why we have the "Hilbert Hotel" example and actually, the axiom of infinity in ZF as it's written.
I think there is as it's the way that set theoretic books describe it and the way I've learned Cantorian set theory.
That there isn't a bijection the set of natural numbers and the set of real numbers is basically why there is all the fuzz about aleph-0 and aleph 1. And here we get to the Continuum Hypothesis already.
Hence infinity is actually very puzzling to us. Still.
Was that early or also late Wittgenstein? Because I suspect late Wittgenstein wouldn't have read any metaphysics into set theory. It's just a useful language game we play, not something that entails the realist existence of abstract mathematical objects.
Big problem with consistency when the use of "=" is not consistent.
Quoting Michael
In the sense that axioms are a representation of what mathematicians are doing, they can be judged as true or false, just like any other description. However, as you rightly describe, a judgement of the truth or falsity of an axiom is not required to judge whether it appears to be useful or not.
So this is where self-deception enters the environment. If a mathematician accepts an axiom because it is useful, but it is not representative of what that individual is doing mathematically (and this I argue is the case with the axiom which makes the claim about the relation between identity and equality), then the usefulness of that axiom must be in relation to something other than mathematics. It has some other purpose than a mathematical purpose.
Quoting Michael
Notice "two things". Equality deals with two things, identity only involves one thing.
Quoting ssu
That's a bijection which cannot be carried out, cannot be completed. It's a nonsensical proposition.
Quoting Michael
Notice early Wittgenstein talking about representing the world in terms of "elements". Notice later Wittgenstein rejecting this as not representative of what is really the case in the world.
In the context of maths, when we say that A = B we are saying that the value of A is equal to the value of B. The value of A is equal to the value of B if and only if A and B have the same value.
A non-identical but equal value makes no sense.
Were not saying that the symbol A is identical to the symbol B. This is where I think you are misunderstanding.
I find that hard to believe. How is it possible to imagine a unicorn by not picturing a unicorn? How is it possible to imagine the number 6 without picturing six things. How is it possible to imagine beauty without picturing something beautiful?
It is true that when I imagine a unicorn I could picture the word "unicorn", but this is still a concrete instantiation.
When you imagine a unicorn, if you are neither picturing a unicorn nor the word "unicorn", what exactly are you imagining? What exactly is your "Intentionality" directed at?
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Quoting Metaphysician Undercover
Problematic. If an artist, no matter whether Monet or Michaelangelo, could create something that hadn't existed before, this would be the same problem as to how something can come from nothing, the same problem as to how there be an effect without a cause.
An artist may reorganise existing parts, a blue line, a tree, a sky or a yellow mark, into a new whole, such as a painting of Water-lilies. An artist may change the relationship between parts that already exist, but the artists cannot create the parts out of nothing.
In fact, your challenge would be to find an artwork which included a part that did not already exist in some previous artwork.
The artist finds new relationships between existing parts. They don't create the parts.
The artist imagines new relationships between "concrete instantiations" of existing parts. The artist pictures new relationships between existing parts, and if successful, then applies them to the canvas.
In natural language, we know the meaning of the word "infinity", yet infinity is unknowable by the finite mind, meaning that "infinity" must refer to something knowable.
Something very large is knowable, such as the number of grains of sand on a beach, the number of water molecules in a glass of water or the number of people living in a city.
We are able to understand "Infinity" as like "the number of grains of sand on a beach".
Therefore, our understanding of "infinity" is not literal but rather as a figure of speech, specifically, a simile.
If "infinity" means "like the number of grains of sand on a beach", then the expression "an infinity of infinities" becomes an "like the number of grains of sand on a beach" of "like the number of grains of sand on a beach". This is ungrammatical
IE, in natural language, the term "infinty of infinites" is ungrammatical.
Nonsensical?
Seems you confusing ideas about set theory. Or ignorance about the subject.
Set theory is part of mathematics, but of course you can be with your idea that it's "nonsensical".
Sorry, but I did assume this thread was about mathematics.
You said, "Problem with Set Theory is that their concept "infinite" means "finite""
What set theory textbook, or any reference in set theory or mathematics, says that 'infinite' means 'finite'?
Meanwhile, many textbooks in mathematics, including set theory, analysis, algebra, topology, computability, probability and discrete mathematics give the definition of 'infinite' as 'not finite'. What is your purpose in asking if you're not thinking of reading one of them?
Two pairs of definitions:
x is finite iff x is one-to-one with a natural number
and
x is infinite iff x is not finite
x is Dedekind finite iff x is not one-to-one with a proper subset of x
and
x is Dedekind infinite iff x is not Dedekind finite
But with the axiom of choice (as with the most common set theory in mathematics, which is ZFC) we have:
x is finite iff x is Dedekind finite
thus
x is infinite iff x is Dedekind infinite.
More specifically:
Without the axiom of choice, we have:
If x is finite then x is Dedekind finite
thus
If x is Dedekind infinite then x is infinite
With the axiom of choice, we have both:
If x is finite then x is Dedekind finite
thus
If x is Dedekind infinite then x is infinite
and
If x is Dedekind finite then x is finite
thus
If x is infinite then x is Dedekind infinite
/
Meanwhile you are posting flat our disinformation when you post ""Problem with Set Theory is that their concept "infinite" means "finite"".
And you'll not show any textbook or article or lecture notes in set theory or mathematics that say 'infinite' means 'finite'.
Indeed, even in plain language, the prefix 'in' with 'infinite' is taken in the sense of 'not'. x is infinite if and only if x is not finite.
You need to stop posting confusions and disinformation.
I have already quoted from Wittgenstein from his writings "infinite" in math means "finite", and he adds that the mathematicians discussions will end. It is obvious you have not read the post.
But my point is not about "infinite" is "finite" or whatever. My point was that the concept "infinite" means something totally different, and math's infinity in set theory doesn't exist. This is not what other folks says, or may some folks did, I don't know. But that is just my idea. I don't need any supporting comments on that from anyone, when I think that is the case.
But you are quoting from the old and outdated mathematician Dedekind on the concept of "infinity", and it means "not finite". To me it just sounds vacuous word game to say infinity is not finite, but "not finite". It is a concept which doesn't exist in reality. It is an abstract concept for describing motions, actions and operations.
Anyway, Dedekind's set theory had faults and limitations. Here is what ChatGPT says about his Set Theory and concept of Infinity.
"Dedekind's set theory, while foundational and influential, does have some limitations and criticisms. Here are a few:
Axiomatic Foundation: Dedekind's set theory lacks a formal axiomatic foundation comparable to other set theories like Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Without a clear set of axioms, Dedekind's set theory may be seen as less rigorous or formal by contemporary standards.
Treatment of Infinity: While Dedekind made significant contributions to the understanding of infinity, his treatment of infinity in set theory may be considered less systematic compared to later developments, such as Cantor's work on transfinite numbers and ZFC set theory. Some critics argue that Dedekind's definition of infinite sets as those that can be put into one-to-one correspondence with proper subsets of themselves is not as precise or comprehensive as later formulations.
Lack of Explicit Axioms: Dedekind's set theory does not provide a set of explicit axioms like those found in ZFC set theory. This lack of a formal axiomatization can make it difficult to establish the foundational principles of Dedekind's theory and to reason rigorously about sets within this framework.
Scope and Development: Dedekind's set theory was developed in the late 19th century and may be seen as lacking some of the conceptual developments and formalizations that occurred in later set theories. While his work laid important groundwork for the development of modern set theory, it may not encompass the full range of concepts and techniques found in more contemporary approaches." -ChatGPT
I would have expected your reply to my question from the reputable and well known modern math textbooks which says "infinite" is "not finite", as you have been insisting as the case. But it doesn't matter. To me, infinity is an abstract concept which has no entity, and shouldn't be used for naming the set elements or sets. It doesn't reflect the reality accurately, and is a vacuous concept. Infinity only makes sense when it is describing motions, actions or operations. Or it can be used in the poetry or metaphor as a figure of speech. That is fine.
I am not claiming anything on the math theory. I am just pointing out the contradictions and false information in your posts, and replying to them. It would be a gross distortion of the fact and over exaggeration to state anything more than that about my replies.
Earlier you said (for example):
and
By a 'mathematical antirealist' I meant someone who thinks maths is invented, not discovered. Or someone who thinks that your "objects" in set theory only exist in our minds, or as pebbles or ink or pixels, etc.
The whole of number theory or set theory can be reduced to a game with pebbles like the one I described. More colours of pebbles, more rules, but just rows of pebbles and precisely defined ways of rearranging them. It is thus possible to do number theory or set theory without mentioning numbers, or sets, or any other mathematical objects, or using a natural language at all. Tricky, but possible.
You can interpret some patterns of pebbles as objects of various sorts, but treat them as mental crutches, vague hand-wavy ideas, expressed in natural language with all its confusions and ambiguities, which can guide your intuition. Or you can believe they really exist somewhere. Either way, I don't see any hypocrisy.
I get the feeling you have no experience working with formal systems, and have no real understanding of metamathematics. I can't explain your inability to see the the relevance of my game otherwise.
Robert was the first known usage in a printed work, but he did not invent it. The symbol was used in Italy before Robert.
Looks a bit like he has 1+1=2 mixed up with somethign like "1+1" ="2"?
Quoting Corvus
I can't find anything of the sort in this thread. You quoted him, in another thread, as saying
Quoting Corvus
Which is very far from what you attribute to him here.
But you will double down, again.
Too many threads on infinity. You found it OK. Anyway, it wasn't far.
Tone was in the thread, and he would have seen it.
Anyway, Cantor and Dedekind wouldn't have opposed to infinity in set theory, because they made them up. It was Frege, Russell, Quine who had reservations on it even if didn't oppose to it. Wittgenstein sounds he was against it.
You misunderstood. It meant that Wittgenstein said that mathematician's infinite means finite in his writings. See the quote above.
Then you said, infinite is not finite, but "not finite". I asked for the textbook definition for infinite in math. Again, my point on it is that, infinity is an abstract concept which has no referent object.
Now go back to this:
Quoting TonesInDeepFreeze
Quoting Corvus
No, Tones took up what you said, asking you to justify it. You are in error, both in claiming "Problem with Set Theory is that their concept "infinite" means "finite" and in attributing anything like that to Wittgenstein.
This is your modus operandi.
Quoting Corvus
Wasn't he saying clearly mathematician's infinite are finite?
Yup, that was my interpretation of Wittgenstein. What is your ground for saying it error?
Describe "infinity" in clear and actual way in understandable language, and I will tell you about your modus operandi.
What? No.
Quoting Corvus
My ground involves reading what Wittgenstein says: "mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end." He is not saying that infinity is finite, but that the discussions of mathematicians are finite.
As I said above, you will double down. You will also seek to obfuscate and change topic. But here, your error is clear. The subject of the quote is not the infinite, but mathematician's discussions of the infinite.
Edit: here it is, posted while I was writing the above - the attemtp to change topic:Quoting Corvus
So which discussion is not finite in that case? Does any discussion under the sun go on forever? It doesn't make sense.
Are you possibly suggesting Wittgenstein would have meant that obvious cliche in his writings?
You are descending into incoherence. No discussion is not finite. A double negative that you deserve. Yes, no discussion goes on forever.
With the possible exception of attempting to have you admit an error.
This part is your usual modus operandi, which is ad hominem and straw man.
:rofl:
I have shown that you misattributed a remark to Wittgenstein. Cheers.
So it is evident your interpretation on W. was wrong.
You haven't even explained what "infinity" means. W. would have said, there is no meaning on which things that cannot be described in words.
How can anyone admit error when the other party is pushing his wrong ideas with the misinterpretation of Wittgenstein, and inability to explain fully what the world "infinity" means, when asked?
How can one admit error when he is not in error but the other party is?
Here is what you quoted:
"Let us not forget: mathematician's discussions of the infinite are clearly finite discussions. By which I mean, they come to an end."
This clearly does not support your contention:
'I have already quoted from Wittgenstein from his writings "infinite" in math means "finite"'
You are flailing about.
What Wittgenstein must have meant was the concept of infinity in mathematics. It was a contentious topic at the time. He didn't agree with it. That is the way I understood him on the point. It was just reflecting my point very nicely for the definition of infinity. I am not trying to change your views or ideas. Just telling you about it because you wanted the argument.
it's not a question of interpretation. It's clear that the subject of "mathematician's discussions of the infinite are clearly finite discussions" is mathematician's discussions of the infinite, and not the infinite. Bolding, to display the distinction.
That clearly does not support your contention that Wittgenstein said mathematicians take the infinite to be finite.
No one, not I, not Wittgenstein, and not, apparently, your good self, is suggesting that mathematical discussions are not finite. Now I do not know if this is an issue of comprehension on your part, or a another attempt at using rhetoric to change the topic. The first point here is that you misrepresented Wittgenstein. The second point here is that you refuse to acknowledge your error. The third point is that this is an approach you have repeated in this thread and elsewhere. And not only you, but various others, many of them having contributed to this thread, adopt a similar lack of accountability.
But now I am kicking the pup. Enough, perhaps.
Not only your reading on Wittgenstein is wrong, but also you seem to be misunderstanding many things in philosophy. It is not just this thread, but also in many other threads you seem to be claiming things from your misunderstandings and misrepresentation of the facts. Therefore you seem to be going around the circles on the points not getting clear to the point with no depth and no accuracy in many occasions.
Plus you seem to be tending to take sides of the posters regardless of right or wrong of the points, but who you think your cliques are. It is visible many times, and hard to miss it.
I would be disappointed with Wittgenstein if what he meant in the quote was truly "mathematician's discussions are finite, and they all end." to mean the discussions as per se, as you keep on insisting.
But I know your insistence comes from your misunderstanding of Wittgenstein, and what he meant was the concept of infinite in mathematics is actually "finite", henceforth his usual aphoristic claim, "their discussion will end."
How?
Here it is again:
Quoting Banno
Set your understanding out, or retract.
(1) Please link to where you quoted Wittgenstein writing that 'infinite' in mathematics means 'finite'.
(2) Wittgenstein doesn't speak for mathematics anyway. Whatever Wittgenstein wrote, it wouldn't change that fact that mathematics does not define 'infinite' as 'finite', which would be utterly ridiculous, as mathematics defines 'infinite' as 'not finite'.
Quoting Corvus
Whatever your point is, what you claimed that in mathematics, 'infinite' means 'finite', which is a wildly ridiculous claim and blatant disinformation.
Quoting Corvus
That's a different claim from the claim that, in mathematics, 'infinite' means 'finite'.
Quoting Corvus
You are hopeless as far as rational discussion.
You asked me what textbooks in mathematics define 'infinite' as 'not finite'. The answer to that question is that just about every textbook in mathematics that gives a mathematical definition of 'infinite' gives the definiens as 'not finite', or sometimes the Dedekind definition that is equivalent with 'not finite' in mathematics. The formal mathematical definition 'not finite' goes back to Tarski and the definition 'one-to-one with a proper subset of itself' goes back to Dedekind; but that in no way vitiates that still, the current definition is 'not finite' or its Dedekind equivalent. You challenged me as to what mathematics textbooks say; my answer to that is not vitiated by the fact that the definition is long standing in mathematics. What is wrong with you?
Moreover, obviously, in even just an ordinary context, 'the 'in' in 'infinite' is a negation, so it's 'not finite' and not 'finite'. Again, what is wrong with you?
And now you are pasting Chat GPT (!) quotes that you don't even understand. You can't learn set theory from Chat GPT! What is wrong with you?
Quoting Corvus
You are such an intellectual incompetent.
I am not talking about Dedekind's theory. I'm talking about a particular definition. And that definition is used is equivalent with 'not finite' in the ZFC that you just mentioned. Indeed, any informal theory is less rigorous than formalized ZFC. So what? That doesn't change the fact that Dedekind's formulations cannot or have not been formalized subsequent to his own writings.
Again, since you missed this:
Definition of 'infinite' in mathematics:
x is finite iff x is one-to-one with a natural number.
x is infinite iff x is not finite (by the way, that is sometimes called 'Tarski's definition)
Another definition of 'infinite' in mathematics:
x is infinite iff x is one-to-one with a proper subset of x (Dedekind's definition)
Those are provably equivalent in set theory with the axiom of choice (such as ZFC). Without the axiom of choice, we can only prove: If x is one-to-one with a proper subset of itself then x is not one-to-one with a natural number.
In any case, with both those definitions, it is blatant that 'x is infinite' is defined as 'x is finite'.
Moreover, now you are taking recourse to the notion of formalization, when just a few posts ago you were trying to dispute me when I mentioned a key advantage of formalization! What is wrong with you?
Quoting Corvus
Cantor was more systematic about sets, but Cantor also was not a formal theory and had problems that needed to be rigorously resolved by formal set theory.
Anyway, this has no bearing on the fact that the set theoretical definition of 'infinite' is not ridiculously, as you claim, 'finite', nor on the fact that both Tarski's and Dedekind's definition obtain in current mathematics.
Quoting Corvus
Ask Chat GPT who it thinks those critics are and to quote them.
Dedekind's definition incorporated into axiomatic set theory is absolutely precise. And in ZFC it is exactly as comprehensive as Tarski's definition, since in ZFC they are equivalent, as I stated that equivalence explicitly in my previous post.
You don't know jack aboutany of this. You just want to be right about disdain for set theory, so you're willing to enter any specious and counterfactual argument you can come up with, including inapposite quotes from.. Chat GPT (!).
Quoting Corvus
Again, that does not vitiate that nevertheless his work has been formalized subsequent to his own writings and that include his definition of 'is infinite'.
You are foolishly quoting Chat GPT without even a basis to understand the quotes, their context or their import or lack thereof for our conversation.
Quoting Corvus
So what? Contemporary mathematics still uses his definition of 'is infinite' as it is equivalent with 'not one-to-one with a natural number', i.e. 'not finite', in ZFC.
Quoting Corvus
The textbooks I can cite you are not just reputable, but they are among the most standard, most used, and most referenced textbooks in current use.
I asked you why you want me to name one if you are not interested in looking at it. Indeed, not interested in looking at any of the many I can cite.
On my desk right now, I have a stack of modern textbooks, some of them regarded as quintessential references, in various mathematical subjects, as they all define 'infinite' as 'not finite'. What do I get if I list their titles and authors for you? You're not going to look them up. So what's the point? Or, how about this: I'll list them all, then you can admit that that you don't know what you're talking about when you say that mathematics defines 'infinite' as 'finite' but rather that mathematics defines 'infinite' as 'not finite'.
Quoting Corvus
You're lying in our face. You claimed that in math 'infinite' means 'finite'.
Quoting Corvus
You have shown no contradiction or false information in my posts. Rather, you have shown that you are ignorant of the subject, confused about the subject, disinformational in things you've said about the subject, specious in your arguments about the subject, willing to resort to ridiculously intellectually incompetent posturing by quoting Chat GPT (!) that you don't understand and as the quotes don't even approach impeaching anything I've said, and flat out lying when you say that you didn't make a claim about mathematics when plainly you did.
Quoting Corvus
Nope. If anything it's understatement to say what I've said about your intellectual incompetence and dishonesty.
So what? It doesn't say that mathematics takes 'infinite' to mean 'finite'. And even if it did (which it does not), it doesn't represent mathematics or mathematicians, since they very certainly do NOT take 'infinite' to mean 'finite'.
You don't seem to know anything about Wittgenstein anyway from your posts. Wittgenstein's whole philosophy is about mathematics and grammar. He was also a student of Russell too, and both were deeply into mathematics and logic.
You are the one who is intellectually incompetent and ignorant, because you just keep on writing disinformation in your posts without even checking it. Just Google Wittgenstein on Math, and Wittgenstein on Infinity. It will list the whole loads of academic articles on the topic. And I am quoting one of them here.
"Abstract
The aim of this paper is to give an overview of Wittgensteins conception of the infinite. One focus of the paper is Wittgensteins rejection of what is dubbed a realist model of our idea of the infinite. On this model our idea is the source of beliefs that we have about an independent reality. Another focus is the way in which Wittgensteins rejection of this model leads him to reject the idea of the infinite itself as it appears in certain mathematical contexts. I argue that these two rejections can be uncoupled: abandonment of the realist model of our idea of the infinite is consonant with full endorsement of the use to which mathematicians put the idea. There remains scope for Wittgenstein to take issue, if not with the use to which mathematicians put the idea, then with their choice of language in doing so, something that he has reason to do precisely because this choice encourages adoption of the realist model." - Wittgenstein and Infinity, by Andrew W. Moore
"Rejection of Different Infinite Cardinalities: Given the non-existence of infinite mathematical extensions, Wittgenstein rejects the standard interpretation of Cantor's diagonal proof as a proof of infinite sets of greater and lesser cardinalities." - SEP, Wittgenstein's Philosophy of Mathematics
Whether one tries to uncouple the idea or not, it was clear that W. had rejected the concept of infinity. Infinite in mathematics means "finite". Hence their discussion will end.
It was just to let you know it was what Wittgenstein was saying, and he was a great philosopher of language, logic and mathematics.
You said that mathematics regards 'infinite' to mean 'not finite'. You didn't say anything about Wittgenstein there. If by saying that mathematics takes 'infinite' to mean 'not finite' you actually mean something different, such as that Wittgenstein notes that mathematical discussions are finite, then you need to write that and not that mathematics takes 'infinite' to mean 'not finite' and not to then blame readers for your error.
Moreover, I don't opine on what Wittgenstein meant in that quote of him, but at least, at face value, saying that discussions are finite is not the same as saying that mathematicians mean 'finite' when they write 'infinite'.
Please do some searches and reading on Wittgenstein's infinity.
I thought when I said that you would know whom I was referring to.
And I gave it to you! In detail. With clear, exact explanation.
Again, if I list for you the titles and authors of the many textbooks that are currently standard, even quintessential references in the subject, will you finally admit that you are incorrect in the ridiculous claim that in mathematics 'infinite' means 'finite?
I addressed that. You SKIPPED it.
If you have something to say specific about those mathematicians/philsophers, then please say what it is.
If he's saying that there, then he's definitely not "clearly" saying it. I don't claim to know what he is driving at. But at least at face value, the sentence does not read to be saying that in mathematics 'infinite' is taken to mean 'finite'. But you're free to provide more context and analysis to justify your interpretation.
What? Are you trolling?
Banno didn't say that discussions are not finite. He is saying that "discussions are finite" doesn't mean that mathematics takes 'infinite' to mean finite'.
It was me who addressed at the very first, which was ignored.
I addressed it again. I still have the details of the reasons somewhere.
Quoting TonesInDeepFreeze
I don't. I have been just responding to your posts making my points.
That might be the case. That might be part of Wittgenstein's argument against the notion of infinity. I don't know. But even if it is, it still is not saying, at least at face value, that mathematics regards 'infinite' to mean 'finite'.
No time for that. You just call anyone trolling if you haven't understood something?
Quoting TonesInDeepFreeze
He seemed to be saying discussions are finite, and all discussions end. What he seems to be saying was that it has nothing to do with mathematics infinity. I didn't agree with that. I will read him again. Are you speaking for him too?
To me, it was clear that Wittgenstein meant infinite in mathematics means finite, hence mathematician's discussions will end. - He denies the concept of infinity in mathematics.
Banno said, it is nothing to do with the infinity in mathematics, but mathematician's discussions will end, like all discussions. I thought that was nonsense.
Ive addressed your post and comments directly.
More misrepresentation. Pathetic.
I haven't made any claims about him, other than that, at least at face value, "discussions are finite" does not mean that mathematics regards 'infinite' as meaning 'finite'.
Quoting Corvus
So what?
As I said, Wittgenstein does not speak for mathematics. Mathematics speaks for itself when it defines 'infinite' as 'not finite' and not, ridiculously 'finite'.
Quoting Corvus
You've not shown any disinformation in what I posted.
Quoting Corvus
That doesn't entail that in mathematics 'infinite' means 'finite'. What in all creation is wrong with you?
Quoting Corvus
You are claiming again that in mathematics 'infinite' means 'finite'.
You have a bizarre idea that because Wittgenstein was critical of the notion of infinity in mathematics that therefore mathematics takes 'infinite' to mean 'finite'. Amazing.
Your claim was out of point from the start, because you see the discussion in the quote as discussion in talking. It is the concept of infinity in Mathematics he was meaning, which doesn't exist, hence not speakable and is meaningless. If you are still hanging on that "discussion" and make song and dance about it, you are not in the game.
You sounded as if Wittgenstein was irrelevant in math. That sounded not intelligent or read in philosophy.
Quoting TonesInDeepFreeze
You keep misunderstanding which was the part of the main problem here. It was said by Wittgenstein, and I just used his sayings to support my own point.
You better ask Wittgenstein what he meant by that. I have my own point. What with you?
First, you said that mathematics takes 'infinite' to mean 'finite'. You didn't say anything about Wittgenstein there. Then, you said that you expected me to infer that you meant Wittgenstein, though there was no hint even about him there. Then, you mentioned a post in another thread where you quoted Wittgenstein commenting that discussions about infinity are finite. While, it may be that he meant that as part of his argument against the notion of infinity, at least at face value, it is not an assertion that mathematics regards 'infinite' to mean 'finite'. Finally, even if that was what he meant, he does not speak for mathematics, which speaks for itself when it defined 'infinite' as 'not finite, and not ludicrously as 'finite'.
And then your quoting of Chat GPT as part of your bizarrely specious attempt to dispute my explanation of how mathematics actually does define 'infinite'.
And your really foolish dispute against the fact that modern, current, authoritative, standard and widely referenced textbooks in the main areas of mathematics define 'infinite' as 'not finite'.
And that's all just recent posting by you, not mentioning all the other garbage you've posted in this thread and at least another.
You don't seem to even know who said what, and what was whose points, and just get into ad hominem all the time. Would you say your postings are high standard? Read them yourself. They are full of disrespects to the others. You don't even know what Wittgenstein was up to. If you thought he had little to do with math, then it tells you where you are in the discussions.
You are amazing!
You said that mathematics takes 'infinite' to mean 'finite'. Then you said that actually you meant that Wittgenstein said that. Then when you are offered that you can provide any context or explanation to support that, you say that I should I should ask Wittgenstein rather than indeed, you providing an argument that when Wittgenstein said that discussions about infinity are finite, he implied that mathematics takes 'infinite' to mean 'finite'. And then you totally reversed to say as much as that whatever Wittgenstein's point is, you have your own point. So which of these things that you've already said not is that you mean?:
(1) Mathematics regards 'infinite' as meaning 'finite'.
(2) Wittgenstein implies that mathematics regards 'infinite' as meaning 'finite' and you agree with that.
(3) Maybe Wittgenstein did not imply that mathematics regards 'infinite as meaning 'finite', but that you do claim that mathematics regards 'infinite' as meaning 'finite'.
And it's not been at issue that Wittgenstein was critical of the notion of infinity in mathematics.
Quoting Corvus
It's difficult to make anything sensible from this. The point I am making is simple, you misrepresented Wittgenstein's view. He is saying that mathematical discussions are finite, not that infinity is finite - an absurdity that seems peculiar to you.
Quoting TonesInDeepFreeze
"Incorrigible" would be more accurate.
You have not shown that I've failed to know what has been said.
Quoting Corvus
When you say "just", you're lying.
I post extensive arguments that are not ad hominem, and give extensive information and explanation that is not ad hominem.
Anyway, I do comment on the ignorance, confusion and dishonesty of cranks, but not as an ad hominem argument in the sense of something like "you are [fill in the personal remarks here] therefore your argument is not sound." Rather, I have given you fulsome information, explanation, counter-arguments and refutations, no matter what else I have to say about your ignorance, confusion and dishonesty. Pointing out that you are a crank is merely lagniappe to my substantive comments.
Quoting Corvus
They're never good enough for me, for a number of reasons. But they do provide a lot of information, explanation, and cogent arguments, and sometimes have other virtues too. Posting is hard, because it's impromptu and usually under the constraint of being time diverted from other things one wants and needs to do. So, that considered, all and all, I think I've written some really good posts, while others are just okay, but in just about all cases, I wish that I had time to make them a lot better still. And too many typos in them.
Quoting Corvus
You don't know what I know about Wittgenstein, including what I've forgotten and would need to refresh myself on. And it's aside the point anyway, as my arguments are not about Wittgenstein, not even to try to dissuade you from sharing his views about mathematics. Whatever the case about Wittgenstein, it is not the case that mathematics regards 'infinite' as meaning 'finite', for example.
You're lying about me. (Though you weasel with 'if'.)
I never said that Wittgenstein has little to do with mathematics.
It's overwhelmingly the case that Wittgenstein is one of the very most central philosophers in the subject of logic and mathematics.
That I say he does not speak for mathematics in the sense that mathematics speaks for itself in its definitions, I am not at all saying that it is not relevant to mention what he says about mathematics. He has things to say about mathematics, but he cannot be fairly regarded as speaking for mathematics, especially as the fact that he was critical of the notion of infinity in mathematics does not imply that mathematics regards 'infinite' to mean 'finite'.
Amazing in the forms of incorrigibility.
I don't speak for Banno, but I have said that there is no set named with the noun 'infinity', but rather there is the adjective 'is infinite' defined:
x is infinite iff x is not finite
Even if the other party were in error (which is not the case here anyway), if you are also in error, then you could admit it.
Actually, it seems you can't.
You compound your errors now by claiming that you've not been in error, when its overwhelmingly clear that you have been, and in so many ways.
Of course, we are not talking about the symbols, we are talking about what the symbols represent. In your example, "A" represents something, and "B" represents something. The issue is, what "=" represents
Quoting Michael
Right, A=B means that the value of A is equal to the value of B. This does not mean that A is identical to B, so the "=" signifies a relationship of equality, it does not signify a relationship of identity.
Quoting Michael
How could this be true? Two dollar bills are non-identical, but equal value. There is however, a very special relationship, which a thing has with itself, expressed by the law of identity (a thing is the same as itself), which is known as the identity relation.
Would you agree with me, that every identity relation (the relation a thing has with itself as expressed by the law of identity) is also an equality relation, such that a thing is equal to itself, but not every equality relation is an identity relation? In fact, in the vast majority of cases when things are said to have equal value (like two different dollar bills) they are two distinct things, and it is very rare, because it is rather useless, for a person to say that a thing is equal with itself.
We can skip right to the matter at hand, if you're prepared. Let's propose your example in slightly different terms, unambiguous terms which might better expose the issue. instead of saying "the value of A", and "the value of B", let's simply say that "A" represents "a value", and "B" represents "a value". Then when we say "A=B" we might claim that A and B both represent the same identical value.
But this creates a procedural problem in practice. Let's take the example "1+1=2". The value represented by "1+1" would be exactly the same, identical, to the value represented by "2". The problem is that "1+1"contains the representation of an operation, and "2" does not. And in order that an operation can fulfill what is intended by the operator, the operation must have a very special type of value. Because it is necessary to recognize this special type of value, that signified by the operator, it is impossible that "1+1" signifies the exact same value as "2", because there is no operation represented by "2". In other words the value represented by "1+1" consists of an operation, and the value represented by "2" does not, therefore they are not representations of the exact same value.
Quoting GrahamJ
The issue is a complex one, but here's the simple explanation. If a numeral such as "2" signifies an object, then every time that symbol is used it must refer to the exact same object. However, if a "mathematical antirealist" believes that math is invented and these concepts exist only in human minds, then one must accept that the conception of "2" varies depending on the circumstance, or use. This is very evident from the multitude of different number systems. So for example, when a person uses, "2" it might refer to a group two things, or it might refer to the second in a series, or order. These are two very distinct conceptions referred to by "2". So, since "2" has at least two referents, it cannot refer to a single object. We could however propose a third referent, an object named "2", but what would be the point in that? The object would be something completely distinct from normal usage of the symbol.
Pure nonsense from the pair. You two have been degrading the whole discussions into a comedy.
You speak for Banno, and now trying to speak for me?
It seems obvious your whole purpose of coming into the forum is forcing people to admit errors when the error is on your side.
You do. But of course you won't admit it.
The concept of infinity is for description of motions, actions and operations.
The use of infinity in the set theory is ambiguity.
I explicitly said I do not speak for Banno.
You say that in mathematics 'infinite' means 'finite', but 'infinite' means 'not finite'. Then I say that I do not speak for Banno and you say that I do. I think the problem might be that you don't know what the word 'not' means.
Quoting Corvus
I haven't presumed to speak for you.
Meanwhile, you've put words in my mouth, and failed to recognize that when I caught you doing it.
Quoting Corvus
That's a stupid thing to say.
I haven't presumed to speak for Banno.
You're lying again.
Your sayings and actions are totally different. You don't even know what you have been saying, but denying it. That is truly incorrigible.
You're lying again. I committed no action that constitutes speaking for Banno.
The value represented by the symbol "A" is identical to the value represented by the symbol "B".
Quoting Metaphysician Undercover
They are of identical value.
Quoting Metaphysician Undercover
Given that 1 + 1 = 3 - 1, the value given by the procedure "add 1 to 1" is identical to the value given by the procedure "subtract 1 from 3" that value being 2.
It's not the case that there are two equal but non-identical values of 2.
It sounds like you are a little string controlled doll in Banno's pocket.
Stop distorting the facts, and be your own man and honest to yourself.
First you say I speak for Banno, then you say that Banno controls me. But if Banno controls me, and I speak for him, then I speak for him at his control, so then it should be just fine for me to speak for him. (Though I don't speak for him and he doesn't control me.)
I am only replying to your posts, the way they are. But you two Laurel and Hardy are not worth the time. All the best.
You've not shown that I've distorted any fact. Meanwhile, you've been distorting all over the place, as I have shown.
Whatever that might mean in your own mind.
Quoting Corvus
As long as I can be Laurel. Stan Laurel is a great hero of mine. Right up there with Buster Keaton.
Most of your own posts are filled with distortions. See that's what I meant. You don't recall you have been writing in your own posts.
You argue by mere assertion.
Anyway, you said this in not worth your time and signed off with "All the best", yet you're still going at it.
Ok whatever. Have a good day. cheers.
A = B
A is B.
The value named by 'A' is the value named by 'B'.
A is equal to B.
The value named by 'A' is equal to the value named by 'B'.
A is identical to B.
The value named by 'A' is identical to the value named by 'B'.
Seven ways of saying the same thing.
But you will never bring the crank to understand that.
No that is clearly not the case, because these two procedures are completely different. They are said to result in the same value, 2, but the operations represented do not have the same value, nor are they identical.
Look at the two operations claimed to have an equal value. One is to take two distinct individuals and unite them producing a group of two. The other is to take a group of three and remove one individual, producing a group of two. Surely you cannot believe that these two procedures could have the same value. For example, if you had one dollar and someone gave you a dollar, that would be a far more valuable operation than if you had three dollars and someone took one dollar from you, even though they both result in you having two dollars.
And it is not the case that I equivocate with "value" here, because as I explained in the last post, the reality is that operators signify a different type of value from numerals. And, we must account for this if we are to assert that the value represented on the left side of the equation is identical to the value represented on the right side.
What we can see is that the conclusion of these two different operations results in the same value, 2. But it is clear that we do not have that "same value" unless we come to the correct conclusions in carrying out the procedures. So we have two very different operations each concluding with the same value as one another. The value, which is the same for both, is assigned to the conclusion, not the operation itself. But the operations are what is signified on the right and left sides.
If we assert that the two operations "1+1", and "3-1", each themselves have the same value, we neglect the very important fact that having the same value is really dependent on correctly carrying out the operations which are signified. Therefore "the same value" is attributed to the two conclusions, not to the two operations, themselves.
I propose that what you present here is a very sloppy analysis of what an equation actually is. The operation presented on the right side does not inherently have the same value as the operation presented on the left side, as you propose. What is really the case is that correctly carrying out the two operations, to their respective conclusions, produces the same value. I say it is very sloppy because it neglects the essential aspect of applied mathematics, which is to produce conclusions.
This sloppiness appears to be endemic to the philosophy of mathematics, and is very relevant to the issue of "infinite". The very meaning of "infinite" implies that there can be no conclusion to the operation. But the tendency in the philosophy of mathematics is to ignore the need for the human task of carrying out the operation (the consequence of Platonism which removes the requirement of human conception, I would argue), as you demonstrate with your example. So we find this mistake commonly with examples such as what @ssu suggested a bijection between the natural numbers. Obviously, by the conception of "the natural numbers", that they are infinite, it is impossible to conclude such an operation. Therefore it is impossible that there is such a bijection, or that it could produce a quantitative value.
Operations don't have a value. Operations return a value. The value returned by the operation of adding 1 to 1 is identical to the value returned by the operation of subtracting 1 from 3.
We can go with that position if you want. It is irrelevant to the rest of the post, which demonstrates that "the value" of the right side, and of the left side is only produced by carrying out the procedure to its correct conclusion.
Yes, and the values returned by both sides are identical.
You're conflating an extensional and intensional reading. To hopefully make the distinction clear, consider the below:
1. The President of the United States is identical to the husband of Jill Biden.
Under an intensional reading (1) is false because being the President of the United States isn't identical to being the husband of Jill Biden.
Under an extensional reading (1) is true because the person referred to by the term "the President of the United States" is the person referred to by the term "the husband of Jill Biden".
The intensional reading of "1 + 1" is the operation, the extensional reading is the value returned by that operation. Under that extensional reading, 1 + 1 = 3 - 1 where the "=" symbol is used to mean "is identical to".
Sorry Michael, I cannot follow you. You've strayed from mathematics, just like Tones did with the example of Twain=Clemens. Your example, like Tones' appears to be completely irrelevant. To me, you've changed the subject and I cannot follow the terms of the change. If you want to continue this course, please demonstrate how it is relevant to mathematics. However, in the meantime I ask that you consider the following
Quoting Michael
Because of the issue with Platonism, It is not even proper to designate these values, the one produced by the right side, and the one produced by the left side, as "identical". Identity is what is assigned to an object, by the law of identity, "a thing is the same as itself". Notice it is a thing which is the same as itself, "identical".
When we recognize that the value produced by carrying out the procedure on the right side is "equal" to the value produced by carrying out the procedure on the left side, we implicitly acknowledge with the use of "value", that this is something within the mind, dependent on that mental activity of carrying out the procedure. If we use use "identical", instead of "equal" it is implied that what is really a value (something mind dependent) is an object with an identity. This is why Platonism is implied when we replace "equal value" with "identical value". It is implied that the value is an object with an identity.
Well, I can't explaining the mistake you're making in any simpler terms, so if you don't understand that then I can't help you further.
Quoting Metaphysician Undercover
You really read too much into words. There's just no substantial metaphysical implications in saying that the value returned by one operation is identical to the value returned by some other operation. It's just language and just maths. We don't need to believe in the mind-independent existence of abstract entities.
Like Tones' you refuse to stick to mathematics, committing the folly @Banno pointed to, a pretense of mathematics. Until you define and demonstrate how the distinction between extensional and intensional is relevant to a discussion of mathematical values, your reference to physical objects is completely irrelevant.
Quoting Michael
It's not maths, as both you and Tones have clearly demonstrated, by needing to refer to physical objects rather than mathematical values to support your claims of "identical".
Quoting Metaphysician Undercover
It's an analogy to explain to you the mistake you're making.
a. 1 + 1 is identical to 3 - 1.
Under an intensional reading (a) is false because adding one to one isn't identical to subtracting 1 from 3.
Under an extensional reading (a) is true because the value returned by adding one to one is identical to the value returned by subtracting 1 from 3.
Compare with:
b. The President of the United States is identical to the husband of Jill Biden.
Under an intensional reading (b) is false because being the President of the United States isn't identical to being the husband of Jill Biden.
Under an extensional reading (b) is true because the person who is the President of the United States is identical to the person who is the husband of Jill Biden.
You may try for, literally, years and he will not understand.
No, you are making a mistake.
I suggest you to read an elementary school book on set theory. There indeed are infinite sets and there can be a bijection between these sets. It's not just "mistake" like you think.
From "cuemath" describes this perfectly well:
And furthermore, just how important is a bijection in the definition of cardinality:
See Cuemath: cardinality
Perhaps you should start here:
Lecture on infinity and countability
Or here, just what is an infinite set:
Mathworld Wolfram: Infinite set
Or simply the axiom of infinity in ZF-logic:
Axiom of Infinity
Or if you think that there is no set of the natural numbers N, I think your contribution to any set theoretic discussion or to the subject of infinity is quite limited, to say at least. Otherwise I do value your opinions and remarks on various other subjects.
And if one objects to calling whatever mathematics refers to as 'objects', then we note that the word 'object' is a convenience but not necessary, as we could say 'thing' instead, or 'value of the term', or 'denotation of the term', or even none of that, and just say 'members of the domain of discourse' so that 'T = S' is interpreted as, for any model M for the language, M(T) is M(S).
It is not required to have any particular ontological view of what mathematical terms refer to just to understand that '=' stands for the identity relation. That is, whatever the terms T and S refer to (no matter what one regards mathematical terms as referring to), we understand that 'T = S' stands for the statement that whatever 'T' stands for is the same as what 'S' stands for.
Moreover, there is a difference between what is meant in mathematics by '=' and what one thinks mathematics should mean by '='. Whatever one thinks mathematics should mean by '=' doesn't change the fact that in mathematics '=' stands for identity.
It's not intended to be, and it's not required for the efficacy of mathematics. Detailed explanations of that point have been given over the years in this forum.
Even if there is no set of all the natural numbers and no set of all the real numbers (and even if real numbers are not infinitistically conceived as equivalence classes of Cauchy sequences or Dedekind cuts), still we may note that there is no procedure such that its outputs are increasingly longer finite sequences of real numbers (however real numbers are finitistically conceived or represented such as a real number itself as a procedure for outputting increasingly longer finite decimal sequences) and such that, for any real number, it will eventually be an output. (I think that's right, but perhaps there could be objections?)
??
Of course there are many conceptions of "2". I don't know what you mean by objects, why you're talking about objects, or what point you are attempting to make. I don't know what you mean by the normal usage of "2".
I think it's right. But anyway, even the notion of reals would go against this argument that mathematical objects "cannot be carried out, cannot be completed" and hence are "nonsensical". And when you throw out real numbers as "nonsensical", your mathematics is quite illogical. We do need number like pi!
Quoting TonesInDeepFreeze
The most irritating answer type is that if you ask something about mathematics and mathematical objects, people answer by referring to physics and for instance quantum physics. No, an observation of the physical reality, that we model by a mathematical model, doesn't tell if a mathematical object is true or false.
We start from the need to move something and then build a carriage to move it, not that we just build a carriage without requirements and then try to find something that can be moved with it. Furthermore, it's even more wrong to start arguing that our primary task, need to move something in the first place, is wrong, we shouldn't even think of it, because our carriage can't move it.
Indeed; I did; he doesn't.
Quoting Banno
Even I was not expecting such recalcitrance. That was 24 hours and three pages ago. Those three pages are replete with Corvus' squirming and flailing.
There are interesting and controversial ideas in Wittgenstein's anti-platonism, which could make an excellent thread. But an attempt at any such conversation in these fora would quickly be derailed by those who cannot grasp equality and those who misattribute and fabricate willy-nilly.
That's a limitation on @Jamal's otherwise excellent forums. A more proactive moderation might improve the philosophy being done hereabouts. But so many of the better posts are, as and have shown in this thread, responses to ineptitude.
And so it goes.
Your problem is that you make out as if what you and your cliques say are the only truth, and the rest of the world are false. Many would believe that your posts should be under the moderations for the extremely biased and misunderstood posts and Clequism you have been trying to pursue in this forum.
Trace back all your posts and Tone's in this thread, and you will see who started throwing unfounded posts and ad hominem posts before me, and degraded the discussion into a comedy. All your posts have no grounds for your claims. My posts are based on the philosophy of mathematics (Putnam) and set theories (C. C. Pinter), and various published academic articles.
Sad that the "clique" with which you are in disagreement is that of the mathematicians. Hm.
Anyway, time to move on. Long ago.
If you trace back Tone's posts, he starts with ad hominem before getting into philosophy. And you blindly take his side condoning his absurd and incorrect points, as if they are the only truths on the earth. How petty and juvenile. That's too visible, even a 10 year old would sense it. That is not Philosophy. That is a blatant clequism.
This article in SEP outlines and supports my point in this thread. I can drag out all my other books on Philosophy of Math, and Set theories, but it would be too cumbersome. If you wanted, I can do that, but it doesn't seem necessary. You would still keep saying I will double down. No. You are wrong. See how your whole focus of your posts are "You" "Me" "Him", leaving out the matter under discussion in the deep freeze?
I agree with Wittgenstein's Philosophy of Math. I disagree with all those who take Infinity as real entity, and the Infinite Set theorists, whether mathematicians or not.
If infinity was real, then Zeno's Achilles would be still chasing the tortoise in the race track at this very moment. But is he? It is a paradox. You know that.
Set theory's infinity is a tongue in cheek theory taking nonexistent infinity as if it does exist, hence a vacuous theory, which only seems to be making sense in the textbooks. Fine so be it. But if you used it for solving real world problems, you would end up in a deep ditch.
You are blatantly lying about me. Again. Stop lying about me.
Moreover, I addressed the issue of ad hominem in detail. Of course, you SKIP that.
First post of mine in this thread:
https://thephilosophyforum.com/discussion/comment/879009
Quoting Corvus
You keep saying that, but have not shown anything incorrect in what I've said.
Quoting Corvus
Banno and I have no allegiance or bond or anything like that. We've disagreed at times too. Merely that we happen to agree on a number of points doesn't make us a "clique". And your silly argument could be turned around. I could say that the opposition I've received to my posts comes from a "clique" of cranks. But I don't, because it would be a foolish thing to assert that they form a clique merely because they disagree with me.
Quoting Corvus
Please quote any passage in that article that you think claims that Wittgenstein said that in mathematics 'infinite' means 'finite'.
Quoting Corvus
Please name one that you think defines 'infinite' as 'finite'.
Quoting Corvus
Set theory axiomatizes the infinitistic classical mathematics, such as calculus, that is used for the sciences. All of the technology that you depend on to survive and flourish uses mathematics involving infinite sets. The very computer you are typing on comes from the work of mathematicians who were steeped in the mathematics of infinite sets. Meanwhile, you do dig yourself deeper and deeper into a ditch.
And stop lying about me.
I'm referring to a notion in which there are only finite "approximations". That is, that the real number is taken to be the algorithm for generating successive partial finite "approximations".
It is his metaphor, meaning that even if you claim it is "infinite", it is actually "finite". It is a type of cynicism. He uses aphorism a lot in his writings. Please don't take it literally. Obviously you have not read Wittgenstein at all.
But the point is not about the word games. The critical point is that "infinity" doesn't exist. When you say "infinite", it actually means "finite" in real life. Even if you keep on counting something infinitely, you must stop counting at some point. You cannot keep going on till the eternity. You stopped counting, and what you have is a finite number.
My point was just to point out that if you use the concept for nonexistence as real existence, and use it in your premises, then you will arrive at contradiction misleading yourself and others who believe you are correct.
Here, very early in this thread, you imparted an insult snidely couched as a rhetorical question:
Quoting Corvus
Here is my first post in response to you:
Quoting TonesInDeepFreeze
There is no ad hominem there.
Then after more posts in which you continued to dogmatically insist that you are right, blithe to the (not ad hominem) substance of the replies to you, I said:
Quoting TonesInDeepFreeze
Then, as it got even worse and worse with your strawmen, ignorance of the subject, getting things backwards, etc., I made clear that you're a crank:
https://thephilosophyforum.com/discussion/comment/880933
And, still, you SKIP my remarks about ad hominem, most especially that I don't say that my arguments are supported by ad hominem but rather that, in addition to my arguments on the substantive points, you are indeed ignorant, dogmatic, confused and dishonest. At a certain juncture in threads such as this, the perniciousness of the ignorance, dogmatism, confusion and dishonesty of cranks deserves highlighting.
The ridiculousness is courtesy of you. Maybe not comedy, but still risible is the claim that set theory takes 'infinite' to mean 'finite'.
Asking a second time, what quote in the article do you claim supports your claim that Wittgenstein said that mathematics takes 'infinite' to mean 'finite'?
Whose word games? The point is that you claimed that mathematics takes 'infinite' to mean 'finite', and you support that by claiming that Wittgenstein said that mathematics regards 'infinite' to mean 'finite', and you support that by quoting Wittgenstein saying that discussions about 'infinity' are finite. And now you've said that the Stanford article supports you in this.
So what specifically in the Stanford article do you claim supports you in any of this?
This is not word games. The Stanford article is not word games. You claim it supports you, so if it does, you could quote where it does.
Even if we agreed that there are no infinite sets, it still wouldn't be the case that 'infinite' means 'finite'.
And even if we agreed that the use of the word 'infinite' breaks down because there are no infinite sets, it still wouldn't be the case that the mathematical meaning of 'infinite' is 'finite'.
And your challenge to me to name books in mathematics that define 'infinite' as 'not finite' was specious, gratuitous and ridiculous. As if that it is not the case that indeed books in mathematics define 'infinite' as 'not finite' but instead absurdly as 'finite'!
Anyway, still would like to read the quotes that you think say that mathematics regards 'infinite' to mean 'finite'.
No one counts infinitely. To say "counting infinitely and stopping", in this context, is a contradiction.
The theory of infinite sets is not premised on the supposition that a person can count infinitely.
This has been gone over and over and over already...
I'm glad I don't do that.
I don't know what that would be, but I disfavor censoring cranks or admins using "chilling effects". On the other hand, it is indeed disheartening when admins censor or use chilling effects against posters who are calling out cranks and saying forthrightly that they are ignorant, confused, dogmatic and dishonest. And highly irrational for admins to use chilling effects to slow discussion about mathematics on the basis that it is not philosophy, when cranks are posting confusions and falsehoods about the mathematics as part of their criticisms of it. Moreover, we do not find that sentiment of clamping down against other subjects that are not philosophical or even being discussed from a philosophical point of view.
Hilary Putnam?
How do your views square with indispensability?
The blatant misrepresentation seen here is a very different thing to the psycoceramics. The latter on occasion does force one to explain or re think.
All by way of repeating that the bad posts do elicit good replies.
But I wonder if the general reader is able to tell the one from the other.
Quoting TonesInDeepFreeze
To some extent the misunderstanding of various authors may be the result of our friends being autodidactic. The supposition that somehow the SEP article on Wittgenstein's philosophy of mathematics supports psycoceramic views might be a result of shallow reading of such tertiary sources. These topics are vast, needing careers, rather than degrees, to understand the topic, let alone make a significant contribution.
Anyway, respect to the mods for what they do. While it might be a little bit better, it could easily be a whole lot worse.
And Chat GPT.
It is bewildering why challenged me to show a book that defines 'infinite' as 'not finite' when you could have looked yourself at the book by C.C. Pinter in which he writes:
"A set A is said to be finite if A is in one-to-one correspondence with a natural number n; otherwise, A is said to be infinite."
Exactly the definition I gave, and exactly the definition found in many many books on set theory and fields of mathematics!
And the book is, as any ordinary textbook in set theory, chock full of use of infinite sets and infinite sets of different cardinalities from one another.
A poster who starts out in a thread by declaring "end of story" does not bode well.
I am getting a good laugh though at that poster challenging me to show a book that gives the very definition that is in the book he says he "bases" his posting on!
That paragraph kinda set up for the gross oversimplification that was to come though.
Of course it is a book of Set Theory. However, it explains the historical background of the concept of infinity how controversial the concept was in detail. You only picked out the usage of the infinity in the book for insisting your point in this thread. I read it from the start to the end.
You obviously have problem understanding metaphors and ordinary use of English language. You seem to bite into a little words in the expressions, and as if one has to stick to the every word and comma in the sentence in the legal contract. I tend to write with metaphorical and simile expressions and idioms a lot just like other ordinary English users. You can't seem to understand that.
You start your post with throwing insults to others before even going into the points under discussion. What courtesy are you talking about?
It is a metaphor from my point of view. It is obvious, and I have kindly explained it to you above.
It a way of expression saying, when something is so bad, one could say "Well it's f***ing great." Anyone can see and use it to describe the situation with cynicism. If you have problem understanding it, I cannot help you.
If you are asking in which article he said it, I recall it was from a book I don't own. But I saw it in the internet somewhere. I will try to find it, and update on the book title and page. I don't have the information off hand. I couldn't have made the quote from my own imagination. To me, it sounded a genius in the expression at the time of reading it.
It was controversial when they didn't know better. It's not controversial now because they know better. Those opposed to set theory now are, for the most part, non-mathematicians who don't know better but think they do.
Let mathematicians argue about set theory. Anyone else just isn't equipped to understand the matter.
Many still believe it is controversial, and I do too. No one is saying it is illegal to use it, but just pointing out the existence of the controversy and also reservation on the theory. No one can deny that.
In real life infinite set doesn't exist. I start putting something in a box forming a set, soon the box gets full or the object runs out. It doesn't go on filling the box forever.
Many mathematicians?
Quoting Corvus
This goes back to what I said here:
Infinite sets have a use in mathematics. That's all that matters. Reading more into them is a mistake.
Not sure on Mathematicians, but if they are logical, I would presume they would.
Quoting Michael
Maybe. I don't see much practical point apart from filling in and adding more pages of the textbooks making them heavier.
The problem being, that contrary to your claim, there are no things denoted in mathematics therefore mathematics is not "extensional" in the way of your analogy. @Michael agrees that mathematics deals with values rather than things. And since values are inherently intensional the mistake you made ought to be easily avoided by Michael.
Quoting TonesInDeepFreeze
It is not matter of whether abstractions exist as physical objects, it is a matter of whether abstractions exist as "objects", or "things" in any rational, coherent sense of the word. The law of identity states that a thing is the same as itself, and we can satisfactorily replace "thing" with "object", or vise versa, making them interchangeable for the sake of discussion. Now the issue is whether there is an identity relation (consistent with the law of identity) expressed by "=" in mathematics.
So, the demonstration and reason why, there is not a "thing" or "object" which is referred to by a numeral such as "1" or "2", and why that supposed "thing" would be incoherent and irrational if it was a thing which is referred to, is explained by my example of "1+1=2". If the two 1's both refer to the very same thing, then there is only one thing represent by those two 1's. Therefore no matter how many times we represent that same thing, we cannot have an equivalence with 2. So it ought to be very clear to you that "1" cannot refer to an object or thing because this would render mathematics as incoherent. Even the simple minded ChatGPT understood this example, and in the other thread where Banno presented this to it, it was very clear to say that in mathematics "=" commonly represents equality, "not identity".
Quoting TonesInDeepFreeze
This is exactly the problem which I've been repeating over and over. In common usage of mathematics, "=" signifies equality. GPT corroborated, even though you dispute its authority on common usage of mathematics. However, some mathematical theory, such as set theory defines "=" as signifying identity, regardless of how it is actually used in mathematics. This produces the problem you mention. Some people such as yourself, think that "=" should signify identity, because this would make it consistent with the theory they support, even though the fact remains that in mathematical usage "=" continues to represent equality rather than identity.
Do you agree, that when it is the mathematicians themselves, who are insisting on what "=" should mean, with complete disregard for how it is actually used in mathematics, there is a problem? This is a common epistemological problem demonstrated by Plato in the Theaetetus. Epistemologists have an idea of what "knowledge" should mean, 'JTB', and this supports their epistemological theory. However, as Plato demonstrated we cannot actually exclude the possibility of falsity pervading knowledge, so the T of JTB doesn't actually represent a true definition of "knowledge" according to what the word is actually used for. It simply represent what some epistemologists think "knowledge" should mean. Likewise, "=" does not mean identity in mathematics, it represents equality, despite the fact that some mathematicians think it should represent identity because that's what their theory states.
My criticism remains unaddressed. Let me put it more clearly. Since we are discussing values, not physical objects as in the case of your example, there is no such thing as an extensional reading of "1+1 = 3-1". That constitutes a misinterpretation.
Quoting ssu
Bijection is a specific procedure. If you think that an infinite bijection can be carried out, such that you can produce a conclusion about the cardinality of a supposed infinite set, then you ought to be able to demonstrate this bijection. This would demonstrate that you have made a valid conclusion concerning the set's cardinality. And by "demonstrate" I mean to actually perform this bijection, not to simply represent it with a symbol or symbols, as if it has been performed. The latter does not qualify as a demonstration because one can make a symbol to represent any impossible conception, like a square circle, or whatever. Are you prepared to make that demonstration?
Quoting Banno
This I agree with. There is a serious problem with those who conflate equality and identity to "fabricate willy-nilly". We seem to be in much agreement in this thread, which is unusual. You have already pointed out the problem with people like Tones and Michael who claim to be doing mathematics when they are not. These two have displayed a need to refer to non-mathematical examples like Twain=Clemens, and the president of the United States, to demonstrate their supposedly "mathematical" principles.
There is. The extensional reading of "1 + 1" is the number 2. The extensional reading of "3 - 1" is also the number 2. And the number 2 is identical to the number 2.
Also and correct me if I'm wrong @TonesInDeepFreeze but "1 + 1" doesn't actually mean "add 1 to 1". Rather, it means "the number that comes after the number 1". And "3 - 1" means "the number that comes before the number 3".
The number that comes after the number 1 is identical to the number that comes before the number 3.
Jane is standing between John and Jack, with John on our left and Jack on our right
Inference
The person to the right of John is identical to the person to the left of Jack
The inference is valid even though Jane, John, and Jack are not physical people and are not abstract entities that exist in some Platonic realm.
It seems very straightforward to me.
Wouldn't his views triangle with indispensability even? Corvus seems to be arguing for some kind of anti-realism about at least some mathematical entities, it seems.
Quoting Metaphysician Undercover
When you say "values" it seems you refer exactly to what is supposed to be the extensional reading of 1+1 or 3-1. So, if we are discussing values, saying that 1+1 is the same as 3-1 is correct, as both represent the same value, even if not the same operation.
I have some sympathy for anti-realist views in maths, I've expressed this elsewhere over several years. These stem from reading Wittgenstein. The problem is that both you and @Corvus badly misrepresent Wittgenstein in an attempt to subjugate his name to your psycoceramics.
So far neither of you have been able to cite anything like an endorsement of either your eccentric and unsound view of equity nor Corvus' confusing finite and infinite. Nor will you.
But the result is that we are unable to have a significant discussion of constructivist views of maths.
That's nonsense, you cannot read "1+1" as "2" because that's obviously a misreading. There is an operation signified by "1+1" and this implies that the reading of it must be intentional. It would absolutely be a misreading of "1+1" to read it as "2". And to get 2 out of 1+1 is intensional as well.
Quoting Michael
See, this is proof that your reading of "1+1" is intensional. "The number that comes after the number 1" is clearly intensional, and that's how you read "1+1". You cannot read "1+1" as two because that would be a misreading. Only "2" gets read as two.
Quoting Lionino
That's right, but Michael and I already went through this discussion. The values which are produced by "1+1"and "3-1" are only created by carrying out the operations referred to by "-", and "+". The expressions "1+1" and "3-1" refer to those procedures, not the values produced as a conclusion to the procedures. To conclude that "1+1" and "3-1" both produce the same value requires that the operations referred to be carried out correctly. Therefore, that "1+1", and "3-1" each produce the same value is dependent on correctly carrying out the operations which are represented by the expressions. What is represented by the expressions is the operations, not the values which result as a conclusion.
Quoting Banno
I like that description "psychoceramics". It makes me feel like I belong to a group, the psychoceramicists, rather than just a lone wolf.
Quoting Banno
Oh you poor little boys, can't keep yourselves from being distracted by the antics of a couple of psychocermacists.
Fair enough. I doubt, were you to get together, that you would find much agreement apart from the "cliques" being wrong, and your martyrdom.
What about our interest in crackpots like Tones?
I read the chapter about the history of set theory and philosophy about it. I haven't posted anything to dispute of it nor, in certain parts, anything to affirm it.
Included in that chapter, the author explains the importance of formalization, very much along the lines I did earlier in this thread, on which point you disputed.
I read much of the rest of the book, as it interests me in the particular way that it develops a class theory.
Anyway, the bulk of the book is an intro to set theory, covering material I have studied in similar textbooks, though, as mentioned, I'm tempted to go back over that material with this book, as I am interested in the authors particular way it develops a class theory.
Quoting Corvus
But you missed the definition of 'infinite' that completely agrees with the one I mentioned but that you challenged me to cite a textbook that uses that definition. So, I am still baffled why you challenged me to cite a textbook when your own favorite book on set theory, which you claim to have read, is one of many many textbooks that give the definition you challenged me to show that it is in a textbook.
And I highly recommend that you reread that chapter on the history of set theory and philosophy about it, so you will see how the author and I are aligned on the subject of formalization, as you instead displayed that you don't understand it and as you objected to my remarks about it in your usual style of confusion, strawman and non sequitur.
That's a beam calling the mote a beam.
I understand metaphor.
I didn't demand perfection in what you said.
You said that mathematics regards 'infinite' to mean 'finite'. That's not a metaphor. If you meant that it was a metaphor or that you didn't actually mean to say that mathematics regards 'infinite' to mean 'finite', then you could have conveyed that the first time I told you that mathematics does not regard 'infinite' to mean 'finite'.
Then you deflected to say that it's something that Wittgenstein said. So, that deflects from you making the claim to you claiming that Wittgenstein made the claim. But the Wittgenstein quote, whiles perhaps ironic and acerbic, does not as presented without more context, say that mathematics regards 'infinite' to mean 'finite.
So then you deflected again to say that the Stanford article supports your claim about Wittgenstein. But you fail to give any quote from the Stanford article.
It might be that Wittgenstein meant that mathematics regards 'infinite' to mean 'finite', but you have not shown that he did, and even if he did, merely that he did would not show that mathematics regards 'infinite' to mean 'finite'.
So now you deflect to a claim that is false (that I don't understand metaphor) and one that is both false and a strawman (that the reason I don't agree with you is that I demand perfection of expression). And, this is while you've been complaining about ad hominem, as your quote above is itself ad hominem. And while you've completely skipped my detailed points about ad hominem in posting.
On the contrary, when I check Tone's arguments, they are very mainstream. Almost painfully so. I find that admirable; Tones has corrected my excesses.
This post: by way of example, setting out the issues clearly and historically.
You lied about me when you said I started with insults. I gave you the links that prove that you're lying about that. And even showed that you first made an insult against another poster.
The record of posts shows that I posted without personal remarks, and for a while, until it became clear that you are posting in bad faith - from ignorance, confusion, strawman, evasion of refutations.
For the second time, you are lying when you claim that I started posts in this thread (for that matter, any thread) with insults. Meanwhile look in the mirror for a change - there's a huge steel beam across your eye.
And the courtesy I'm talking about, Mr. Metaphor who can't discern irony, is just what I said it is: that you provide comedic relief when you go through all the ridiculous contortions you do just to avoid simply recognizing that set theory does not define 'infinite' as 'finite'.
I searched 'Wittgenstein mathematics infinite means finite'. Of course, there are hits with all those terms, but the one I saw come up with a preview close to your claim is this thread itself.
And, again, the levels:
You claimed that mathematics regards 'infinite' to mean 'finite'. You claimed falsely.
Then you claimed that Wittgenstein said it. But the quote you adduced did not say that mathematics regards 'infinite' to mean 'finite'.
Then you said that Wittgenstein meant it as metaphor. But, at least without context, it is not clear what the metaphor would be there. And even if the claim that mathematics regards 'infinite' to mean 'finite' is
metaphor, it doesn't relieve that you did not present as metaphor yourself.
Then you claimed that the Stanford article supports that Wittgenstein said that mathematics regards 'infinite' to mean 'finite'. But there you cannot give a quote in which the article says that or even implies it.
Then you claim that your original claim that mathematics regards 'infinite' to mean 'finite' was merely metaphorical. But since it doesn't read as metaphor, when you were first told that mathematics does not regard 'infinite' to mean 'finite', you could at that time just say that indeed you do not claim that mathematics regards 'infinite' to mean 'finite'.
Meanwhile, you challenged me to cite a book that defines 'infinite' as 'not finite', while your own favorite book itself gives that definition.
Meanwhile, you lie about the very record of posts in this thread.
And you hypocritically decry ad hominem, while you use ad hominem. And you ignore the detailed remarks I said about ad hominem.
And all of that is in your pursuit to take down set theory, while you know virtually nothing about it, are confused about it, misrepresent it, and ignore explanations given you about it.
/
So someone might say, "Oh, but Tones, why are you going on, prosecuting this one little item?"
Because every time I catch this crank in his intellectual dishonesty (even to the extent of lying about the record of posts) he comes back with even more intellectual dishonesty. It is worth making that clear as yet another object lesson about the perniciousness of Internet crankery.
One may reject ideation and communication premised in abstract objects. But the notion of identity is not even limited to abstract objects. Whatever things one does countenance as existing, named by, say, T and S, we have T = S if and only if T is S. That is what '=' means when it is used in contexts of ordinary identity theory, logic, mathematics and other contexts to. If one wishes to use it with another meaning in another context, then, of course, fine. But that doesn't justify saying that in logic and mathematics it is not used just as logic and mathematics says it is used.
'=' doesn't even require a mathematical theory as its context or even the acceptance of abstractions, but rather that in identity theory, for whatever things, abstract, concrete, physical or are being looked at on your desk right now, the statement of identity is that of being the same thing.
Again, more exactly:
If 'T' and 'S' are terms, then
'T = S' is true if and only if T is S.
And whether 'T' and 'S' stand for abstract things, abstract objects, values that are abstract things, values that are abstract objects, concrete things, physical things, or whatever things you are looking at right now on your desk.
And how can anyone, even a crank, not understand:
1+1 = 2
'1' refers to the number one. And since '1' and '1' are the same numeral, it would be redundant, though obviously true, to say that both '1' and '1' refer to the number one.
Then, '1+1' refers the SUM of the number one with the number one. And that SUM is the number two. Or in more formulated mathematics, '1+1' refers to the successor of the number one; and the successor of the number one is the number two.
The denotation of '1+1' is not two of the number one, but rather it is the SUM of the number one with the number one.
'2' refers to the number two, and '1+1' refers to the sum of the number one and the number one, which is the number two. So '2' and '1+1' both refer to the number two, so '2' and '1+1' refer to the same number. That is, 1+1 is 2, which is expressed as:
1+1 = 2.
It is difficult to reason with someone about mathematics who doesn't understand that 1+1 is 2.
(Yes, I can hear sane and rational people saying, "Really, Tones? You spent your precious time tonight explaining to a grown person that 1+1 is 2?")
/
Taking Chat GPT as an authority, or even remotely reliable, as does the crank is pathetic.
Anyway, I couldn't resist. Chat GPT told me that:
1+1 equals 2.
1+1 is 2.
/
In mathematics and logic 'equality' and 'identity' mean the same.
/
The reason I say that '=' stands for the identity relation is not that "it would make it consistent with the theory I support". Rather, in mathematics, not just in set theory, ordinarily '=' stands for equality, which is identity.
But the crank is now arguing that in mathematics it's equality, which is not identity, which is untrue.
/
In ordinary mathematics, bijection is not a "procedure". Rather a bijection is a certain kind of function.
And we do prove the existence of certain bijections.
And one does not "perform" a bijection.
Here's a bijection:
{<1 1>}
One does not "perform" it.
Here's a bijection:
{
One does not "perform" it.
(But maybe there's no point in explaining this to someone who does not understand 1+1 = 2.)
/
The crank misrepresents again by claiming I mentioned intensionality regarding a person's name to support a mathematical claim. I explained exactly the role of mentioning the example of intensionality and that it was not an argument that such an example applies mathematically. But the crank skipped that so that he could misrepresent my point.
Just to be clear:
I enjoy reading classical mathematics; I find great wisdom in mathematical logic; I admire the rigor of logic and mathematics; I admire the astounding creativity in logic and mathematics; I recognize that classical mathematics is the basic mathematics used for the sciences; I recognize the objectivity in mechanical checking of proofs (and generally that at least in principle, if time were taken to fully formalize then proofs are mechanically checkable); I admire the intellectual honesty of logicians, mathematicians and many philosophers of logic, mathematics and language; I admire the simplicity of the axiomatization of mathematics from the set theory axioms, especially the relative simplicity, as axiomatizing alternative mathematics is often much more complicated; and I enjoy, though I am haunted by, the philosophical problems that arise from classical mathematics.
But I do not claim that classical mathematics is the only "true" mathematics; or that there can't be a better mathematics; or that it is wrong to have philosophical objections to classical mathematics. Indeed, with my limited time and limited talent for mathematics, I do very much enjoy learning about alternative logics and alternative mathematics, and I very much admire and relish the wisdom, creativity, and productivity of the alternatives, and also the great philosophical debates around classical and non-classical mathematics.
Each of these is true if and only if each of the others is true:
S = T
S equals T
S is identical with T
S is T
the denotation of 'S' = the denotation of 'T'
the denotation of 'S' equals the denotation of 'T'
the denotation of 'S' is identical with the denotation of 'T'
the denotation of 'S' is the denotation of 'T'
/
All of the below are identical with one another. All of the below are equal to one another. All of the below are the same as one another.
1+1
the sum of 1 and 1
1 added to 1
1 plus 1
the successor of 1
3-1
the difference of 3 and 1
1 subtracted from 3
3 minus 1
the predecessor of 3
2
two
the denotation of '1+1' [but not the Godard movie '1+1']
the denotation of 'the sum of 1 and 1'
the denotation of '1 added to 1'
the denotation of '1 plus 1'
the denotation of 'the successor of 1'
the denotation of '3-1'
the denotation of 'the difference of 3 and 1'
the denotation of '1 subtracted from 3'
the denotation of 'the predecessor of 3'
the denotation of '2'
the denotation of 'two'
Yep.
That reminds me, I still am interested in how he thinks Putnam's indispensability view jibes with his own views.
That quote could be written only by someone who does not understand what is meant by 'extensional' and 'intensional'. A name is not just one of extensional or intentional. Rather, a sentence has both an extensional aspect (the denotation) and an intensional aspect (the connotation).
/
'1+1' does not stand for an operation. It stands for the result of an operation applied to an argument.
'+' stands for a function ('operation' if you insist).
'1+1' stands for the value of the function applied to the argument <1 1>.
My way of making sense of it is that in the modern sense we understand the extension of "2+3" as 5; and the intension of "2+3" as the algorithm, or perhaps the program, it prescribes for us to follow. Hence "2+3" and "4+1" give different algorithms for us to follow, but each will give the same answer - different intension, same extension.
Wittgenstein worked with a more directly platonic notion of extension - the thing that "2+3" points two - and it was at least partially his rejection of this Platonism - what could such a "thing" be? - that led him to his somewhat more contentious views. Roughly, his view prevented him from accepting that there are infinite mathematical extensions.
My own suspicion, which is without a strong formal argument, is that all mathematical entities might be best understood as sets of instructions - that in effect there are no extensions in mathematics. I hold to this view on Tuesdays and Thursdays, the remainder of the time thinking that it makes no nevermind if we do treat the results of these processes as if they are real; in a fashion not unlike how we treat money as real despite it being only a series of transactions.
I had a go at articualting this in the thread "'1' does not refer to anything" four years ago: that mathematical entities are things we do, not things we find.
But here we are getting into the sort of discussion that I think will prove impossible with present company.
Indeed it is a fine idea that we may talk about a program that outputs successively longer finite sequences rather than talking about an infinite sequence. But easier handwaved than axiomatized. There have been proposals for intensional mathematics (especially, for example, Church), but it's not an easy thing, the devil is in the details; it's not realized by crank handwaving, confusions and illogic.
There's some hint here of Wittgenstein's idea of following a rule as implementing a practice, of "continuing in the same way", but this is very speculative. An area well worth keeping one eye on, I think.
I'm not suggesting mathematics should be done intensionally, so much as puzzling over what the distinction between intension and extension amounts to. It hints at something pivotal, but well beyond my ken.
Mathematics does not pretend to be isomorphic with all the concrete objects and particles of science. Nor that mathematics is a factual report about concretes. Rather, mathematics provides an idealized framework that we can choose to use in different ways, including providing an axiomatization for the formulas we do use for the sciences. Mathematics is an armature for knowledge about concretes; it is not supposed to be itself a report of those concretes. The armature is not itself the things you put in it.
And the way I understand - other mileages may vary - frameworks, whether mathematical, philosophical or conceptual in any field of study, is that they should facilitate fluid thinking and communicating, and to avoid, if possible, having to stretch oneself in contortions such as having to grasp for convoluted expressions to avoid saying the word 'object' in an utterly natural way when talking about things such as numbers, or to have to eschew the economy of conceptualizing numbers as things rather than to commit to imagining that a number is born and dies, off and on and off and on, every time someone thinks of it and then stops thinking of it or that there even is no 'it' they are thinking of but only physical events in a brain, or that, wait, what is the notion of 'event' anyway without abstraction?
On the other hand, if one wants to try to think of mathematics and formulate it and communicate it but without reference to abstractions or abstract objects, I say have at it. But that doesn't make everybody else wrong for thinking of numbers as things and saying such ordinary things as "the sum of two and two is four". And especially classical mathematics is not crippled by the mere wish of a crank, without a concrete proposed alternative, that there is an unannounced, unarticulated physicalist replacement.
I find it a crude notion that each mathematical mention must correspond to represent each, every and any of the concretes and particles that are themselves present to us mentally as constructs in a conceptual framework. A framework is not an assertion, and its value is being able to conceptually and/or practically cope with or predict experience. Such frameworks may be preferred or not in how well they conceptually and/or practically cope with or predict experience, but also in the satisfaction derived from the conceptual order and beauty they provide. When I am confused by too many facts all at once, or about, for example, how things work, I am relieved of that confusion by a framework that allows me to put that experience in order, to process it. I may be confused by the behaviors of other people, for example. But then I may posit such things as traits, goals, etc. I don't posit that those are concrete things. They are abstractions, they are posited as a conceptual armature so that a person's actions don't appear to me as a random jumble but rather my framework allows me to think of those actions in a narrative and to make predictions about them on that basis. When I there are of numbers mentioned, I don't have to think of them as popping in and out of existence each time they are mentioned or not, but rather I have an armature in which numbers don't do that. And when I there are a lot of numbers involving some problem, either conceptual or practical, that I want to solve, I have a system of principles about numbers that allows me to find the answers I want. That system is an abstract armature, not a concrete thing. It's not required that each concept, each abstraction itself corresponds to a particular concrete.
Meanwhile, maybe there is a way, but I don't know of it, to avoid that thought and language themselves presuppose that 'object', 'thing', 'entity', 'is', 'exists', etc. are basic and that explication of them cannot be done without invoking them anyway. When I say "What is that thing in the sink?" I presuppose even the concept, which itself is an abstraction, that there are things, that concretes are things, and even the notion of 'concrete' is an abstraction. And I don't see anyone who can talk about experience ('experience' also an abstraction) without eventually invoking utter abstractions such as 'object' and 'is', whether referring to abstractions or concretes.
The best introduction to the subject I have found is in the introduction to Church's 'Introduction To Mathematical Logic', as indeed that whole introduction is a quintessential primer for the basics of logic.
I appreciate your post.
Yes, the mind has a framework of concepts, such as beauty, infinity, pain, mountain, house, government, addition, multiplication, sky, salt, which we use to organise our concrete experiences.
On the one hand, for example, our concept of mountain refers to an abstract object, in that it does not refer to one particular mountain, but mountains in general. But on the other hand, there must be an intentionality to our concepts, in that the mind is not able to comprehend a concept without thinking about something concrete, whether a particular object, such as the Mont Blanc, a particular process, such as adding more height to a high hill or the particular word itself, "mountain".
That is not to say that each concept can only have one concrete instantiation, but rather each time I think of a mountain my concrete instantiation may be different, and different again for anyone else who thinks of a mountain.
My point is that I agree that it is not the case that an abstract concept corresponds to one particular concrete instantiation, but rather we can only understand an abstract concept by thinking of some concrete instantiation of it, which may be a concrete object (Mont Blanc), a concrete process (addition) or a concrete word ("mountain").
Increasingly I find learning novel logic systems quite difficult, as if there is too much background missing. The topic is surprisingly different to when I first studied it, much more of an emphasis on computing, far more integrated than it once was, and my short term memory is not what it was fifty years since. When in a masochistic frame of mind I'll work through bits of the Open Logic text. It's often very simple things that hold one up - it seems to be the text of choice for undergrad logic courses and for some of the more advanced stuff. There is a lot to be said for the discipline of having a tutor to help work through examples, especially where small bits of jargon catch one out.
Church is an interesting choice.
The sense of "identity" I am concerned with is that stated by the law of identity, "a thing is the same as itself". Do you agree with this formulation of the law of identity, and that if logic and mathematics uses "identity" in a way which is inconsistent with this, then logic and mathematics violate the law of identity?
Quoting TonesInDeepFreeze
I would accept this as consistent with the law of identity, if we're careful to clarify that what we are talking about is the thing which "T" and "S" each signify. Clearly T itself, as a symbol, is not the same as S as a symbol.
Quoting TonesInDeepFreeze
The problem with this statement, is that a careful analysis and thorough understanding of what is here called "abstract things" will reveal that abstractions cannot be adequately understood as things with identity. So all these so-called "internal objects", conceptions, ideas, values, emotions, feelings, and everything else in this category, cannot be assumed to have an identity. This issue is extensively reviewed by Wittgenstein in The Philosophical Investigations. Particularly relevant is the part commonly known as the private language argument, where Wittgenstein provides the example of an attempt to assign the symbol "S" to a sensation. What is revealed is that "the sensation" cannot be known as having an identity. And this principle is extended by Wittgenstein to include all supposed "internal objects".
Due to what has been revealed by a large body of philosophical work in the past, I propose to you that if "T" and "S" are intended to stand for abstractions, conceptions, or anything else in this category commonly known as "internal", then "T" and "S" have no proper identity, as demonstrated by Wittgenstein. This was extensively covered by Aristotle under the concept of "substance", when he noticed the need to apply the law of identity against the sophistical arguments of Pythagorean idealists. Allowing that abstractions are identifiable things breaks down the categorical separation between ideas and things, allowing that the universe is composed of ideas.
Quoting TonesInDeepFreeze
This is incorrect, and this incorrectness I already explained to Michael. It is very clear that "1+1" refers to a specific operation which is indicated by "+". If we ignore this, and take a shortcut, assuming that the operation has already been carried out, and assume that "1+1" refers to the sum, then we ignore the role of "correctness" in the carrying out of the operation. Then one could stipulate any arbitrary expressions as referring to the same thing. I could say "1+1 = 8-2", and have my own private operations which produce this identity. In reality it is only through the means of carrying out the correct operation which is specifically signified by "+", that "1+1" can be said to be equal to "2". Therefore the meaning of "+" in that expression "1+1" is extremely significant to the meaning of the expression. It is intensional, and this intensionality cannot simply be taken for granted in the interpretation, to claim that the expression is extensional.
Quoting TonesInDeepFreeze
Obvious falsity. We read "1+1" as it is written, we don't read the implied result, "2". If what you said is true, then there would be no place for the learning of mathematics. We would not be able to account for the person who can read a mathematical expression, but cannot properly apply the principles required to produce the correct answers.
The truth of the matter is that the ability to correctly produce the answer, from the expressed operation, must be accounted for. It is simply not the case that a person goes from reading "1+1" as one plus one, to reading it as two, without a learning process, and that means acquiring the intensionality. The reality of this learning process, and how to properly account for it, is what Plato looked at in his theory of recollection, and what Wittgenstein looked at in The Philosophical Investigations.
And, I request that you please be honest with yourself. Do you really believe that you read the left side of an equation as the result of the expressed operation? That's simply not true, it's impossible because some operations are not carried out in the order that they appear. That is why I request honesty from you, and recognition that what is expressed by "1+1" is an operation to be carried out, not the SUM of that operation.
Quoting TonesInDeepFreeze
That goes two ways. When a person such as yourself, unwaveringly insists that the right side of an equation signifies the very same thing as the left side, despite a world full of applied mathematics as evidence to the contrary, then it becomes very difficult to reason with this person. The person simply refuses to look at all the evidence, and denies the evidential status of the evidence. The simple fact is that all the mathematical evidence supports what I say, so I am justified in my stance. But there is nothing but a stipulated "axiom" which supports your stance.
@Banno@MichaelThe issue we've encountered is that the axiom of extensionality is simply false. Of course, some will say that truth and falsity are not applicable judgements for mathematical axioms, and that is exactly why the axiom of extensionality is an ontological principle rather than a mathematical axiom.
What this so-called axiom attempts to do is to introduce truth and falsity into mathematics in the form of correspondence. It implies that there is an identified object which corresponds with the expressions of "1+1" and "3-1", replacing the true representation of 'correct answer' with this proposed corresponding "object". Now we'd have an "object" which corresponds with "1+1", as a form of truth, just like there is an object which correspond with "Mark Twain", as a form of truth.
This is why the axiom of extensionality is not a mathematical axiom, it is an ontological principle. Therefore it ought to be judged in a way which is appropriate to ontology.
Quoting TonesInDeepFreeze
I don't see any strawmen, you just demonstrate a simple misunderstanding of how "=" is used in mathematics, and an equally simple refusal to seriously consider the evidence, resulting in a simple denial. Perhaps it would help you if we move on to more complex equations. Do you really believe that "2?r" signifies the very same concept as "the circumference of a circle"? Surely you recognize that "r" signifies a straight line, and "circumference" signifies a curved line, and by no stretch of the mathematical imagination do these two expressions represent the exact same thing. A curved line cannot be made to be compatible with a straight line, as indicated by the fact that pi is irrational.
Which is correct.
Quoting Metaphysician Undercover
Maybe for mathematical realism, but then thats a problem with mathematical realism. Just be a mathematical antirealist and accept that true in the context of maths just means something like follows from the axioms, with the axioms themselves not being truth-apt.
Referring back to this, it makes no sense to say that the axiom is either true or false. It just is an axiom, and the inference follows.
Youre making a mountain out of nothing.
Under your thinking, anyone not thinking the same as you is misattributing everything. That is just nonsense. Under your eyes, people shouldn't be thinking differently from you.
Anyone thinking differently from you are downright wrong, and misattributing. What is the point of your philosophy? Forcing others to think the same as you do? That is not right.
Many would believe that your posts should be under proactive moderation for the low quality posts you have been spewing out with the meaningless quibblings stemming from your misunderstandings, and forcing people to believe and think exactly the same as yourself.
Putnam edited a book called Philosophy of Mathematics Selected readings. He put in there various articles by different people. It is not a book solely written by Putnam. You obviously have no idea about the book, or what the Edited book means.
If I really lied, then I would have told you that I lied, which is true. But you claim that I lied, which is false, and a lie.
I didn't lie, but you claim that I lied. Clearly and obviously you are telling a lie.
Therefore in whatever the case, you are the one who lied.
I was just telling you about Pinter's book to say that even classic Set theory books admit the historical controversies with the concept of infinity. I wasn't meaning to say the book is denying, accepting or defining on the infinity as per my view.
If you still don't understand what the point is, then you need to read a book called "The Oxford Handbook of Philosophy of Mathematics and Logic" Edited by Shapiro. Again in that book, there are various different articles with different views on the topic. But one that you must read about is "Quine".
The axiom of extensionality is
Quoting Open Logic
It tells us how to use the "=" sign. It is an instruction, and so is not the sort of thing that can be false. You either follow the instruction or you do not. If you do not follow the instruction you are not participating in the logic of sets.
The law of identity has various forms, but in set theory it is that
Quoting Open Logic
This is a consequence of extensionality, not an axiom.
What Meta is doing is refusing to use "=" in the way the rest of us do. It's as if someone were to insist that the Rook move along a diagonal, all the while pretending that they had made a profound discovery about chess in doing so. Meta is simply not playing the game right.
To this sin Meta adds that of mischaracterising Wittgenstein. The private language argument is not that a symbol cannot be identical to an internal sensation, But that internal sensations cannot be treated in the way we treat other objects.
Well, one cannot play chess if there is disagreement as to the rules. A chess player expects their opponent not to think differently, at least in that regard.
You have made claims about the ideas espoused by various philosophers, but when challenged you have not produced citations or produced citations that do not support your claims.
You are not playing the game right.
And that is worth pointing out.
When I think about questions like 'what is mathematics really?' I tend to consider three different ways. How did mathematical skills arise in evolution? How do they develop during the lifetime of an organism? How could we make a machine that learns these skills 'without being told'? I won't say anything here about that third one.
Let's start with bees. Bees are capable of using numerical quantities in at least three different ways. Firstly, they can learn to recognise the number of objects that are present in a particular place. For example, they can learn to associate three objects with the presence of nectar, regardless of the shape, size, colour of the objects. They can also be trained to find their way around a simple maze where they have to learn to take the third turning on the left, for example. They can learn to do this even if the third turning is in different places. These are two different ways in which they can work with 'threeness': three things separated spatially or three things separated temporally. Bees can use oneness, twoness, threeness, fourness, fiveness, but things start to go wobbly there. Arguably they can use zeroness. Thirdly, they can use their waggle dance to communicate an approximate distance and direction. This is innate, inherited behaviour, and hence inflexible.
Next, some quotes from What Babies Know, ELIZABETH S. SPELKE
OBJECTS
PLACE
NUMBER
One might wonder at this point ask what it is that we've got that bees haven't. Perhaps they can't combine numbers. I don't think they have fully abstracted numbers from their environment. They can use threeness as a property in two different ways, but can they unify these notions of threeness? Could they be trained to take the nth turning after having seen n objects (for n <= 5)? That would be another step towards abstraction.
My own feeling is that for an agent to achieve full abstraction from its environment it needs to find some part of that environment where it can exert intricate control. A good way is making sequences of marks (or making rows of 'bodies that are cohesive, bounded, solid, persisting, and movable on contact'), and then looking at them. I think bees could make marks in wax and look at them easily enough, but I guess their environment does not give them sufficient motivation to do so.
Marks are made one after another in time in the sequence, but once made they are spatially separated. This helps unify notions of 'n-ness'. They persist in time, so extend memory capabilities. Sequences of marks can be created and modified by the agent, and by modeling this behaviour internally, the agent can make another step towards abstraction. The agent can start to predict what would happen if marks were modified this way or that. I would say that once an agent starts this sort of imagining, it has started thinking mathematically.
I have not made many claims quoting hundreds of philosophers. That is just another distortion of the truth with exaggeration. My point was simple, and I quoted one philosopher, from which was the Wittgenstein's writing, and mentioned 2-3 others. If you still cannot understand the point, you can look them up yourself, and find out. No one has to spoon feed you.
Quoting Banno
I said this before, but will say again. Your problem is that you blindly say that others' points are wrong before presenting your arguments with evidence supporting your claims. That appears to be your trademark modus operandi of philosophy.
But because you keep on doing it firstly and unfairly to others, the other party will quite rightly try to argue against your wrong points and the style of your absurd claims dissecting the faults in your modus operandi. It is a vicious circle in your philosophy.
Quoting Corvus
:rofl:
indeed, it is. By you.
Quoting Corvus
I don't now actually recall what your point was. It wasn't very clear to start with, and is now buried in the clamour of your protest.
See? That was what I meant. You don't even understand the point, but rubbish it as wrong. How absurd is that. By the way, you are still in deep illusion. I was not protesting on anything. I was just pointing out problems in your inaccurate posts.
I baulk at this. Bees only do bee things, and numbers are a people thing. I'd say that bees do things that people describe using numbers.
A small pedantry.
That deserves consideration, though I'm not sure about it while also I don't have an argument in disagreement to give at this time.
What is a concrete example of the concept of 'does not exist'? What is a concrete example of 'there are things that do not exist irrespective of any list of properties that no existing thing have'?
Yet, the notion of 'concrete instantiation' is itself an abstraction made of the the two abstractions 'concrete' and 'instantiation'.
Also, the lines I'm thinking along is that certain utterly basic abstractions, such as 'object', 'thing', 'entity', 'is', and 'exists' themselves presuppose abstraction no matter what concretes are involved or not.
Anyway, to say that thinking of abstractions requires thinking of a concrete examples does not say that we don't think of abstract objects; or at least that to demand that we utterly eschew a notion of abstract objects brings even everyday conceptualization to a screeching halt or at least a devastating slowdown.
When you said that you base on philosophy of mathematics and mentioned Putnam in particular, naturally I thought you meant that you base on views of Putnam. From what you said, one couldn't be expected to think that what you actually meant is that you have read a particular book edited by Putnam.
But now that you have added to your original statement, fair enough, your views are informed by reading that book.
So now, what are the articles in that book that you base your views on?
That book is full of great stuff. One article in particular that I think helps a lot is Boolos's 'The Iterative concept of set'.
Quoting Corvus
It is a really stupid inference from (1) I took you to mean that you base your views on Putnam when you said you base on philosophy of mathematics and mentioned Putnam in particular to (2) I don't know what an edited book is.
That is false, since you didn't say that you lied but you did lie.
The plain record of the posts in this thread prove that you lied, as I explicitly linked to the posts. But you skip that.
You're confused as usual. I didn't say anything about your view of infinity regarding the book. Rather I note that you challenged me to show you a book in which 'infinite' is defined as I said it is defined in mathematics, while that definition is in the very book you mention as your reference. Again: It's not a matter of whether you agree or not with the definition, but rather that the definition is given in that book while you challenged me to cite such a book.
Meanwhile, my point stands that I think the chapter on the history and philosophy discusses the very point I made a while ago, but which you rejected, about the benefits of axiomatization.
Quoting Corvus
I have read Quine. Not enough though, since logic and mathematics not topics to which I devote my primary attention. If there is something you have to say about Quine, then you can say it.
Did Banno ever say or imply that he believes that?
Who are those many people?
That's an interesting idea.
The quote above, written to Banno, is exaggeration thus distortion.
Banno didn't say that you have made claims by quoting hundreds of philosophers..
And that quote doesn't support the claim you made that mathematics regards 'infinite' as meaning 'finite'. Though, lately, you say that claim is only a metaphor for something. If you like, you may remind me what exactly you intend it to be a metaphor for.
He did not say that anyone is wrong merely by the fact of disagreement.
'1+1' is not '2'
the denotation of '1+1' equals the denotation of '2'
the denotation of '1+1' is the denotation of '2'
1+1 equals 2
1+1 is 2
One may say that they don't themselves construe that way or that we shouldn't construe that way. But it is counterfactual to say that we don't construe that way.
/
One may claim that abstract objects should not be referred to in identity statements. But thought and communication, not just about mathematics, but even about everyday ideas, pretty much crashes if we are not permitted to apply the identity relation to abstract objects.
/
What passages in Aristotle are being referred to?
/
Saying that '1+1' names an operation rather than the result of the operation with the arguments is merely assertion. Again, one may say that one thinks that '1+1' should be regarded as a name of an operation, but that does not entail that in fact that is now how '1+1' is construed in ordinary mathematics.
/
The order of evaluation of terms and formulas is recursive. There is no ambiguity in that regard.
/
Yes, one can utter "1+1 = 8-1", but if the denotation of '+' is the addition operation on integers, the denotation of '1' is the integer one, the denotation of '8' is the integer eight, the denotation of '=' is the identity relation, and the denotation of '-' is the subtraction operation on integers, then "1+1 = 8-1" is false. This in no way vitiates that '=' stands for the identity relation. On the contrary, it is an example of the mathematics working just as we want it to work, just as it works even for the crank when he adds the number of pens on his desk, figures out his finances, relies on the entire body of science that allows him to stay alive, and uses the very computer he types on to send his ignorant, illogical and confused posts to this forum.
\
If one wishes to take '+' as intensional and not extenstional, then one should have a ball doing that. But that doesn't require that ordinary mathematics does that, espcially as, without some alternative rigorous framework, it would render thought and communication about even everyday mathematics unmanageable.
Several years ago, I moved to an apartment across town. Everything was very nice on the grounds of the apartment and in the neighborhood. Except the litter on the sidewalks. The first day, I picked up all the litter in front of my building, thinking that I wouldn't have to pick up for maybe another week or two. But the very next day, there was even more litter than I picked up the day before. And over time I started noticing that it was the same candy wrappers and soda cans every day. So it seemed that much of it was coming from a certain person. Someone daily spewing the same trash.
Shopkeeper: Here you go, eight cans of crank juice.
Crank: No, I said I want six plus two cans.
Shopkeeper. But six plus two is eight.
Crank: No, you must be reading those lying fool mathematicians with their extensional identity nonsense. I bet you even recite that awful axiom of extensionality every night before you go to bed. Six plus two means that I want you to get six cans then two more.
Shopkeeper: But I didn't have to do that. I already knew that I had eleven cans on the shelf, so I gave you all the cans on the shelf except three, so I subtracted three from eleven to give you eight cans.
Crank: No! No! No! I told, you just the way I posted at The Philosophy Forum, that six plus two is not eleven minus three. You're taking addition and subtraction extensionally! They are intensional, you dumb cluck! When I tell you what I want, I want it intensionally not extensionally! How can I be any clearer that six plus two is not eight?!
Shopkeeper: I'm very sorry, sir, but do you want the eight cans of crank juice or not?
Crank: Just forget about it. I want six plus two cans, not eight cans and definitely not eleven minus three cans! I guess I'll have to take my business elsewhere, even if I have to drive a potentially infinite number of miles to get there!
For my part, I hope he's right...
Captures the theme admirably.
Quoting TonesInDeepFreeze
The human mind can discover both abstract and concrete concepts in physical object
I agree. The ability of being able to think about abstract concepts, such as independence, beauty, love, anger and infinity, is a crucial part of the human mind.
When travelling to a foreign country, and hearing the word "hasira" for the first time, how in practice can we learn its meaning. If it were a concrete noun, the local could point out several physical examples that could be described by it in the hope that the foreigner was able discover what they had in common. However if it were an abstract noun, would the same approach be possible, as in pointing out several physical examples that could be described by the abstract noun?
If I visited a university with a guide, and the guide pointed at the buildings and people moving between the buildings carrying books, and said "this is a university", would I understand the meaning of "university" just as the concrete concept "a set of buildings" or the abstract concept "a place of teaching"?
I believe that the human mind is inherently capable of discovering when looking at a physical object both concrete and abstract concepts
IE, the human mind knows the meaning of words such as "beauty" and "infinity" and can think about the abstract concepts beauty and infinity, and could have learnt what concept is connected to what word by being shown several physical examples of each.
===============================================================================
Quoting TonesInDeepFreeze
The human mind can discover abstract concepts in physical objects
You are a foreigner in a foreign land learning the language. You have already learnt that the word "tufaha" means "apple", and you are now trying to learn the concepts expressed in the words "ni" and "sivyo". To make life more difficult, these are abstract concepts. But what is your best guess as the the meanings of "ni" and "sivyo"? The fact that are able to make an educated guess shows that abstract concepts can be instantiated in concrete examples.
===============================================================================
Quoting TonesInDeepFreeze
Abstract concepts don't of necessity refer to physical things, but wouldn't exist without physical things
Yes, the meaning of the abstract concept "exists" is independent of any particular physical thing referred to, whether an apple, a house or a government. However, the abstract concept "exists" would not exist in the absence of physical things.
The concept "Beauty" would not exist if there were no beautiful things. The concept "anger" would not exist if there were no angry people. The concept "governments" would not exist if there were no societies of people. The concept "infinity" would not exist if there were no physical objects.
IE, an abstract concept does not refer to a particular concrete entity, but abstract concepts wouldn't exist if there were no concrete entities.
===============================================================================
Quoting TonesInDeepFreeze
New words are learnt either within language or within a metalanguage
On the one hand, words can be learnt by description, where a new word is learnt from known existing words. For example, if I know the words "a part that is added to something to enlarge or prolong it", then I can learn the new word "extension".
However, if only words were learnt by description, there would be the problem of infinite regress. Sooner or later some words must be learnt by acquaintance, where a word such as "beauty" can be learnt by looking at particular "concrete instantiations", such as a Monet painting of water-lilies or a red rose in the garden.
Similarly, I can learn the meaning of the word "concrete" by looking at particular "concrete instantiations", such as a bridge over a river or a skyscraper in a city.
But, as you point out, this suggests an infinite regression, in that it seems that I can only learn the word "concrete" if I already know the meaning of "concrete instantiation".
However, this is not the case, as whilst words by description are learnt within language, words by acquaintance are learnt outside language and within a metalanguage.
IE, the new word "beauty", although it becomes part of language does not require language to be learnt. This avoids the infinite regress problem of learning the meaning of the word "concrete", if in order to learn the meaning of "concrete" I must already have to know the meaning of "concrete instantiation"
Quoting Michael
Then you'd have to reject the axiom of extensionality, and all axioms which follow from it, and set theory in general. As I explained, allowing that there is an object (abstraction, conception, or whatever you want to call it), which is referred to by a description like "1+1" is "truth" by correspondence. So it would be hypocritical to accept axioms which are demonstrably based in "truth", correspondence, yet claim that they are not truth-apt.
Quoting Michael
I'm not trying to make a mountain, just arguing a point, and points are "nothing". You are making points into a mountain by implicitly accepting Platonic realism.
Quoting Banno
The problem occurs when that axiom is interpreted as indicated that when A=B, then A is "identical" to B, in the sense that "A" and "B" each signify the same thing, as @TonesInDeepFreezeargues. This would mean that there is a "thing", with an identity, which is represented by both "A" and "B", such as in the examples provided by @Michael and @TonesInDeepFreeze. However, since there is no necessity of order within a set, and also there is such a thing as an empty set, it is very evident that it would violate the law of identity to interpret the axiom of extensionality as indicating "identity".
Incidentally, I argued extensively with @fishfry, that to read the axiom of extensionality as indicating identity rather than as indicating equality is a misinterpretation. However, it seems like identity is the conventional interpretation, and there are further aspects of set theory which require that equal sets are the same set And that produces a problem.
[Quoting Banno
My argument is a very simple one, and I am not trying to build it into a mountain. The point is that the sense of "identity" employed in set theory is not consistent with, therefore violates, a proper formulation of "the law of identity" expressed as an ontological principle. That itself is not a big deal, many philosophers like Hegel for example, have argued that there is no good reason for logicians to have respect for that ontological law. Leibniz, on the other hand, for example, argued that this law, along with the related principle of sufficient reason, ought to be respected. You may portray this as "the law of identity has various forms", but if the forms are inconsistent with each other, that implies inconsistency in what we believe constitutes "identity".
The thing which irks me as a metaphysician, (and why I argue this point fervently), is when philosophers of mathematics insist that the sense of "identity" employed in set theory is consistent with the "law of identity", as stated in ontology. These philosophers will employ examples like Tones and Michael did, of the "identity" of a physical object, implying that the "identity" of an abstraction is analogous, through some misinterpretation of "extensionality".
The reason it bothers me is that the law of identity is the principal tool employed by Aristotle against the sophistry of Pythagorean/Platonic realism. If we allow corruption of that "law", and ignore the difference between "identity" as employed by set theorists, and "identity" as stated in the law, we give up the front line in that defence, effectively surrendering to Eleatic sophistry (ref. Plato, The Sophist)
Quoting Banno
If any one of you would look at the evidence of what I've presented, the use of the equation in mathematics, they would see that "the rest of us" use "=" in the way that I describe. The common way, that of the applied arithmetic of the common people, and the applied mathematics of architects, engineers, and scientists, is the way I describe. It is only a select few, those immersed in the advanced mathematics of set theory, who desire, for the sake of this theory, that mathematical objects have an "identity", who choose to make "=" signify something different.
You ignore what ChatGPT told you in the other thread, common arithmetic and mathematics use "=" as equality not identity. And, in this thread GP said, that sometimes in "advanced mathematical contexts like set theory" ... "from the need to express relationships between objects", "=" will signify identity. Why do you refuse to accept what GP told you? That's because Tones told you 'don't to listen to that machine it doesn't support me', or something like that. But what is GP's account really based on? The "way the rest of us" use "=". Clearly, it's Tones who is "refusing to use "=" in the way the rest of us do", not Meta.
Quoting Banno
That's exactly the point. Objects each have an unique "identity", like Wittgenstein shows with the chair example. Even if two chairs might appear to be the same so that we couldn't readily decide which is which, we'd still know that through some temporal continuity each maintains its own unique identity. This ontological belief is expressed by the law of identity. Whether that law is actually true or not is not the point, it's just an ontological belief, and by believing it we assume that it's true. Internal sensations cannot be treated as if they have such an "identity". Therefore we make your conclusion, "internal sensations cannot be treated in the way we treat other objects". You seem to readily accept the conclusion which Wittgenstein comes to, without understanding the argument that he presents which produces it.
I see no philosophy nor mathematics in your latest replies to me. It appears you've simply gone off the rails in your crackpot ways. Oh well, maybe next time you'll be able to stay on track and manage a reasonable discussion.
Why do you hope that many people want your posts to be monitored by the moderators?
Make an appointment with an ophthalmologist.
In this part of the discussion, as in some recent posts, that is not directly about what mathematics itself in fact says, I am not trying to convince anyone else to see it the way I do. Rather, these are explanations of my best attempts to, for myself, have a framework to understand abstraction, truth and related questions. As a very broad generalization, I think of at least these two categories: (1) Matters of fact. (2) Matters of frameworks for facts. With (1), truth and falsity apply. With (2), coherence and explanatory robustness applies.
In this thread, for example: That ordinary mathematics says "1+1 is 2" is matter of fact. But whether ordinary mathematics should say that 1+1 is 2 is a matter of framework. And I take ordinary mathematics not as an account of facts about concretes, but rather as a framework for such facts. And, again, we need to distinguish between what ordinary mathematics does in fact say with what one may think it should say and what one thinks should be the framework for mathematics.
Frameworks involving abstractions sometimes work in an "as if" way. There is a wide range of thinking about what mathematical objects are - platonic, fictional, in some special sense concrete, or even extreme nominalism. But whatever we take mathematics to be talking about, at least we may speak of abstractions "as if" they are things or objects. Not concrete objects, but "as if" they are handled grammatically similar to they way concretes are handled.
Common examples, to the point of cliche, abound, such as that the knight in chess is not a concrete knight, not any particular piece of wood or plastic resting on a particular piece of wood or cardboard, but rather an abstract concept that we speak of similarly to the way we speak of concretes. This similarity does not imply that the abstract chess object that is the knight is a certain piece of wood. And so we use such locutions as "It either moves up or down one square vertically and over two squares horizontally, or up or down two squares vertically and over one square horizontally" The 'it' there must refer to something, and it sure as shootin' don't gotta a physical object. Also, 'moves' and 'square'. Moves can even be made by telephone and the piece that one player moves on his board at home is a different piece from the one the other player moves on his board at home, but it is understood (courtesy of ABSTRACTION) that those pieces represent the SAME knight. One can even perform a game of chess purely mentally, as chess masters actually do. And just as one can perform arithmetic and even more complicated mathematics purely mentally. That is courtesy of ABSTRACTION.
And the number 1 in mathematics is an abstract mathematical object that we speak of in a similar way to the way we speak of concretes, but that does not imply that the number 1 is a concrete object.
Going back to even more fundamental considerations, my starting premise is that experience is occurring. And as 'experience' and 'occurring' are the notions I start with, I must take them as primitive.
Notice that I didn't say 'experiences' plural, because I had not yet gotten to saying that not only does experience occur but that experience has parts and thus there are a multiplicity of experiences. But I do go on to say that if I do not allow there are many experiences, it would be intractable for me to talk about the experience that occurs.
Notice that I didn't say "I am having experiences", since I had not yet gotten to a premise that there is a thing that is named 'I'. But then I do refer to 'I' and as a thing, as it would be intractable for me to go beyond 'experiences are occurring' without being able to couch my experiences with reference to 'I', thus I as a thing.
As I go on, I find that certain other notions such as 'is', 'exists', 'thing' or 'object', 'same' 'multiple'. etc. are such that I don't see a way to define them strictly from the primitives I've allowed myself. So I then take such notions as themselves "built in" to whatever thinking I'm going to be doing.
So, by adding more concepts, I eventually - pretty crudely, without all the steps filled in, since I don't claim to have provided even for myself a rigorous philosophical system - get to the idea that there are other people having experiences, and that enough of these experiences have commonality among people such that we can submit claims about them to a process of judging claims as publicly factual or not, and with finer and finer standards of judgement such as those of the sciences. But the very determinations of fact, let alone the conceptual organization of facts, are vis-a-vis frameworks, and it is not disallowed that one may use different frameworks for different purposes.
For me, the value and wisdom of philosophy is not in the determination of facts, but rather in providing rich, thoughtful, and creative conceptual frameworks for making sense of the relations among facts. And, again, different frameworks may be used for different purposes.
Meanwhile, I would not contest that formation of concepts relies on first approaching an understanding of words ostensively.
As I say, I do not propose this as prescriptive, but rather only that it describes my own humble attempt to make sense of stuff for myself. Hopefully it might be a heuristically useful for others, but I don't insist that it must be.
Here is the most trivial model of both the axiom of extensionality and identity theory ('e' for the epsilon symbol):
The domain of discourse is {0 {0}}
'=' for the identity relation on the domain of discourse, i.e., {< 0 0> <{0} {0}>}
'e' for the membership relation on the domain of discourse, i.e., {<0 {0}>}
In that that model, all the axioms of identity theory and the axiom of extensionality are true.
Along the lines of "Bad publicity is at least publicity".
In mathematics, there are 'points of infinity' (which are not necessarily infinite sets), and the expression 'to infinity' (which is a figure of speech that when we unpack it we don't have an 'infinity' that is referred to), etc. But ordinarily there is no object that we name with the noun 'infinity', rather there is the adjective 'is infinite'. So, at least in a mathematical context, "Infinity is unknowable" doesn't have an apparent meaning to me. On the other hand the meaning of 'is infinite' is quite clear, as it means 'not finite'.
Quoting RussellA
That requires a framework that defines 'physical thing' or takes it as primitive. Then, depending on whatever framework is proposed, we could examine whether there cannot be abstractions independent of physical things.
It does seem to me that concepts are formed from prior ostensive inferences. But that is epistemological, not necessarily ontological.
I'm just glad that when I think of mathematics, there is the abstract mathematical object 1 and that there are not as many of the number 1 as there are each of certain physical events in brains starting and stopping.
Here is the axiom of extensionality:
Here is the law of identity
Set out for us exactly how these are not consistent.
Again, the crank LIES about me.
I never said that Chat GPT is not to be trusted because it doesn't agree with me.
Rather, it is not to be trusted because, over and over and over, it has been demonstrated, by many people, to spout blatant falsehoods. Indeed, even the makers of AI themeleves stress that AI is not necessarily a source of information but rather its primary role is as a composition tool.
Anyone can see for themselves that Chat AI outright fabricates, easily by asking it questions that it does not have ready answers to.
Moreover, in my conversation with Chat AI, it did say "1+1 is 2". But the crank SKIPS that.
To advance his illogical and confused argument, the crank resorts to LYING about what I said. He is pathetic.
Which is to say, he cannot do the impossible.
The axiom of extensionality is not used to prove axioms. Rather, the axiom of extensionality is used to prove theorems.
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The denotion of '1+1' is not truth by correspondence. '1+1' is not a statement, thus it doesn't have a truth value.
/
There are non-platonistic senses in which we may speak of mathematical objects.
/
Identity theory is congruent with Leibnitz's principles of the identity of indiscernibles and the indiscernabilily of identicals.
/
Chat GPT is not even remotely a reliable reference. Even a lowly fortune cookie can tell you: "He who cites Chat GPT as authority makes of himself a fool". Anyway, just to see, I asked Chat GPT, and it said that 1+1 is 2, and it did not confine that to set theory. And, of course, the cranks SKIPS that. Anyway, that a supposed philosopher is citing the famously "hallucinating" Chat GPT as an authority indexes how pathetic the Internet has become, not merely how pathetic he has always been.
So the reply will consist in an obfuscation of the law of identity by confusing it with an "ontological" principle. Mistaking a language act for a thing in the world.
Support Rep: Hello, thank you for calling MetaCard, your card to be used anywhere for making purchases of things about things. How can I help you today?
Crank: It says on my statement that I have 10000 rewards points. Please send them to my address: 666 South Georg Wilhelm Friedrich Hegel Alley, Apartment 0, Unreality Village, Planet Mars. No zip code on that.
Support Rep: I'm sorry, but they are rewards points. We don't send them to you. You use them for discounts on things.
Crank: How can I use them if I don't have them here? If they're real, then you can send them to me.
Support Rep: They are real. But they're not like physical goods that we ship to you.
Crank: Please don't tell me that you too have been infected by set theory! I'm asking you again to send me my points!
Support Rep: I'd like to help. Let me explain it this way: The rewards points are a number. It is noted on your account. Your points may change, but when we say "Metafizzled Blunderbound's points on February 18, 2024" we mean whatever number of points is on your account at that date. So it's a number, not a physical thing. However, depending on what that number is, you may apply that number for discounts on purchases you make. For each point you get ten cents off your puchase. There's nowhere for me to go get your number of points and ship them to you.
Crank: Look, what is that number now?
Support Rep: Metafizzled Blunderbound's number of points is 10000.
Crank. No, no, no! Not is. Equals! The number of my points equals 10000. Not the number of points is 10000. It is nonsense to say that the number of my points is 10000!
Support Rep: I can say 'equals' if you like. But I'm just telling you what your number of points is.
Crank: You are talking nonsense, and these are not even real things, so don't even bother!
Support Rep: So you don't want to apply your points to any purchases?
Crank: How can I apply them if they're not even real?! I surely am not going to be fooled into trying to use something that is not real! And I definitely am not going along with your bad philsophy where you say is instead of equals! Good day to you! (Hangs up.)
Two weeks later:
Bob (Blunderbound's neighbor): Hey, Blunderbound, how's it hangin'?
Crank: Okay, Bob. You should see how I'm tearing apart those set theory guys at The Philosophy Forum. Ha, I even showed them that Chat GPT says I'm right!
Bob: That's great. Hey, check out this waterproof watch I got. Got a discount with my MetaCard points. Have you used any of your points?
Crank: No, I don't use things that are not real. Besides, I don't need a watch. I have my sundial in the backyard. (Music cue, trombone "wamp wamp wamp")
That's great. Some kids really liked it, even though their parents didn't. I only tried teaching it in a typical college algebra course using a book by Vance. The first chapter was elementary set theory. My students had the ordinary curriculum in elementary through high school and for the most part were aghast at having to reason that a0=0.
Don't know that book, but
Ax x*0 = 0 is an axiom of first order PA, so it's easy to prove x*0 = 0
and in set theory, the PA axioms are theorems.
/
It was wonderful me in the 5th grade to learn about reasoning in mathematics and seeing things like different base numbering systems and modular arithmetic rather than just memorizing multiplication tables, executing steps in long division and reducing fractions to lowest terms.
Maybe it was aI=I. I don't recall. (In the recovery annex of the hospital recuperating from a broken leg at age 87) :sad:
I really hope you get well soon.
To begin with, the obvious. "Every element of A is also an element of B" is insufficient for identity by the law of identity because "A=A" implies that not only the elements, but also the order to the elements of A and B would need to be the same. Furthermore, every aspect of what is named A, and what is named B, must be precisely the same, even the unknown aspects.
Quite simply, stating some feature such as "every element is the same", is insufficient to qualify as identity by the law of identity, because the law of identity, as "a thing is the same as itself", or "A=A", implies that every aspect of the thing must be the same to qualify as "identity.
What?
Why would A=A imply that the order of the elements in B would need to be the same as A?
I think you've lost the plot entirely.
An axiomatization of identity theory:
Axiom:
Ax x=x
Axiom Schema:
If P is an atomic formula
and
Q is the same as P except y occurs in zero or more places in Q where x occurs in P
then all closures of the following are axioms:
(x=y & P) -> Q
/
Set theory has the primitive 2-place relation symbol 'e'.
We can formulate set theory with the 2-place relation symbol '=' taken as prmitive, or we can formulate set theory with the 2-place relation symbol as defined. The formulations are equivalant.
Primitive:
Adopt the axioms of identity theory and add the axiom of extensionality. But in the language of set theory, the only atomic formulas are of the form xez for any variables x and z (so also zex, yez and zey).
So the axioms of identity theory for the language of set theory include:
Ax x=x
Axyz((x=y & zex) -> zey))
Axyz((x=y & zey) -> zex))
Axyz((x=y & xez) -> yez))
Axyz((x=y & yez) -> xez))
Axiom of extensionality:
Axy(Az(zex <-> zey) -> x=y)
So we have the theorem:
Axy(x=y <-> (Az((zex <-> zey) & (xez <-> yez))
Defined:
Axy(x=y <-> (Az((zex <-> zey) & (xez <-> yez))
So the two formulations are equivalent.
Notice that we have these theorems:
Axy(x=y -> (Az((zex <-> zey) & (xez <-> yez)), which carries, for the language of set theory, Leibniz's indiscernability of identicals.
and
Axy((Az((zex <-> zey) & (xez <-> yez) -> x=y), which carries, for the language of set hteory, Leibniz's identity of indiscernabiles.
/
Moreover, the consistency of the axiom of extensionality with idenity theory is trivally proven by the trivial model I mentioned:
The domain of discourse is {0 {0}}
'=' for the identity relation on the domain of discourse, i.e., {< 0 0> <{0} {0}>}
'e' for the membership relation on the domain of discourse, i.e., {<0 {0}>}
In that that model, all the axioms of identity theory and the axiom of extensionality are true.
The crank can't properly respond to that, because he doesn't know anything about models and consistency.
An ordering is a certain kind of relation on a set.
The axiom of extensionality pertains no matter what orderings are on a set.
{0 1 } = {1 0}
{<0 1>} is an ordering on {0 1}
and
{<1 0>} is an ordering on {0 1}
{<0 1>} not= {<1 0>}
{<0 0> <1 1>} is a sequence whose range is {0 1}
and
{<0 1> <1 0>} is a sequence whose range is {0 1}
{<0 0> <1 1>} not= {<0 1> <1 0>}
Orders and sequences are rigorous in set theory. And the axiom of extensionality is not inconsistent with them.
See your ad hominem attacks on other interlocutors from the beginning of your posts? That is not a good manner at all. Please just discuss the philosophy. Have some respect. Don't throw insults to the other interlocutors.
This is exactly how you have started in this thread on many of your previous posts. If you track back your posts, you will see them clearly unless you have edited them out. My statements on the point is proven to be true here.
You don't present any logic in your claims and statements. You just imagine that I didn't say something, and that is the only ground for your false claim. Where is your logic and evidence for your claim?
Still speaking on behalf of Banno? Look I am not interested in your ad hominem posts begging for attention. I am here to read and discuss philosophy.
I posted the links. That's the evidence. The logic is pretty much inferring that what is posted at the links says just what it says.
No, I don't speak on his behalf. I speak on my own behalf to say that it is a plain fact that Banno did not exaggerate by saying 'hundreds' but that you exaggerated by saying that he did say 'hundreds'.
Again it's in the plain record of the posts.
That is not logic. Logic must have premises and conclusions. The premises must be backed up by the evidence. You don't seem to know even what Logic means.
Jesus Banno, if A is the same as B, as implied by "A=B", (if "=" signifies identity, or "the same"), then the order of A's elements is the same as the order of B's elements, necessarily, as this is a part of "being the same"..
Quoting TonesInDeepFreeze
That is the first, and most obvious piece of evidence which indicates that the axiom of extensionality does not state identity. Clearly "identity" by the law of identity includes the order of a thing's elements, as it includes all aspect of the thing, even the unknown aspects. So the ordering of the thing's elements is therefore included in the thing's identity, unlike the supposed (fake) "identity" stated by the axiom of extensionality.
Yes, he made his post sounding like that. Do you still not understand any simile or metaphor expressions in English?
Hey look, if you don't have any meaningful philosophy to write down, please remain silent. We want to discuss philosophy here.
First, you lied that earlier I began with ad hominems.
Second, you skipped my reply about that my arguments are not ad hominem, but rather I give arguments that are not ad hominem but also add the needed observation that the interlocuter is indeed ignorant, confused, and dishonest, as at a certain point it deserves remarking that he is ignorant, confused and dishonest.
Indeed, you can see that my first posts in this thread, and others like it, are devoid of personal remarks, and my first posts to new interlocuters are devoid of personal remarks. But, eventually stubbornly ignorant, confused and dishonest posting deserves to be pointed out for being what it is.
Meanwhile, as you take such umbrage to disparaging remarks, you're free to cut them out of your own posting. I think it's your prerogative to make them, but yours happen to be quite inapposite, which is putting it mildly.
That is ridiculous. You accused him of exaggeration. That's not simile or metaphor. So you exaggerated, not him.
Look at the links and my remarks about them.
At least, I presented the logic that I have never lied. And I have now the evidence of your post quote, you starting your post with ad hominem insults to the other interlocutors.
You, have no logic, no evidence, no ground for your claims. But just make up false statements and claims on the others.
Now I am only asking you to stop your nonsense, and let us get on with the philosophical discussions with the basic manners, respects and rationality.
There are different orderings on sets.
There is no such thing as "THE" ordering for sets with at least two members. There are often what we call 'standard orderings' but still there is not just "THE" ordering of a set with at least two members.
Again for the crank:
The axiom of extensionality pertains no matter what orderings are on a set.
{0 1 } = {1 0}
{<0 1>} is an ordering on {0 1}
and
{<1 0>} is an ordering on {0 1}
{<0 1>} not= {<1 0>}
{<0 0> <1 1>} is a sequence whose range is {0 1}
and
{<0 1> <1 0>} is a sequence whose range is {0 1}
{<0 0> <1 1>} not= {<0 1> <1 0>}
The treatment of orders and sequences is rigorous in set theory. And the axiom of extensionality is not inconsistent with the theorems about them.
Of course, you can't deal with the plain fact of the record of posts, which document not only that you've been lying (which itself is insulting) but which also includes relatively detailed remarks by me about mathematics and philosophy.
See above post.
Christ, Meta, sets are not order.
Quoting Metaphysician Undercover
The order of the elements is not part of what a set is. See
But we are at the point where further discussion is without purpose. Again, you have shown that there is no value in discourse with you.
You havn't posted anything of philosophical merit for page after page; just bitchin'.
Here's the link that proves it.
Quoting Banno
Your posts are biased and full of distortion of the facts as usual. I don't see a point in philosophical discourse with you either. You claim that you care for philosophy, but in reality you distort the truths with your bias, prejudice and false judgement.
That is not an insult but "ignorant and confused" is?
I am not sure what planet you live, and say that. But it is an insult, and definitely needless thing to say to your interlocutors without valid reasons.
If someone came on, and replied to your post starting "You are ignorant and confused ... intellectually incompetent" without any evidence or ground, then I am sure you wouldn't feel pleasant.
Then, please, don't feel any need to reply to my posts. For page after page. :wink:
But if you do want to get back into a conversation that is on topic, you might re-phrase whatever your position is, taking into account the various responses hereabouts.
I wonder what makes of this. They haven't responded at all, but that seems to be their way; they are in the unusual position of having less comments (8) than Discussions (11)...
I think I've had enough of this. There's been no progress for days. Thanks for sharing your insights.
Exactly, the ordering of the elements which make up "a thing" is essential to the identity of the thing. Therefore "identity" in set theory is not consistent with "identity" as stated by the law of identity, which is a statement about things.
We might go on and consider the supposed identity of an empty set as well. What type of "thing" has no elements in its composition? Well, that's not a thing at all, and it has no identity, because "identity" by the law of identity is a statement about things.
Further, we might consider whether a thing with infinite elements could really have an identity. That's a difficult philosophical question, which you might just take the answer for granted, because there's a serious lack of rigour in your concept of "identity".
Quoting Banno
I know, and that's exactly the point, because order of a thing's elements is an essential aspect of identity. That's why if two sets are said to be "the same", they are not the same by the conditions of the law of identity, because the order of the elements is not included in that supposed (fake) identity..
How do you suppose that there is a thing which has an identity, yet that thing has no order to its elements? That's not a thing at all. And if it's not a thing it has no identity, by the law of identity, which is a statement about things.
Quoting Banno
Yes, as usual, I prove you to be wrong in your belief, and then you go off and ignore me for a period of time. The problem though, is that you never learn, and will come back later to argue what has already been demonstrated to you as wrong. Oh well, its no loss to me.
As usual, you evaded the point. Again:
"distortion [...] bias, prejudice and false judgement."
That is not an insult but "ignorant and confused" is?
As to what is not pleasant, it is not pleasant to have one's posts misrepresented, strawmanned and outright lied about, as you regularly do, and to confront supposed arguments against them that skip their key points, as you regularly skip the key points.
You can look back in this thread to see that I posted back and forth with you with my not saying anything remotely personal, until I pointed out that you were skipping the points.
And copious evidence and argument have been given showing that the main crank in this thread is ignorant and confused about this subject - including right up to this very moment.
...unless it isn't, as is the case with sets...
Quoting Metaphysician Undercover
No; and that's why the order is irrelevant when determining if two sets are the same...
Fucksake.
I sincerely regret having entered into a direct discussion with you. I will try not to make the same error again.
Yes, "the order is irrelevant when determining if two sets are the same". But the order of the elements is essential to determining the identity of a thing. And the law of identity is a statement about the identity of things. Therefore the identity of sets is not consistent with the law of identity. Understand?
/
Set theory is an axiomatic system with one non-logical primitive. From the axioms we prove there is a unique object such that there is no x such that x bears the relation denoted by 'e' to said object. If the nickname 'the empty set' does' comport with one's notions about set, then it can be called 'the zempty zet' or 'the-thing-that-has-no-things-on-the-left-of-it'. The nicknames would not alter the formal theory.
Generally as to what things don't have members other than the empty set: urelements.
This is truth. He just comes along says that the others interpretations are wrong, and there is no arguments or logical ground for that judgement. No one can think differently from him.
Quoting TonesInDeepFreeze
If you call someone ignorant and confused from the start of your posts, when it is you who are ignorant and confused, then that is an insult to the person. You may not know that, because he is not saying anything, but just thinking about it. It is also unnecessary to say things like that in philosophical discussions.
I am sure if someone said that to you, it would be because you said it to him first. I know you said something like that first to me, and wasn't pleasant.
Quoting TonesInDeepFreeze
I have not been replying to all of your walls and walls of off topic posts to me. I don't see a point in ad hominem posts. I have no time or inclination for getting involved in non-philosophical quibbles with you. I was just pointing out problems in your posts for the inaccuracies and personal comments you were putting out.
The issue I'm discussing is identity. It's only indirectly related to the op, so if you do not want to discuss this, that's fine. What you can "prove from the axioms" is irrelevant, when it is the acceptability of the axioms which is being questioned.
First, it's not. Second, it is also truth that the main crank here is ignorant and confused about the subject.
So it seems you think that ""distortion [...] bias, prejudice and false judgement" is not an insult, because you think it is true, but "ignorant and confused" is an insult.
As to start of posts, there are different starting points: The start of a single post, the entry posts in a thread, and the first posts between posters in this forum. In the start of this thread, I did not make personal remarks. Over time, as the main cranks misrepresents, strawmans, posts in ignorance and confusion on the subject in this thread, then I remark on that. And this is in context of a MASSIVE amount of that kind of insulting dishonesty in many other threads in this forum.
It's not a question of replying to all of what poster writes, but it deserves remarking when you criticize posts while skipping their key points and misrepresent what they say.
So an hourglass changes its identity as each sand grain drops.
A few pages back I said:
Quoting Banno
...and here it is. Thanks.
You're free not to!
You don't see a point in them, but that doesn't stop you from posting insults.
And, again, it is very important to distinguish between an ad hominem ARGUMENT and, on the other hand, stating an non-ad hominem argument but in addition remarking that a poster is confused, ignorant and dishonest, especially when detailed explanation is given the poster as to what his ignorance, confusion and dishonesty are.
I wasn't in the direct discussions on the topic of identity you have been discussing. But when I connected to the thread, the first thing I saw in your post was you throwing out the sentence saying "ignorance and confusion" to the other interlocutor. I immediately recalled what you have been saying to me in the similar way previously, and it gave a strong impression, that you have been insulting not just me, but the others who don't agree with your opinions.
Bottom line is, that you should try to avoid doing that if possible. It can happen during the discussions in the heat of the moment unintentionally. But if you keep doing that constantly, and especially at the start of your posts, then your posts will look as if they are intentionally meant for insulting others.
You're making claims about the axiom vis-a-vis identity. So it is very relevant what the axiom proves regarding identity.
And as I said: There is a difference between what mathematics says and what one thinks mathematics should say. So anyone is welcome to say how they think mathematics should be formulated, and better yet, to provide an actual formulation. And anyone is welcome to say why they think the ordinary axioms are not acceptable. But to do that, one should at least understand what those axioms are and how such mathematics is formulated. And not to continually strawman about them. Also it always helps to understand context, which here includes why mathematics adopts axioms, what the axioms of classical mathematics are intended to provide, and the perspective of the use of classical mathematics for the sciences and computing.
Again, for the hundredth time, I don't remark on the ignorance, confusion and dishonesty of posters merely because they disagree with me.
And, of course, you read the first line, but not the context that justifies that first line.
Meanwhile, look at your own posts.
And as I said, nobody's stopping you from not talking about it.
Quoting Corvus
While you are free to not post in an insulting manner.
Quoting Corvus
You were replying to me in an insulting manner (criticizing my arguments while not even addressing their key points and taking me for a fool with ignorant and false red herrings that study of mathematics is just a bunch of regurgitating what is in book, or however you actually phrased it) well before I said even a word about you personally. And as I observed you replying insultingly to another poster in this thread in even the earliest part of this thread.
Oh, yes, the main crank's insistence that set theory handle order in the way he thinks it should be handled, even though he is ignorant of how set theory does handle order.
No, the law states "a thing is the same as itself". Nothing here says that the thing cannot change as time passes. But all those changes are necessarily a part of the thing's identity. That's one of the important features of the law of identity, it allows for a true understanding of the temporal continuity of things, and the reality of change itself, by allowing that a thing maintains its identity despite changing.
Have you no familiarity with the law of identity? It seems to me that you've only been exposed to misrepresentations, proposed by logicians who want to reformulate it to support their own proposals. I do not argue that it is without problems, like the one presented by The Ship of Theseus example. And as I said earlier, some philosophers propose that we reject the law of identity altogether. But that would give us no principles for understanding the reality of temporal continuity. You see, there is an incompatibility between eternal unchanging Platonic "Ideas", and the temporal continuity of objects which are constantly changing. The law of identity refers to the latter, and the identity which a set is said to have refers to the former.
Quoting TonesInDeepFreeze
I don't think so. A proposition (or axiom) needs to be judged by the principle it states, not based on what can be proven through the use of it. If you accept an axiom because it can prove what you want it to prove, that is just begging the question.
Quoting TonesInDeepFreeze
I'm starting to like that handle, it makes me feel powerful like the driving part of a magnificent machine. Do you think it would be suitable for me to change my name?
But, of course, you will resort to any argument you can to evade actually learning anything about the subject on which you dogmatically declare.
Let's refresh just recent matters alone:
You claimed that axiom of extensionality is inconsistent with identity theory. I proved it is not. You evade that, because you know virtually nothing about identity theory, the axiom of extensionality or consistency.
Most basically, you haven't a clue what the axiomatic method is about.
I explained for you how set theory does provide for the identity of indiscernibles and the indiscernibility of identicals.
You claim that a set and an ordering on the set determine the set. You were saying that years ago in this forum and it was debunked then. You still don't get it. Still don't get what even a young child can understand.
/
I don't care what you call yourself, but if you do change your name, at least I'll know that the denotation of 'Main Crank' is the denotation of 'Metaphysical Underground', which is the same foolish, arrogant, ignorant, illogical, irrational, dogmatic, confused, intellectually blocked, dishonest, lying poster he always was.
That's a good example of a crackpot reply. You are avoiding the issue, by switching to "identity theory" rather than the law of identity. The problem I brought up is that "identity" in set theory is not consistent with "identity" in the law of identity. Whether or not "identity" in set theory is consistent with "identity" in "identity theory" has no bearing on the problem I've exposed.
Ax x=x
That is one of the axioms of identity theory.
I posted that earlier today, but of course you SKIPPED it.
Since the axiom of extensionality is consistent with identity theory, perforce it is consistent with the law of identity.
So I am not at all avoiding the issue or switching the issue.
/
You keep using the word 'consistent' while you show no inconsistency; we only learn from your bungled arguments that the axiom of extensionality does not accord with your confused and dogmatic views about mathematics. As I've said an uncountably infinite number of times if I've said it a countably infinite number of times, no one disallows you from positing your own framework of understanding, but the mere fact that mathematics does not adopt your framework (which is, meanwhile, confused) does not make mathematics incorrect. And you keep evading many of the other arguments and considerations about set theory, such as the application of classical mathematics to the sciences and computability, including the existence of the computer you're typing on right now. It is funny how not only is it the case that cranks can never answer that point, but they will never even recognize it.
Moreover, when you resort to dogmatically insisting that a set is determined by a particular ordering, you resort to silliness that was debunked years ago in posts to you in this forum. We went over it and over it back then, and you still don't get it. I think you don't get it because you have a mental block about such things. So I suggest that not only should you make an appointment with an ophthalmologist to find out why you can't see things right in front of your eyes, but you should see a cognitive psychologist to find out about the mental blocks you have that disallow you form understanding even such basic ideas that can be understood by a young child.
It seems that "infinity" as an object is more a problem of natural language than mathematics and set theory.
As you also say:
Quoting TonesInDeepFreeze
As I indicated earlier, the issue is with the way that x=x is interpreted. Unless the interpretation employed by "identity theory" is consistent with the way that the law of identity is stated in its original formulation, "a thing is the same as itself", then the meaning of "x=x" which is employed by "identity theory", is not consistent with the law of identity.
What is required now, is that you state the interpretation of "x=x" which is employed by "identity theory", and more specifically "set theory", such that we can judge it for consistency with the law of identity, "a thing is the same as itself".
That's better, it's more relaxed in the lounge, and may serve to lower the tension by a few foot-pounds or something like that. I hope there's no drinks available here though, or things might go the opposite way.
I wonder what "nicknamed" would imply in supposed rigorous logic.
Quoting TonesInDeepFreeze
:up: For me, the statement "Monet's Water-lilies is an example of beauty" is a fact and is true. However, I am speaking within the framework of a European Modernist. Within a different framework, say that of a Californian Post-Modernist, the statement, may be neither a fact nor true.
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Quoting TonesInDeepFreeze
:up: Within a different framework, say that of binary numeral system, 1 + 1 = 10
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Quoting TonesInDeepFreeze
Is this an example of Putnam's Modalism, the assertion that an object exists is equivalent to the assertion that it possibly exists?
If I said "I am going to buy an apple", I am not referring to "an apple" as a particular concrete thing or object, but rather referring to "an apple" "as if" it were a particular concrete thing or object.
Whilst the definite article refers to a particular concrete thing "a house" "a mountain" or "a cat", the indefinite article, "a house", "a mountain" or "a cat", doesn't refer to a particular concrete thing, but rather refers to a particular concrete thing that possibly exists.
In language also, we can refer to things that exist, "I want this cat", and refer to things that possibly exist, "as if" they exist, such as "I want a unicorn".
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Quoting TonesInDeepFreeze
What does "it, the knight on a chess board, refer to?
"It" must refer in part to a physical object that exist in the world and in part to rules that exist in the world.
The game of chess is played between two people, and as neither player can look into the other's mind, the rules must exist in the world in order to be accessible to both players. For example, "the knight either moves up or down one square vertically and over two squares horizontally, or up or down two squares vertically and over one square horizontally". However, as rules cannot refer to themselves, in that rules cannot be self-referential, they must refer to something external to the rules, in this case, a physical object.
IE, if there were no rules there would be no game of chess, and if there there were no physical objects the rules would have nothing to refer to.
There are therefore two aspects to "it". The extension, the physical object of a knight, and the intension, the rules that the knight must follow.
Such an approach to understanding "it" is supported by Wittgenstein's Finitism. Wittgenstein was careful to distinguish between the intensional (the rules) and the extensional (the answer). Mathematics is the process of using rules contained within an intension to generate propositions displayed within the extension. For example, the intension of 5 + 7 is the rule as to how 5 and 7 are combined, and 12 is the extension. (Victor Rodych - Wittgenstein's Anti-Modal Finitism - Logique et Analyse)
Such as approach to understanding "it" also follows from natural language. The intension of the word "beauty" is a rule that determines what is beautiful and the extension of the word "beauty" are concrete instantiations, such as Monet's Water-Lilies or a red rose in a garden.
There may be many possible rules for what is beautiful. Francis Hutcheson asserted that Uniformity in variety always makes an object beautiful.. Augustine concluded that beautiful things delight us. Hegel wrote that The sensuous and the spiritual which struggle as opposites in the common understanding are revealed as reconciled in the truth as expressed by art .
It is impossible for a finite mind to have a list of all beautiful things in the world, yet can recognise when they see something is beautiful. The human mind has the concept of beauty prior to seeing a beautiful thing. Such a rule is probably innate, the consequence of millions of years of evolution existing in synergy with the outside world.
Such a rule is the intension of the word "beauty" and physical examples, such as a Monet Water-Lily are the extensions of the word "beauty".
IE, "it" refers to the intension and extension of the word "knight". The intension being the rule the "knight" follows and the extension being the physical object ,whether made of wood or plastic.
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Quoting TonesInDeepFreeze
However, if there were no concrete objects in the world, there would be no concept of the number "1".
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Quoting TonesInDeepFreeze
:up: Yes, there are some concepts, such as "beauty", that we cannot learn the meaning of by description from the dictionary, but are probably innate within us. Innatism is the view that the mind is born with already-formed ideas, knowledge, and beliefs.
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Quoting TonesInDeepFreeze
In my terms, thinking about the concept "beauty", which is probably innate, and therefore primitive within us, there only needs to be one intrinsic rule able to generate numerous extrinsic examples.
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Quoting TonesInDeepFreeze
In Kant's terms, we have a unity of apperception. The mystery is why.
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Quoting TonesInDeepFreeze
Certain words such as "house" can be defined as "a building for human habitation, especially one that consists of a ground floor and one or more upper storeys". We can learn these concepts from the dictionary using definitions. But sooner or later, we come across other words, such as "is", "exists" and "thing" that are primitive terms, cannot be defined, but only learnt from acquaintance.
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Quoting TonesInDeepFreeze
:up: For me, a Modernist, the statement "Monet's Water-lilies is a beautiful painting" is true, but for others, the Post-Modernists, the same statement is false.
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Quoting TonesInDeepFreeze
But how can there be wisdom in the absence of facts. How can we understand the wisdom of Kant without first knowing those facts he applied his wisdom to?
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Quoting TonesInDeepFreeze
:up:
A very rare case on the internet of someone who wants to listen more than they want to talk.
The logic is not merely supposed to be rigorous. It is rigorous in these senses: (1) The axioms and rules of inference are recursive, thus, for a purported proof given in full formality, it is mechanical to check whether it is indeed a proof, i.e., merely an application of the inference rules to the axioms. (2) It is proven that the logic is sound, i.e. that a formula is is provable from a given set of formulas only if the formulas is entailed from the set of formulas.
Moreover, the nicknaming (my word) I mentioned is not so much about the logic but rather about defined symbols in a theory such as set theory.
Set theory, as formalized, uses only formal symbols, not natural language words. A formal proof is not allowed to use connotations, associations or any of the suggested notions that natural language words have. However, for everyday communication of proofs among mathematicians and students it is unwieldy to recite exactly each formal symbol in the formulas that are sequenced for a proof. Moreover, it aids picturing the content of the theory to informally use words. I call that 'nicknaming'. For example, in set theory in all formality, there is no constant term 'the empty set'; instead there is a 1-place operation symbol, a pure symbol, with a purely formal definition. Moreover, as I've mentioned, the adjective 'is a set' or even a formal predicate for 'is a set' are not even required as in formal theories such as ZFC. So, as concerns the formal theory, it doesn't matter whether or not one's personal notion of sets allows that there is one special set that has no members that is called 'the empty set' but rather, the theory has a formal theorem, such as:
E!xAy ~yex
thus a definition
0 = x <-> Ay ~yex
So, in that particular regard, we could just as well use the nickname "zee zempty set". It would not change the "structure" of the mathematics, which is the relationships of the definitions and theorems.
Note that my remarks about this are not necessarily a commitment to extreme formalism expressed as "mathematics is just a formal game of symbols". Rather, in this context, we may note that, no matter what framework or philosophy one has for understanding mathematics, at least we have the formalization, even just the fact of that formalization, as a component in our understanding - whether a fully self-contained and isolated component (i.e. extreme formalism) or as merely a point of reference and a rigorous constraint against handwaving.
I agree. There was developing an interesting discussion on the law of identity and (non-ordered) sets. Or so it seems, I just glanced at it.
A moment of clarity.
That's not an example of what I was talking about. I'm talking about general frameworks such as hold one's intuitions, perspective or philosophy, not matters such as variations in base numbering systems.
Quoting RussellA
No.
Quoting RussellA
When I wrote 'it', I was not referring to a particular piece of wood or particular array of pixels on a screen. I'm referring to the idea, the thing that players who are not even in each other's presence - thus not moving the same piece of wood or even seeing the same array of pixels - can still refer to as "the knight".
Quoting RussellA
But it doesn't. Again, one may argue that leading up to the formation of the concept, there were particular pieces of wood or ivory or whatever that were carved to resemble a knight and that were moved around on a two-colored board. But soon enough, we have the abstraction that can be referred to. Indeed, 'chess' itself can be defined mathematically without even mentioning particular characters such as 'knight', but rather only a purely mathematical construct. We could say there are four objects, called 'WKL', 'WKR', 'BKL', 'BKR' ("white" and "black" "knights" on "left" and "right") and the other "pieces", then define a C-sequence (sequence of "chess moves") to be a certain sequence of matrices with those objects associated with cells in a matrix and successive matrices having a property that the "pieces" are in different cells only according to certain allowed ordered pairs of cells ("moves"), etc. So you see that when I say 'it', I'm talking about an abstract object, even though attaining that abstraction required previous concrete or ostensive understanding.
Quoting RussellA
I am not opining whether or not the basic concepts 'is', 'exists', 'same', etc. are innate. Indeed, I have no ready argument that they are not first understood only ostensively. I'm only reporting that I don't know how I could arrive at successively more involved frameworks without them.
Quoting RussellA
I don't say that there can be.
More a painfully needed, though unsuccessful, intervention than a discussion.
The points are simple:
* In mathematics, in ordinary context, 'x=y' is true if and only if x and y are the same object, which is to say 'x=y' is true if and only if what 'x' stands for is the same as what 'y' stands for. The claim that there are no such objects is not properly given as an objection to the fact that '=' stands for identity, since we would still have '=' standing for identity if the objects were physical, concrete, fictional, hypothetical, 'as if', abstract, platonic, etc.
* Sets are not determined by an order in which the members happen to be mentioned. If I say, "What are the members of the set of books on your desk", then if you say, the set of books on my desk is all and only the books 'The Maltese Falcon', 'Light In August' and 'The Stranger', then no one could say "No, that's wrong, the set of books on your desk is actually all and only the books 'Light In August', 'The Stranger' and 'The Maltese Falcon'!"
{'The Maltese Falcon', 'Light In August', 'The Stranger'} = {'Light In August', 'The Stranger', 'The Maltese Falcon'}
{8, 5, 9} = {5, 9, 8}
And '=' reads fine whether 'equals', 'is identical with' or 'is'.
No law of identity is violated there.
/
Boss: Jake, tell me what is the set of items on our shipping clerk's desk?
Jake: It's the set whose members are a pen, a ruler, and a stapler.
Maria: But he also has another set on his desk! It's the set whose members are a ruler, a stapler, and a pen.
Boss and Jake: Wha?
Boss: Maria, take the rest of the day off. You're not quite with it lately.
/
{pen, ruler, stapler} = {ruler, stapler, pen}
Nobody says that the set of items on a desk is different depending on the order you list them.
On the other hand, mathematics does have ordered pairs and triples. For example:
=
With ordered tuples, yes, order does matter.
Since soundness requires true premises, and the logic you are talking about proceeds from axioms, which are not truth-apt, instead of from true propositions, how do you propose that it could be "proven that the logic is sound"?
Quoting TonesInDeepFreeze
The point I made, is that the sense of "identity" you use here, is not consistent with the sense of "identity" used in the law of identity. So it doesn't really matter that you insist that "=" stands for "identity". Anyone can make up one's own personal sense of "identity" and have a symbol for it, state the axiom, and persuade others to use the axiom, and even create a whole "identity theory", but that doesn't make that sense of "identity" consistent with the law of identity.
Quoting TonesInDeepFreeze
This is an excellent example of why your sense of "identity" is not consistent with the law of identity. By the law of identity, if the identified thing is "the books on your desk", then everything about that thing, including the order of the parts, must be precisely as the books on your desk, to satisfy the criteria of "identity". Stating an order other than what the books on your desk actually have, would not qualify as an identity statement, because that specific aspect, the order of the parts, would not be consistent with the thing's true identity.
Quoting TonesInDeepFreeze
I have become fully aware that you are not at all familiar with the law of identity. Therefore your statements about the law of identity, and whether it is violated under specific conditions, I simply take as off-the-cuff remarks of a crackpot.
Quoting TonesInDeepFreeze
Anyone with any degree of common sense recognizes that the identity of the specified thing, "the items on a desk" includes the ordering of the mentioned items. If describing those items as a "set" means that the ordering of the mentioned items is no longer relevant, then you are obviously not talking about the "identity" of the specified thing, which is "the items on the desk". You are talking about something other than "identity" as defined by the law of identity.
[math]2(x+5)\equiv 2x+10[/math] Identity
[math]2(x+5)=3[/math] Conditional
Here's how I would put it:
2(x+5)= 2x+10
is understood to be implicitly universally quantified:
Ax 2(x+5) = 2x+10
and that is true
Then, by universal instantiation, we have:
2(x+5) = 2x+10
and that is true for every assignment of a value to the variable 'x'
2(x+5) = 3
is understood to be implicity existentially quantified:
Ex 2(x+5) = 3
and that is true
In high school algebra, we are asked to state the members of the "solution set", which is to say:
{x | 2(x+5) = 3} = {-7/2}
and that is true
(1) With logical axioms and rules of inference. (Known as 'Hilbert style'.)
or
(2) With only rules of inference. (E.g., natural deduction.)
In either case, we have the soundness theorem, since the logical axioms are true in every model and the rules of inference are truth preserving.
/
This point has been posted already, but, alas, the posts come back around full circle when the correct explanations are skipped again and again:
The law of identity is that a thing is identical with itself. Using '=' to stand for 'identical with' is not inconsistent with the law of identity. Indeed, the law of identity is stated:
Ax x=x
That is, for all x, x is identical with x. That is, for all x, x is x.
Not only is '=' as used in mathematics consistent with the law of identity, but the law of identity is itself an axiom of identity theory that is taken as part of the logic used for mathematics.
/
This point has been posted already, but, alas, the posts come back around full circle when the correct explanations are skipped again and again:
"the ordering of the set" is meaningful only when there is only one ordering of the set (in this context, by 'ordering' we mean a strict linear ordering). Any set with at least two members has more than one ordering.
The set whose elements are all and only the members of the Beatles in 1965 has 4 members, and there are 24 orderings of that set.
In other words, there is no "the" ordering of the 4 membered set {x | x was a member of the Beatles in 1965} since that set has 24 orderings.
/
"crackpot" is said by the crank calling the pot cracked.
After much back and forth, you finally revealed to me that you do not understand the basic material implication [math]P \implies Q[/math] of propositional logic; and that you are incapable of understanding the statement of the axiom of extensionality on its Wiki page. I'm afraid I can't dialog with you further till you remedy these very basic misunderstandings.
Ha, ha, very funny.
From that wiki page:
"or in words:
Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B."
I see the phrase "A is equal to B", but where does it indicate that A is the same as B?
It's better for me if I defer continuing this discussion at this time.
But for what it's worth, the symbol string "same" has no meaning in ZF. I do not know what it means, and I do not need to know what it means in order to do set theory.
The symbol "=" is being defined by the axiom of extensionality. You're adding things that aren't in the game. It's as if I'm trying to teach you chess and you say, "Where are the zebras?"
I wasn't being funny.
"Same" has a meaning in the law of identity. So when you say that the axiom of extensionality is a statement of "identity", you are employing the concept of "same", where it does not belong. "Same" is implied by "identity". That's why I argued that to interpret the axiom of extensionality in this way, as a statement of identity, is a faulty interpretation.
Quoting fishfry
Actually, it is you who is adding things that aren't in the game, with your faulty interpretation. You are adding "identity", when the law of extensionality is really a definition of "equal". As I've been telling you, equality and identity are not the same concept. This is because "identity" implies "the same" whereas "equal" does not. So, we can have two different things which are equal, but two different things cannot have the same identity.
Therefore it is incorrect to interpret the axiom of extensionality, which is clearly an expression of equality, as an expression of identity. To interpret as a statement of identity is to add something which is not in the game, sameness, when the statement really concerns equality instead, which does not imply "same".
What do you think identity in mathematics / set theory is?
So the axiom extensionality is that sets are equal if they have the same elements, if I understand it correctly.
So I think then the question for you, @Metaphysician Undercover, is how is the identity different between two sets that have the same elements?
Because you say "to read the axiom of extensionality as indicating identity rather than as indicating equality is a misinterpretation", it seems that you think this is different. A lay person would think that a set defined by it's elements.
And please just look how identity is defined in mathematics, and you'll notice what @fishfry is talking about.
__ __ __
Nice to see you, @fishfry on the forum again! It's been a while. :grin:
The article states that the axiom of extensionality uses '=' with regard to predicate logic, with a link to an article on 'First-order logic'. And that article correctly states that the most common convention is that 'equals' means 'the same'. Moreover, the article on 'Equality (mathematics)' defines equality as sameness, and the article on the equals sign refers to equality, and the article on 'Identity (mathematics)' refers to equality.
In ordinary contexts in mathematics, including mathematical logic, including set theory, 'equals' means 'is the same as', which means the same as 'is identical with'. This is formalized by identity theory, which extends first order logic without identity, and adds a primitive binary predicate symbol '=' with axioms and a semantics.
More specifically:
/
Identity theory is first order logic plus:
Axiom: Ax x=x
Axiom schema:
For all formulas P,
Axy((x=y & P(x)) -> P(y))
Semantics:
For every model M, for all terms T and S,
T = S
is true if and only if M assigns T and S to the same member of the universe.
/
Set theory can be developed in at least two ways:
(1) First adopt identity theory. This gives us:
Theorem: Axy(x=y -> Az((xez <-> yez) & (zex <-> zey)))
Then add the axiom of extensionality:
Axiom: Axy(Az(zex <-> zey) -> x=y)
This gives us:
Theorem: Axy(x=y <-> Az(zex -> zey))
Thus, with identity theory and the axiom of extensionality, every model of
Az(zeT <-> zeS)
is a model that assigns T and S to the same member of the universe.
(2) Don't adopt identity theory. Instead:
Definition: Axy(x=y <-> Az((xez <-> yez) & (zex <-> zey)))
That gives us as theorems all the axioms of identity theory.
However, if we don't also stipulate the semantics of identity theory, the axioms of identity theory along with the axiom of extensionality do not provide that every model in which S=T is true is a model that assigns S and T to same member of the universe.
Thank you. This forum drives me to extended vacations sometimes.
I'm pretty sure I never said that, but if I did, please supply a reference to my quote.
Quoting Metaphysician Undercover
Now you're getting it.
I don't think mathematics/set theory deals with identity at all. I think that identity is an ontological principle defined by the law of identity. Mathematics deals with equality, which is distinct from identity. Some people on this forum have argued against this, claiming that there is an identity within set theory. But then they only seem to be able to argue that "equal" means "the same", when clearly this is false.
Quoting ssu
Having the same elements does not mean being the same as. Having a different order for example would make two sets with the same elements not "the same" by the law of identity which indicates that a thing is the same as itself only. So, even identifying them as "two sets" indicates that they have a different identity.
Quoting ssu
Yes, identity is very different from equal. By the law of identity, we only call it "the same" if it is one and the same thing. The computer I typed on this morning is the same computer as the one I type on now. Two equal things are not necessarily the same. There were other computers at the store which are equal to mine but each one of them is different, i.e. not the same computer as the one I brought home. I think a lay person would agree with that.
Quoting ssu
If you can find that definition for me, I'll take a look. Then we can discuss whether "identity" in mathematics is consistent with the law of identity.
Quoting fishfry
My apologies, for misrepresenting what we argued about. I thought you argued that the axiom of extensionality indicated identity.
As to sets and order, as has been demonstrated very many times on this forum, sets with at least two members have different orderings, so there is not "the" ordering of a set.
A while ago, I gave this example: The set whose members are all and only the bandmates in The Beatles is a set. But there is not "the" ordering of that 4 member set. Indeed there are 24 orderings of that set:
https://thephilosophyforum.com/discussion/comment/884421
Or put it this way, if every set has an order that is "the" order of the set, then the set whose members are all and only the bandmates in The Beatles has an order that is "the" order. If one will venture to state which of the 24 orders of that set is "the" order, then I can ensure that we could find at least 23 Beatles fans who would disagree with that being "the" order.
A definition of 'identity' was requested and the poster said he will look at it. In identity theory in mathematics, '=' is not primitive. But the semantics require that S=T is true if and only if 'S' and 'T' name the same thing. To look at this in more detail and with all the groundwork for it provided, one may look one of many introductory textbooks in mathematical logic.
(1) If we use identity theory at the base of set theory, then the axiom of extensionality merely adds a sufficient condition for '='. And the semantics of identity theory provide that '=' means 'the same as' or 'is identical with'.
(2) If we do not use identity theory at the base of set theory, then then we may use the axiom of extensionality but augmented with an additional clause to define '='. However, without the semantics of identity theory, it is not the case that such an axiom alone proves that '=' means 'the same as' or 'is identical with'.
Apology accepted. I do see how my view may have seemed that way to you. For example I am certain that I'd have maintained that if A and B are sets, and we can write A = B, then A and B are the same set.
That is certainly true in set theory. But I think it's really more true in the metatheory or the way we talk about set theory, than set theory itself.
In set theory, "same" is a shorthand for "satisfies the premises of the axiom of extensionality." You are trying to overload the word with metaphysical baggage that it simply does not have in math. The axiom of extentionality is syntax. You are imbuing it with semantics that you are making up or bringing over from other meanings of the word you may know. You need to take things on their own terms when studying any technical field.
What I mean is, formal set theory says:
"If such-and-so, then we can write A = B."
In casual, everyday talk about set theory, we say, "A and B are the same set if they have exactly the same elements."
So you see there is a gap between those. Set theory is a purely syntactic exercise. If, given the definition of "=" as in the axiom of extensionality, we can derive a formal proof from the axioms that A = B, then we can write A = B from now on.
But all this talk of "sameness" is really a very loose casual adaptation of the axiom of extensionality. And in so doing, we seem to add semantics to it. As if we are making a metaphysical claim that A = B.
But in actuality we are not doing that!! Rather, we're simply claiming that the symbol "=" is to henceforth be defined as this other condition.
So the set theory is syntactic; and it's a mistake to confuse our everyday casual talk about set theory, with some kind of ontological claim.
tl;dr: When a set theorist says two sets are "the same," there is a formal derivation from first principles that A and B satisfy the premises of the axiom of extensionality. It's a purely syntactic exercise.
They are NOT implying any kind of metaphysical baggage for the word "same." If pressed, they'd retreat to the formal syntax.
Make sense? You are using "same" with metaphysical meaning. Set theorists use "same" as a casual shorthand for the condition expressed by the axiom of extensionality. It's a synonym by definition. The set theorist's "same" is a casual synonym; your "same" is some kind of ontological commitment. So all this is just confusion about two different meanings of the same word.
Also, meta: This thread, "Infinity," is active, and I keep getting mentions for it and replying. But this thread does not show up in my front-page feed! Anyone seeing this or know what's going on?
What I've argued is that this use of "same", is not consistent with "same" as defined by the law of identity. And if this sense of "same" is claimed to be constitutive of "identity", as Tones argues, then this is a violation of the law of identity. If this is what you want to call "metaphysical baggage", that's ok with me. There are many words with similar forms of "metaphysical baggage". The use of a specific word in one field may contradict its use in another field, and it only becomes a problem if people start to believe that the two uses are consistent with one another.
Quoting fishfry
You may want to take this up with Tones, and his notion of "identity theory", which obviously is a kind of metaphysical baggage.
Quoting fishfry
That completely makes sense. However, not every mathematician is as reasonable as you are. If you look at what TonesoffTheDeepEnd is writing here, you'll see great effort to support some kind of formal identity theory. That is not a "casual shorthand for the condition expressed by the axiom of extensionality".
Quoting fishfry
What has happened is that there was a new policy initiated which was to relegate a whole lot of these rambling, bickering, blah. blah, blah, type of threads to The Lounge. And threads in The Lounge don't show on the front page. You'll find The Lounge in the list of Categories on the left side of the front page.
And it is exactly my point that use of terminologies in different fields are often not compatible with one another, and, as I have said many times in this forum, and again in this thread, mathematics makes no claim that '=', 'equals' and 'is identical with' are used in mathematics in the same senses as in all those in everyday life and in other fields of study.
And I don't have a personal sense of 'identity theory'. I am merely referring to a publicly studied formal theory.
And I don't claim to "support" identity theory. I am merely saying what it is, what some of its theorems are, something about the semantics that goes with it, and how it relates in certain ways to set theory.
There is nothing "off the deep end" about anything I've said here. Barely clever putdowns by means of renaming posters might be at least minimally apropos if they were based on at least something.
That is just one of myriad examples. Without an answer, the notion that every set has its "the" ordering is dead in the water.
The poster challenged by asking where in a certain Wikipedia article it says that 'equals' means 'the same'. I pointed out: The article states that the axiom of extensionality uses '=' with regard to predicate logic, with a link to an article on 'First-order logic'. And that article correctly states that the most common convention is that 'equals' means 'the same'. Moreover, the article on 'Equality (mathematics)' defines equality as sameness, and the article on the equals sign refers to equality, and the article on 'Identity (mathematics)' refers to equality.
I don't usually reference Wikipedia, but at least it is abundantly clear that the poster's challenge regarding what Wikipedia does happen to say is answered, when he could have found out it out for himself. It's typical of the poster to stick with his method of railroading full speed ahead with his own claims and challenges while hardly ever granting that they have been answered.
Again, usually set theory presupposes identity theory, in which case it is by semantics that the interpretation of '=' is stipulated, and in which case '=' means 'is the same as'. And if set theory does not presuppose identity theory, then the axiom of extensionality is not enough syntactically, as we need the axiom of extensionality with an added clause. And still that is not enough to have that '=' means 'is the same as'. The details were given here:
https://thephilosophyforum.com/discussion/comment/897006
Quoting Metaphysician Undercover
Ok,
Even if the discussion has moved on, I'll just point out this, what identity in math is and why math does deal with identity:
or in an other way:
And what it isn't:
It's in the Lounge.
It was deemed not Philosophical enough, or just math. Or lousy math. :yikes:
The reason is that the Ukraine crisis thread and the Israel-Palestine thread (Israel killing civilians in Gaza and the West Bank) along other political threads got so heated and ugly, the admins decided to put them into the Lounge (meaning not Philosophical debates). Having to do with the appearance that a Philosophy Forum site would discuss eloquently Philsophy, I guess. :snicker:
All right, you just confirmed what I thought. "Identity" in mathematics is equality. That clearly violates the law of identity. The law of identity allows that identity is a relation between a thing and itself, so there is only one thing involved. Equality on the other hand allows that two distinct things may be equal to each other. So unless you can provide this distinction between equality and identity, or else show that to be equal necessarily means to be one and the same thing, in set theory, then you ought to accept this violation. And, as I mentioned already, we know that the latter cannot be done, because set theory allows that two sets with the same elements in different orders are equal. Therefore I think we ought to just conclude that "identity" in mathematics is a violation of the law of identity. Agree?
A certain kind of equality, identity is an equation that us true for all values of its variables.
An equation might be true for some variables, like x+1=3 is true if (iff?) x=2. There's also equality, but not identity. Hence equality isn't always identity.
Law of identity, that each thing is identical with itself, isn't actually math, but general philosophy. So I guess the law of identity is simply a=a or 1=1. Yet math it's actually crucial to compare mathematical objects to other (or all other) mathematical objects. Hence defining a set "ssu" by saying "ssu" = "ssu" doesn't say much if anything. Hence the usual equations c=a+b.
It's very hard to think here that math would go against logic, so this is more of a mixture of definitions here. It's like comparing what in Physics is work and what in economics / sociology is work. The definitions are totally different.
It is adopted in mathematics.
Ax x=x
is math.
/
Using '=', 'equals', and 'is identical with' interchangeably does not violate the law of identity.
Suppose I owe a creditor a certain amount of money, and ask them, "I have record of my balance as being 582 dollars plus 37 dollars. Do you have the same number?" They say, "Yes, your balance is 619 dollars and 0 cents." It would be ridiculous for me to say, "No! 582 plus 37 is not the same number as 619.00!"
582+37 is the same number as 619.00.
582+37 is identical with 619.00.
582+37 is equal to 619.00.
582+37 = 619.00.
That is not vitiated by the fact that:
'582+37' is not the same expression as '619.00'
Even a child can understand that
2+2 = 4 means that '2+2' and '4' name the same number.
/
In set theory, there are no two different sets with the same elements and different orderings.
However, for any set with at least two elements, there are different orderings on that set. To express a set S with an ordering R on S and a different ordering Y on S, we may simply say:
R is an ordering on S & Y is an ordering on S & ~R=Y
To talk about a set S and a particular ordering R on S we may mention;
Millions of people who have studied mathematics understand that. Including those who have built the digital computers we are using at this moment.
/
There are 24 orderings of the set whose members are the bandmates in the Beatles. That doesn't entail that there is more than one set whose members are the bandmates in the Beatles. There is only one such set. It is the set whose members are the bandmates in the Beatles no matter how you order them.
So, I'm still curious what "the" ordering of the Beatles is supposed to be.
If one cannot answer that, then one ought not claim that for every set there is "the" order of that set.
And, in this particular case, by a lack of response to the question, I take it that the poster who makes that claim has no answer.
"To the Lounge with this rubbish" indeed!
Thanks.
Quoting ssu
I'm sure that might happen someday .... /s
I'm not a mathematician. I studied math in school, long ago.
Thanks for calling me reasonable.
I can't defend the views of other posters, and I can't engage with what someone else might have said.
Quoting TonesInDeepFreeze
For sure. The mods have an aversion to math-related topics that I don't understand.
This is the crucial point, which I'd like to bring to your attention. The law of identity is an ontological principle which deals which "objects" as we meet them in our daily lives, all the different objects which we sense around us, and it forms a defining principle of what it means to be an object. It is meant to recognize the reality of these objects, in the face of skepticism, and offer guidance as to the type of existence which they have. It is probably the most universal of general principles, applying to all the different types of objects that we might possibly encounter.
On the other hand, supposed "mathematical objects" are distinctly different, and the law of identity does not apply. Now we might simply consider that the so-called mathematical objects are not even objects, therefore they don't have any identity as such, and they are really just thoughts and ideas. That seems like a reasonable approach. However, some people want to compose a special type of "identity", specifically designed for these thoughts and ideas, and through reference to this special type of identity they argue that these thoughts and ideas actually are a special type of object, mathematical objects.
However, it ought to be clear to you, that this is just smoke and mirrors sophistry. The special type of identity is formulated for the special purpose of creating the illusion that these thoughts and ideas are a type of object. But it is actually impossible that these ideas are a type of object because the law of identity applies to all types of objects which we might possibly encounter, and this supposed special type of object requires a distinct form of identity which is incompatible with the law of identity.
Quoting ssu
I don't think that's true true, "work" in physics is consistent with "work" in economics/sociology, physics is simply a broader sense, and "work" in sociology is narrowed down to work is done by human beings.
In the case of "identity" in mathematics, it is inconsistent with, (in violation of), "identity" in the law of identity.
Quoting TonesInDeepFreeze
Obviously, you do not understand the law of identity. By the law of identity, the symbols printed here, 582, are distinctly different from the symbols printed over here, 582. Although they appear very similar there is a different position, context, etc., so it is very clear that this is not two instances of one and the same thing, 582. By the law of identity two separate occurrences of the printed symbols, are not the same thing, they are two similar things. Under the conditions of the law of identity we cannot say that 582 printed here is the same thing as, or has the same identity as, 582 printed here.
Therefore I recommend that any wise person would completely disregard the following statement, as coming from the mind of someone who does not know what they are talking about.
Quoting TonesInDeepFreeze
(2) Indeed, one should not conflate symbols and occurrences of symbols. In the formula:
1 = 1
the only symbols that occur are '1' and '=', but there are two different occurrences of '1'. Indeed, the first occurrence of '1' is not the same as the second occurrence of '1', but still the symbol '1' is the symbol '1'.
The law of identity is: "a thing is itself"
Or: "a thing is identical with itself"
Or: for all x, x is identical with x
Or: for all x, x is x
Symbolized: Ax x=x
Indeed we do not thereby claim that the three occurrences of 'x' in that formula are the same occurrence. To argue that we do is a strawman.
And the point stands that we well understand that '2+2' and '4' refer to the same number. There are not two different numbers, one named '2+2' and one named '4'. There are two different expressions: '2+2' and '4', but they name the same number.
And, for example, the procedure of adding 2 to 2 is different from the procedure of subtracting 2 from 6, but the result of those procedures is the same. And '2'+2' and '6-2' do not stand for procedures but rather for the result of the procedures. So '2+2' and '6-2' stand for the same number. So mathematics writes: 2+2 = 6-2.
/
We still have not heard a reply to the challenge to state "the" ordering of The Beatles, pertinent to the claim that every set has only one ordering. After several requests to even address the challenge to state "the" ordering, I take it that the poster has no answer and no willingness even to address the challenge. And so it stands that the poster has no viable claim that sets have only one ordering.
If this is addressed to me, please clarify what you are asking..
By the way, I never said that a set has an inherent order. I acknowledge that a set does not have an inherent order, and that is a problem for the "identity" of a set. A thing has an ordering of its parts as a feature of its identity, covered by the law of identity.. And that's why the "identity" of a set is inconsistent with "identity" by the law of identity. Do you remember now?
What is supposed to be THE order of the set whose members are all and only the famous four bandmates in the Beatles?
And woah! The poster is now saying the exact opposite of what he said for dozens of posts in this forum.
Over and over, the poster argued against the axiom of extensionality on the grounds that there is THE ordering of a set. Yes, I do remember.
And now he's denying he said that sets do have a certain ordering that is the ordering of the set.
The poster is flat out lying.
And completely reversing his previous argument. He says now that sets do not have an inherent order so that the law of identity does not apply to them. But he had been saying that sets do have an inherent order so that that order must be regarded when we evaluate self-identity. But it is the axiom of extensionality, which the poster denies, that provides that sets do not have an inherent order. And the poster denied the axiom of extensionality on the basis that sets do have an inherent order.
Then the poster mentions "things", though he denies that mathematical abstractions are things. But now he talks about sets as things. And he says their identity depends on their ordering but that a set can't be identical with itself. (What? A set can't be identical with itself?) Why can't a set be identical with itself? Now he says it's because a set does not have a certain ordering. Not only does it make no sense that a set is not identical with itself, it makes no sense that the reason a set is not identical with itself is that it does not have a certain ordering, especially when the poster's argument was that the reason sets fail the law of identity is that they do have a certain ordering.
The poster is not only lying, but he's very confused.
For any set that has more than one thing as a member (whether mathematical things or material things), there is more than one ordering of that set. That does not contradict the law of identity.
The set whose members are all and only the famous four bandmates in the Beatles.
That set is identical with itself (the law of identity applied to that set). But there is more than one ordering of that set. If a set being identical with itself required having one particular ordering, then no set with more than one member could be identical with itself. But the set that whose members are the bandmates in the Beatles is identical with itself.
So again the question for the poster: What is supposed to be THE one particular ordering of the Beatles that permits the set whose members are the bandmates in the Beatles to be identical with itself? Or, it seems that the poster's answer now is that sets are not identical with themselves. But that itself violates the law of identity, since the law of identity is that every thing is identical with itself - and whether material things or mathematical things.
MU (many times a while back): The axiom of extensionality is wrong because sets have a certain order but the axiom allows that sets are equal without regard to their order.
TIDF (then and now): Sets of more than one member have more than one order. If every set has just one order, then, for example, what is the order of the set whose members are all and only the famous bandmates in the Beatles?
MU (many times): [silence]
TIDF (many times): Still interested in an answer.
MU (now): Sets do not have a certain order. And that is why sets are not identical with themselves.
TIDF (now): Many times you said that sets do have a certain order so that the axiom of extensionality is wrong. Now you say that sets do not have a certain order and that that is why they are not identical with themselves. But it's a violation of the law of identity that there are things that are not identical with themselves. So, sets, whatever they are, are identical with themselves. And from identity theory with the axiom of extensionality we have that sets are identical if and only if they have the same members, which denies that sets have only one certain order, and the law of identity is upheld not contravened. You are dishonest, self-contradictory and confused, and you don't know anything about this subject.
Quoting TonesInDeepFreeze
Tones, I argued that the axiom of extensionality does not indicate identity in a way which is consistent with the law of identity, because the identity of a thing (by the law of identity) includes the order of the constituent elements, while the identity of a set (by the axiom of extensionality) does not include the order of the elements. Therefore, i conclude that "identity" in set theory, as indicated by the axiom of extensionality is inconsistent with the law of identity. That sort of "identity", found in set theory, is a violation of the law of identity. If you really think that I was arguing the opposite to this, then I'm sure you can provide a reference.
If anyone is lying, it is you in your misrepresentation of what I argued in the past. However, I do not think you are lying, I think you have difficulty understanding English. Perhaps you have been overworking yourself, being immersed in mathematical symbols for so long that you no longer understand common language.
[s]This is revisionist:[/s] [strikethrough in edit, since 'revisionist' is not the right word there.]
"the identity of a thing (by the law of identity) includes the order of the constituent elements, while the identity of a set (by the axiom of extensionality) does not include the order of the elements."
Yes, by the axiom of extensionality, sets do not have an inherent order. But in the past the poster argued that therefore the axiom of extensionality is wrong, because there IS the ordering of a set. That is why I asked: What is THE ordering of the set whose members are the bandmates in the Beatles.
Moreover:
A thing that has elements (or at least is made up only of its elements) is a set. So if things have an ordering that is "THE" [emphasis added] order of its elements, then, AGAIN, that is to say that, for sets, there is THE ordering of a set.
As I said:
Originally, his longstanding claim was that there is "THE" [emphasis added] ordering of a set . And that was the basis for him rejecting the axiom of extensionality. So I challenged by asking what does he claim is THE ordering of the set whose members are all and only bandmates in the Beatles (keeping in mind that there are 24 different orderings of that set, as, in general any set with n number of members has n! orderings).
But yesterday he said that sets do "NOT" [emphasis added] have an inherent ordering. So that was a reversal from his previous longstanding claim. But, even worse, yesterday he contradicted himself on the matter of whether there is "THE" ordering of a set.
Today he's back to:
There IS "THE" ordering of "things [with] constituent elements".
And that is what I said that he had said originally.
/
And contrary to the poster with his wildly ersatz, ignorant, stubborn, and mixed up ideas not just of mathematics but of everyday notions, I do recognize the sense of sets in everyday discourse. For example:
There is the set whose members are the bandmates in the Beatles. Suppose a painter asks for the Beatles to sit for a portrait - left to right in their order. Then they would say, "WHAT order?" There is no THE order of the Beatles. There are 24 of them.
Even if a person doesn't know to calculate that there are twenty-four orderings of a set with four members, a person does understand that there is not just one ordering of the set.
If a schoolmaster says, "Now, children, line up in your order." Even children would have the sense to say, "What order? Order by height? Order by age? Order by grade point average? Or what?".
But the poster still cannot grasp this fact of common understanding, though he presumes to take refuge in a sense of "common" language.
Sheesh!
Blah, blah, blah., so much hot air. Show us your evidence. Fishfry showed me years that there is no necessary order to the elements of a set. That's definitional, why would I argue against it?
What I argue, as I've argued for years, is that the so-called "identity" of set theory, is inconsistent with, therefore in violation of, the law of identity. And, the fact that there is no order to the elements of a set is very good evidence for what I argue. However, I do not argue that set theory is "wrong" on account of this violation, because some philosophers suppose the law of identity to be unacceptable. I argue that people like you, who insist that identity in set theory (or what you call identity theory), is consistent with the law of identity, are wrong.
So fly away now, Mr. Balloon, because you expulsions of hot air are threatening to blow the thread off track. .
Evidence includes the poster's penultimate post: "I argued that the axiom of extensionality does not indicate identity in a way which is consistent with the law of identity, because the identity of a thing (by the law of identity) includes the order of the constituent elements, while the identity of a set (by the axiom of extensionality) does not include the order of the elements."
[s]And revisionist:[/s] [strikethrough in edit, since 'revisionist' is not the right word there.]
"I do not argue that set theory is "wrong" on account of this violation, because some philosophers suppose the law of identity to be unacceptable."
It is not at all typical that philosophers of mathematics who are interested in set theory suppose the law of identity to be unacceptable.
What specific philosophers is the poster referring to?
And what the poster says makes no sense, again. He endorses the law of identity (no?). And even one post ago he falsely claimed that set theory is not consistent with the law of identity. But now, revisionistically, he says he doesn't fault set theory, and that the reason he doesn't fault set theory is that some philosophers deny the law of identity. But if he endorses the law of identity, then it would make no sense for him to let set theory off the hook on the basis that there are philosophers who disagree with his endorsement of the law of identity. The poster is brazenly illogical, again.
/
The law of identity is that a thing is identical with itself. It is an axiom of identity theory, which is presupposed by set theory. Not only is set theory consistent with the law of identity, but the law of identity is one of the pre-axioms of set theory.
/
And the poster is back to claiming that a set has an ordering that is THE ordering of the set. And still he does not address the natural rejoinder: If a set has an ordering that is THE ordering of the set, then which of the 24 orderings of the set whose members are the bandmates in the Beatles is THE ordering of that set?
/
As to the track here, the poster's position regarding mathematics in this context is fairly paraphrased as:
1/2 + 1/2 is not 1, because you can't cut pie without some crumbs falling around so that the resulting two pieces of pie are not precisely the same size.
No exaggeration. The poster's notion is that that 1/2 + 1/2 is not 1, since the '1/2 + 1/2' is not '1' (though the particular example was different). (And even the most utterly ridiculous irrelevancy that with '1 = 1', the first occurrence of '1' is not the second occurrence of '1'.)
And now, moreover that the identity assertion fails, because, for example, pie cutting is not exact.
Hegel for example:
https://thephilosophyforum.com/discussion/9078/hegel-versus-aristotle-and-the-law-of-identity/p1
Moreover, the context is the law of identity vis-a-vis mathematics. As I said, it is not at all typical that philosophers of mathematics who are interested in set theory suppose the law of identity to be unacceptable. So it is not typical for mathematicians and philosophers of mathematics to vindicate set theory on the basis of denying the law of identity. Quite the contrary, it is typical for mathematicians and philosophers of mathematics to accept or endorse the law of identity, especially as the law of identity is one of the axioms used in set theory.
The law of identity in its historical form is ontological, not mathematical. Mathematics might have its own "law of identity", based in what you call "identity theory", but it's clearly inconsistent with the historical law of identity derived from Aristotle. He proposed this principle as a means of refuting the arguments of sophists such as those from of Elea, (of which Zeno was one), who could use logic to produce absurd conclusions.
Discussion with you about this is pointless because you make statements like the one above, where you acknowledge the difference between the mathematical concept of "identity" and the ontological concept of "identity", but you claim that the only relevant concept of "identity" is the mathematical one.
Of course, relevance depends on one's goals, and truth is clearly not one of yours.
"you claim that the only relevant concept of "identity" is the mathematical one"
That is false. I've said very much the opposite in this forum. Of course identity is treated in philosophy aside from mathematics and as an everyday notion. And especially in philosophy and in certain alternative mathematics there are a great many differing views of the subject, all of which are we may benefit by study and comparison.
I wrote:
"Moreover, the context is the law of identity vis-a-vis mathematics."
That was in response to the poster's own claims about the law of identity vis-a-vis mathematics. That is, the context of this part of the discussion with the poster has been his attack that mathematics is incompatible with the law of identity. I have never at all claimed that mathematics has sole authority regard the subject of identity. Rather, I have shown (in posts in this forum) how the poster's attacks on mathematics vis-a-vis identity are ill-founded.
The poster disputes that mathematics upholds that law of identity. But the law of identity, both symbolically and as understood informally, is an axiom of classical mathematics. The poster has two prongs in reply:
One prong in the poster's reply is (as best I can summarize): Mathematicians may claim to state the law of identity, but those statements are incompatible with the actual law of identity, since mathematics regards numbers and such* as objects but those are not objects, and the axiom of identity pertains only to objects.
* If I say 'numbers and other things' or 'mathematical things', then that is tantamount to referring to objects since 'object' and 'thing' are synonyms. So, if we may not refer to mathematical objects then we should not even use the word 'thing' regarding, well, mathematical things.
The other prong in the poster's reply is: Mathematics regards sets as identical if and only they have the same members, but that ignores the orderings of the members of the sets, and the ordering of a set is crucial to the identity of a set, so sets are not identical merely on account of having the same members.
(1) OBJECTS.
If I'm not mistaken, objection to referring to numbers and such as objects is something like this: There are only physical, material or concrete objects, but mathematics regards numbers and such as abstractions.
But, without prejudice as to whether numbers and things are not physical, material or concrete, in mathematics, philosophy and in everyday life, people do refer to numbers and such as objects. It's built into the way we speak, as numbers and such are referred to by nouns and are the subject in sentences. If we weren't allowed to speak of numbers and such as objects then discourse about them would be unduly unwieldy.
But one might counter, "People talk like that, or think they need to talk like that, but that doesn't entail that it is correct that they do." To that we might say, "Fair enough. So when we say 'mathematical object' we may be regarded, at the very least, as using the word 'object' as "place holder" in sentences where it would be unduly unwieldy otherwise. The mathematical formulations themselves do not use the word 'object'. One could study formal mathematics for a lifetime without invoking the word 'object'. But to communicate informally about mathematics, it would be unwieldy to not be allowed to use the word 'object'. Moreover, when 'object' is used in that "place holder" way, one may stipulate that one does so without prejudice as to whether mathematical objects are to be regarded as more than abstractions, concepts, ideals, fictions, hypothetical "as if" things, platonic things, values of a variable, members of a domain of discourse, etc.
And it seems to me that the notion of 'object' itself may be regarded as primitive - basic itself to thought and communication. Any explication of the notion of 'object' would seem fated to eventually relying on the notion of 'object'.
In regards all of the above: Philosophy of mathematics enriches understanding and appreciation of mathematics, but one can study mathematics for a lifetime without committing to any particular philosophy about it. Moreover, one can study philosophy for a lifetime without committing to any particular philosophy. One may critically appreciate different philosophies without having to declare allegiance to a certain one. And one may use the word 'object' in a most general sense, even in a "place holder" role, without saying more about then that we may regard one's usage without prejudice as to how it should or should not be explicated beyond saying, "whatever sense of object that you may have about the "things" mathematics talks about".
The poster wrote:
"The law of identity in its historical form is ontological, not mathematical. Mathematics might have its own "law of identity", based in what you call "identity theory", but it's clearly inconsistent with the historical law of identity derived from Aristotle."
(2) LAW OF IDENTITY
To start, from the above quote, should the poster be charged here with argumentum ad antiquitatem? Even if not, there are more things to say.
Just to note, if I'm not mistaken, Aristotle's main comments about identity are in 'Metaphysics'. I don't have an opinion whether that's properly considered ontology.
The law of identity is usually stated as "A thing is identical with itself", or "A thing is itself" or similar.
Through history identity became an important subject in logic, philosophy and mathematics.
In logic, two central ideas emerged: The law of identity and Leibniz's identity of indiscernable and indiscernibility of identicals.
Eventually, mathematical logic provided a formal first order identity theory:
Axiom. The law of identity.
Axiom schema. The indiscernibility of identicals.
(The identity of indiscernbiles cannot be formulated in a first order language if there are infinitely many predicates, but it can be formulated in a first order language if there are only finitely many predicates.)
Along with the axioms, a semantics is given that requires that '=' stand for 'is identical with'. That is taken as 'is equal to', 'equals', 'is', 'is the same as' or any cognate of those.
So when we write:
x = y
we mean:
x is identical with y
x is equal to y
x equals y
x is y
x is the same as y
However, the poster, in all his crank glory, continues to not understand:
x = y
does NOT mean:
'x' is identical with 'y'
'x' is equal to 'y'
'x' equals 'y'
'x' is 'y'
'x' is the same as 'y'
but it DOES mean:
what 'x' stands for is identical with what 'y' stands for
what 'x' stands for is equal to what 'y' stands for
what 'x' stands for equals what 'y' stands for
what 'x' stands for is what 'y' stands for
what 'x' stands for is the same as what 'y' stands for
The law of identity is:
For all x, x = x
And by that we mean:
For all x, x is identical with x
And that does not depend on what kind of objects x ranges over. WHATEVER you regard a term 't of mathematics to refer to, the referent of 't' is identical with itself.
Classical mathematics does uphold the law of identity as it has been ordinarily understood in philosophy and as it came through Aristotle.
(3) EXTENSIONALITY
Still, the poster cannot say what THE ordering is of the set whose members are the bandmates in the Beatles. So, still, his claim (every set has and order that is THE order of the set) is not sustained, thus still unsustained is his second prong mentioned above.
I can't believe this thread is still going. I see you've hijacked it to your hobby horse.
For my edification, can you explain the above paragraph?
My understanding of your point is this, and do correct me if I'm wrong.
My concept of your thesis: The law of identity says that a thing is equal to itself. Mathematical equality is not metaphysical identity. Therefore math is wrong. Or something.
But nobody claims mathematical equality is identity. It may be spoken of that way in casual conversation, and by mathematicians who have not given the matter any thought and mean nothing by it.
When pressed, a mathematician would readily admit that mathematical equality is nothing more than a formal symbol defined within ZF set theory in the logical system of first order predicate logic. It's not actually the same as mathematical identity. It's not the same as anything.
If we like, we can visualize that equality is identity. Why not? Everything in our mathematical world is a set; if there are things that are identical but not equal as sets, they're entirely out our consciousness. So they often think of it that way. But they don't mean it as any kind of metaphysical thesis. It's just a manner of speaking, like jargon in any field.
Any logically or philosophically-oriented mathematician will immediately concede the point; and the rest would have no opinion at all. Most working mathematicians don't spend any times thinking about whether mathematical equality is logical identity. The question doesn't enter their minds.
So that's what I understand about your thesis, and I don't know anything about the Eleatics. What is your point with all this?
Quoting Metaphysician Undercover
I'm just jumping into this convo between you and @TonesInDeepFreeze, and I can't speak for why he wrote what he did.
But I can tell you that I would agree with what he said, in the context of mathematics.
That is, if I'm a mathematician, all of the objects I deal with in my life are sets of one kind or another, and we know what equality for sets, we defined it via the axiom of extension.
You would be fun in set theory class. You're entirely hung up on the very first axiom. "Class, Axiom 1 is the axiom of extensionality. It tells us when two sets are equal." You, three years later: "But that's not metaphysical identity! You mathematicians are bad people. And you don't understand anything!" And your professor goes, Meta, We still have a countable infinitely of axioms to get through! Can we please move on?
But axioms don'g mean anything. They're just rules in a formal game, like chess. As I say, if you asked a mathematician if mathematical equality is metaphysical identity, a few of them would have an educated opinion about the matter and they'd immediately agree with you. The rest, the vast majority, wouldn't understand the question and would be annoyed that you interrupted them.
Quoting Metaphysician Undercover
Such a civil one. It would be impolite even if you weren't also completely wrong.
Quoting Metaphysician Undercover
I found this in an old post of yours. It it exactly your misunderstanding.
Nobody claims that math refers. That's your straw man.
Now math can be useful. So we often play the game of (1) Assume math refers to this particular situation; and (2) Use math to improve your control of the situation, whatever it is.
Just because we constantly apply math as if it refers to the world; and just because it so often turns out to be really useful; does not mean that math itself refers to the world. That's the brilliant essence of math. Math refers to nothing; but is locally useful in almost everything.
You just don't get that. You're fighting a straw man of your own creation.
Quoting Metaphysician Undercover
I remember fondly when I spent weeks trying to explain order theory to you, back when I thought you were trying to understand anything. You are still at this. If two sets have the same elements and the same order, they are equal as ordered sets. It's just about layers of abstraction, separating out concepts. First you have things, then you place them in order.
Somehow this offends you. Why?
That was said to Metaphysician Undercover.
Actually, I am the one who took up his misconception that sets have an inherent order. I don't consider that "hijacking", since his posts in this thread need to be taken in context of his basic confusions about mathematics, as mathematics has been discussed here.
Quoting fishfry
In ordinary mathematics, '=' does stand for identity. It stands for the identity relation on the universe.
For terms 't' and 's', 't = s' is true if and only if what 't' stands for is identical with what 's' stands for.
Quoting fishfry
An extreme formalist would say that. There is no evidence I know of that more than a very few mathematicians take such an extreme formalist view. Indeed, mathematicians and philosophers of mathematics often convey that they regard mathematics as not just formulas. Not even Hilbert, contrary to a false meme about him, said that mathematics is just a game of symbols.
And the semantics of first order logic with identity usually do require that '=' stands for the identity relation.
Yes I understood that. I haven't been in this thread in a couple of weeks and when I checked it out, @Metaphysician Undercover had evidently introduced his favorite theme, that mathematical equality is not metaphysical identity, which seemed a little afar from the original topic.
Quoting TonesInDeepFreeze
Yes I did actually understand that! I was just startled that @Meta was still going on about order being an inseparable and inherent aspect of a set, when I had already had such a detailed conversation with him on this subject several years ago. I did actually realize you were quoting him -- I was just surprised to see him still hung up on that topic.
Quoting TonesInDeepFreeze
@Meta's basic confusions in math are too big to to be placed within context. He's completely unwilling to meet any mathematical idea on its own terms.
Quoting TonesInDeepFreeze
Ah ... you said that? Well I understand why my friend Meta is unhappy with you then. I am not sure if I agree with your statement.
After all if = is the identity relation on the universe, why does ZF need to redefine it then? Is the = of ZF the same as the identity relation on the universe?
I do not think so. Because the ZF version is a definition. It's a defined symbol that wasn't defined before.
Answer me this, maybe I'll learn something. Suppose X and Y are objects in the universe, but they are not sets?
Well in ZF they don't exist. And even if they did, how would you define X = Y?
There are three ways we could approach for set theory:
(1) Take '=' from identity theory, with the axioms of identity theory, and add the axiom of extensionality. In that case, '=' is still undefined but we happen to have an additional axiom about it. The axiom of extensionality is not a definition there. And, with the usual semantics, '=' stands for the identity relation. It seems to me that this is the most common approach.
(2) Don't take '=' from identity theory.
Definition: x = y <-> Az(z e x <-> z e y)
Axiom: x = y -> Az(x e z -> y e z)
(3) Don't take '=' from identity theory.
Definition: x = y <-> Az(z e x <-> y e z)
Axiom: Az(z e x <-> z e y) -> Az(x e z -> y e z)
With (2) and (3), yes, '=' could stand for an equivalence relation on the universe that is not the identity relation. But it seems to me that even in this case, we'd stipulate a semantics that requires that '=' stands for the identity relation. And I think it's safe to say that usually mathematicians still regard '1+1 = 2' to mean that '1+1' stands for the same number that '2' stands for, and not merely that they stand for members in some equivalence relation, and especially not that it's just all uninterpreted symbols.
Quoting fishfry
Depends on what you mean by 'set' and what meta-theory is doing the models.
In set theory, contrary to a popular notion, we can define 'set':
x is a class <-> (x =0 or Ez z e x)
x is a set <-> (x is a class & Ez x e z)
x is a proper class <-> (x is a class & x is not a set)
x is an urelement <-> x is not a class
Then in ordinary set theory we have these theorems:
Ax x is a class
Ax x is set
Ax x is not a proper class
Ax x is not an urelement
If our meta-theory for doing models has only sets, then all members of universes are sets.
If our meta-theory for doing models has proper classes, still a universe is a set (proof is easy by the definition of 'model'). And no proper class is a member of a set.
If our meta-theory for doing models has urelements, and '=' stands for the identity relation, then the axiom of extensionality is false in any model that has two or more urelements in the universe or has the empty set and one or more urelements in the universe.
Meta, once I understood that Tones was arguing that set equality is the law of identity, I realized why you're arguing this point. I entirely agree with you. I apologize to you for jumping to multiple wrong conclusions.
I noticed that you posted then deleted a response to me, so perhaps you at first objected to my post then realized that by the end, I was in agreement with you. I'll think of it that way.
I haven't yet worked through Tones's reply to me outlining his argument, so I should reserve judgment. But at this moment it seems to me that set equality is a defined symbol in a particular axiomatic system. As such has no referent at all, any more than the chess bishop refers to Bishop Berkeley. It doesn't refer to anything concrete, nor anything abstract. It simply stands for a certain predicate in ZF. It can't possibly "know" about logic or metaphysics. It can't refer to "sets" since nobody knows what a set is. A set is whatever satisfies the axioms. And set equality is a relation between sets, which have no existence outside the axioms; and have no meaning even within the axioms.
Meta I agree with you on this point and had no idea that I was jumping into an ongoing conversation in a thread I didn't realize I was in. My bad all around. Tones, I look forward to working through your post. I'm sure many clever people must have considered this very problem of set equality and identity and there's a lot I don't know.
I didn't get the business about the Eleatics. Did they have this conversation back in the day?
Quoting Michael
Explains a lot ...
Ok I finally made a run at this, and I am having a little trouble understanding your meaning.
I believe you are trying to convince me that logical identity is the same thing as set equality as given by extensionality.
I'm a little unclear on what you mean by logical idenity. Do you mean the law of identity, everything is equal to itself? or identical to itself? Or did you mean something else?
I could use some specifics to help anchor my understanding.
Quoting TonesInDeepFreeze
I already got in trouble here! I looked up "identity theory." Both Wiki and IEP say it's a theory of mind. So that's not what you're talking about. Wiki has a disambiguation page that led me to Pure identity theories, a linkable paragraph within an article called List of first-order theories.
So if you could just define "identity theory" for me, and tell me what "=" means in that theory, I'll understand you better.
Quoting TonesInDeepFreeze
Even if I knew what you mean by identity theory, I still did not understand this. Still undefined but additional axiom. Sorry I don't follow. I'm probably missing your point I'm sure.
I have a question for you.
In ZF, I define [math]R = \{x \notin x\}[/math] and [math]S = \{x \notin x\}[/math], two definitions of the Russell set.
I ask: What is the truth value, if any, of [math]R = S[/math]?
How should I think of this? In ZF? In set theories with classes?
Quoting TonesInDeepFreeze
Ok, I hope I will understand this when you tell me what the identity relation is. But if extensionality is not an axiom, what is it? Axioms and definitions are the same thing. You can take them as "assumed true," or you can take them as definitional classifiers, separating the universe into things that satisfy the definition and things that don't.
For example, take the axioms for group theory. As axioms, they are assumed true for every group. But we actually use the axioms as a definition. If a mathematical structure obeys the axioms it's a group; and if not, not. So we can use axioms as a definitional boundary between everything we're interested in, and everything we're not. Axioms and definitions are the same thing viewed from different perspectives.
Quoting TonesInDeepFreeze
I don't really understand this. What are you trying to say in the axiom? That if two sets satisfy extensionality (the definition) then any set one of them is an element of, the other is also an element of? Am I getting that right? I think that already follows from the definition. In fact I convinced myself I could prove it, but did not work out the details. So I could be wrong about this.
But what is the intent?
Quoting TonesInDeepFreeze
I didn't get all this, what's the intent of the axiom, what does it all mean?
As far as 1 + 1 = 2, I've explained to @Metaphysician Undercover that these are two expressions that refer to the same set. By extensionality there is only one set, and two different representations of it. Other posters have mentioned that the intentional meanings of 1 + 1 and 2 are different, and the extensional meanings are the same. Mathematicians use the extentional meaning of a symbol. 1 + 1 = 2 "point" to the same abstract number in abstract number land, wherever that may be.
Mathematicians don't think of numbers as uninterpreted symbols. But perhaps if pressed to explain where numbers live, what kind of existence they have, it's safer to revert to formalism.
Quoting TonesInDeepFreeze
I don't know much about set theory with classes. I'm just a humble ZF guy. I have no choice! (set theory joke).
Quoting TonesInDeepFreeze
Didn't get to read much of this closely. Perhaps we can revisit later.
I did note one thing I disagreed with. You wrote:
"If our meta-theory for doing models has proper classes, still a universe is a set (proof is easy by the definition of 'model')"
Perhaps we're using different terminology. When they do independence proofs, models are sets. So for example to prove that ZF is consistent, we are required to produce a set that satisfies the axioms. It's no good to just provide a proper class, since the class of all sets satisfies the axioms but isn't sufficient to prove the consistency of ZF. (As usual I'm still confused about how ZF can see this, since it doesn't have proper classes).
Secondly, I know universes that are not sets. For example:
* The von Neumann universe and Gödel's constructible universe, both of which are proper classes (however you regard them) and are commonly called universes.
So perhaps I am not sure what is your definition of a universe.
Even informally, in ZF the universe is "all the sets there are." The axioms quantify over all the sets. And the universe of sets is not a set.
Summary of all this
These are some of my thoughts on reading your post. I'm sure I missed a lot. I'm still interested in understanding why you think that the logical identity (whatever that is, I'm still a little unclear) is the same thing as set equality under extensionality. I just can't see how that could even be. Once we get outside of the world of sets, we need a new definition. Because extensionality only applies to sets. And if we have a new definition, by definition we do not have the SAME definition. We have a DIFFERENT definition, which is to be consulted whenever we are wondering about the equality of two things that are not both sets.
It's like if I'm a computer multiplying integers, I use one algorithm. If I'm multiplying floats, I use another. So if we have two sets, we use extensionality to tell if A = B. If at least one of them is not a set, we have to use some OTHER way of telling. Which is your logical identity. It's a different thing, not the same thing.
Since I've convinced myself from first principles that logical identity and set equality are different things, I have a bit of a hard time following your arguments, since they must be wrong, or we must not be talking about the same thing.
Perhaps you can walk me through this slowly and clearly.
Just to humour you Tones, I read this post. So here's a question for you. When you state the law of identity as "a thing is identical with itself", would this identity include not only all of the thing's constituent elements, but also the ordering of those elements? For example, if I say that this rock is identical with itself, not only would this indicate that all the elements which compose the rock are identical with the rock, but also the ordering of those elements. If the ordering of the elements was not the same, then it would not be identical with the rock.
Quoting fishfry
Apology accepted. I wrote most of the following before reading this, so ignore, or read, whatever suits you. The deleted post must have occurred in the transfer from the other thread. Michael was concerned about derailing the thread, and took out certain posts and transferred them.
Quoting fishfry
As I indicate in my latest post in the supertask thread, Tones has a knack for taking highly specialized definitions designed for a particular axiomatic system, and applying them completely out of context. Be aware of that.
Quoting fishfry
Tones does, obviously.
Quoting fishfry
I dropped out of abstract mathematics somewhere around trigonometry, for that very reason. I got hung up in my need to understand everything clearly, and could not get past what was supposed to be simple axioms. I had a similar but slightly different problem in physics. We learned how a wave was a disturbance in a substance, and got to play in wave tanks, using all different sorts of vibrations, to make various waves and interference patterns. Then we moved along to learn about light as a wave without a substance. Wait, what was the point about teaching us how waves are a feature of a substance?
But Tones is a bit different. Tones forges ahead with misunderstanding of fundamental axioms. Tones insists that the axiom of extensionality tells us when two sets are identical. He refers to something he calls "identity theory", which I haven't yet been able to decipher.
Quoting fishfry
If axioms are rules, then they mean something. They dictate how the "formal game" is to be played. If the rules are misunderstood, as is the case with Tones, then the rules will not be properly applied.
Quoting fishfry
Tones is a monster, not of my own creation.
Quoting fishfry
It is self-contradicting, what you say. " First you have things, then you place them in order."
If you have things, there is necessarily an order to those things which you have. To say "I have some things and there is no order to these things which I have, is contradictory, because to exist as "some things" is to have an order. Here we get to the bottom of things, the difference between having things, and imaginary things.
Quoting fishfry
You are taking Tones' misrepresentation. I fully respect, and have repeatedly told Tones, that sets have no inherent order, exactly as you explained to me, years ago. What I argue is that things, have an order to their constituent elements, and this is an essential aspect of a thing's identity. So I've been trying to explain to Tones, that the "identity" of a set (as derived from the axiom of extensionality) is not consistent with the identity of a thing (as stated in the law of identity). But Tones is in denial, and incessantly insists that set theory is based in the law of identity.
Quoting Metaphysician Undercover
Thanks.
Quoting Metaphysician Undercover
I often don't follow the purpose of his symbology.
You guys should stay out of that thread, you're not discussing supertasks and I can see why @Michael moved you.
Quoting Metaphysician Undercover
Yes I didn't realize that. I don't see how it can be, but I'm not aware of how the logicians and set theorists resolve this.
Quoting Metaphysician Undercover
Yeah that business about waves without a medium is pretty murky. I've seen videos where they say, "It's a probability wave!" as if that explains anything. Probability waves are purely abstract mathematical gadgets, they aren't physical. Leaving unexplained the question of what electromagnetic waves are waving.
It's a well known problem that physics is no longer about the physical world, but rather about esoteric models that seem to work, without telling us much about the physical world. The "shut up and calculate" school of quantum physics.
Quoting Metaphysician Undercover
Aha! I just asked him about this very point.
Quoting Metaphysician Undercover
I can't comment on Tones's opinions in that area. I haven't been reading this thread.
Quoting Metaphysician Undercover
I can't tell if he knows a lot of logic but doesn't always explain himself, or is just typing stuff in. I had a very hard time with his last post to me explaining how identity was set equality.
Quoting Metaphysician Undercover
No it's perfectly sensible. You have a class of screaming school kids, eight year olds say, on the playground. They're totally disordered. The only organizing principle is that you have a set of kids.
Then you tell them to line up by height. Now you have an ordered set of kids. Or you tell them to line up in alphabetical order of their last name. Now you have the same set with a different order.
It's an everyday commonplace fact that we can have a set of things in various orders.
Now maybe you are making the point that everything is in SOME order. The kids in the playground could still be ordered by their geographical locations or whatever.
But sets don't have inherent order.
Quoting Metaphysician Undercover
Another way to look at it is that, as you say, perhaps every set has some inherent order, but we are just ignoring the order properties to call it a set. Then we bring in the order properties. It's just a way of abstracting things into layers.
But mathematical sets by themselves have no inherent order till we give them one. It's just part of the abstraction process.
Quoting Metaphysician Undercover
I'm moved that I had an effect. It was not in vain. I'm happy.
Quoting Metaphysician Undercover
Yes, I am starting to come around to your point of view. But tell me this. Since, given a set, there are many different ways to order it, how do you know which one is inherently part of it?
Quoting Metaphysician Undercover
I would go so far as to say that identity isn't set equality, because identity applies also to things that are not sets.
But if you ask me whether I think that two sets that are equal are identical, I'd have to say yes. Because if they're equal, they're the same set. Not because of metaphysics, but because of set theory. Set theory only talks about sets, and doesn't even say what they are. Nobody knows what sets are. They're fictional entities. They obey the axioms and that's all we can know about them.
Quoting Metaphysician Undercover
I can see that you've developed a bit of a, what is the word, obsession? attitude? annoyance? with him.
They're not totally disordered though. At any time you can state the position of each one relative to the others, and that's an order. When you say "they're totally disordered", that's just metaphoric, meaning that you haven't taken the time, or haven't the capacity, to determine the order which they are in.
Quoting fishfry
Those are just 'identified orders'. When the kids are running free, in what we might call a 'random order', what you called "totally disordered", there is still an order to them, it has just not been identified. So, for the principle "height", we could make a map and show at any specific time, the relations of the tallest, second tallest, etc., and that would be their order by height. And we could do the same for alphabetic order. So we number them in the same way that you would number them in a line, first second third etc., then show with the map, the positions of first second third etc., and that is their order. The supposed "random order", or "totally disordered" condition, is simply an order which has not been identified.
Quoting fishfry
Yes, that's very apprehensive of you fishfry, and I commend you on this. Most TPF posters would persist in their opinion (in this case your claim of "totally disordered", which implies absolute lack of order), not willing to accept the possibility that perhaps they misspoke.
So that is the point, everything is in "SOME" order. Now, consider what it means to say "sets don't have inherent order". Would you agree that this sets them apart from real collections of things? A real collection of things, like the children, must have SOME order. And, this order which they do have, is very significant because it places limitations on their capacity to be ordered.
So when you said "first you have things, then you place them in order", we need to allow that the "things" being talked about, come to us in the first place, with an inherent order, and this inherent order restricts their capacity to be ordered. For example, let's say that the things being talked about are numbers. We might say that 1 is first, 2 is second, 3 is third, etc., and this is their "inherent order". This is the way we find these "things", how they come to us, 1 is synonymous with first, 2 is synonymous with second, etc., and that is their inherent order. The proposition of set theory, that there is no inherent order to a set, removes this inherent order, so we can no longer say that one means first, etc.. Now there cannot be any first, second, or anything like that, inherent within the meaning of the numbers themselves. This effectively removes meaning from the symbols, as you've been saying.
Quoting fishfry
Yes, I think I see this. I would say it's a type of formalism, the attempt to totally remove meaning from the symbols. The problem though, is that such attempts are impossible, and some meaning still remains, as hidden, and the fact that it is hidden allows it to be deceptive and misleading. So, by the abstraction process you refer to, we remove all meaning from the symbols, to have "no inherent order". Now, what differentiates "2" from "3"? They are different symbols, with different applicable rules. If what is symbolized by these two, can have "no inherent order", then the rules for what we can do with them cannot have anything to do with order. This allows absolute freedom as to how they may be ordered.
However, we can ask, can the two numbers,2 and 3, be equal? I don't think so. Therefore we can conclude that there actually is a rule concerning their order, and there actually is not absolute freedom as to how they can be ordered. The two symbols cannot have the same place in an order. Therefore, there actually is "SOME" inherent order to the set, a rule concerning an order which is impossible. And this is why I say that these attempts at formalism, to completely remove meaning which inheres within what is symbolized by the symbol itself, are misleading and deceptive. We simply assume that the formalism has been successful, and inherent meaning has been removed (we take what is claimed for granted without justification), and we continue under this assumption, with complete disregard for the possibility of problems which might pop up later, due to the incompleteness of the abstraction process which is assumed to be complete. Then when a problem does pop up, we are inclined to analyze the application as what is causing the problem, and the last thing we would do is look back for faults in the fundamental assumptions, as cause of the problem.
Quoting fishfry
As described above, you need to look for what is inherent within the meaning of the symbol. Formalism attempts the perfect, "ideal" abstraction, as you say, which is to give the imagination complete freedom to make the symbol mean absolutely anything. However, there is always vestiges of meaning which remain, such as the one I showed, it is impossible that 2=3. The vestiges of meaning usually manifest as impossibilities. Any impossibility limits possibility, which denies the "ideal abstraction", by limiting freedom.
So to answer your question, the order which is inherent is not one of the orders you can give the set, it is a preexisting limitation to the orders which you can give. When we receive the items, what you express as "first we have the items", there is always something within the nature of the items themselves (what you call "SOME order"), as received, which restricts your freedom to order them in anyway whatsoever.
Quoting fishfry
There are many different ways that "same" is used. You and I might both have "the same book". The word "set" used here is "the same word" as someone using "set" somewhere else. So it's like any other word of convenience, it derives a different meaning in a different sort of context. In common parlance, mathematicians might say "they are the same set", but I think that what it really means is that they have the same members. So that's really a qualified "same".
Quoting fishfry
Actually I got annoyed with Tones rapidly, when we first met, but now he just amuses me.
Ok. For things in the real world, they are already in some order, even if it's a complete state of disorder. Even a completely disordered collection of gas molecules in a container, at every instant each molecule is wherever it is. And that set of coordinates, locating every molecule in space, is the order.
I get that. But by the same token, there is no preferred order. Suppose for example that I got my schoolkids from the playground to line up single-file in order of height. And now YOU come along and say, "Ah, that is the inherent order, and all other orders are disorders of that."
But of course your observation was a complete accident. I could have lined them up alphabetically by last name.
So even among physical objects, if we allow that they are always in some order, even if it's disorderly; but nevertheless, there is no preferred or inherent order.
I believe you are saying there's an inherent order, have I got that right?
Quoting Metaphysician Undercover
Yes we see that the same way. Totally disordered gas molecules in a container, at every instant there is a list of all atoms and where they are in the box (more or less, quantum effects notwithstanding, but that's not the point I'm making). I'm agreeing with you that even the most disordered state is still an order. It's just the set of facts about where everything is.
Quoting Metaphysician Undercover
Yes perfectly happy to regard whatever physical positions and attributes -- their state -- is regarded as the order they're in at that moment. I'm fine with that.
Quoting Metaphysician Undercover
You have indeed persuaded me that every collection of physical things has an order, even if they are apparently disordered. But sets, well you know ...
Quoting Metaphysician Undercover
Everything in the physical world.
Quoting Metaphysician Undercover
I have obviously spent much time considering that. As much time as I've spent explaining to you that sets don't have inherent order!
Quoting Metaphysician Undercover
Of course. Sets aren't real. They're a mathematical abstraction. I've never asserted otherwise.
Quoting Metaphysician Undercover
No, I did not say that. I said that the things come to us WITHOUT an inherent order; and we place one on them.
And secondly, I was not talking about things in the world. I was talking about abstract mathematical structures. Objects of thought. Not of the world.
Quoting Metaphysician Undercover
Yeah it's harder to line the kids up by height depending on how they happen to be arranged on the playground. But sets aren't kids on the playground. Sets are a conceptual gadget for thinking, what would it be like to have a collection of things that don't have any order? What could we still say about them?
It's a philosophical game like that. And it's useful. It lets us separate out the facts about sets that depend on their order, from the facts that don't.
This is just the essence of mathematical abstraction.
Quoting Metaphysician Undercover
Well now that you mention it, no. 1, 2, 3, ... is NOT the inherent order of the set [math]\mathbb N[/math], believe it or not. On the other hand it sort of is, in a sneaky way. Von Neumann defined the symbols 1, 2, 3, ... in such a way that [math]n \in m[/math] if it happens to be the case that we want n < m to be true.
But we still have to (1) define what 1, 2, 3, ... are as set; and then (2) define that what we mean by n < m, is that [math]n \in m[/math] as sets.
So even though it's kind of rigged for 1, 2, 3, ... to be the natural order -- in fact that's what they call it, the natural order -- it is still explicitly defined. And without that definition, the natural numbers have no inherent order.
I know this is hard for normal humans to accept, since it's pretty obvious that 1 < 2 < 3 and so on. But mathematicians insist on being picky about how numbers and other things are defined. In the set-theoretic view of modern math, the numbers 1, 2, 3, ... are defined as particular sets, with no inherent order; and then we impose their order by leveraging the [math]\in[/math] operator.
Now I'm going to meet you halfway on something. I admitted that the set definitions of 1, 2, 3,... are already set up to leverage the [math]\in[/math]. But you could say -- and I am going to agree with you -- that when John von Neumann invented the modern set-theoretic definition of the natural numbers; he already had a pre-intuition of the inherent order 1 < 2 < 3 ... and that's why he defined things to work out that way.
So even though formally we've removed the scaffolding and there's no inherent order in the finished mathematical theory; the inspiration for designing the theory that way was in fact the inherent order of the natural numbers, no matter what set theory says.
I am going to agree with you about that. Mathematics is mysteriously influenced by something "real" about even the most abstract things, like numbers. Formalism is defeated in the end. It's NOT just about the symbols. Math is expressing something real about the world.
Is any of this along the lines of your thinking?
Quoting Metaphysician Undercover
I just talked myself into (almost) agreeing with what I take to be your point of view:
* Even though the numbers 1, 2, 3, ... have an inherent order; when we do set theory we pretend they don't, purely for the sake of the formalism. But the formalism is missing something important. The set-theoretic formalism denies that the numbers have an inherent order. But the counting numbers DO have an inherent order that is obvious to every school child. Therefore the set-theoretic natural numbers do not capture the full metaphysical properties of the natural numbers.
Have I got any of that right?
Quoting Metaphysician Undercover
Only to quickly put it back. And as I just acknowledge, the method of defining numbers as sets, and then being able to "define" the order 1 < 2 < 3 ... was obviously set up to facilitate just that. Showing that von Neumann had a priori knowledge of the order he was formalizing.
But that's not surprising, really. Even formalists don't think everyone's just making everything up. Math is "about" something "out there" in the world. Right? Am I making any sense to you?
Quoting Metaphysician Undercover
From the formal perspective of set theory. But not to deny that numbers don't have inherent order. Our formal model of numbers has no inherent order. But that's a virtue. It lets us study those aspects of sets that don't depend on order. I think I said that earlier. It's a process of abstraction, not lying for ill intent or metaphysical error.
Quoting Metaphysician Undercover
Yes if you'll accept that it's really all that's going on. Like when you go to the dermatologist to remove a mole, he doesn't send you in for a bunch of tests on your pancreas. He deals with one particular level of your entire being. He's not denying you have all these other organs. He's just focussing on one thing at a time.
It's perfectly ok to think of sets as having an inherent order and all their other possible orders, but we're just not concerned about them today. We only want to look at the property of membership.
Quoting Metaphysician Undercover
Yes, so that we can reason precisely about the objects the symbols represent.
Quoting Metaphysician Undercover
Maybe you mean that von Neuman secretly knew that 1 < 2 when he formalized the natural numbers in such a way to make 1 < 2 come out true.
But of course he did! So maybe you are getting at the intuition that guides mathematicians to do things the way they do.
In other words math is discovered inductively; and only presented deductively.
Quoting Metaphysician Undercover
What differentiates any set from any other set? All together: The axiom of extensionality!
In von Neumann's clever encoding, we make the following symbolic definitions:
[math]0 = \emptyset[/math]
[math]1 = \{0\}[/math]
[math]2 = \{0, 1\}[/math]
[math]3 = \{0, 1, 2\}[/math]
and so forth. One virtue of this encoding is that the cardinality of each number is what it "should" be. The set representing 3 has cardinality 3. Which of course isn't within the theory, it's outside the theory. We secretly already know was 3 is even before we define it.
Is that one of the thing's you're getting at?
Anyway, back to the question. How do we know that 2 and 3 are not the same set?
Well [math]2 \in 3[/math], but [math]2 \notin 2[/math].
Therefore by extensionality, [math]2 \neq 3[/math], because they don't have exactly the same elements.
Perhaps you can begin to see the virtues of working a the set level separately from its order properties. We can see the mechanics of how to use the axiom of extensionality. No order properties are needed to determine that 2 and 3 are different sets. It's just a matter of ignoring hypotheses that you don't need for a particular argument.
Nobody is saying that a given set doesn't have an order, as well as a lot of other stuff. A topology, some algebraic operations, a manifold structure perhaps. But we can learn a lot just from restricting our attention to the membership relation and seeing what we can learn just about that.
Quoting Metaphysician Undercover
Yes, math is often concerned with the most general kind of structure or conceptual framework.
What most people think of the order of the counting numbers: 1, 2, 3, ..., mathematicians call the "standard order" or "usual order," in contrast with many other interesting orders we could define.
While normal people only think about the usual order, mathematicians are people who think about all the different ways a set like the natural numbers can be ordered. In fact there's a beautiful order to the ways that the natural numbers can be well-ordered. Those are the ordinal numbers.
In other words the collection of all the ways a set can be ordered ... can itself have a natural order that we can study.
So that's the kind of thought process mathematicians enjoy, when they go from order to orders. From an order like 1 < 2 < 3, to the concept of order itself in its most abstract form.
Quoting Metaphysician Undercover
I believe I already demonstrated from first principles, from the ZF axioms, that 2 and 3 are not the same set; which, in the von Neumann encoding, proves that they are not the same number.
Quoting Metaphysician Undercover
I just did. 3 contains an element, namely 2, that is not an element of 2. Therefore they're not the same set by the axiom of extensionality.
Quoting Metaphysician Undercover
I hope you have taken the foregoing to heart. We define 2 and 3 as particular sets, and by design they are different sets under extensionality. They do not have the same elements therefore they are not the same set. Under the identification of sets as representatives of numbers, they are not the same number.
Quoting Metaphysician Undercover
I hope I've already convinced you that 2 and 3 are not the same number, and that we can demonstrate that using nothing more than the axiom of extensionality. In other words we do not need to use any order properties to prove that 2 and 3 are not the same number.
Quoting Metaphysician Undercover
No. I proved [math]2 \neq 3[/math] using only their set membership properties and without any need to invoke their order properties, inherent or assigned.
Quoting Metaphysician Undercover
Then you have been proven wrong. I don't need to mention or consider or use any of the order properties of 2 and 3 to determine that they're different numbers.
Quoting Metaphysician Undercover
Entirely without rational basis. This para is a wild generalization of your complaint about 2 and 3, but I already showed how we can distinguish 2 and 3 using only their membership properties and not their order properties. If that was the basis for this paragraph, you have no basis. But even then, the para makes wild unsupported accusations.
You will need a much better example -- well an example, period -- of the formalism failing. It certainly held up to your first test. I used the axiom of extensionality to prove that 2 is not 3. And a good thing, too!
The set theoretic abstractions have held up for a century, from Zermelo's 1922 axiomitization to today.
Quoting Metaphysician Undercover
You are thrashing away at a strawman you've created out of your imagination, and under the mistaken belief that we can't tell 2 from 3 without their order properties. But we can.
Quoting Metaphysician Undercover
You know you have stopped being clear and coherent in the last few paras. All based on a mistaken belief. This last para does not parse. Not for me anyway. Formalism is a tool, it's not the goal.
Quoting Metaphysician Undercover
That works for numbers. What about kids? What is the inherent order, the playground or the single file?
Quoting Metaphysician Undercover
No, you are consistently wrong about this. If A and B are sets and I can prove that A = B, then A and B are the same set. They are in fact the identical set, of which there is only one instance in the entire universe. They are NOT "two copies" or two distinct entities that we are calling the same by changing the meaning of the word "same."
"mathematicians might say "they are the same set", but I think that what it really means is that they have the same members."
DUH that is what it MEANS to be the same set. That is the ONLY thing it means to be the same set. Sets don't have any existence other that what the axioms say. There is nothing else to know about sets.
I can't believe you wrote that. Yes that is what it MEANS for two sets to be the same. That they have the same members. That's ALL it means and EVERYTHING it means.
You simply can't accept that and I don't know why. The knight in chess moves the way it does. Not for any reason other than those are the rules. Likewise two sets are the same when they have the same members. Period, end of story. That's it.
There is ONE SET {1, 2, 3}. That's the only one. There is exactly one instance of every set. If two supposed sets are equal they are the same sets. Like the morning star and the evening star, they're the same object. I don't know how many times I've explained this over the years and I can't understand why it eludes you or troubles you.
Quoting Metaphysician Undercover
If you say so ...
I did not say that.
I said that classical mathematics has the law of identity as an axiom and that classical mathematics abides by the law of identity.
Quoting fishfry
I addressed that. Written up in another way:
Ordinarily, set theory is formulated with first order logic with identity (aka 'identity theory') in which '=' is primitive not defined, and the only other primitive is 'e' ("is a member of").
But we can take a different approach in which we don't assume identity theory but instead define '='. I don't see that approach taken often.
But both approaches are equivalent in the sense that they result in the exact same set of theorems written with '=' and'e'.
Quoting fishfry
No, I am not saying any such thing.
(1) I don't think I used the locution 'logical identity'.
But maybe 'logical identity' means the law of identity and Leibniz's two principles.
Classical mathematics adheres to the law of identity and Leibniz's two principles.
The identity relation on a universe U is {
Identity theory (first order) is axiomatized:
Axiom:
Ax x = x (law of identity)
Axiom schema (I'm leaving out some technical details):
For any formula P(x):
Axy((P(x) & x = y) -> P(y)) (Leibniz's indiscernibility of identicals)
But identity theory, merely syntactically, can't require that '=' be interpreted as standing for the identity relation on the universe as opposed to standing for some other equivalence relation on the universe.
And Leibniz's identity of indiscernibles cannot be captured in first order unless there are only finitely many predicate symbols.
So we make the standard semantics for idenity theory require that '=' does stand for the identity relation. And then (I think this is correct:) the identity of indiscernibles holds as follows: Suppose members of the universe x and y agree on all predicates. Then they agree on the predicate '=', but then they are identical.
(2) The axiom of extensionality is a non-logical axiom, as it is true in some models for the language and false in other models for the language.
As mentioned, suppose we have identity theory. Then we add the axiom of extensionality. Then we still have all the theorems of identity theory and the standard semantics that interprets '=' as standing for the identity relation, but with axiom of extensionality, we have more theorems. The axiom of extensionality does not contradict identity theory and identity theory is still adhered to. All the axiom of extensionality does is add that a sufficient condition for x being identical with y is that x and y have the same members. That is not a logical statement, since it is not true for all interepretations of the language. Most saliently, the axiom of extensionality is false when there are at least two urelements in the domain.
In sum: Set theory adopts identity theory and the standard semantics for identity theory, and also the axiom of extensionality. With that, we get the theorem:
Axy(x = y <-> Az((z e x <-> z e y) & (x e z <-> y e z)))
And semantically we get that '=' stands for the identity relation.
And if we didn't base on identity theory, then we would have the above not as a non-definitional theorem but as a definition (definitional axiom) for '='; and we would still stipulate that we have use the standard semantics for '='.
Think of it this way: No matter what theory we have, if it is is built on identity theory, then the law of identity holds for that theory, and that applies to set theory in particular. But set theory, with its axiom of extensionality, has an additional requirement so that set theory is true only in models where having the same members is a sufficient condition for identity.
As I said much earlier in this thread, it is the first order theory axiomatized by:
Axiom:
Ax x = x (law of identity)
Axiom schema (I'm leaving out some technical details):
For all formulas P(x):
Axy((P(x) & x = y) -> P(y)) (indiscernibility of identicals)
The meaning of '=' is given by semantics, and the standard semantics is that '=' maps to the identity relation on the universe. So, for any terms 't' and 's', 't = s' is true if and only if 't' stands for the same member of the universe that 's' stands for.
Quoting fishfry
'=' is a primitive symbol. The axiom of extensionality is an additional axiom, not an axiom of identity theory.
Quoting fishfry
Maybe you mean {x | ~ x e x} (you left out 'x |').
In Z we prove there is no set R such that Ax(x e R <-> ~ x e x). Therefore, the abstraction notation {x | ~ x e x} is not justified. How we handle that depends on our approach to abstraction notation. Personally, as a matter of style, I prefer the Fregean method, but reference-less abstraction notation is a whole other subject.
Quoting fishfry
It is an axiom in ordinary set theory. I was describing a different approach, much less common, in which we don't have the axiom of extensionality.
Quoting fishfry
There are two different senses:
(1) Syntactical definitions. These define symbols added to a language. In a theory, they are regarded as definitional axioms. But they are not like non-definitional axioms, in the sense that definitional axioms only provide for use of new symbols and don't add to the theory otherwise (criteria of eliminability and non-creativity).
This is the sense I'm using in my remarks about approaches to '=' in set theory.
(2) A set of axioms induces the class of models of the axioms. For example, we say first order group theory "defines" 'group'.
That is the sense that goes with the notion you mention
Quoting fishfry
Yes.
Quoting fishfry
Do the details. Remember that you don't have the identity axioms, so you can't use anything prior about '='. For example, you can't use substitutivity.
Quoting fishfry
To fulfill the other approach where we don't start with identity theory.
(3) Don't take '=' from identity theory.
Definition: x = y <-> Az(z e x <-> y e z)
Axiom: Az(z e x <-> z e y) -> Az(x e z -> y e z)
With (2) and (3), yes, '=' could stand for an equivalence relation on the universe that is not the identity relation. But it seems to me that even in this case, we'd stipulate a semantics that requires that '=' stands for the identity relation. And I think it's safe to say that usually mathematicians still regard '1+1 = 2' to mean that '1+1' stands for the same number that '2' stands for, and not merely that they stand for members in some equivalence relation, and especially not that it's just all uninterpreted symbols.
TonesInDeepFreeze
Quoting fishfry
The intent is to arrive at the theorems of set theory but without adopting identity theory.
Quoting fishfry
What you quoted by me agrees with that. A universe for a model is a set. But that doesn't entail that our meta-theory cannot be a class theory, as long as universes for models are sets.
Quoting fishfry
That's a different sense from 'universe for a model'.
Quoting fishfry
A model M for a language is a pair such that U is a non-empty set and F is an interpretation function from the set of non-logical symbols. U is referred to as 'the universe for M'.
Quoting fishfry
I'm not talking about informal usage such as that.
If set theory has a model (which we believe it does), then the universe for that model is a set. That universe doesn't have to be "all the sets" (which is an informal notion anyway). It's a purely technical point: A model for a language has a universe that is a set.
Quoting fishfry
I think no such thing.
/
Rather than sorting out your questions in this disparate manner, it would be better - a lot easier - to share a common reference such as one of the widely used textbooks in mathematical logic. I think Enderton's 'A Mathematical Introduction To Logic' is as good as can be found. And for set theory, his 'Elements Of Set Theory'.
You know, I studied math and I never heard that said, anywhere. I don't think it's true. I'd be glad for a mathematical reference. Pick up a text book on set theory and you won't find it.
Quoting TonesInDeepFreeze
I asked for a reference to "identity theory," since a Google search brings up many different meanings, none of them bearing on this topic.
Quoting TonesInDeepFreeze
Yes, there's a first-order theory of equality. Is that what you mean?
https://en.wikipedia.org/wiki/Theory_of_pure_equality
Quoting TonesInDeepFreeze
Maybe, I don't know. You haven't convinced me that set equality has anything to do with the law of identity.
Quoting TonesInDeepFreeze
You seem to be saying that.
Quoting TonesInDeepFreeze
More rabbit holes. Nothing to do with set theory. I believe you have said logical identity. But if not, what do you mean? Define your terms please.
Quoting TonesInDeepFreeze
None of those come up in set theory. Can you tell me, where did you come up with this line of discourse? Logic class? Logic in the philosophy department or the math department? Your own original ideas?
I was literally shocked the other day when you claimed that set equality is "identity," no matter how you define it. You still haven't made a case.
Quoting TonesInDeepFreeze
We're not having the same conversation now. I couldn't even parse that. It certainly doesn't tell me when two sets are equal.
Quoting TonesInDeepFreeze
You're just typing stuff in and not addressing the issue. How does set equality relate?
Quoting TonesInDeepFreeze
Losing me big time. I can't make sense of any of this in the context of set equality.
Quoting TonesInDeepFreeze
There is no model of set theory in which extensionality is false. None whatsoever.
Quoting TonesInDeepFreeze
I don't believe that, but I don't know anything about sets with urelements. Have a reference by any chance? Or give me an example.
Quoting TonesInDeepFreeze
Can't possibly be.
Quoting TonesInDeepFreeze
You speak a funny language. What subject is this? Where did you learn this? Have you a reference?
Quoting TonesInDeepFreeze
Nonsense. Models of what? How do you know whether having the same members is sufficient for identity unless you already have the axiom of extensionality? In which case you're doing set theory, not identity theory.
I'll stipulate that maybe you have a point to make. You have not communicated it to me. I didn't have the heart to tackle this long post tonight.
(edit) I see this was a response to my earlier long post, so I really should read this. I skimmed a bit. I'm sure you have something in mind. You could even be right. Maybe I'll get to this at some point.
I checked my copy of Kunen, Set Theory: An Introduction to Independence Proofs. It's a standard graduate text in set theory.
He states that "=" is one of the symbols of the theory, and that "informally it stands for equality." He doesn't go any deeper in that direction. So if someone else has explained these matters more deeply, and if that person is Enderton, I'll have to take your word for it.
From your previous post, I wonder if you can give an example of a set theory with urelements in which extensionality doesn't apply.
I found a pdf of that here:
https://docs.ufpr.br/~hoefel/ensino/CM304_CompleMat_PE3/livros/Enderton_Elements%20of%20set%20theory_%281977%29.pdf
I looked at it and he starts with the axiom of extensionality, which he states in the conventional manner, without reference to identity.
Please let me know what page I should look at in this book to see any kind of discussion of the issues you bring up. I'll wait for your response before looking for a copy of his logic book, because I can skim a set theory book and have an idea what's going on, but not so for logic.
Classical mathematics is regarded as being formalized by ZFC. ZFC starts with a base of first order logic with identity. Whether called 'first order logic with identity', or 'first order logic with equality', or 'identity theory', the usual axioms, whether named as axioms for 'identity' or axioms for 'equality' are as I mentioned. For example, Enderton's 'A Mathematical Introduction To Logic'.
One of the axioms of identity theory is the law of identity formalized: Ax x=x. So the law of identity pertains to ZFC and, if ZFC is consistent, then ZFC does not contradict identity theory.
Ask anyone who studies set theory, whether ZFC is a first order theory with identity. I can't help that the Google doesn't help to find this.
The Wikipedia article you mentioned is not well written. (1) It doesn't give an axiomatization and (2) It doesn't mention that we can have other symbols in the signature and that by a schema we can generalize beyond a signature with only '='.
But, yes, of course we can use the word 'equality' or the word 'identity', as I said so many times. Perhaps 'equality' is used more often, but the exact formal theory is the same, and is axiomatized as I mentioned.
Quoting fishfry
I gave you fulsome explication. Said yet another way:
The axiom of extensionality gives a sufficient condition for equality. But it doesn't give a necessary condition. So it is not a mathematical definition. A formal mathematical definition for a binary predicate R is of the form:
x R y if and only if (something here about x and y)
A definition of '=' would be of the form:
x = y if and only if (something about x and y)
But the axiom of extensionality is usually given:
If, for all z, z is a member of x if and only if z is a member of y, then x= y.
That is of the form:
If (something about x and y) then x = y. And that is only a sufficient condition, not a biconditional.
But I also mentioned that we could have this variation:
x = y if and only if (for all z, z is a member of x if and only if z is a member of y; and for all z, x is a member of z if and only if y is a member of z).
And that would be a definition, since it gives both a sufficient and necessary condition for x = y.
However, I'll say it yet another way:
Ordinarily, set theory is written with a signature of '=' and 'e', where '=' is logical and 'e' is non-logical, and we have the axioms for first order logic with identity (aka 'idenity theory' or 'first order logic with equality'). Then we add the axiom of extensionality. And then we get (1) as a theorem.
No matter which approach we take, we end up with the same theorems.
Quoting fishfry
For the third time, I did not say that. And I again told you what I do say. Please stop saying that I said something that I did not say.
I said that set theory adheres to the law of identity but that set theory, with its axiom of extensionality, adds an additional sufficient condition for identity.
I don't know why you don't grasp this:
Ax x= x is an axiom of first order logic with identity. And set theory is a theory that subsumes first order logic with identity. So Ax x=x is also an axiom incorporated into set theory. But also set theory adds the axiom of extensionality.
Quoting fishfry
You are egregiously glossing over what I exactly say. So you form incorrect "seems".
If, for whatever reason, there is a point of my terminology that requires definition for you, then, time permitting, I would supply the definition. Or if there is an argument you can't see to be logical, then, time permitting, I would explain it in even more detail if there even is more detail that could be reasonably provided. However, that can lead to a long chain of definitions back to primitive notions, so it would be better to start at the beginning such as in some chosen textbook. But that does not justify you claiming that I said things that I did not say.
Quoting fishfry
It parses perfectly even if it seems difficult when one is not familiar with the basic mathematical logic in which symbols are interpreted with models. I'll put it this way:
'=' is a 2-place predicate symbol.
A model (an interpretation) for a language assigns a 2-place relation on the universe to a 2-place relation symbol. In other words, that is an assignment of the meaning, per the model, of the 2-place relation symbol. We call that 'the interpretation of the symbol'. That is semantical.
In general, relation symbols are interpreted differently with different models. But in the special case of '=', we stipulate that, with all models, '=' is stands for the identity relation on the universe for the model.
So my point was that from the mere syntactical presence of '=' in a formula, we can't ensure that '=' stands for the identity relation, and we have to turn to semantics (models) for that.
Quoting fishfry
Correct. There are no models of the axiom of extensionality in which the axiom of extensionality is false. But there are models in which the axiom of extensionality is false; they are not models of the axiom of extensionality. Take this is steps:
A model M is for a language.
Theories are written in languages.
So if a theory T is written in a language L, then a model for M for L is model for the language of T.
Given a model M, some sentences in the language L are true in M and some sentences in the language L are false in M.
A theory is a set of sentences closed under provability.
If every sentence in theory T is true in a given model M, then we say "M is a model of T".
So, notice that that there is a difference in meaning between "M is a model for the language L" and "M is a model of the theory T".
That is a crucial thing to understand and keep in mind.
Now, back to the axiom of extensionality.
Let T be any theory that is axiomatized with a set of axioms that includes the axiom of extensionality. If M is a model of T, then the axiom of extensionality is true in M.
But there are models for the language for set theory that are not models of set theory. For example
Let M have universe U = {0 1} and let 'e' be interpreted as the empty relation.
M is a model for the language for set theory, and M is not a model of set theory.
Again, to stress:
There is a difference between a language and a theory writtten in that language.
Let L be a language, and T be a theory written in L, and M be a model for L. It is not entailed that M is a model of T.
/
Urelements. Even though search engines are often deficient, I bet that you can find articles about urelements.
Df. x is an urelement <-> (~Ey yex & ~ x = 0) ('0' here standing for 'the empty sety')
A theorem of Z:
~Ex x is an urelement.
But we could have other axioms where
~Ex x is an urlement
is not a theorem.
And we could have axioms where
Ex x is an urlement
is a theorem.
But a theory that has the theorem:
Exy (x is an urelement & y is an urelement & ~ x = y)
is obviously inconsistent with the axiom of extensionality.
But we could do this:
Axiom: Axy((~ x is an urlement & ~ y is an urelement & Az(z e x <-> z e y)) -> x = y)
Quoting fishfry
It be's. Just as I've explained again. And I'll explain yet another way:
Certain proofs, including in set theory, use the identity axioms I mentioned. Set theory also has the axiom of extensionality, which allows for even more proofs. And we have this theorem of set theory:
Axy(x = y <-> Az((z e x <-> z e y) & (x e z <-> y e z)))
That's all purely syntactical.
Meanwhile, we interpret '=' to stand for the identity relation on the universe of any model for the language for set theory. And that is semantical.
Quoting fishfry
Models for the language of set theory. Some of them are models of set theory.
Always keep in mind the distinction between "model M for a language L" and "model M of a theory T" (or, mentioning the language also, "model M of a theory T written in a language L").
Quoting fishfry
You're very mixed up thinking that you're somehow disagreeing with me on those points.
Indeed, I am saying that the axiom of extensionality is what makes Az(z e x <-> z e y) sufficient for x = y.
And, indeed, identity theory is not set theory. Rather, identity theory is a sub-theory of set theory:
Every theorem of identity theory is a theorem of set theory. But not every theorem of set theory is a theorem of identity theory: Right off the bat, the axiom of extensionality is not a theorem of identity theory.
And again, we needn't quibble that "pure" identity theory does not have 'e' in its signature. I'm talking about the axioms of identity theory written in a signature that includes 'e'.
I'm only trying to figure out what you're talking about. There are many theories of identity. Wiki actually has a disambiguation page on the subject. Give me a reference to this identity theory you keep talking about.
You pointed me to Enderton. I pulled a pdf of his set theory book and found nothing beyond the standard explication of the axiom of extensionality, with no reference to "identity" in any context.
I gave you the pdf. Please tell me what page to read to understand your point.
Or if it's in his logic book, give me a reference for that.
I appreciate that you wrote me another lengthy post, but I'm about two or three lengthy posts behind you, and it would be infinitely helpful if, having said that it's all in Enderton, if you'd give me a page reference in his set theory book; and failing that, his logic book.
I can't deal with the rest of this now. You said Enderton, I pulled Enderton. Give me a page ref please. His statement of extensionality is just like everyone else's and absolutely nothing like yours.
Quoting TonesInDeepFreeze
Not in Enderton. Not in Kunen (grad level). Not in Halmos. Not in any set theory book I know, though I don't know many. Used to have a copy of Shoenfield but that was a long time ago and I don't remember what he said on the subject.
The Enderton reference was to the identity axioms. See page 112 in the logic book. And also, on page 83, he specifies satisfaction regarding '=' so that it adheres to interpreting '=' as the identity relation.
For set theory, an example is Hindman's 'Fundamentals Of Mathematical Logic' in which he is explicit that set theory is a first order theory. And earlier in the book, he gives the logical axioms as including the axioms regarding '=' similarly to the way I did. And he also mentions that the interpretation of '=' is the identity relation.
In general, even if many texts don't belabor that set theory is a first order theory, surely you don't dispute that it is? And that is first order theory with identity. To demonstrate:
Suppose you have only the axioms of set theory and no axioms for identity, then how do you suppose you would derive:
Axyz((x = y & y =z) -> x = z)
Watch out: You can't use "substitution of equals for equals", since that principle is derived from the identity axioms.
More generally, let P be a formula with only x free. How would you derive?:
(P(x) & x = y) -> P(y)
For example:
How would you derive?:
(x is finite & x = y) -> y is finite
Basically, the axiom of extensionality lets you infer equality, but it usually doesn't help in inferring from equality.
Yes, from x and y having the same members, we can infer that x is y. But from "x is y", and without identity axioms, how would we infer very much else?
You need axioms for '=' other than the axiom of extensionality to deduce all the theorems of set theory. And those other axioms are axioms of identity, such as the axiomatization I've been mentioning. You need the identity axioms and the axiom of extensionality to get all the theorems.
Sure, in informal expositions, even as found in many set theory textbooks, we don't belabor our use of the axioms and theorems of identity theory, but take them as implicit in our arguments, especially since the principles (such as substitution of equals for equals) are engrained in mathematical reasoning. But when we rigorously formalize, we need the identity axioms. Meanwhile, such a book as Hinman, which is quite scrupulous does mention the identity axioms explicitly as among the first order axioms and does mention that set theory is a first order theory.
Also: That textbooks might not mention something, especially as textbooks often don't belabor every formal detail (and some hardly even glance upon even basic formal considerations), it is not entailed that we can't specify details that are left out. And even if terminology used is not common (such as, 'identity theory' seems less common than 'first order logic with equality') we should allow the terminology as long as it is defined. And I did exactly define it, as I defined it as the theory axiomatized by the exact axioms I specified.
In discussions about languages, models and theories, the prepositions 'for' and 'of' might get overlooked if one is not reading closely. But we can eschew those prepositions:
language
model (aka 'interpretation' or 'structure' (different from another sense of 'structure' in mathematics))
theory
are key concepts in mathematical logic.
We can stipulate this terminology:
M interprets a language L iff M is a pair where U is a non-empty universe and F is an interpretation function from the non-logical symbols of L
A theory T is written in a language L
M models a theory T iff every sentence in T is true in M
Then we have:
There are languages L, models M and theories T such that: T is written in L, and M interprets L, but M does not model T.
I think we have to look at context here. What is our subject of discussion, what are we talking about here? Are we talking about things (individuals), of which there is a multitude, or are we talking about a group (set) of individuals, of which there is one? Your description above, seems to imply the former. You are talking about separate things, many schoolkids, and there is many possibilities as to the order they could have. On the other hand, if you were talking about the group as a whole, as your subject, then the parts of that group, the individuals, must have the order that they have at that time, even though it could be different in past or future times. If you were talking about the same individuals in a different order, this would require a change to that specific group, so you would be talking about that group, at a different time, because you'd be talking about the individuals, changing places.
You might understand this better through what is known as internal and external properties. To each individual, as a subject, its relations to other individuals are external properties. To the group, as a unit, and the subject, the relations between the individuals is an internal property.
You talk about the schoolkids as distinct individuals, where the various relations between them are the external properties of each and everyone of them. There are no internal relations here. Each schoolkid is a subject to predication, age, height, etc.. and you might produce an order according to those predications. The order is external to each schoolkid, people say it transcends, and changing the transcendent order does not change any of the schoolkids in anyway.
Now, let's take the group as a whole, as an object, and produce a corresponding subject, the set, and make that our subject. Since the whole group is our object of study, any change to the order of the individuals is an internal change to that object, therefore a change to that object itself. The order of the individuals (as the parts of the whole) is an internal property of that object, and a change to that order constitutes a change to the object, which we must respect as predicable to the corresponding subject. Therefore we can say that the order of the individuals, as the parts of the whole, is an intrinsic property of the whole, which is represented as the set.
Notice however, the switch from "subject" to "object", and this I believe is the key to understanding these principles. There is an implicit gap, a separation, between the meaning of "logical subject" and "physical object". When we make a predication, "the sky is blue" for example, "the sky" is the subject, and if there is an object which corresponds with that subject, the predication may be judged for truth. However, we can manufacture subjects and predications with complete disregard for any physical objects, and so long as we have consistency, we have a valid "subject", with no corresponding object.
Consider the following proposition, "There is a group of schoolkids". We have a propositional subject, without a corresponding object, what some people would call "a possible world". Since there is no assumed corresponding object which would cause a need for conformity, we can predicate any possible order we want, so long as it is not contradictory. The hidden problem of formalism which I referred to lies in the naming of the group, "schoolkids". That name needs to be clearly defined and the definition will place restrictions on what can be predicated without contradiction.
Perhaps, we can remove these restrictions, by making the things within the group, the elements of the set that is, completely nondescript. "There is a group of nondescript things". We still have the name "things", with implied meaning, so this name has to be defined, and this would put restrictions on what we can predicate. So we go to a simple symbol, "x" for example, and assume that the symbol on its own, has absolutely no meaning, and this would allow any individual predication whatsoever without any risk of self-contradiction. X is a subject which has absolutely no inherent properties.
It might appear like we have resolved the problem in this way, we have a subject "x" which can hold absolutely any predication, so long as the predications don't contradict.. However, when we assume that the subject has no inherent properties, we disallow any predication because the predication would be a property and this would contradict the initial assumption. So this starting point allows no procedure without contradiction.
Now look what happens when we say "there is a group of x's". There is actually something implied about x, which is implied simply by saying that there is a group of them. It is implied that x has a boundary, separation, etc.. We may start with the assumption that there is no intrinsic properties of X, but as soon as we start to predicate, we negate that assumption. And the symbol, x, without any predications is absolutely useless.
Quoting fishfry
I agree, what I meant is that this appears to be the inherent order, but it's not necessarily, that's why I went on to say that we can deny that order.
Quoting fishfry
I think I see the need for this, and so I understand it.
Quoting fishfry
I think so, but I also think, that sort of inherent order has minimal effect, and the real issue comes up with the restrictions, or limitations to order which are constructed. What I am arguing is that how the inherent order manifests, is as a limitation to the order which one can select. If there is absolutely no inherent order, then we can select any order, but if there is limitations to what can be selected, we cannot choose any order. The examples you give are, I believe, selected, therefore they're probably no true inherent order. The example I gave, is that we cannot give 2 and 3 the same place in the order, they cannot be equal, so we need to proceed toward understanding how this limitation exists.
Quoting fishfry
So this is where the real problem lies, in defining a symbol, such as 2 or 3, as a set. Check back to what I said about the difference between internal and external properties. The subject now is a set, say 2, and a set necessarily has internal properties. We have the elements which compose the set, 0,1, which are also sets. As the set is also related to other sets, it has external properties, represented by the ?
operator. The external properties are not necessary, and are a matter of choice, but whatever choice is made, that choice dictates the nature of the internal properties.
Now here's where I think the illusion lies. A set necessarily has internal properties, even though there may be infinite possibility as to the nature of the internal properties, making the specific nature of the internal properties dependent on choice, in this case von Neumann's choice. The illusion is that since the specific nature of the internal properties is dependent on a choice from infinite possibilities, it would therefore be possible to have a set with no internal properties. Clarification of the illusion implies that a set cannot exist prior to the choice of external properties, which dictate the internal properties. Internal properties are essential to "a set", and so a set has no existence prior to the choice of external properties, which determine the internal properties. This makes the empty set, as a set with no internal properties, impossible. The problem now, is what is zero? It can't be a number, because numbers are sets, and an empty set is impossible.
Quoting fishfry
I think you misunderstand. As I explain above, you refer exactly to the internal (intrinsic) properties of 2 and 3, as sets, to show that they are different numbers. What the set theory has done is denied order as an external property of those things, 2 and 3, as numbers with order relative to other numbers, and made it into an internal property of those things, as sets. An internal property is an intrinsic order. The fact that the intrinsic order is ultimately dependent on choice is irrelevant, because some order must be chosen for, or else the system would be meaningless.
Quoting fishfry
No, you've simply shown how external order has been switched for internal order. And now I've shown the problem which arises from this switch, the contradictory, therefore impossible "empty set", which makes the inclusion of zero an inconsistency.
Quoting fishfry
As I say, the idea that you've gotten rid of the order properties is just an illusion. The order inheres within each individual number, as the definition of that specific set. Rather than simply being an external property of a number, as an object, and how it relates to other numbers, order is now an internal property of the number itself, as a set..
Quoting fishfry
I argue the exact opposite, that you are consistently wrong about this. It is exactly "two copies", just like the word "same" here, and the word "same" here, are two distinct copies, even though we say it's the same word. Look, we are talking the meaning of symbols here. "A=B" means that that symbol A has the same meaning as B, it does not mean that A signifies the same entity as B, without additional information. However, the additional information in this case indicates that what is signified by A and B is a set, "the same set". But a set is not a thing, it is a group of things, grouped by a categorization such as type. Therefore this is an instance of "the same meaning", signified by A and B (indicated by "type"), not an instance of the same entity signified by A and B. This is just like when we use the same word twice when the word has meaning, rather than referencing a particular object. We say that the word has the same meaning, just like we might say A and B have the same meaning, in your example.
Quoting fishfry
Exactly, and this is a different meaning of "same" from the meaning of "same" in the law of identity. That is the point. In the law of identity "same" means a lot more than simply having the same members (what I called a qualified "same"), it means to be the same in every possible way ("same" in an absolute, unqualified way),
Quoting fishfry
I totally agree with that, that's what "same" means in this context. The problem is that it does not mean what you stated above: "They are in fact the identical set, of which there is only one instance in the entire universe". The set is an imaginary thing, indicated by meaning, it is not something in the universe. So it's not even coherent to say that there is one instance of that set, it's not even a thing which has an instance of existence, it's just the meaning of a symbol. So you speak of "the same set", and claim there is only one instance of that set, but this would be taking a different meaning of "same", which refers to instantiated things, and applying it to "same set", which really means having the same meaning, and not referring to one instantiated thing. Do you see the difference between referring to one and the same thing with a name, "MU", and using a word which has meaning, like "person", without any particular thing referred to? Person refers to a type, so it has meaning, just like "set" refers to a type, so it has meaning. These do not refer to instantiated things of which we could say there is one instance of, they refer to ideas.
You're the one who gave me Enderton's set theory book as a reference. Ok whatever. Nevermind that.
I will agree with you that identity is implicitly in extensionality, in the sense that two sets are equal if they have the "same" elements. We need identity to know when two elements are the same.
Now, since the elements of sets are other sets (barring urelements for the moment), I can see that there's a bit of a pickle. I''m not sure how this pickle is resolved.
Perhaps this is what you're trying to explain to me.
Is it?
Good question. Nothing, really. Can we work on getting these posts shorter?
Quoting Metaphysician Undercover
Physical collections have inherent order. Sets don't. That's all I'm saying. You seem to agree. What are you concerned with then?
Quoting Metaphysician Undercover
Is this related to the intensional and extensional meaning of symbols as has been discussed previously?
I have no idea what topic you are discussing at this point. I've agreed with you about the playground and I thought I'd explained to you about sets. What is left to discuss?
Quoting Metaphysician Undercover
This is lost on me. I've already conceded your point about the school kids being ordered, even if in a state of disorder.
Quoting Metaphysician Undercover
School kids are not sets. This para and the point of view it represents is totally lost on me. I understand what you're saying, I just don't have much interest in the subject. You think order is an inherent part of a set, so that if I line the kids up differently, it's a different set of kids. That doesn't sound reasonable.
Quoting Metaphysician Undercover
Can't argue with that! Why are you telling me this?
Quoting Metaphysician Undercover
I'm sorry, I know you put some thought into this and wrote a lot of words, but I don't feel part of this conversation.
Quoting Metaphysician Undercover
Absent from the convo.
Quoting Metaphysician Undercover
I am not your philosophy professor and this is not going to get you a good grade in my class. Why are you telling me all this? Honestly. I don't get it. I'm sorry.
Quoting Metaphysician Undercover
Boundaries and separations are topological properties about sets. it's amply covered in mathematical treatises on topology.
Quoting Metaphysician Undercover
Then you agree with me about sets. So we're good on that. I have no opinion about the other matters you've discussed.
Quoting Metaphysician Undercover
Did you read that back to yourself before you committed it? What am I supposed to make of this?
Quoting Metaphysician Undercover
2 and 3 are different sets per extensionality. I thought I explained that.
Quoting Metaphysician Undercover
Take it up with von Neumann. It's his idea. I'm only a humble student of these concepts from many years ago.
Quoting Metaphysician Undercover
Can't understand a word of that.
Quoting Metaphysician Undercover
The empty set is given by the axiom of the empty set.
https://en.wikipedia.org/wiki/Axiom_of_empty_set
Quoting Metaphysician Undercover
I'm not referring to internal properties of anything. 2 and 3 are MODELED within set theory as sets. There are other encodings. Maybe you are trying to get at Benacerraf's concerns in his famous paper, What Numbers Could Not Be, in which he points out that number's aren't the same as their encoding. This paper is regarded as having sparked the the movement toward structuralism in modern math, in which numbers are what they do, not how we encode them.
Quoting Metaphysician Undercover
Word salad.
Quoting Metaphysician Undercover
It's quite pointless to deny the empty set. It shows that you utterly fail to understand the nature of abstraction.
Quoting Metaphysician Undercover
Lost me again.
Quoting Metaphysician Undercover
You're wrong about that. You might as well argue that the knight in chess doesn't really move that way.
Quoting Metaphysician Undercover
I think I already agreed with you about this point.
Quoting Metaphysician Undercover
Now that we're agreed can we stop? You lost me with your theme of this post.
Quoting Metaphysician Undercover
There's only one instance of each set. You seem to disagree. Don't know what to say.
ps -- I know these ideas are important to you and my post was dismissive. Perhaps if you could give me your overall point it would help.
Are you saying set theory's a poor model for reality? Well of course it is, nobody claims it's a model for reality, only for math.
You say you don't believe in the empty set? But that's like saying you reject the way the knight moves. The empty set and the knight move are each rules in their respective formal games.
How can you disbelieve in a rule in a game?
Nobody but you is making ontological or metaphysical claims about sets.
If you could just clearly summarize your concerns, it would help. The internal and external stuff, I'm sure it's interesting, but I was not able to relate it to anything we've ever talked about. So just toss me a clue if you would.
Enderton's set theory text is a great book. But, as with many excellent set theory books, it doesn't mention all the technical details.
Quoting fishfry
I didn't say that identity is implicitly in extensionality, whatever that might mean.
I've said that usually set theory is based on first order logic with identity. That includes the identity axioms (such as found in Enderton's logic book). Then set theory adds the axiom of extensionality that provides a sufficient condition for identity that is not in identity theory.
I don't know how I could be more clear about that. Explicity:
Start with these identity axioms:
Ax x=x (a thing is identical with itself)
and (roughly stated) for all formulas P(x):
Axy((P(x) & x=y) -> P(y) (if x is y, then whatever holds of x then holds of y, i.e. "the indiscernibility of identicals")
Then add the axiom of extensionality:
Axy(Az(z e x <-> z e y) -> x = y) (x and y having the same members is a sufficient condition for x and y being identical)
Quoting fishfry
We need identity axioms to prove things we want to prove about identity, including such things as:
x = y <-> y = x
(x = y & y = z) -> x = z
(P(x) & x = y) -> P(y) (for example, (x is finite & x = y) -> y is finite)
etc.
/
Suggestion: Learn the details of the axioms and rules of inference of first order logic with identity. Then start with the very first semi-formal proofs in set theory (such as a set theory textbook usually gives semi-formal proofs), and confirm how those proofs would be if actually formalized in first order logic with identity. Then you would see how the axioms and rules of inference of first order logic with identity play a crucial role in set theory.
/
Quoting fishfry
I have no idea what pickle you see.
Quoting fishfry
No.
If you read again the first post in this thread on this particular subject, with regard to exactly what I've said, step by step, then it may become clearer for you. But also, as mentioned, learning the axioms and inference rules of first order logic with identity would be of great benefit. My suggestion would be to start with:
Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar
(but you could skip it if you feel strong enough already in doing formal proofs in symbolic logic and making simple models for proofs of consistency and proofs of invalidity by counterexample)
Then:
A Mathematical Introduction To Logic - Enderton
I found both of those books to be a special pleasure and profoundly enlightening. The Enderton book especially blew my mind, as I saw in it how mathematical logic so ingeniously, rigorously and elegantly gets to the heart of the fundamental considerations of logic while making sure that no technical loose ends are left dangling.
Right. Only mentioning it because when I asked for references you mentioned it.
I might look at his logic book. I confess, as you insightfully noted, that I may know a little set theory, but my first-order predicate logic is for sh*t. You got it. In fact I had a great course in sentential logic, basic stuff; then I became a math major and just sort of picked up the idea of logic.
I took undergrad mathematical logic, but it made my eyes glaze so I dropped it. The professor was a famous logician but I couldn't understand his lectures. So I dropped the course. Then I took grad level mathematical logic from Schoenfield and I was fairly lost, from not having taken the undergrad version.
So you are right, I suck at logic. I hope you'll keep that in mind and keep it simple :-)
Quoting TonesInDeepFreeze
Best I can interpret your thesis. Else I have no idea what your overarching point is. Maybe you can state that. What one thing do you want me to know about this?
Quoting TonesInDeepFreeze
I believe you to be saying that the axioms of set theory implicitly incorporate the axioms and rules of first order predicate logic. Is that what you're saying? If so, I agree.
But if set theory adds an axiom, then clearly it is not the same thing. It's something else, a new thing. It's a new algorithm that we apply when we try to determine if two things are equal.
Have I got that right? So they're different things, they're principles that operate at different levels of the abstraction. Yes? Maybe?
Quoting TonesInDeepFreeze
Ok at this point, I am wondering: Why are you telling me this? I don't understand what you want me to know about this. What problem are we trying to solve?
Quoting TonesInDeepFreeze
I am certain I never said we don't need identity! Did I give the impression I'm part of a committee to ban the law of identity? I'm all for the law of identity. A thing is equal to itself. That's good do know. In fact it helps make equality an equivalence relation with exactly one item per equivalence class.
Quoting TonesInDeepFreeze
If I could but dispatch a clone for that job. You, clone, go spend two years learning mathematical logic and report back to me.
It's a worth aspiration, but not something I'm likely to do. I actually have a bit of a research interest of a historical nature, that's where my study time should go. I'm pretty lazy at that too. So the logic has no chance. Alas.
Insightful of you to notice, though. My ignorance laid bare for the world to see. I am ignorant of many things.
Quoting TonesInDeepFreeze
Then my attempt to explain my take on the subject we're discussing failed.
Leaving me to wonder what we are talking about.
Quoting TonesInDeepFreeze
I am at a loss then. Mystified.
Quoting TonesInDeepFreeze
If that is the price of conversing further with you on these matters, I must confess that I'm not worthy. I will not be reading these nor studying first order predicate logic. Not because I would not dearly love to. But because time is finite.
Quoting TonesInDeepFreeze
I am happy you had that profound intellectual experience.
Now if you will just tell me what we're talking about, then I will have a profound intellectual experience.
Who was the famous logician?
Shoenfield's logic textbook is rich and has lots of stuff not ordinarily in such a book. But it is difficult, and he uses some terminology inconsistent with ordinary use in the field.
As I recall, many posts ago, my initial point was that, contrary to your assertion, the axiom of extensionality, as ordinarily given, is not a definition. Then I went on to explain how there are other ways to set up the logic and the set theory axioms so that a different version of extensionality would be a definition.
An ordinary presentation of set theory either explicitly or implicitly has set theory based upon first order logic with identity theory.
Yes, of course, set theory has non-logical axioms, so set theory is not just first order logic with identity.
Quoting fishfry
That's not right.
In set theory, we use both the logic axioms (which include the identity axioms) and the set theory axioms (which include the axiom of extensionality). lf our focus now speaking is identity theory and the axiom of extensionality, then it suffices to say that we use both.
Quoting fishfry
I was trying to solve the problem that you had not been understanding me as you characterized my point again incorrectly, so I tried to state it in as simple terms as I could.
Quoting fishfry
I didn't say that you did. Rather you gave your reason that we need identity. And I take it that 'identity' in that context is short for 'the axioms and semantics regarding identity', and I gave better reasons that we need them.
However, several posts ago you did indicate (as best I could tell) that you think the axiom of extensionality is all we need for proving things about identity in set theory, which would comport with your view that the axiom of extensionality is a definition. So, I mentioned that, for example, from the axiom of extensionality alone we cannot prove:
(x = y & y = z) -> x = z
Quoting fishfry
We need the law of identity, but we also need the indiscernibility of identicals.
(But Wang has an axiomatization in a single scheme.)
Yet, interestingly, from the axiom of extensionality we can derive the law of identity:
(1) Az(z e x <-> z e x) logic
(2) x = x from (1) and the axiom of extensionality
But the law of identity does not ensure that '=' stands for an equivalence class. It only provides
x = x
It does not entail
x = y -> y = x
nor
(x =y & y = z) -> x = z
To get all the needed identity theorems, we need both the law of identity and the indiscernibility of identicals.
Moreover, we want to ensure that '=' stands not just for an equivalence relation but for, indeed, the identity relation. And to do that we have to make the semantic stipulation that '=' is interpreted as standing for the identity relation on the universe.
Quoting fishfry
Me too. There is so much I didn't learn a long time ago but should have learned. I never got past a pretty basic level. And now I am very rusty in what I did learn, and don't have very much time to re-learn, let alone go beyond where I was a long time ago.
Quoting fishfry
You said that sets have sets as members and that there is a pickle about that viv-a-vis identity.
But I've not had any pickle in that way and I have not read about such a pickle. So I can know what pickle you have in mind only if you tell me.
Quoting fishfry
Of course it is not.
Ok, thanks.
Quoting TonesInDeepFreeze
Rather not say.
Quoting TonesInDeepFreeze
It did me in, sadly.
Quoting TonesInDeepFreeze
Of course it is. It's an axiom. It says what is true about all the things we call sets. Therefore we can characterize the world of things into sets and non-sets, according to whether they satisfy the axiom. So axioms serve as definitions and vice versa. They are the same thing.
Quoting TonesInDeepFreeze
That's fine, but that's one of the points where you lose me. Why do you care, or why do you think your doing so will make me understand something I didn't understand before?
Quoting TonesInDeepFreeze
I've never heard of identity theory except in the context of many of the Wiki disambiguations. And when I showed you the most likely meaning, you rejected it. So I have no idea what identity theory is.
Quoting TonesInDeepFreeze
I don't recall even having an opinion about this, let alone expressing it in this thread.
Quoting TonesInDeepFreeze
Every attempt I make to understand you is wrong. So maybe just give up because I don't get it.
Quoting TonesInDeepFreeze
Give me an example.
Quoting TonesInDeepFreeze
You can see that you could be more clear, because EVERY idea I toss out to try to relate to what you're saying, you reject.
Quoting TonesInDeepFreeze
Apparently not simple enough. Perhaps I'm not capable.
Quoting TonesInDeepFreeze
I said reasons and you said better reasons? Ok. Your reasons are much better than my reasons for believing things we both agree on.
Quoting TonesInDeepFreeze
There is no such thing as identity in set theory. It's not defined as any part of set theory in Enderton or Kunen.
Quoting TonesInDeepFreeze
I already explained why it's both an axiom and a definition.
Quoting TonesInDeepFreeze
Of course we can, straight from the axiom.
Quoting TonesInDeepFreeze
Not really.
Quoting TonesInDeepFreeze
Irrelevant to anything I can relate to, in this conversation or in general.
Quoting TonesInDeepFreeze
The identity relation is an equivalence relation.
Quoting TonesInDeepFreeze
I never stipulated to it.
Quoting TonesInDeepFreeze
Even so, logic would be way down my list. I think I already passed on a couple of excellent opportunities.
Quoting TonesInDeepFreeze
I think I talked myself out of my pickle. No pickle.
Quoting TonesInDeepFreeze
Ok.
I addressed that in detail. You could reread what I wrote.
Quoting fishfry
I post for at least as an end in and of itself, and also meaningful record for whomever may read it, no matter how few people or even accepting that it might be none at all. It would be good if my best efforts in explanation were understood, but I cannot ensure that they are, especially given that they are ad hoc and out of context of the required material they depend on.
Quoting fishfry
We're going around full circle.
(1) I said it may be more commonly called 'first order logic with equality'.
(2) For about the fourth time, a only a few posts ago I gave the axioms. And you responded by asking why I posted it!
(3) And I gave you a reference to Enderton where he stated an axiomatization equivalent with the one I gave. And Hinman also, and moreover as he states set theory as based on first order logic (which is to say, first order logic with equality).
(4) You said yourself that you recognize that set theory is based on first order logic. Set theory is based on first order logic with equality. That is what identity theory is, as I've said before. Whether called 'identity theory' or 'first order logic with equlality', it's the same set of axioms.
Quoting fishfry
You had written.
Quoting fishfry
And I agreed with that, but wished to phrase it as I prefer.
Quoting fishfry
Yes, because the reasons I mentioned go the heart of the motivation for the axioms.
Quoting fishfry
That's up to you. But I am not errant for correcting things that are wrong.
Quoting fishfry
You just say "not really" without basis and without even taking a moment to reflect on it.
Again, without the indiscernibility of identicals we wouldn't be able to prove the theorems regarding identity that are required for our formalized mathematics.
Quoting fishfry
Then you could show it. Give the proof steps, using only the axiom of extensionality and the law of identity (and even also any number of the other set theory axioms). This is the second time I've suggested that you show what you think is a proof:
And this is the second time I say:
WARNING: You may not use substitution of equals for equals unless you prove that principle from only the law of identity and the set theory axioms.
And now I'll add:
HINT: You won't be able to prove the principle of substitution of equals for equals, since it is itself tantamount to the indiscernibilithy of identicals.
And you studied with Shoenfield. On page 21 lines 13 and 15 of his book you will see the equality axioms that are the indiscernibility of identicals, similar to the way I formalized and that you asked why I posted it.
Quoting fishfry
Of course it is. My remarks imply that I know that it is. My point though is that it is a particular equivalence relation, which is that x is identical with y iff x is y and not merely that they
Quoting fishfry
'EVERY' (all caps no less) is an overgeneralization. It is proven wrong even in my previous post where I agreed with you that set theory is more than identity theory.
But, yes, I do comment often where I find errors.
Quoting fishfry
Relate to what you will, but the statement I made corrects your false claim that that we do not need the indiscernibility of identicals for doing even ordinary math, even if it is a reasoning principle that mathematicians don't bother to log in their arguments. It is, as they say, "the water the fish lives in". The substitution of equals for equals is ubiquitous.
Quoting fishfry
So what? In logic it is ordinarily stipulated.
(x = y & y = z) -> x = z
You may use only the law of identity Ax x = x and the axioms of set theory. You may not use substitution of equals for equals, unless you prove it from only the law of identity and the axioms of set theory.
For reference, here's the axiom of extensionality:
Axy(Av(v e x <-> v e y) -> x = y)
Here's a head start to a dead end:
Assume x = y & y = z
Show x = z
To show x = z it suffices to show
Av(v e x <-> v e z)
Go to town on it! I'd like to see what you got!
/
Or try proving:
x = y -> y = x
The reason why physical collections are different from sets, in this way, is that physical objects are different from intelligible (including mathematical) objects. What I am concerned about is that the law of identity, as formulated from Aristotle, is specifically designed from a recognition of this difference, and intentionally designed to protect, and maintain the understanding and acceptance of that difference. To put it simply, an abstraction, intelligible object, is a universal, and a physical object is a particular. The law of identity refers to the identity of a particular. And, because intelligible objects are different from physical objects, as you recognize and acknowledge, they cannot be held to this law. So mathematical ideas, if they are called "objects", are objects which naturally violate the law of identity. In short, that's how we distinguish physical objects from ideas, with the law of identity.
In classical sophistry physical objects are confused, mixed up, and conflated with intelligible objects. The difference between the particular and the universal, as "objects" is ignored. This allows sophists to logically prove things which are absurd. The law of identity is intended to enforce that difference, and expose the faults of the sophist. The head sophist at TPF, TIDF, continues to defend sophistry by arguing that intelligible objects are consistent with the law of identity.
Quoting fishfry
I'd have to say, no, not really. Internal/external properties is a distinction we make concerning the properties of particular physical objects, the object's internal relations, and the object's external relations. Intensional/extensional meaning is a distinction concerning the meaning of a word, how the word relates to ideas, and possibly physical objects. This is a matter of semiotics, and Charles Peirce provides some very good insight into the use of symbols. But that is a completely different matter from what I was discussing, as the internal/external properties of a physical object.
Quoting fishfry
The problem, is that you continually cross the boundary of separation between physical objects, and intelligible objects, in your manner of speaking, in the sophistic way, without even noticing it. That's what happened with your example of schoolkids. In order for the example to work, "schoolkids" must refer to a multitude of particular physical objects. Yet "set" must refer to an intelligible object. So in speaking the example you cross the category separation, back and forth in the way of sophistry, without even realizing it.
Imagine if we were to maintain the boundary. Instead of having schoolkids in a playground, we would be talking about the idea of "schoolkid", or an imaginary schoolkid. This appears to deny the possibility of any extensional meaning. Further, if we want a number of schoolkids, then we need a principle of separation to distinguish one from the other. But that principle of separation would either create an order amongst the imaginary schoolkids, or else produce a complete separation of type, making distinct types of schoolkids.
Quoting fishfry
OK, you have no interest in the difference between a subject to be studied and an object to be studied. That's fine by me, but until you learn this difference you are likely to continue to speak in a way which mixes these two up, and makes your examples and arguments appear like nothing more than sophistry, and arguing by equivocation, just like Tones. This is what happens when a subject is called an object (mathematical) and the difference between the physical object and the mathematical object, (as defended by the law of identity) is ignored.
Quoting fishfry
That's right, you are not my philosophy professor, that would reverse credentials. I am your philosophy professor, and your lack of interest deserves a failing grade.
Quoting fishfry
Right, this is why a set is not an object, objects have internal properties and external properties, sets have meaning.
Quoting fishfry
There is no "instance" of any set. You recognize that there is a difference between physical objects an sets, why do you not see that there is no such thing as an instance of a set? Sets are not the type of thing which have an instantiation. "Instance" refers to a particular, a set is a universal. That sort of misleading statement is where the sophistry kicks in, even though I know you are not intending to be misleading..
Quoting fishfry
That's a simple question with a simple answer. When a rule in a game contradicts another rule in a game, this is cause for disbelief in the whole game. That was the point of the example I gave you of waves in physics.
Quoting fishfry
That has become obvious to me. But in a philosophy forum, things ought to be the other way around. We ought to be discussing the ontology of sets and working through the problems which arise.
Quoting fishfry
There's too many concerns to summarize. But let's look at a most fundamental problem of set theory as an example. You recognize the difference between physical objects, and sets, so let's start there. Now, consider the elements of a set, these might be sets as well. The elements of a set are not physical objects, just like sets are not physical objects. The elements are ideas, universals, they are not particulars or individuals. Since they are not particulars the set cannot be measured as particulars. A set cannot have a cardinality. That's a basic problem.
So not necessarily for me. Ok good to know. Maybe this site should have a @Whoever generalized user so that people can direct their rantings to the universe.
Quoting TonesInDeepFreeze
Many times. Very high winding number.
Quoting TonesInDeepFreeze
Sorry I asked. I don't think I can continue to hold up my end of this conversation.
Quoting TonesInDeepFreeze
You gave me a ref to Enderton's set theory book, then retracted the reference when I took the trouble to check it out.
Quoting TonesInDeepFreeze
Ok
Quoting TonesInDeepFreeze
You have MUCH BETTER REASONS than I do. Ok.
Quoting TonesInDeepFreeze
You're right, I'm wrong.
Quoting TonesInDeepFreeze
I admitted to being a logic slacker.
Quoting TonesInDeepFreeze
Ok. I have no response. I no longer know what we were talking about. Definitely regretting getting into the middle of this. You're right, I'm wrong.
Ok wrong question. I asked why are you concerned, and you wrote that para. What I should have asked is, why do you think I care? What is this to me? I'm not involved in this conversation.
Quoting Metaphysician Undercover
Drat those sophists. Are they in the room with us right now?
Oh I see. Tones. Well my fundamental error was accidentally getting between you and @TonesInDeepFreeze, which has caused me to recall the saying, Act in haste, repent at leisure.
Quoting Metaphysician Undercover
Ok.
Quoting Metaphysician Undercover
I make many errors.
Quoting Metaphysician Undercover
Well discussing set theory with you on its own terms has proved futile in the past.
So I gave an informal example of real world objects, and you have been hammering me at length about it now for several posts. So forget the school kids. The elements of sets have no inherent order. The purpose of setting things up that way is so that we can abstract the qualities of belonging and order from each other. End of story.
Quoting Metaphysician Undercover
For God's sake it was an informal example, which I had to resort to because you dislike my saying sets have no inherent order. Except when you occasionally say that you accept the point.
Quoting Metaphysician Undercover
LOL That's like that famous conversation between Jordan Peterson and Cathy Newman. "I think the world is round." "Why do you hate minorities and gays?" Not the exact quotes but same general rhetorical technique.
Quoting Metaphysician Undercover
I used the school kids example because in the past you've had trouble understanding set theory. From now on, pure set theory only. No real world examples, since sets aren't real and you needn't further belabor that point.
So am I now on the same sh*t list as Tones in your book?
Quoting Metaphysician Undercover
Give it a rest, man.
Quoting Metaphysician Undercover
Well Tones has already flunked me in logic, and now having been flunked by you in philosophy, my academic career is complete.
I'm not referring to internal properties of anything. 2 and 3 are MODELED within set theory as sets.
fishfry
Quoting Metaphysician Undercover
Sets have no meaning whatsoever, other than that they obey the axioms of set theory. You still don't understand that.
Quoting Metaphysician Undercover
There is exactly one instance of every set.
Quoting Metaphysician Undercover
Because I know set theory.
Quoting Metaphysician Undercover
You are beyond help. You refuse to understand.
Quoting Metaphysician Undercover
This was in response to your denial of the empty set. Tell me exactly -- and be extremely clear and specific, please -- tell me what other rule of set theory is contradicted by the empty set.
Quoting Metaphysician Undercover
I'd be happy to do that, but since you aggressively refuse to engage with set theory on its own terms, we cannot have that discussion.
I have explained to you the ontology of sets many times. They are mathematical abstractions.
Quoting Metaphysician Undercover
Ok.
Quoting Metaphysician Undercover
Yes. They generally are, since set theories with urelements are mostly for specialists.
Quoting Metaphysician Undercover
Meta you are on a roll. You've said several correct things in a row.
Quoting Metaphysician Undercover
You know, I am not sure I agree that sets are universals. My understanding is that "fish" is a universal, and the particular tuna that ended up in this particular can of tuna I bought at the store today is a particular instance of the category or class of fish.
Sets are not like that at all.
I did ask you a long time ago to explain what you meant by universals, and you snarked off at me. And now you come back at me claiming that sets are universals. Explain to me what you mean by that.
The concept of a set is a universal. The set of rational numbers is a particular set, of which there is exactly one instance.
Quoting Metaphysician Undercover
LOL. Oh man you're crackin' me up. The set of rational numbers most definitely has a cardinality of [math]\aleph_0[/math], because of Cantor's discovery of a bijection between the rational numbers and the natural numbers.
Making clear corrections, giving generous explanations, commenting the deplorable methods of cranks, and posting ideas in general is not ranting.
/
Looking back:
Quoting TonesInDeepFreeze
So I see now that I recommended Enderton's set theory book in general. I didn't say that it is specifically a reference to the fact that set theory is based on identity theory (first order logic with equality).
And by starting with Enderton's logic book, which does present the axioms for '=', you would see how they work in set theory even if not explicitly stated in his set theory book.
But when you complained that it does not mention identity theory, I said that I would have been mistaken if I offered it for reference on that matter. And, now that I see the context, I grant that, since the context was general, it would not be entirely unreasonable for you to take it that at least part of the reason for my recommending the book is that it mentions identity theory, so, in that respect, and to that extent, my recommendation was faulty.
But then I followed up by pointing to Enderton specifying the equality axioms in his logic book (though he doesn't mention in that book the fact that set theory is based on first order logic with equality). And that was pertinent to your complaint that you couldn't find anything on that topic.
And I cited Hinman's book that both gives the axioms for equality as part of first order logic, equivalent to the axioms I posted, and he says that set theory is based on first order logic.
And I referred you to Shoenfield's book that specifies the axioms for '=', equivalent to the axioms I posted.
And, you yourself agree that set theory is based in first order logic. So, all that is needed is to show citations that first order logic ordinarily includes identity theory (i.e. first order logic with equality) and that was accomplished by citing Enderton's logic book, Hinman, and Shoenfield. But I guess that, despite my sin of overlooking that a certain book doesn't supply reference to a particular point (though it still is an excellent reference for the context of this subject and on other points) it seems I am finally past needing to explain over and over and over that the identity axioms are in first order logic and set theory is based in first order logic, as you post:
Quoting fishfry
/
Quoting fishfry
In that instance, yes, and that is made clear by what I wrote. But rather than address the substance of the matter, you opt for ill-premised sarcasm about the exchange.
Quoting fishfry
Of course, that's hardly even a foible. But it's at least odd that someone who knows nothing about the matter would categorically say that it false that the indiscernibility of identicals is not included in first order logic with '=' as primitive.
Quoting fishfry
We were talking about how '=' is interpreted.
Was referring to all of us, not anyone in particular.
https://en.wikipedia.org/wiki/Mostowski_collapse_lemma
So I see now that I recommended Enderton's set theory book in general. I didn't say that it is specifically a reference to the fact that set theory is based on identity theory (first order logic with equality).[/quote]
Too late to wriggle out, you've already been found guilty by the court of Me.
Quoting TonesInDeepFreeze
Well maybe I'll see if I can find the pdf and sort this out for myself someday.
Quoting TonesInDeepFreeze
Ah, you have 'fessed up after all. Good. Let's speak of the matter no more. I'm sure you're right about some aspect of this.
Quoting TonesInDeepFreeze
Having already been found guilty, are you now preparing your appeal? Won't help, I'm the appellate court too :-)
Quoting TonesInDeepFreeze
I'm sure it is. I'll concede the point (if I even knew what the point was) for the sake of keeping the peace.
Quoting TonesInDeepFreeze
Have you ever been accused of taking things too literally and too seriously?
Quoting TonesInDeepFreeze
Yes.
Quoting TonesInDeepFreeze
I have been so cited.
Quoting TonesInDeepFreeze
You are making more of this than I intended for you to make.
Quoting TonesInDeepFreeze
à la Walter White: I am the one who rants.
Quoting TonesInDeepFreeze
I'm mocking you for saying that you agreed with a point I made, but that your reasons were better; and now for doubling down on that silliness.
I have a bad habit of tweaking and needling people who take things too seriously, and I better put a stop to this before it goes too far.
Quoting TonesInDeepFreeze
I don't know "nothing" about the matter. I know logic as it's used in math, but did not study enough formal predicate logic. Indiscernibility of identicals I know of in other contexts, and am genuinely surprised to hear that it's incorporated into set theory.
Quoting TonesInDeepFreeze
It's interpreted as the axiom of extensionality in set theory. Which doesn't actually require identity, and I've asked for a specific example to prove otherwise.
If I have two sets, and I want to know if they're equal, I apply extensionality. Not identity. And if I have an two objects that are not necessarily sets, I don't see them because I'm doing set theory. This is my point. I ask for a clear clear clear clear clear refutation or counterexample. I could be wrong. I'd like to understand. Explain better please.
ps -- I downloaded a pdf of Enderton's book on mathematical logic. Toss me a page ref please and I'll look it up.
No wiggling. It was faulty of me to reference that book without specifying that I do not claim it discusses the identity axioms.
Quoting fishfry
I had previously admitted that I would have been mistaken if I referenced the book regarding identity theory.
So, no, not "after all".
Quoting fishfry
You could have stopped the first time I recognized my lapse.
And I make of it what it is worth: Being clear as to what was actually posted.
Quoting fishfry
I know that. And your mocking is sophomoric. There is nothing amiss in agreeing on a point with someone but commenting that nonetheless their reasoning about it is poor.
If you cared more about the subject at hand then in prevailing with lame smart-aleckisms, then you could go back to the post to see my substantive point.
Quoting fishfry
You're not good at it. And I don't buy that your motive is just to josh but not also imbued with putdown as a kind of trump card.
Quoting fishfry
You don't know enough to know that when we use the principle of substitution of equals for equals in mathematics, including set theory, we are in effect using the principle of the indiscernibility of identicals, whether we explicitly recognize it or not. And, with first order logic, which codifies and formalizes the reasoning for classical mathematics, we do explicitly formulate the principle in an axiom schema.
As I recall, the reason I mentioned the subject lately, and with that fancy name, is that the law of identity had been mentioned as historical and fundamental. My point was that also the indiscernibility of identicals is historical and fundamental. Indeed, with those two historical ideas, we axiomatize first order identity theory.
Quoting fishfry
I have to repeat myself:
(1) Interpretation is semantical. The axiom of extensionality is syntactical.
(2) Even just syntactically, the axiom of extensionality is not a definition, in the sense of a syntactical definition.
(3) If set theory didn't have the identity axioms, then, even with the axiom of extensionality, set theory would wouldn't even get very far off the ground.
(4) I already gave you specific examples about three times!
Without the indiscernabilty of identicals you can't prove:
(x = y & y = z) -> x = z
x = y -> y = x
(n is even & j = n) -> j is even
Quoting fishfry
I've said, extensionality gives a sufficient condition for equality, but not a necessary condition for equality.
Yes, to prove x = y, then it suffices to prove Ax(z e x<-> z e y)
But that's not the only thing we do with '='.
We also use equal to reason this way:
x has property P, and y = x, so why has property P.
And that uses the indiscernibilithy of identicals.
Sometimes it's called "substitution of equals for equals".
When a school kid says:
"2 = 1+1, so 1/(1+1) = 1/2"
that is using substitution of equals for equals.
So, to codify and formalize use of subtsiution of equals for equals, we make it an axiom. And that axiom is a fomalization of the identity of indiscernibles,
Quoting fishfry
I gave three, three times!
But you claimed that you could prove:
(x = y & y = z) -> x = z
using just the axiom of extensionality (or, even all the set theory axioms but not identity theory axioms), and I said:
THEN DO IT.
That exchange deserves nothing more than a snort.
The crank still can't vindicate his claims sets by answering what is the inherent order of the set whose member are the bandmates in the Beatles.
And fishfry is lately asking me to give examples regarding identity when I've given three of them three times already, while he has not shown a proof of the transitivity of equality without using sub of equals for equals, though he claims he can do it.
Spare me. I just looked through Enderton's logic book, The word indiscernible does not appear in the index. I looked up identity and did not find any kind of description of what you're talking about. Page ef please.
Quoting TonesInDeepFreeze
Don't become unpleasant.
Quoting TonesInDeepFreeze
And I gave you page number and line numbers for Shoenfield:
Quoting TonesInDeepFreeze
I said that the indiscernibility of identicals is formalized in identity theory. I didn't say that any particular formalization mentions it with the phrase 'the indiscernibility of identicals'. The principle was enunciated by Leibniz. But in mathematics, it's often called 'the principle of substitution of equals for equals'. And in modern logic, it is an axiom schema in the manner I've posted, which is equivalent (though notation and details differ) to Enderton and Shoenfield, for example.
I'm giving you a lot of the same information and explanation over and over, since you skip over it over and over.
I hope you know that 'the crank' does not refer to you. If that was not clear in the context, then I should have made it clear.
Oh thanks. I dropped by the site and saw I had 6 mentions and that they were all from you so I was snapping back pretty quickly without actually reading much.
So I saw a ref to equality on 112 of enderton that had nothing to do with set theory, and can't find anything at all on page 83. But on 112 he said that we can take as a rule x = x. But we don't need that for set theory! This is my point, or point of confusion. If I want to know if x = x for some set x, I can just apply extensionality and check to see if if for all z, z in x iff z in x. Which is of course true. So x = x. I don't need the law of identity to determine if x = x if x is a set. This is my point.
Quoting TonesInDeepFreeze
It gets lost in all the verbiage and symbology.
Quoting TonesInDeepFreeze
Ok. But you say this isn't written down anywhere?
Quoting TonesInDeepFreeze
This pushed hard against my understanding. The identity of indiscernibles says (afaik) that two things are the same if they share all properties.
Substitution says you can plug in things that are equal in expressions. I'm not sure how that relates.
Quoting TonesInDeepFreeze
I suppose I'll have to take your word for it, because it's not in Enderton or I missed your explanation earlier. I just glanced at the SEP entry for ident of indisc. and it doesn't say anything about substitution. If there's no written reference, is this perhaps an idea of your own?
In any event, if I want to know if two sets are equal I apply extensionality. And that's another reason I get lost in your posts. I don't see your point. Extensionality tells you everything you need to know about when two sets are equal. You don't need anything else.
Leibnizs Law, the principle of the indiscernibility of identicals, that if x is identical with y then everything true of x is true of y.
Ok thank you for that specific reference. You should know that I generally respond to my mentions and don't always monitor the threads. Please give me a mention when you want me to see your posts. Of course that doesn't guarantee I'll see everything you want me too, but at least I'll know you said something to me.
Yes, I said that it doesn't mention set theory, but rather it is a place to see the logical axiom schema for first order logic with equality.
However, as you have agreed, set theory uses first order logic with equality. So there you have it.
And, yes, at least twice, I said myself that Ax x=x can be derived from the axiom of extensionality. So in set theory Ax x=x is redundant. But the axiom schema for the indiscernibility of identicals is not redundant in set theory.
Quoting fishfry
But that is just a part of the picture. And you keep slapping about saying that we need only the axiom of extensionality and don't need the indiscernibility of identicals. And when I pointed out we can't prove
(x = y & y = z) -> x = z
from the set theory axioms without the indiscernibly of identicals, you claimed that you easily can.
So, again, I say that I'd love to see your attempt.
That is where he gives the semantics for '=', as I mentioned that '=' is given a fixed interpretation.
If you mean that it would help for my posts to link to yours, then I'll hope not to forget doing that each time.
My preference regarding you is that you don't gloss my posts and jump to conclusions that I've said something I didn't say but that you think I must have said in you own confusions or lack of familiarity with the concepts or terminology.
Ok I'll check again. I'm reading the SEP article on identity, and it's interesting reading. Puts some of what you've been saying in context. They did say that "Leibnizs Law must be clearly distinguished from the substitutivity principle ..." so perhaps that's pushback to your claim.
But there are actually two principles, identity of indiscernibles and indiscernibilility of iten ...
oh man i'm typing and you are replying back. I can't keep up. Let me just say that the SEP article is interesting and I'll get to the rest of this tomorrow.
However! You just said
Quoting TonesInDeepFreeze
in which case you agree with my main point and there is nothing more to say. That's why I'm confused. Once you concede that identity is not necessary for set theory, then I don't know why you are going on about set theory.
Quoting TonesInDeepFreeze
Challenge accepted, I will get to this tomorrow or day after, I am a little busy tomorrow.
So I appreciate that you are now writing much shorter posts, making it possible for me to read them. But you are compensating by writing very quickly, so that right now I'm two posts behind and you have one or two already ahead of me! I can't keep up.
So let me work on (x = y & y = z) -> x = z and I'll read through your posts tomorrow. It's after midnight right now where I live.
Quoting TonesInDeepFreeze
That only works if either (a) I happen to read all the recent posts in a given thread, which I rarely do; or (b) I happen to be posting at the same time as you.
I'm sure you can see this leaves a window where I might not see your posts.
That might be. I'm speaking in broad terms about them in that regard. If the article draws a needed distinction then I should say that they are at least akin.
Quoting fishfry
Right. I discussed that in about my first post on the subject in this thread.
Quoting fishfry
A few posts ago, I explained exactly why there is more to say.
Again:
Whatever was your main point, one of your points, and the one we've been thrashing over for dozens of posts, is that we only need the axiom of extensionality for identity. And I've explained and explained for you, a trillion ways to Sunday, exactly why that is not true.
Quoting fishfry
I write long posts because you post so many incorrect claims and confusions about the subject and confusions about what I've posted, and, often enough, I give meaty explanations.
Quoting fishfry
I am hopelessly behind composing posts in at least a few threads. Even years behind in threads that I just had to let go because I really should be spending my time on other things more important than posting.
Nope. I am consistent with the SEP article. The context in this discussion is plain predicate logic where substitution works, not intensional contexts.
I didn't get to your most recent yet. But I did have a bit of an epiphany and it's possible you may be steering me to enlightenment.
I started working on (x = y & y = z) -> x = z, which seems an easy consequence of extensionality.
So I started by writing down the statement of extensionality, and right away I see that I'm in trouble! I don't need extensionality ... I need the converse of extensionality. I need to go from x = y to saying that for all z, z in x iff z in y. That is not given by the axiom. So you have actually taught me something.
I did a quick search, and found this: The converse of the axiom of extensionality where he says that the converse "follows from the substitution property of equality."
So this is quite a bit more subtle than I thought, and I will have to work on this some more. I do think you have made your point, at least provisionally. I can't assume the converse of extensionality. Who knew, right?
In which case ... an axiom is NOT the same as a definition, because definitions are reversible, and extensionality is not. So I do believe you may have made a couple of good points with this example.
ps -- Ah ... the Wiki page on extensionality explains your point that if equality is not a primitive symbol in predicate logic, then extensionality is taken as a definition rather than an axiom. You did say this to me several times. I now begin to see your point.
This example did it for me. I have to study this a bit. I did not realize that extensionality goes only in one direction, and that the = symbol is not being defined, but is inherited from the underlying predicate logic. You have made your point. I need to understand this better.
Exactly. That goes right with what I've been saying.
Without sarcasm I say that it gives me a good feeling that reason, intellectual curiosity and communication have won the day finally.
I said that
ExAy y e x
is consistent.
You disputed that.
So I pointed out that I am not saying it is consistent with set theory, which has the axiom schema of separation from which we derive:
~ExAy yex.
Rather, it is consistent just as it stands alone.
I said "It is consistent onto itself." Yet, you still disputed me. Much later it dawned on me that you were thinking that I meant 'onto' as with a surjection. But I meant 'onto itself' to mean 'in and of itself'. And later I found out that people don't usually say 'onto itself' that way. So I saw that I had lapsed in English.
So here we are, and I am hoping that you see that I was correct that
ExAy yex
is consistent in and of itself, even though not consistent with the axiom schema of separation.
At the time I proved by adducing this model:
U = {0}
'e' stands for {<0 0>}
A typo there? I think you meant 'z in x iff z in y'?
Yes. Quite the epiphany. I've actually just found several web pages and articles explaining all this. One even mentioned that the converse of extensionality follows from Leibniz (either ident of indisc. or other way 'round). Evidently I'm the last person to find this out. Even the Wiki page on extensionality mentions this, and I thought I'd read it several times but evidently not that part.
Late here way past bedtime I actually need to be somewhere tomorrow morning I'm going to regret this. Will be offline till tomorrow evening or day after.. Thanks for the insight. It was the proof of the transitive property that did it. Once I realized I needed the converse, the floodgates opened. Great example.
Ok more later. Thanks again.
And we can state the identity of indiscernibles as a first order schema if there are only finitely many nonlogical symbols in the language.
But it's interesting that we cannot state the identity of indiscernibles as a first order schema if there are infinitely many nonlogical symbols in the language.
Not link, quote. Either quote a fragment of my post, as I just did to yours; or else just mention me as @fishfry, where you have to type "" around the handle name.
Linking posts is something else, at the bottom you can get a hard link to the post, but you don't need to do that.
Quoting TonesInDeepFreeze
Something about the scorpion and the frog. You expect me to stop having the many flaws I have? I will do my best, but interact with me at your discretion.
I have a large pile of mentions, so I'll get to them and save yours for later. I'm still in the afterglow of my set theoretic epiphany. I understood your point. You're right and I was confused, but now, thanks to your untiring efforts, you have unconfused me. Actually I think I was just hallucinating, because I do know that extensionality is an implication and not a bi-implication. I just never thought about the converse. But the converse is the "portal to the next level down," predicate logic.
I'm happy to have clarified this, it makes a lot of sense.
I think we can jump forward past the extensionality. The moment I saw the problem with proving the transitivity of set equality, I was enlightened. I swear, I almost literally smacked my head. "I can't use extensionality. I need the converse. So you picked the perfect puzzle to get through to me.
So going forward, I stand educated on this point. And although I do try my best not to exhibit my flaws, well, I may yet leap to an unwarranted conclusion now and then.
Shouldn't we all!
I wasn't clear; I didn't mean a URL link; I meant a reply link. Does the link in this post do what you want?
Now that we got the axiom of extensionality straightened out, it's apropos to get the rest of the dissension worked out.
It starts with these good posts:
https://thephilosophyforum.com/discussion/comment/911857
https://thephilosophyforum.com/discussion/comment/913150
{0 1} = {1 0}
There are two orderings of that set:
{<0 1>} and {<1 0>}
So there is not "THE" ordering of that set, since there are two of orderings of the set.
But we may indicate the set with regards to a particular ordering. The notation is:
where S is the set and R is a particular ordering. For example"
, }
<{0 1} {<1 0>}>
is the set {0 1} along with the ordering that is the greater-than relation on the set.
For example, the set whose members are all and only the bandmates in the Beatles has 24 orderings. So there is not "THE" ordering of that set.
But we may indicate that set with regards to a particular ordering. For example, the alphabetical ordering by first name:
{
As a sequence: {<1 George Harrison> <2 John Lennon> <3 Paul McCartney> <4 Ringo Starr>}
As a list: George Harrison, John Lennon, Paul McCartney, Ringo Starr.
But, obviously there are many other ways to order the Beatles: by age from youngest to oldest, by age from oldest to youngest, by height from tallest to shortest, by height from shortest to tallest, by wealth number of record sales as an artist after the Beatles, ...
So there is not "THE" ordering of the set whose members are the bandmates in the Beatles.
But what about that rock? If it's the one that is the crank's head, then it is indeed empty and there is only one ordering of the set of its particles, which is the empty ordering.
But what about more complicated, more intelligent rocks? The rock is not a set. However, we may speak of the set of particles of the rock. And in that case, again, there is no "THE" ordering of that set. But the crank mentions structure. Yes, we may describe the rock in terms of a certain structure. But the rock, even as described per a certain structure is not a set; it's a rock. Moreover, we may describe a rock as different isomorphic structures. Your structure is based on rock's pointy tip facing up, and my structure, isomorphic to your structure is based on the rock's pointy tip facing down.
/
The crank says I use definitions out of context. The crank confuses self-description with outward observation.
/
The crank says that he doesn't know what I mean by 'identity theory' even though I've stated and explained the axioms of identity theory at least a few times. (Or if I hadn't done that prior to the crank's post, then nothing was stopping him from asking me to do it.)
/
The crank makes the ridiculous claim that I misunderstand the rules of axiom systems. I understand the formation syntax of the formal languages, the formation syntax of the formulas, the formation syntax of the axioms, the formation syntax of the inference rules - all recursively. And the formation of the semantics for the meaning of the formulas - all by inductive definition. I understand exactly how to check that a purported formal proof is a proof and also I understand exactly how to interpret the meaning of formulas.
The crank doesn't know what he's talking about regarding mathematics or the axiomatic method or regarding me. Then he says that I annoyed him when we met but now I merely amuse him. Ah, the classic arch line, "You merely amuse me". The crank is not only a feeble thinker, he's a lame flamer. And why was he initially annoyed? Because as he was freely spewing confusion, ignorance and disinformation on this forum, I corrected him.
/
The crank repeats his argument that the notion of identity in mathematics is wrong since mathematics regards objects that don't exist. So, yet again, the crank just ignores the responses I've given to that. Just to start: He ignores even the examples I've given of sets of non-abstract objects, such as the set of pencils strewn on my desk, etc.
I explain in detail. And it's a stupid thing to say that I just type stuff. But in post or even a series of them, I can't fit in an explanation all the way back to the basics of the subject, so if one doesn't have the benefit of a context of adequate knowledge, it's not my fault that I can't supply all that needed context in even several posts.
Quoting fishfry
Exactly and well put. I've given the crank that same explanation. He will never understand it, because he wants to not understand it. If he found himself understanding it one day, then he would face the crisis of seeing that he's been confused and in the dark for years and years (decades?).
The crank's rejoinder is that we may state the positions and that that is "an order".
He is exactly right there. It is AN order. He said it himself! It is not "THE" order since there are different orders, each of them AN order.
And perhaps I am wrong: checking ArXiv.org I see that in the past week there have been around 25 new logic papers submitted - about the same number as those in my area, complex analysis. And the axiom of extentionality on Wikipedia garners about 60 views per day - a healthy enough following.
Just passing thoughts when reflecting on the current discussion. Kudos to the three or four involved. :clap:
Excellent, I love it. TPF's head sophist has a sense of humour.
Quoting fishfry
OK, so here we have the issue. Remove the examples of real world objects (schoolkids etc.) as "the elements", and what exactly is an element? It cannot be a particular thing, because it does not obey the law of identity, so it is some sort of universal, an abstraction. But what type of abstraction is it, one which we pretend is a particular? Why is it pretended that these are particulars? Maybe so that the set can be subjected to bijection, and have cardinality. The question then is whether the elements are truly individuals, or just pretend individuals.
Quoting fishfry
Isn't that exactly what meaning is, obeyance of some rules? Now, we know what a set is, something which obeys the rules of set theory, the real issue though is what is an element of a set.
Quoting fishfry
It seems you are having problems understanding the inherent difficulty of the empty set. I think we'd better have clear agreement on what an element is before we approach that more difficult problem of the empty set.
Quoting fishfry
Yes, but you also claim that sets have no meaning. An abstraction with no meaning is contradictory. That's why I can't understand your teachings about set theory.
Quoting fishfry
Any abstraction is a universal because its applicable to more than one particular set of circumstances. Whatever it is that any multitude of particulars has in common, is a universal.
You appear to be suggesting a third category other than particular and universal, an abstraction which is not a universal. Care to explain?
Quoting fishfry
Bijection is a problem, because it requires that the elements are individuals, particulars, which I argue they are not. This is why we need to clear up, and agree upon the ontological status of an "element" before we proceed.
Quoting TonesInDeepFreeze
I suggest we adhere to the principle you stated, the elements of a set are not things, like schoolkids, rocks or anything else. TPF's head sophist doesn't respect this principle.
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Etc..
So the sophist crank finally comes close to a true sentence, but still only half true. I'm not a sophist, neither philosophically nor rhetorically.
The sophist crank says, "the principle you stated, the elements of a set are not things".
I never said any such thing. I've said the opposite. The sophist crank again lies about me, as a function of his abysmal confusion.
The sophist crank says, as a paragraph, "Etc.."
More eloquent than a rock, by a word.
In general, excepting the somewhat lesser-known example of set theories with urlements, the elements are other sets. If we are justified, given the axioms (whichever we choose) of set theory, to write:
[math]x \in y[/math]
then we may colloquially read this as, "x is an element of y." That's what an element is.
Quoting Metaphysician Undercover
Actually I am wrong about that @TonesInDeepFreeze showed me the error of my ways. All sets satisfy the law of identity. If I have a set X, I may write X = X by way of the law of identity. I do not need the axiom of extensionality for that. Perfectly clear to me now.
Quoting Metaphysician Undercover
The law of identity applies to sets. So this line of argument is null and void.
Quoting Metaphysician Undercover
If your criterion is that they satisfy the law of identity, they do. So your concern is addressed.
Quoting Metaphysician Undercover
Yes, very good. A group is any mathematical structure that obeys the axioms for groups. A set is any mathematical object that obeys the axioms for sets.
Quoting Metaphysician Undercover
Typically it's another set. Sets are subject to the law of identity. This should satisfy your concerns.
Quoting Metaphysician Undercover
I believe in the field of psychology, this is known as projection. YOU have problems with the empty set. I have no such problems. The empty set is the set of purple flying elephants in my left pocket. Oh wait you don't like "real life" examples. Never mind.
The empty set is the set of things that violate the law of identity. In symbols:
[math]\emptyset = \{x : x \neq x \}[/math]
Happy now? (Of course you're not!) There are other formulations.
https://en.wikipedia.org/wiki/Axiom_of_empty_set
Quoting Metaphysician Undercover
An element of a set is a the left side of an expression [math]x \in y[/math] that can be deduced from the axioms of set theory. x is the element, and y is a set. But x is typically a set as well. Think paper bags inside of paper bags. Oops there I go with real world analogies again.
Quoting Metaphysician Undercover
They can be viewed that way from a formalist perspective.
It's of no importance to set theory. Certainly sets don't necessarily have real-world referents, since sets are quite a bit stranger than paper bags or collections in general.
What of it?
Quoting Metaphysician Undercover
What does chess mean?
See https://plato.stanford.edu/entries/abstract-objects/ and tell me if you find anything interesting in there.
Quoting Metaphysician Undercover
Do you see the difference between the concept of set, and the concept of the set {1, 2, 3}?
One's a general set, and the other's a particular set.
Since you won't define a universal in such a way that you can sort this terminology out, I think your idea of universals must be vacuous. Fish is to this particular tuna on the end of my fishing line, as sets are to the set {1,2,3}. There is nothing problematic about that.
Quoting Metaphysician Undercover
Me? I'm making no such suggestion.
Quoting Metaphysician Undercover
They are, as far as I understand your use of the terminology, which you refuse to explain.
Quoting Metaphysician Undercover
An element is a set in a set theory without urelements. We say x is an element of y if we can legally write [math]x \in y[/math]. Nothing could be simpler.
Here is another real world example.
Fairy tale characters are an abstract universal. They are general, and they don't actually exist.
Cinderella is a particular fairy tale character. She doesn't exist either, but she is an INSTANCE of the category of fairy tale characters.
Fairy tale characters are abstract universals, and Cinderella is an abstract particular.
In your world you don't have any abstraction at all. I think you're taking a point too far.
I would say that your communication style, with me at least, tends to be confusing. The only thing you wrote that made sense to me was the challenge to prove the transitivity of set equality. Once I realized I needed the converse of extensionality, I was enlightened.
Many other things you wrote were lost on me. I know this frustrates you, but it's like fishing. You had to go through a whole container of worms to finally hook the fish(fry). You should be happy, instead of complaining about the wasted worms.
I apologize for the typing things in remark. I must have written that before I understood your point.
Quoting TonesInDeepFreeze
What dissension? I'm happy I understood your point. I prefer not to go back into the old posts.
Yes, point being that if I'm away from the board for a while I have no recollection of what threads o conversations I'm involved in. I look up my mentions and work through them. If I don't see a mention, I may miss your post.
AN ordering of the children is not the ONLY ordering of the children.
And back to 'The Adventures Of The Crank Radio Hour':
Crank: Hey boss, I put our sales products in this spreadsheet in the order.
Boss: Which order?
Crank: The order.
Boss: Order by revenue or by items sold or by catalog number or what?
Crank: You know, the order.
Boss: Remind me how you got this job.
x is an element iff Ey xey
x is a class iff (x=0 or Ey yex)
x is a proper class iff (x is a class & ~Ey xey)
x is a set iff (x is a class & Ey xey)
x is an urelement iff x is not a class
Classical set theory theorem: Ax x is a set
Classical class theory theorem: Ax x is a class & Ex x is a set & Ex x is a proper class
Set theory with urelements theorem: Ex x is a set & Ex x is an urelement
Nicely said.
I know you never said such a thing. You mix up physical objects and mathematical objects as if there is no difference between them, and as if the law of identity would apply to both equally. That's why I call you a sophist. It was fishfry's principle, that elements of a set are not physical objects.
Quoting fishfry
No, Tones was referring to the principle called "the identity of indiscernibles", which is completely different from the law of identity. The law of identity makes a thing's identity the thing itself, the identity of indiscernibles associates a thing's identity with the thing's properties. These are fundamentally different principles.
Quoting fishfry
No, you simply fell for the sophistry. Tones is very good at it, and apt to convince others, earning the title "head sophist".
Quoting fishfry
So, a set is a mathematical structure. How do you make this consistent with the head sophist's claim that the members of The Beatles is a set, and the particles which make up a rock is a set? The sophist says "the set whose members are all and only the bandmates in the Beatles has 24 orderings". Notice that this is not stated as possible orderings, it is stated as the "orderings"
Remember your schoolkid example? You recognized that the objects which bear that name have what you called SOME order, and this is an expression of the condition which they are actually in, at any point in time. I would call this their "actual order". Can you see what the head sophist has done? The sophist has removed any distinction of an actual order, to say that the group, or set, has 24 orderings, and all these orderings are equal, or the same, being in each case a different presentation of the same set. But you and I recognize, that in reality there is "SOME order", an actual order, which is the order that the objects are actually in, at any given point in time. The sophist might talk about 24 orderings, but you and I recognize that if these 24 account for all the possibilities, only one of those possibilities represents the very special "actual order", and, that since these elements are physical objects, there must be an actual order which they are in, at any given time.
The law of identity is very important to recognize the actual existence of a thing, and its temporal extension. Through time a thing changes, and the law of noncontradiction stipulates that contradicting properties cannot be attributed to the same thing at the same time. So if a specific group has ordering A at a specified time, that is a property of that group, and it surely cannot have ordering B at the same time. The head sophist claims that the specified group has 24 orderings, all the time (as time is irrelevant in that fantasy land of sophistry). Obviously the head sophist has no respect for the law of noncontradiction, and is just making contradictory statements, in that sophistic fantasy.
That is what happens when we allow that abstractions such as mathematical structures have an identity. Inevitably the law of noncontradiction and/or the law of excluded middle will be violated. Charles Peirce did some excellent work on this subject. It's a difficult read, and you've already expressed a lack of interest in this subject/object distinction, so you probably don't really care. Anyway, here's a passage which begins to state what Peirce was up to.
[quote=Digital companion to C. S. Peirce] The relevance of all this to the principles of excluded middle and contradiction is as follows. Peirce wrote that anything is general in so far as the principle of excluded middle does not apply to it, e.g., the proposition Man is mortal, and that anything is indefinite in so far as the principle of contradiction does not apply to it, e.g., the proposition A man whom I could mention seems to be a little conceited (5.447-8, 1905). If we take Peirce to have meant LEM and LNC, then it appears that he wanted to deny the principle of bivalence (according to which all propositions are true or else false) with regard to universally quantified propositions, and that he meant to claim that existentially quantified propositions are both true and false. But why think that Man is mortal, which seems to be straightforwardly true, is neither true nor false? And why think that one and the same proposition, A man whom I could mention seems to be a little conceited, is both true and false? Once we see what Peirce meant by principles of excluded middle and contradiction, we see that this is not what he was claiming.[/quote]
http://www.commens.org/encyclopedia/article/lane-robert-principles-excluded-middle-and-contradiction
Quoting fishfry
This is blatantly untrue, and as demonstrated above, if we assign "identity" to a set, the law of non-contradiction will be violated. The law of identity enables us to understand an object as changing with the passing of time, while still maintaining its identity as the thing which it is. Sets have distinct formulations existing all the time, which would cause a violation of the law of noncontradiction if we allow that a set is subject to the law of identity. Therefore we must conclude that sets are not subject to the law of identity. The type of thing which the law of identity applies to is physical objects. And there is obviously a big difference between physical objects and sets, despite what head sophist claims.
Quoting fishfry
You also have no problem with contradiction, it seems.
Quoting fishfry
This tells me nothing until you explain precisely what ? means. To me, you are simply saying that x is an element of y if x is an element of y. What I am asking is what does it mean "to be an element".
Quoting fishfry
If we go with this definition, you ought to se very clearly that sets, as categories, abstract universals, do not have an identity according to the law of identity. A category is not a thing with an identity.
Obviously this does not work. As you said already, elements are often sets. Therefore you cannot characterize the set as an abstract universal, and the element as an abstract particular, because they're both both, and you have no real distinction between universal and particular. There's no point in trying to justify the head sophist's denial of reality. If "Cinderella" refers to a particular, an instance of the category "fairy take characters", then that is a physical object. If "Cinderella" refers to a further abstract category, like in the case of "red is an instance of colour", then it does not refer to a particular. The head sophist seems to have convinced you that you can ignore the difference between a physical object and an abstraction, but you and I both know that would be a mistake.
Then the sophist crank says "I know you never said such a thing. You mix up physical objects and mathematical objects as if there is no difference between them, and as if the law of identity would apply to both equally."
That's another LIE from the crank.
That the law of identity applies to both numbers and rocks does not entail that there is no difference between numbers and rocks! It does not entail that there is no difference between abstractions and concretes. The laws of traffic apply to both domestic vehicles and foreign vehicles, but that doesn't entail that there's no difference between domestic vehicles and foreign vehicles! The crank can't reason successfully in even the most basic ways!
Moreover, I did not say that an element of a set cannot be a concrete thing. The set of pencils on my desk has only concrete things as members.
The sophist crank is as usual abysmally confused and making false claims about what I've said.
/
When I first used the term 'sophist crank' I knew I was indulging redundancy', since cranks are by nature sophists. But I've been doing it anyway, to stress the point. It's clear enough by now, though it's been clear enough about him for years.
/
Then the crank, in his usual manner of self-serving sophistry, misconstrues @fishfry. fishfry didn't contradict that the law of identity is different from the identity of indiscernibles.
/
The crank says that the bandmates in the Beatles don't provide for a set. But they do, as they provide for the set {George, Ringo, John, Paul}. The crank can't understand what even a child can understand.
{the pencil on my desk, the pen on my desk} is a set whose members are of concretes and it has two orderings.
{1, 2} is a set whose members are mathematical objects and it has two orderings.
And even if we demurred from saying that such things as number are abstract objects, then still the principle that there is more than one ordering of a set obtains, since we may adduce sets whose members are concrete objects.
So, what example would the crank give of a set with more than one member? Whatever example the sophist gives, that set has more than one ordering.
Then the crank says "a set is a mathematical structure". That is an example of arguing by mere insistence on one's personal definition. Typical sophistry. Of course, one may stipulate any definition one wants to stipulate. But that carries no argumentative import in context of use of the word with a different definition. In mathematics and even in everyday life, the word 'set' is not ordinarily used to mean 'a structure'. However, mathematics does also address the notion of structure, and provides rigorous definition; but the crank, in his obdurate willful ignorance knows nothing about that, as he knows nothing about the mathematics he incessantly gets completely wrong.
Then the crank points out that I said the set has 24 orderings and that I did not say it has 24 possible orderings. That is, typically, an inane objection by the crank. (1) Extensional mathematics does not use intensional modalities. (But there are systems of intensional mathematics too.) (2) Even if we do speak instead of 'possible orderings', any particular one of those possibilities is not the only possibility, so it is still not THE ordering. It is merely one of the "possible" orderings chosen for our consideration. Any other "possible" ordering could be chosen and then, following the crank's notion, it would have to be considered to be THE ordering. So there would a different THE ordering depending on which ordering we happen to choose for consideration, which is still incoherent.
So, even deferring to the crank's insistence about "possible", which of the 24 possible orderings of the set whose members are the bandmates in the Beatles is THE ordering of that set?
I've given the crank the following information about half a dozen times already, but like the horse led to water who will not drink, the crank will not think (apologies to D. Parker):
Yes, we can specify a particular ordering of a set and refer to that set vis-a-vis that specified ordering. For example, let B be the set whose members are all and only the bandmates in the Beatles, and let R be the ordering of B alphabetically by first name. Then we have the STRUCTURE . That accords with the notion of a set along with a particular ordering.
The crank says, "there is "SOME order", an actual order, which is the order that the objects are actually in, at any given point in time.
At this exact moment of time, there are two orderings of the set of writing tools on my desk:
{
and
{
At this exact moment of time, there are two orderings of the two kids on the playground:
{
and
{
/
Then the crank goes on with yet more confusions. Reading his posts, I am reminded of a character in 'The Office' saying about the on and on, full of it fool, Michael Scott, "Where's the off button on this moron?"
The crank says, "[Tones In Deep Freeze] has removed any distinction of an actual order, to say that the group, or set, has 24 orderings, and all these orderings are equal, or the same"
That is yet another flat LIE from the crank, and a really stupid lie.
I never said that 24 orderings are the same or that they are equal. That would be a ridiculous thing to say. Indeed it is my point that they are not the same. There are 24 different orderings. Of course they are not all the same orderings. The crank is so mentally inept that he can't distinguish between (1) there are 24 different orderings that each have the property of being an ordering of a certain set and (2) all those orderings are the same.
The cranks daily posts rank garbage on a ... PHILOSOPHY forum!
I know, that's the problem. For you, a set may consist of concrete things, or it may consist of abstractions, because in your sophistry you do not differentiate between the two. Then you claim that there is no order to the concrete things which compose a set, when in reality there is.
Quoting TonesInDeepFreeze
Right, continue in your violation of the law of noncontradiction. The same set has contradicting properties, i.e. different orderings. Good one bro, I hope that's just your sense of humour again.
Stop lying, crank.
So which is it then? Does a set consist of concrete things, or does it consist of abstractions?
The crank mindlessly replies "Right, continue in your violation of the law of noncontradiction."
The crank is so mentally deficient that he can't see that it's not a contradiction that "there are 24 orderings of a set" does not imply "all those orderings are the same". It's an incorrect implication, because, indeed it is a contradiction to say that 24 different orderings are all the same ordering. It seems it is in the crank's imagination that, somehow 'different' implies 'same'. That's his problem, not mine, since indeed, for me 'different' does not mean 'the same'.
The crank's illogic and utter obtuseness are not less than stunning. And in a philosophy forum!
The orderings are different, and contradictory properties of the set. And, it is a violation of the law of noncontradiction for that set to have those contradictory orderings.
Quoting TonesInDeepFreeze
Exactly as I said, you fail to provide a differentiation between concrete objects and abstractions. Why did you say I lied about this?
If a set consists of concrete objects, then it has the order that those concrete objects have, and no other order. To say that the set has other orderings is to mix up concrete objects with abstract objects in the way of sophistry.
It's not a matter of whether I explicate the difference between concrete and abstract. Rather, whatever one's explication of the difference, abstract objects can be elements and concrete objects can be elements.
The crank argues by persistently ignoring the rebuttals, examples and explanations given him:
The crank says, "If a set consists of concrete objects, then it has the order that those concrete objects have, and no other order."
That was answered posts ago by me. The crank can't or won't read the posts he replies to.
The crank asks, "Why did you say I lied about this?"
Here are the lies:
"You mix up physical objects and mathematical objects as if there is no difference between them"
I explicitly mentioned that a number is abstract and a rock is concrete. And I reiterated that. The crank is a liar.
"For you, a set may consist of concrete things, or it may consist of abstractions, because in your sophistry you do not differentiate between the two." [the lie bolded]
I have never conflated abstractions with concrete. The crank is a foolish liar.
"[Tones In Deep Freeze] has removed any distinction of an actual order, to say that the group, or set, has 24 orderings, and all these orderings are equal, or the same"
I do not at all make any such utterly ridiculous statement that the orderings are the same. And I reiterated that. The crank is a profoundly illogical liar.
I say, in clear, emphatic, and unequivocal terms that the 24 orderings are different orderings. It is at the heart of my point that they are different orderings. It would be absurd to say that they are not different orderings. But the crank says that I say that they are the same ordering.
Illogic doesn't get much more dire than the crank's.
Set consisting of three balls colored red, white and blue. They also have differing weights. What is THE order? Just curious.
Yes I understand that. I didn't realize @TonesInDeepFreeze was talking about IofI. Actually I just learned there's an identity of indiscernibles and an indiscernibility of identicals.
Quoting Metaphysician Undercover
Ok this is interesting. My quote was, "Sets are subject to the law of identity." So that if X is a set, I can say X = X without appealing to any principle of set theory.
Tones convinced me of that. Now you say that he only sophisted me. How so please? If X is a set, how is X = X not given by the law of identity? You have me curious.
You think I'm a victim of Tones's sophistry. That is an interesting remark.
Quoting Metaphysician Undercover
Set theory is a mathematical structure. The analogy is:
Set theory is to group theory as a particular set is to a particular group.
But a set is a mathematical structure too, since the elements of sets are other sets.
Quoting Metaphysician Undercover
I take no responsibility for anyone else's posts. I barely take responsibility for my own. You've already told me you don't like real world examples about playgrounds so I don't use those anymore with you.
Quoting Metaphysician Undercover
Yes. I agreed with you that this was only a casual analogy, and once you told me that you don't like it, I stopped using it.
Quoting Metaphysician Undercover
This is true about kids in playgrounds, NOT mathematical sets. You have informed me that you don't like real-world analogies so I no longer use them. Mathematical sets have no inherent order.
Quoting Metaphysician Undercover
Yes I understand that. I didn't realize @TonesInDeepFreeze was talking about IofI. Actually I just learned there's an identity of indiscernibles and an indiscernibility of identicals.
Quoting Metaphysician Undercover
Ok this is interesting. My quote was, "Sets are subject to the law of identity." So that if X is a set, I can say X = X without appealing to any principle of set theory.
Tones convinced me of that. Now you say that he only sophisted me. How so please? If X is a set, how is X = X not given by the law of identity? You have me curious.
You think I'm a victim of Tones's sophistry. That is an interesting remark.
Quoting Metaphysician Undercover
Set theory is a mathematical structure. The analogy is:
Set theory is to group theory as a particular set is to a particular group.
But a set is a mathematical structure too, since the elements of sets are other sets.
Quoting Metaphysician Undercover
I take no responsibility for anyone else's posts. I barely take responsibility for my own. You've already told me you don't like real world examples about playgrounds so I don't use those anymore with you.
Quoting Metaphysician Undercover
Yes. I agreed with you that this was only a casual analogy, and once you told me that you don't like it, I stopped using it.
Quoting Metaphysician Undercover
School kids and physical objects in general do. Mathematical sets don't. You explained to me that you don't like physical examples so I no longer use them. Mathematical sets have no inherent order. The purpose of defining things that way is so that we may study the abstract notion of order.
Quoting Metaphysician Undercover
I can't comment on what anyone else has said. This has nothing to do with the conversation you and I are having.
Quoting Metaphysician Undercover
A temporal extension. You are saying it only applies to things that exist in time? Meaning not sets? I don't think that's right. Any set is identical to itself and also equal to itself by virtue of the law of identity.
Tones did explain that to me, but not via sophistry. He asked me to prove the transitivity of set equality. Once I attempted to do that, I realized that I needed not the axiom of extensionality, but its converse. And that converse is true by way of the law of identity from the underlying predicate logic. This I discovered for myself when Tones pointed me to it.
Quoting Metaphysician Undercover
That may or may not be true about physical objects. You say the kids in height order is not identical to the kids in alphabetical order. I say the set of kids is the same. But I do not argue this point and od not care about it.
I tell you that a set has no inherent order; and that the set of natural numbers in its usual order; and the set of natural numbers in the even-odd order say -- 0, 2, 4, 6, ...; 1, 3, 5, 7, ... is exactly the same set. It is a different ordered set, because in an ordered set, the order is part of the identity of the set. In a plain set, it's not. This is how mathematicians play their abstraction game.
Quoting Metaphysician Undercover
On the contrary, I've expressed great interest in the ideas of Pearce when members of this forum have mentioned them to me.
Quoting Metaphysician Undercover
Yes, well, discussions of denying LEM don't interest me much. I'll agree with that. But I've come by it honestly. I've made a run at constructivism and intuitionism more than once. I've read Andrej Brauer's "Five Stages of Accepting Constructive Mathematics." It doesn't speak to me. The paragraph you quoted is a little above my philosophical pay grade. Perhaps you can explain its relevance to the topic at hand.
Quoting Metaphysician Undercover
I'll take a look as the spirit moves me, but I don't think this is a particularly productive line of conversation for me. I don't know what you are trying to tell me.
Quoting Metaphysician Undercover
I don't see why. If X is a set, then X = X by identity.
Now if you are trying to say that the order properties might differ or whatever, I say you are just being your usual anti-abstraction self. I don't understand your aversion to mathematical abstraction. But it doesn't effect my mathematical ontology in the least.
Or maybe you're saying something else. If so, please explain.
Quoting Metaphysician Undercover
There is no time in set theory. Mathematics is outside of time, or talks about things that are outside of time.
Quoting Metaphysician Undercover
You are just wrong about this. But give me a more specific example, if you would, so that I may understand you better.
Quoting Metaphysician Undercover
I can always tell when you can't defend your point. The insults come out. You can do better, can't you?
Quoting Metaphysician Undercover
It doesn't "mean" anything. It's an undefined primitive in set theory, as point, line, and plane are undefined primitives of Euclidean geometry. Its behavior and usage are defined by the axioms.
Quoting Metaphysician Undercover
I have no problem with that. If you want to say that set theory, as the universal, is not subject to identity, that's fine. I can' say that "set theory equals set theory." I'm perfectly fine with that. Nobody ever says it.
But given particular instances of set theory; that is, sets; we can ask if they are equal to each other or not.
So I promise not to say that the universe of sets is equal to the universe of sets. Though the category theorists will probably disagree with you.
Quoting Metaphysician Undercover
You are distorting what I said. ANY particular set is a particular instance of the concept of set, as any particular apple is an instance of the concept (or category) of apple. That causes no problem.
You are willfully obfuscating this point. It's a very clear point. The set of integers is a particular. Sets in general, or the concept of sets, are generalities or a category or a universal.
Quoting Metaphysician Undercover
It's pointless to keep referring to conversations you've had that I haven't seen, with people who aren't me.
Quoting Metaphysician Undercover
Clearly Cinderella is not a physical object. That's exactly why I used a fictional character as an example.
Quoting Metaphysician Undercover
Knock it off, will you please? Take your complaints WITH the other party, TO the other party.
I have long ago agreed that physical objects and mathematical objects are not the same.
Cinderella refers to the individual fictional character of Cinderella, as Captain Ahab refers to the fictional character of Captain Ahab.
Quoting tim wood
Set and order are very interesting concepts that I discussed in other threads, and it is not always clear what we can include in them. If I am not mistaken, I guess MU refers to those objects that are logically attached to an order and, therefore, make a set. For example, ground, bricks, walls, ceiling, windows, and a door altogether make a set, which is the house.
Please keep in mind that I am not arguing about whether those are necessary order objects or not. However, it is evident that they create the order and set.
Quoting jgill
The order is how items are organised with one another based on a specific attribute. The only distinguishing feature is that they are spherical. The weight and colours are only accessories. The set would be spheres, and the order would be the three balls. Right?
Quoting TonesInDeepFreeze
cc: @Metaphysician Undercover
You guys get a room! LOL. FWIW you convinced me that X = X for sets follows from the definition of = in the underlying predicate logic. How the = of logic relates to the law of identity, I have no idea.
Having X hat does not exclude having Y hat, that's obvious. The two do not contradict. But if X order contradicts Y order (e.g. John is closer to the front of the stage than Paul, contradicts Paul is closer to the front of the stage than John), then X order excludes Y order.
Your analogy is not relevant because having one property clearly does not exclude the possibility of having another property, but having the property of one order clearly contradicts having the property of the contradictory order.
I think it's time for you to go back to grade school and learn some fundamentals of logical thinking.
Quoting jgill
Show me your balls and I will tell you their order.
Quoting tim wood
However those objects relate to other objects, the context, or environment they are in, dictates their order.
Quoting fishfry
What is the case, is that "X=X" is an ambiguous and misleading representation of the law of identity. This is because "=" must mean "is the same as", to represent that law, but it could be taken as "is equal to". Notice that in the axiom of extensionality it is taken to mean "is equal to". Therefore when Tones takes "X=X" to be an indication of the law of identity there is most likely equivocation involved.
Quoting fishfry
So, do you recognize, and respect the fact that group theory is separate from, as a theoretical representation of, the objects which are said to be members of a specified "group"? And, I'm sure you understand that just like there is a theoretical representation of the group, there is also a theoretical representation of each member of the group. In set theory therefore, there is a theoretical "set", and also theoretical "elements".
So when Tones says that a set may consist of concrete objects, this is explicitly false, because the set is the theoretical representation, and the elements of the set are theoretical representations as well. Through such false assertions, Tones misleads people and earns the title of sophist.
When Tones speaks about the set "George, Ringo, John, Paul", these names signify an abstract representation of those people, as the members of that set, the names do not signify the concrete individuals. You, Fishfry, have shown me very clearly that you know this. So there is an imaginary "George", "Ringo" etc., which are referred to as members of the set. The imaginary representation is known in classical logic as "the subject". We make predications of the subject, and the subject may or may not be assumed to represent a physical object. Comparison between what is predicated of the subject, and how the object supposedly represented by the subject appears, is how we judge truth, as correspondence.
What is important to understand in mathematics, is that the subject need not represent an object at all. It may be purely imaginary, like your example Cinderella. This allows mathematicians to manipulate subjects freely, without concern for any "correspondence" with objects. Beware the sophist though. I believe that when the sophist says that the members of a set may be abstractions, or they may be concrete objects, what is really meant if we get behind the sophistry, is that in some cases the imaginary, abstract "element", may be assumed to have a corresponding concrete object, and sometimes it may not. Notice though, that in all cases, as you've been insisting in discussions with me, the elements of the sets are abstractions, as part of the theory, and never are they the actual physical objects. Failure to uphold this distinction results in an inability to determine truth as correspondence. And that is the effect of Tones' sophistry
Quoting fishfry
I'll return to the schoolkids example briefly to tell you why I didn't like it. Using that example made it unclear whether "schoolkids" referred to assumed actual physical objects, or imaginary representations. That's why "real-world analogies" are difficult and misleading. The names, "George", "Paul", etc., appear to refer to real-world physical objects, and Tones even claims that they do, but within the theory, they do not, they are simply theoretical objects. If we maintain the principle that the supposed "schoolkids" are simply imaginary, then they have no inherent order unless one is stipulated as part of the rules for creating the imaginary scenario. Set theory ensures that the elements have no inherent order, but this also ensures that the elements are imaginary.
Quoting fishfry
This is wrong, and where Tones mislead you in sophistry. A set is not identical to itself by the law of identity. The set has multiple contradictory orderings, and this implies violation of the law of identity. We allow that "a thing", a physical object has contradictory properties with the principle of temporal extension. At one time the thing has a property contradictory to what it has at another time, by virtue of what is known as "change", and this requires time. But set theory has no such principle of temporality, and the set simply has multiple (contradictory) orderings.
Quoting fishfry
As I said, the reference was to the identity of indiscernibles, not the law of identity. You recognize that these two are different. The proof was not by way of the law of identity. If you still believe it was, show me the proof, and I will point out where it is inconsistent with the law of identity.
Quoting fishfry
We agree on this very well. The principle we need to adhere to, is that this is always an "abstraction game". If we start using names like "Ringo" etc., where it appears like the named elements of the set are concrete objects, then we invite ambiguity and equivocation. And if we assert that the elements are concrete objects, like Tones did, this is blatantly incorrect.
Quoting fishfry
The three fundamental laws of logic, identity, noncontradiction, and excluded middle, are inextricably tied together. Therefore one cannot discuss identity without expecting some reference to the other two. There has been some philosophical discussion as to which comes first, or is most basic. Aristotle seemed to believe that noncontradiction is the most basic, and identity was developed to support noncontradiction.
What C.S. Peirce noticed, is that if we allow abstract objects to have "identity" like physical objects do, as Tones seems to be insisting on, then necessarily the validity of the other two laws is compromised. Instead of denying identity to abstract objects, as I do in the Aristotelian tradition of a crusade against sophistry, Peirce sets up a structure outlining the conditions under which noncontradiction, and excluded middle ought to be violated.
Quoting fishfry
You are missing the point. The law of identity refers explicitly to things, "a thing is the same as itself". A "set" is explicitly a group of things. Therefore when you say X = X, and X is a set, rather than a thing, then "=" does not signify identity by the law of identity.
Quoting fishfry
Right, this is the point. "Time", or temporal extension allows that a thing may have contradictory properties, at a different time, yet maintain its identity as the same thing, all the while. This is fundamental to the law of identity. Without time (as in mathematics), the multiple orderings of a set, which Tones referred to, are simply contradictory properties. That is a good example of the issue Peirce was looking at.
Quoting fishfry
Fine, but can you respect the fact that "equal" does not imply "identical", despite the sophistical tricks that Tones is so adept at.
Quoting fishfry
No, that's simply wrong. A particular apple is a physical object. A set is an abstraction. An instance of an apple is a physical object. Your supposed "instance" of a set is an abstraction, a concept. The two are not analogous, and I argue that this is a faulty, deceptive use of "instance".
An instance is an example, and understanding of concepts or abstractions by example does not work that way. Assume the concept "colour" for example. If I present you with the concept "red", this does not provide you with an instance of the concept "colour". An instance of the concept "colour" would be the idea of colour which you have in your mind, or the idea of colour which I have in my mind, expressed through the means of definition. Each of those would provide you with an example of the concept of "colour", an instance of that concept. The concept "red" does not provide you with an example of the concept of "colour". Nor does a specific "set" provide you with an example or instance of the concept "set".
What you are saying in this case is completely mixed up and confused.
I think you and I agree substantially on the difference between abstractions and physical objects, and that the elements of a set are always abstractions and never physical objects. So you might avoid a long reply on that subject. It's only the head sophist who disagrees with us on this, claiming that the elements of a set may be concrete objects.
We do have significant disagreement concerning your claim to a proof that "X=X", when X signifies a set, means that X is the same as itself by virtue of the law of identity. You have not provided that proof in any form which I could understand.
Quoting tim wood
Sorry tim, I'm not picking up what you're putting down.
Yikes.
I never spoke about "one and one", nor about "many". I have no idea what you are making reference to, or how you draw the conclusion that "one and one" are a requirement for "many".
I'm not trying to be toxic, only I have no idea of what you are trying to express. What I said was an expression meaning that I am not understanding what you are saying.
The sense of humour leaves the head sophist exposed, revealing no control over the inclination to equivocate.
Quoting tim wood
This is not more than one order, it is just different aspects of one order, like one object has numerous properties, but the properties are all just different aspects of the one object. That we say a thing has volume, weight, colour, etc., is just a feature of how we describe the thing. For example, saying B is prior to A in the smallness scale, and A is prior to B in the largeness scale, does not mean that the two objects have more than one order, it's just a feature of the way we describe things, i.e. our way of imposing a conceptual order.
You appear to be mixing up the natural order which things have by being the things which they are, in the circumstances which they are in, and the conceptual order which we artificially impose on the thing in abstract understanding. That is the same mistake the head sophist makes, failing to distinguish the concrete thing itself, from the conception of it.
Judging by what you argued in the other thread on "purpose", you believe that the only kind of order is conceptual order. Until you realize that this idea is faulty, you will never understand the natural order that things have simply by being the things that they are, in the circumstances that they are in. Discussion seems pointless right now.
As I said, the context dictates their order, and context is singular. An object does not exist in a multitude of distinct contexts at the same time, despite the fact that the context may change over time. I covered change and temporality in my reply to fishfry. You might go back and read that.
Sorry tim, I have no interest in engaging with you here in the Lounge. You have demonstrated that you are very steadfast with an extremely closed mind. Banter with TIDF is at least somewhat amusing. That sophist actually has a sense of humour and some degree of conscience, which you seem to be fully and completely lacking in. Furthermore, Tones actually listens to what I say, and sometimes makes an attempt at understanding, whereas you simply dismiss it as "toxic", "dishonest", and "a waste of time". No point in wasting time if it doesn't come with entertainment.
Seems like a peculiar use of the word "order".
Deep stuff, here. :roll:
I believe that's what The Lounge is for. The deep stuff gets booted off the main page, being for most, undistinguishable from shit.
Quoting TonesInDeepFreeze
I agree, it's no compliment to say that you're higher in the order of virtue than tim is. I should have just spoke the truth, tim is even lower down than you are. And both of you make a snake appear like an angel.
Frame that.
The crank can't discern irony, even when it is declared with an emoji.
/
The crank says, "tim is even lower down than you are. And both of you make a snake appear like an angel."
That's lower than lame.
The converse of extensionality is not provided by the law of identity. It is provided by the indiscernibility of identicals.
There's plenty of detailed information and explanation posted in this thread.
If you have any questions, or wish to learn more, then it's as simple as asking (and not asking someone who doesn't know anything about the subject).
Quoting javi2541997
Which positions? You think it's a good position to deny that a set with more than one member has more than one ordering?
I know. This long thread is very informative. I just didnt want to ask because it is obvious that I dont have the same high level of math and/or logic as you do, and my posts would interrupt the debate. But it is dreary to read such negative comments loaded with animosity.
When someone lies about your posts and incessantly posts disinformation about the subject, then it is appropriate to comment on that and it is pertinent too to comment on the modus operandi behind it. After correcting a crank over and over and over, with the crank still continuing to post the disinformation and indeed adding even more, the more salient subject becomes not the topic but the deleterious effects of the crank. The point is to not normalize cranks. I see forums ruined by being inundated by cranks, not ruined by informed people posting back against cranks. Then, also, search engines include posts with disinformation from cranks near the top of search results on various subjects. And that is added to the disinformation and confused presentations about mathematics on Wikipedia. And that's along with the outrageous disinformation and confusion transmitted at the speed of light by AI answer bots. Thus the torrent of effluvia from which the Internet spirals down and down into a cesspool of disinformation.
I would think of those as aspects of the house, not members of the house. I wouldn't think of a house as being a set. There are sets of aspects of a house. But that set is not a house. A house is something you live in and pay a mortgage on. You don't live in a set and pay mortgage on a set. There is a housing market and a housing shortage; there is not a sets market and a sets shortage.
If the attribute is color then there are six orderings based on that attribute:
red ball, white ball, blue ball
red ball, blue ball, white ball
blue ball, red ball, white ball
blue ball, white ball, red ball
white ball, red ball, blue ball
white ball, blue ball, red ball
There is not just one ordering that we can call "THE" ordering.
If the attribute is size, and it is not the case that there are two balls with the same size, then there are six orderings based on that attribute:
the largest ball, the middle sized ball, the smallest ball
the largest ball, the smallest ball, the middle sized ball
the middle sized ball, the largest ball, the smallest ball
the middle sized ball, the smallest ball, the largest ball
the smallest ball, the largest ball, the middle sized ball
the smallest ball, the middle sized ball, the largest ball
There is not just one ordering that we can call "THE" ordering.
Do you see that?
By the way, this pertains to linear orderings (aka 'total orderings'). There are other kinds of orderings, especially partial orderings, but here the context is linear orderings.
If the elements cannot be concretes and can't be abstractions, then what can they be?
Or does the crank reject even the notion of sets and elements?
Is a rock not a concrete? If a rock is not a concrete, then what is an example of a concrete?
If there is no such set that is the set of rocks on my table, then what are examples of sets?
/
The crank writes, "The sense of humour leaves the [s]head sophist[/s] [TonesInDeepFreeze] exposed, revealing no control over the inclination to equivocate."
I dare not ask what in the world that is supposed to mean.
By the way, though I am a magnitude of light years away from being a sophist, I would rather be merely a sophist than a crank, since being a crank includes being a sophist and a lot worse too.
If that isn't a house, what set are you talking about? Assume they are all 'aspects' of a set called furniture. We could agree on that. But what is the sense of doing those things separately? All of the 'aspects' I mentioned in my example follow a common logic. They'll end up in construction. A house or building. I can't envision a house without a wall or a ceiling as structural elements. Otherwise, this type of construction would be unsustainable. Perhaps I am misunderstanding the concepts of "set," "order," "members," and so on. I am aware of my limited understanding on the subject. But I still believe they are members a house.
Quoting TonesInDeepFreeze
Yes, I do. I never claimed there was one and only THE order. I referred to the balls in the example of jgill because that was what I thought when trying to use logic. But I hadnt in mind only one ordering.
I didn't say that a house is not a house. I said a house is not a set.
Quoting javi2541997
What things separately?
Quoting javi2541997
Nor can I. That doesn't entail that a house is a set. Again, a house is a thing you live in. You don't live in a set; you live in a house.
Quoting javi2541997
Sets:
S = {the door of the house, the roof of the house, the floor of the house ... the balcony of the house}.
S is not a house. It is a set whose members are features of the house.
Members:
The members of S are the features of the house.
Order:
I don't make a distinction between "same as" and "is equal to." In math they're the same. If you have different meanings for them, it does not bear on anything I know or care about.
Quoting Metaphysician Undercover
I don't think so. I don't agree with you.
Quoting Metaphysician Undercover
Yes, as ichthyology is different from any particular fish or school of fish or class of fish.
Quoting Metaphysician Undercover
I don't know what you mean by "theoretical" elements. The integers form a group under addition. Is the number 5 theoretical? Well it's abstract, as numbers are. What of it?
Quoting Metaphysician Undercover
Is concrete different or the same as physical? 5 is a concrete mathematical object, I guess. "Concrete" is not a term of art in this context, although there's a thing called a concrete category.
You're making up your own definitions of words and arguing with me about them. I am lost.
Quoting Metaphysician Undercover
Too deep for me. I take Tones's point that = in set theory derives from = in the underlying logic. I have no problem with that and it was perfectly obvious as soon as he drew my attention to it.
Quoting Metaphysician Undercover
Ok you should take this up with Tones. You failed to convince me that I am a victim of anyone's sophistry.
Quoting Metaphysician Undercover
Jeez man you already told me you don't like it, so I stopped using it. To me it's a good illustration of how a collection may be ordered in many different ways.
Quoting Metaphysician Undercover
Sets are mathematical abstractions. I don't know what you mean by "imaginary," which is a term of art in math referring to complex numbers with real part 0.
Quoting Metaphysician Undercover
No you are as wrong as can be about that.
Quoting Metaphysician Undercover
You have been misunderstanding this point for years, and I surely have nothing new to say on the topic.
Quoting Metaphysician Undercover
I'd walk you through the Wiki page on the axiom of extensionality, but this also is something I've done for years with you, to little productive effect.
Quoting Metaphysician Undercover
You must be misrepresenting what Tones said, since he made his point with me; and after that, I found several clear references supporting his point.
Quoting Metaphysician Undercover
Aristotle thought bowling balls fall down because they're "like the earth," and fire goes up because it's "like the air." But never mind that. You are swimming in murky logical waters that have absolutely nothing to do with anything I'm saying.
Quoting Metaphysician Undercover
I'm ill-equipped to argue Peirce and Aristotle with you. I don't think your points bear on set theory.
Quoting Metaphysician Undercover
A set is not a "thing?" A set is a thing in set theory. It's an abstract thing, to be sure. But it's still a thing.
Quoting Metaphysician Undercover
No, orderings are not "contradictory properties." Technically, an order on a set is another set, namely the set of pairs (x,y) for which we mean to denote that x < y in the ordering. The ordering is distinctly and noticeably separate from the set it applies to.
Quoting Metaphysician Undercover
That distinction has no meaning or relevance in my understanding of the world. "equal" and "the same as" are entirely synonymous. I do take your point that 2 + 2 = 4 does not mean that they are the same as strings. We've been over this many times, as have many philosophers. The morning star and the evening star are the "same object," (which turns out to be a planet and not even a star) but they have different senses. What of it, this is not news to anyone.
Quoting Metaphysician Undercover
Would you agree that "number" is a general abstraction and that 5 is a partcular instance of number? Isn't that the most commonplace observation ever?
Quoting Metaphysician Undercover
5 is a terrific example of a number. One of the best I know.
Quoting Metaphysician Undercover
It doesn't? Red is not an instance of the concept of color? How do you figure that?
Quoting Metaphysician Undercover
Incoherent. Red is a color. Red is an instance of the concept of color.
Quoting Metaphysician Undercover
What??
Quoting Metaphysician Undercover
Of course it does.
Quoting Metaphysician Undercover
Same back atcha.
I ain't draggin' your butt through the Wiki page on extensionality again. That was a very dispiriting experience the last time I did it. However, I'll point you at the relevant sentence.
"The axiom given above assumes that equality is a primitive symbol in predicate logic."
https://en.wikipedia.org/wiki/Axiom_of_extensionality
All the things I previously referred to. The objects or elements that constitute a house: walls, ceiling, windows, door, etc.
Quoting TonesInDeepFreeze
Yes, yes. I understand that I live in a thing, but my point was different. I tried to explain that the thing is based on different elements. Without these elements or 'objects', the principal thing (the house) is senseless, in my humble view. Maybe I was wrong in using those concepts in a confusing way. Yet I think we both agree that the house is senseless without furniture, unless you are minimalist. But even a minimalist house needs walls, a door, and a ceiling. Therefore, these three elements are necessarily elements of the house.
Quoting TonesInDeepFreeze
I see. Thanks. But then I wonder: what is the point of that order, or does it arise spontaneously? Obviously not. The house is what they relate to.
Oh. In that case @Metaphysician Undercover is right and you are making a point I can't agree with.
I take your point about set equality as expressed in the Wiki page on extensionality, which says, "The axiom given above assumes that equality is a primitive symbol in predicate logic."
If you mean something else, we're back to square one.
The same set can be specified in two different ways:
{the door, the floor, the roof ... the balcony} [fill in '...' with all the other features of the house .]
{x | x is a feature of the house}
Quoting javi2541997
Okay.
Quoting javi2541997
In casual conversation, the word 'elements' can be used that way. But if we are talking in a focused context about sets, 'elements' refers to members of a set. And the house is not a set. Sure, in some informal way, we could stretch the meaning of 'set' so that in some view a house is a set. But in a focused sense of 'set', a house is not set, just as a rock is not a set.
If anything at all were a set - a house, zebra, rock, cloud - then 'set' wouldn't have any special meaning. You could point to a stop sign and say, "Hey look at that set over there, the stop sign". But that's not the common notion of 'set'.
Ah. Well, I think I have to agree with you. My arguments were based on casual and informal examples, and I cant go further than that. Thanks for your explanations.
I think those examples and a common informal context are okay. They suggest that, for example, a rock is not a set. My point is that it would only be a far stretch of the notion of 'set' that would permit taking a rock to be a set.
'=' is primitive.
But there is more to say.
So indeed, let's go back to square one:
'=' is primitive in logic (first order logic with equality, aka 'identity theory').
And '=' has a fixed interpretation (which is semantical, not part of the axioms) that '=' stands for identity.
So identity theory has axioms so that we can make inferences with '='.
The axioms are:
Ax x=x ... the law of identity
And the axiom schema (I'm leaving out technical details):
For all formulas P:
Axy((P(x) & x= y) -> P(y)) ... the indiscernibly of identicals
Then set theory adds its axiom:
Axy(Az(zex <-> zey) -> x=y) ... extensionality
Now we ask how we derive:
(zex & x=y) > zey
Answer: from the indiscernibility of identicals. Indeed the above is an instance of the indiscernibility of identicals, where P(x) is zex.
I don't think I'm going to get involved in the details at this point.
Quoting TonesInDeepFreeze
"Let's not and say we did," as the saying goes.
Quoting TonesInDeepFreeze
Aha. "Identity theory" is first order predicate logic with equality. Is that your own terminology? Nothing in the Wiki disambiguation page for identity theory refers to it
Quoting TonesInDeepFreeze
I'm going to pass on engaging with this. Just don't have the inclination at the moment.
But one query. If you are doing first order logic, how do you quantify over all propositions P? Maybe I shouldn't ask.
I stated explicitly several times that that is what I mean by 'identity theory'. I recall having seen the term used professionally before, and so I adopted it a long time ago, but I would have to dig to find citations. I like it, because it is a first order theory about one certain predicate that is indeed the identity predicate.
If someone says "I'm talking about blahblah theory'" and they tell me the axioms, then I don't quarrel with them about it. I know the axioms so I know precisely what is meant by 'blahblah theory'.
We quantify over them in the meta-theory not in the object theory.
That is what an axiom schema is.
For example (leaving out some technical details here:):
In first order PA the induction axiom schema:
For all formulas P:
(P(0) & An(P(n) -> P(Sn))) -> An P(n)
In set theory, the axiom schema of separation:
For all formulas P:
AzExAy(y e x <-> (y e z & P(y)))
Those are statements in the meta-theory that describe an infinite set whose members are all axioms that are in the object-theory.
An interesting point is that while we can express the indiscernibility of identicals as a first order schema, we can express the identity of indiscernibiles as a first order schema if and only if there are only finitely many operation and predicate symbols.
It's an interesting exercise to try to express the identity of indiscernibiles as a first order schema with a language of infinitely many non-logical symbols. You'd think you'd just reverse the indiscernibility of identicals. But when you try, it doesn't work! If I'm not mistaken, one of the famous logicians proved it can't be done.
/
Another nice thing: Identity theory can be axiomatized another way, courtesy of Wang:
For all formulas P:
Ax(P(x) <-> Ey(x=y & P(y)))
From that we can derive both the law of identity and the indiscernibility of identicals.
You never said that LOL!
Quoting TonesInDeepFreeze
Now that I know what you mean, it's helpful.
Quoting TonesInDeepFreeze
I see your point. But I don't always catch your meaning from your symbology.
I still have it in my queue to go back to your recent post about the indiscernibles. Maybe if you could make your point in words. I thought the = of set theory is the = from the underlying logic. But now you say it's not. So I'm confused again. If you could explain it clearly in a sentence or two I'd find it helpful
And I know that if I make this request, and you give me a response and I don't related to it, that's frustrating to you. Maybe there's a happy medium of explanatory level.
Ok. I see your point. Lately wishing I'd paid attention in logic class.
Sadly this is all over my head. Maybe I'll crack open a logic book. If I could only dispatch a clone.
Quoting TonesInDeepFreeze
That looks interesting.
Please allow me to clarify, if I wasn't clear enough for you last time. You and I have a completely different understanding of the nature of "a relation". We could not even find grounds to start any agreement, to converse. Consequently you'll understand "relation" in your way, and I'll understand "relation" in my way. Since "order" is a specific type of relation, any discussion about order, between us, will be rife with misunderstanding. Furthermore, I have no inclination to stoop to your level, and utilize your meaning of "relation", so that you might actually understand me, because I see it as nothing but childish closed mindedness.
Quoting fishfry
This is exactly the problem, failing to distinguish between "same as" and "equal to". Because you do not believe that there is a distinction to be made here, you will not notice the effects of such a failure, and you will insist that it doesn't bear on anything you care about. Insisting that it doesn't bear on anything you care about will allow you to be mislead, even tricked by intentional deception (as you were by the sophist's employment of "identity of indiscernibles"), and you may never ever even notice it.
Here is a simple example of where the difference bears in a substantial way, though I am sure there are more complex examples. In quantum physics, a quantum of energy is emitted as a photon, and an equal quantum may be detected as a photon. Since these two quanta of energy are equal, they are said to be "the same" photon. That is the mathematician's use of "same", equal quanta implies one quantum, a photon. By the law of identity "same" implies temporal continuity, such that the photon exists, with that identity, for the entire period between emission and detection. Equivocation between these two senses of "same" inclines some people to believe that the photon exists, as the same "particle" for the entire period of time between emission and detection. However, the electromagnetic energy is observed to exist as waves in the meantime.
This produces significant theoretical problems. Some claim a contradictory wave/particle duality theory, in which the energy travels as both waves and as particles at the same time. Furthermore, since the photon of energy emitted is assumed to be "the same photon" as the photon detected, and it's path cannot be determined, it is claimed to have multiple paths all at the same time. All of this sort of problem is due to equivocation of "same". The mathematical "same", an equal quantum of energy is emitted and detected, is confused with "same" by the law of identity, to conclude that a distinct quantum (particle) of energy, known as the photon, has continuous existence between the time of emission and the time of detection.
You say that this issue doesn't bear on anything you know or care about, but until you recognize and understand the issue you'll never know how it bears. Furthermore, I saw how the head sophist, persuaded you to see a mathematical axiom differently, through reference to the identity of indiscernible, so I know that it really does bear on things that you care about.
Quoting fishfry
What is said about a thing is distinct from the thing itself. Contradiction is not in the thing itself, it is in what is said about the thing. To say that a thing has contradictory orderings is contradiction. The contradiction "is distinctly and noticeably separate from the [thing] it applies to".
Quoting fishfry
You are in denial, just like the sophist. "Equal" means to have the same value within a system of valuation, "same" means identical, not different. Notice that "equal" is a qualified sense of "same" the "same value", meaning identical value, whereas "same" refers to identity itself without such qualification. Two distinct things are said to be equal, being judged according to a specified value system. Two distinct things are not the same. Please tell me that you understand this difference.
Quoting fishfry
This is colloquial vernacular insufficient for logical rigour. The proper classification is like this. The abstraction "number" is more general, and the abstraction "5" is more specific, just like "animal" is more general, and "human being" is more specific, or "colour" is general and "red" specific. Neither is a "particular instance".
One might however say that there is a particular instance of the abstraction "5", and the abstraction "number", in your mind, and another particular instance in my mind. But that would be an ontological stance which would be denying common Platonism. Platonists would say that what I just called particular instances, are really just parts of one unified concept "5".
Quoting fishfry
Fishfry, do you not understand what "instance" means? Here, from OED, "an example or illustration of". How do you think that a specific colour, red, is an example or illustration of the concept of colour? Red cannot exemplify "colour", because all the other colours are absent from it. That's why we go from the more general to the more specific in the act of explaining. Referring to the more specific abstraction, "red" is an instance of "specifying", it is not an instance of "colour".
Quoting fishfry
How do you propose that this indicates that equal implies "same as"? I cannot follow your association. What the head sophist calls "identity theory" is simply an axiom of identity which is inconsistent with the law of identity. The sophist dictates that "=" means identical to, and this is the first principle of the sophistry referred to as "identity theory".
I stated the axioms of identity theory in multiple posts. Not funny, but true.
Quoting fishfry
I did not say that it's not.
I'll say again:
First order logic with identity provides:
(1) law of identity (axiom)
(2) indiscernibility of identicals (axiom schema)
(3) interpretation of '=' as standing for the identity relation (semantics)
Set theory takes (1) - (3) and adds:
(4) extensionality (axiom)
Nearly all of these text symbols are quite common:
~ ... it is not the case that
-> ... implies
<-> ... if and only if
& ... and
v ... or
A ... for all
E ... there exists a/an
E! ... there exists a unique
Axy ... for all x and for all y [for example]
if P(x) is a formula, then, in context, P(y) is the result of replacing all free occurrences of x with y [for example]
= ... equals
< ... is less than
<= ... is less than or equal to
> ... is greater than
>= ... is greater than or equal to
+ ... plus
- ... minus
* ... times
/ ... x divided by y
^ ... raised to the power of
! ... factorial
e ... is an element of
0 ... the empty set (also, zero)
w ... the set of natural numbers [read as 'omega']
{x | P} ... the set of x such that P [for example]
{x y z} ... the set whose members are x, y and z [for example]
(x y) ... the open interval between x and y [for example]
(x y] ... the interval between x and y, including y, not including x [for example]
[x y) ... the interval between x and y, not including x, not including y [for example]
[x y] ... the closed interval between x and y [for example]
U ... the union of
P ... the power set of
/\ ... the intersection of
x u y ... the union of x and y [for example]
x n y ... the intersection of x and y [for example]
x\y ... x without the members of y [for example]
|- ... proves
|/- ... does not prove
|= ... entails
|/= ... does not entail
PA ... first order Peano arithmetic
S ... the successor of
# ... the Godel number of
Z ... Zermelo set theory
ZC ... Zermelo set theory with the axiom of choice
ZF ... Zermelo-Fraenkel set theory
ZFC ... Zermelo Frankel set theory with the axiom choice
Z\I ... Zermelo set theory without the axiom of infinity
(Z\I)+~I ... Zermelo set theory with the axiom of infinity replaced by the negation of the axiom of infinity
Z\R ... Zermelo set theory without the axiom of regularity
ZF\R ... Zermelo-Fraenkel set theory without the axiom of regularity
ZFC\R ... Zermelo Frankel set theory with the axiom choice without the axiom of regularity
p ... possibly
n ... necessarily
when needed for clarity, ' ' indicates an expression not its referent ('Sue' is a name, Sue a person)
It's not a problem to me. I suspect, but have no supporting evidence, that it's not a problem even for most philosophers. It's a problem for you, and I hope you can get it resolved so that it no longer troubles you.
Quoting Metaphysician Undercover
They're two phrases or words for the same thing. I concede that you have some deep or perhaps pseudo-deep reason to make a distinction, but you haven't explained it to my satisfaction.
Quoting Metaphysician Undercover
Is it your contention that if I understood this failure, I would suddenly arise and go over to the math department at the nearest university and give them a piece of my mind? Or renounce my heresy, do penance, confess to a priest? Or what, exactly, would you like me to do?
Quoting Metaphysician Undercover
You're cracking me up. I find your prose very funny tonight. You are going on about this but making no point at all.
Quoting Metaphysician Undercover
AHA!!!!!!! After berating me about the playground, and getting me to stop using real-life examples, you whip out an example from physics. But physics is not math.
Now explain this to me ONCE AND FOR ALL. Are we talking about pure math and set theory? Or are we talking about the physical world of time, space, energy, quantum fields, and bowling balls falling towards earth?
You can not have it both ways.
Quoting Metaphysician Undercover
The physics complaint department is across the street. I'm the math complaint department. In math, "equals" and "the same" and "is identical to" are synonymous. I can't help you with your complaints about physics.
And you see, having BERATED ME ABOUT THE PLAYGROUND, you now give me yet another physical example. But I have already agreed not to use physical analogies or examples anymore, because physical things are different than abstract things.
So your physics example has no bearing on mathematics. In any event, photons are just excitations in the electromagnetic field. It's far from clear what a "thing" or "object" is in physics these days.
Quoting Metaphysician Undercover
You rejected my playground story and now you're resorting to examples in physics. Please stay on topic, You are wasting your keystrokes talking about the wrong thing.
Photons are not sets. I have no idea what physicists mean by "same," "equal," or "identical." I doubt they do either, they don't bother themselves with philosophy these days.
Quoting Metaphysician Undercover
When I'm talking about the foundations of math, of course. When I arrive home in the evening, it makes quite a big difference to me if I return to the same residence or just one that's "equal" to it in value.
You are quite the sophist tonight yourself.
Quoting Metaphysician Undercover
You haven't said a single thing of substance. You've given an example from physics when the subject is math. You've said nothing. As Truman Capote once said of a book he didn't like, "That's not writing. That's typing."
Quoting Metaphysician Undercover
Well that certainly clears things up.
Quoting Metaphysician Undercover
In physics? In playgrounds? Could be, for all I know, to the extent that your word salad communicated anything at all. In math? No. You don't understand how math works, and you continually demostrate that.
Quoting Metaphysician Undercover
When it comes to colors, or colours if you prefer, I admit I'm on shaky ground. I don't know if if philosophers consider red an instance of color, though I think it is.
But when it comes to math. I am sure. The set of real numbers is most definitely a particular set in the universe of sets, and is an instance of the general concept of set, or the universe or sets, or the category of sets,
Quoting Metaphysician Undercover
You finally said something interesting. Is the 5 in your mind the same as the 5 in my mind? I think so, but I might be hard pressed to rigorously argue the point.
So what do you say? Are the 5 in your mind and the 5 in my mind the same? Identical? Equal? Curious to know. You did interest me with this example.
Quoting Metaphysician Undercover
Is an apple an instance of fruit? Apples don't have a peelable yellow skin. 'Splain me this point. By this logic, nothing could ever be a specific instance of anything, since specific things always differ in some particulars from other things in the same class.
Quoting Metaphysician Undercover
Equal is the thing being formally defined or referenced from logic. "Same as" is a colloquial usage with no formal definition.
Quoting Metaphysician Undercover
You demean only yourself and nobody else to continually refer to another member of this forum by a pejorative.
Not true, but funny.
Quoting TonesInDeepFreeze
I'm sure you're right. Just a little beyond my awareness.
Not quite my point, but thanks.
Not not true, but not funny.
At least, if you are ever interested in a formula, but you don't know the use of the symbol, there you have it. Of course, if you're not interested in formulas, though they are the most exact and often the most concise communication, then I can't help that.
Symbols need explanatory words to go with them. This is probably more true in math than in logic.
Quoting TonesInDeepFreeze
From this thread, not addressed specifically to fishfry:
Quoting TonesInDeepFreeze
The first mention that extensionality provides only a sufficient condition (which fishfry now regards as an epiphany for him), though not addressed specifically to fishfry, but there were at least a few others addressed specifically to him:
Quoting TonesInDeepFreeze
In this thread, not addressed specifically to fishfry.
Quoting TonesInDeepFreeze
In this thread. Comments on the subject, not addressed specifically to fishfry:
https://thephilosophyforum.com/discussion/comment/911857
In this thread, addressed specifically to fishfry:
Quoting TonesInDeepFreeze
In this thread, addressed specifically to fishfry.
Quoting TonesInDeepFreeze
In this thread, fishfry himself quoting me:
Quoting fishfry
In this thread, addressed specifically to fishfry:
Quoting TonesInDeepFreeze
In this thread, addressed specifically to fishfry:
Quoting TonesInDeepFreeze
In this thread, fishfry again quotes me stating the axioms:
Quoting fishfry
Then, in this thread, fishfry denies the plain record of the posting, denying that I had never neve said what I mean by identity theory, even though he had not very long before quoted me stating the axioms:
Quoting fishfry
Not only did I indeed state the axioms of identity theory several times, I stated them directly to fishfry, and he even quoted my statement of them.
Later he denied that I stated them, even though I had stated them not very many posts prior.
What explains fishfry's bizarreness?
(1) The symbols I used are common. The formulas I gave are not complicated. If one knows merely basic symbolic logical notation, then one can read right from my formulas into English. For example:
AxEy yex
reads as
For all x, there exists a y such that y is a member of x.
For example:
(P(x) & x=y) -> P(y)
reads as
If P holds for x, and x equals y, then P holds for y.
And now that I've provided a one-stop list of the most common symbols, it's even easier.
(2) I did give lots of explanations in certain contexts.
(3) You complain about the length of my posts, but also say I should give more explanation. You can't have it both ways. And you're hypocritical since your own posts are often long, and often enough have not merely a few symbols.
Quoting fishfry
I don't understand you. I gave you an example of how equivocation of "same" has a considerable effect. Of course it has no effect in "pure mathematics", because by definition "pure" mathematics maintains its purity, and the purity of its definitions. Pure mathematics is not applied, and therefore has no effect in relation to the physical world where "same" means something else.. We live in the physical world, our cares and concerns involve the world we live in, it is impossible that anything in the fantasy world of "pure mathematics" could actually concern us. This is known as the interaction problem of idealism. However, in reality we apply mathematics and this is where the concerns are.
You seem to misunderstand the issue completely. You appear to understand that there is a difference between the use of "same" by mathematicians (synonymous with equal), and the use of "same" in the law of identity (not synonymous with equal). You said that this difference has no bearing on anything you know or care about. The things included in the category of what you know and care about, are not limited to principles of pure mathematics, because you live and act in the physical world. The law of identity applies to things in the physical world which we live and act in.
So, to make myself clear, I do not claim that there is a problem with using "same" as synonymous with equal, within the conceptual structures of mathematics. The problem is in the application of mathematics, as inevitably it is applied, and this use of "same" is brought into the world of physical activity, and taken to be consistent with the use of "same" when referring to physical objects. That is where the problem occurs. Sophjists such as Tones enhance and deepen the problem by arguing that the use of "same" in mathematics(synonymous with equal) is consistent with the use of "same" in the law of identity (not synonymous with equal).
Quoting fishfry
This is exactly the issue, the reality of the situation is that we do have it both ways. There are two very distinct ways for understanding "same". You can dictate "you cannot have it both ways" all you want, but that's not consistent with reality, where we have both ways of using the term. If you think that we ought to reduce this to one, (insisting "we cannot have it both ways"), the two cannot be combined, or reduced to one, because they are fundamentally incompatible (despite what the head sophist claims). This means that we have to choose one or the other. If we choose the one from pure mathematics, then we have nothing left to understand the identity of a physical object in its temporal extension. If we choose the one from the law of identity, then we simply understand "equal" as distinct from "same", and the problem is solved. Obviously the latter makes the most sense, and doing this would support your imperative dictate: "You can not have it both ways."
Quoting fishfry
It is very clearly you are the one who does not understand how math "works". Math only works when it is applied. "Pure mathematics" does nothing, it does not "work", as math only works in application. You are only fooling yourself, with this idea that pure mathematics is completely removed from the physical world, the world of content, and it "works" within its own formal structures. That is the folly of formalism which I explained earlier. To avoid the interaction problem of Platonist idealism, the formalist claims that mathematics "works" in its own realm of existence. But the claim of "works" is sophistic deception, and the formalist really digs deeper into Platonism, hiding behind the smoke and mirrors of the sophistry hidden behind this word "works". That is when the term "mathemagician" is called for.
Quoting fishfry
I believe, the concept of "five" in my mind is completely different from, though similar to, the concept of "five" in your mind. There is a number of ways to demonstrate the truth of this. The first is to get two different people to define the term, and see if they use the exact same expression. Another way is to look at what "five" means in different numbering systems, natural, rational, real, etc.. Another is from the discussions of mathematical principles in general. There is always difference in interpretation of such principles. You and I have significant differences, You and Tones have less significant differences.
Nevertheless, the differences exist and are very real. There is a principle which I've seen argued, and this is to say that this type of difference is a difference which does not make a difference. Aristotle called these differences accidentals, what is nonessential. The problem with that expression though, "difference which does not make a difference", is that to notice something as a difference, it is implied that it has already made a difference. So this argument is really nothing other than veiled contradiction.
Anyway, this is the issue with identity, in a nutshell. When we ignore differences which we designate as not making a difference, and say that two instances are "the same" on that basis, we really violate the meaning of "same". The meaning of "same" is violated because we know that we are noticing differences, yet dismissing them as not making a difference, in order to incorrectly say "same". Therefore we know ourselves to be dishonest with ourselves when we say that the two instances are the same, by ignoring differences which are judged as not making a difference. So when you say that you think the 5 in my mind is the same as the 5 in your mind, I think that this is an instance of dishonesty, you really know that there are differences, and if pressed to argue such a claim, you'd end up in contradiction, dismissing the obvious differences as not making a difference.
Quoting fishfry
Right, particulars are instances, specifics are not. The concept "red" is not an instance of colour, it is a specific type of colour. A particular red thing is an instance of red, and an instance of colour, exemplifying both. The concept "apple" is not an instance of fruit, it is a specific type of fruit. A particular apple is an instance of both. The concept "5" is not an instance of number, it is a specific type of number. A group of five particular things is an instance of both.
Quoting fishfry
Hey fishfry, do you not remember what you said to me? You said " I don't make a distinction between "same as" and "is equal to." In math they're the same. If you have different meanings for them, it does not bear on anything I know or care about." Now you've totally changed your position to say "it makes quite a big difference to me", if the taxi driver took you to a house which had an equal fare as yours, but was not the same house.
(1)
bill 1 is not the same as bill 2
bill 1 does not equal bill 2
bill 1 is not identical with bill 2
Those are true and say the same thing as one another.
(2)
the value of bill 1 is the same as the value of bill 2
the value of bill 1 equals the value of bill 2
the value of bill 1 is identical with the value of bill 2
Those are true and say the same thing as one another.
(1) and (2) together is not a contradiction.
The valuation of a bill is a function, call it 'v'.
bill 1 does not equal bill 2
but
v(bill 1) equals v(bill 2)
Example from math:
Let v be the squaring function. So v(x) = x^2.
1 does not equal -1
but
v(1) = v(-1)
as
1^2 = -1^2
Sales Clerk: That will be five dollars.
/ The crank puts four one dollar bills on the counter /
Crank: There you go, five dollars.
Sales Clerk: That's only four dollars.
Crank: No, it's five dollars.
/ The sales clerk counts the bills by hand /
Sales Clerk: You see, only four dollars.
Crank: Five in your mind is different from five in my mind. I pay based only on what is in my mind. I gave you five dollars, now would you please put that copy of 'Hegel For Dummies' in a bag for me, as I just paid you five dollars for it?
Sales Clerk [on microphone]: Security at register seven. Security at register seven please.
Sorry Tones, but "for any x, x=x" does not say "for any x, x is x", unless "=" is defined as "is". And, in mathematics it is very clear that "=" is not defined as "is".
Explicitly stated by you, the head sophist
are the same -(-5) = 5
2z2 + 4z - 6 = 0
https://www.techtarget.com/searchdatacenter/definition/Mathematical-Symbols
For any terms 'T' and 'S'
T = S
is true
if and only if
the denotation of 'T' is the denotation of 'S'.
Consult any textbook in mathematical logic. The ignorant, confused, arbitrary personal dictates of the crank don't count.
Are you referring to 'is' in terms of identity or value? For example: 5 is 5 in both mathematics and in our understanding of numerical systems. Meanwhile, £5 doesn't equal $5 or 5 because of the disparity in monetary value. Although every bill or note is represented by the payment of x5, it will depend on the value. So, x = x, doesn't equal to "is." To apply this, I need to carefully consider the specific context. Right?
Could it be defined as "equals to..."?
That is not my point. For whatever reason, I've always had difficulty with your posts. Maybe it's the symbols. Maybe it's the words. I don't know.
Quoting TonesInDeepFreeze
I'll concede that either I'm too dense to follow your arguments, or I just lack the logic background. I appreciate your efforts.
Quoting TonesInDeepFreeze
Guilty as charged on all counts.
Then you DO understand me perfectly.
1) With regard to math, "same as", "=", and "is identical with" are synonymous. Period.
2) With respect to everything outside of math, you are probably right, but I take no position on it whatsoever. So if you're right about photons, that's great. Not any point I'm making or anything I care about in the context of this discussion.
Quoting Metaphysician Undercover
This is of course true, but irrelevant to my part of this discussion. If you have some point to make about sameness as it pertains to playgrounds and photons, I'll concede the point without even thinking about it. Because I don't care about the issue. So if you're right, you're right. And even if you're wrong, I don't care, so I won't bother to try to prove you're wrong. Therefore I concede that you are right about a discussion I'm not even having with you.
Quoting Metaphysician Undercover
You agree with me about pure math. And I'm entirely agnostic about any other aspect. I can't misunderstand what I have no interest in understanding. I can be ignorant; but I can't be wrong, because I haven't even taken a position.
Quoting Metaphysician Undercover
Not as it's understood in math. You're just trying to argue with me about something that I have no opinion on and no interest in having an opinion. I know, for an opinionated guy like me that's unusual. But there are a lot of things I have no opinion on.
Quoting Metaphysician Undercover
I don't get worked up about this at the grocery store. For example one onion is pretty much the "same" as any other, and the ways in which they are the same and the ways in which they are different may be of interest to a philosopher. But I don't trouble myself with it. I just pick out some onions.
Quoting Metaphysician Undercover
If only.
Quoting Metaphysician Undercover
Well then we are DONE. You have agreed with my point, with my ONLY point. I have no other point. I have no other opinion. I have no other thesis. I concede everything you say about the subject, not because you're necessarily right, but because I just don't care one way or the other.
So we're done. You agree with me about the only point I'm making. Which I appreciate a lot, actually. It's taken me a long time to get you to agree that "same" and "=" are synonymous in math.
Quoting Metaphysician Undercover
I care not about applications. Nor do I believe that you are correct on this point. Math is math, whether you apply it or not.
Quoting Metaphysician Undercover
I can't comment on the thoughts of others. Nor do I understand much of what @TonesInDeepFreeze says. But he did clarify a point of longstanding confusion on my part about the axiom of extensionality, for which I'm grateful to him, even if he thinks I'm still misunderstanding.
Quoting Metaphysician Undercover
You have conceded my point regarding math. I have no other point. I don't worry about sameness for onions or photons. I can chop up an onion like nobody's business; and I can flip on a light switch. That's as far as I go with onions and photons.
Quoting Metaphysician Undercover
Sadly I came to this conclusion in grad school.
Quoting Metaphysician Undercover
I don't know why you make demonstrably absurd claims like that.
Quoting Metaphysician Undercover
Tens of thousands of professional pure mathematicians would disagree.
Quoting Metaphysician Undercover
Yeah yeah.
Quoting Metaphysician Undercover
I'm only a formalist sometimes, when it helps clarify an argument. Clearly math is "about" something. Just not clear exactly what.
Quoting Metaphysician Undercover
I agree with you that most mathematicians are Platonists. A group theorist does not believe he's merely pushing symbols. He's discovering facts about groups. I concede your point. But I'm not concerned with it.
Quoting Metaphysician Undercover
Any two set theorists will give {0, 1, 2, 3, 4} as the definition of 5. That's due to John von Neumann, who invented game theory, worked on quantum physics, worked on the theory of the hydrogen bomb, and did fundamental work in set theory. Now there was a guy who blended the applied with the pure.
Quoting Metaphysician Undercover
I like that.
Quoting Metaphysician Undercover
Can you give an example? I might have not followed you.
Quoting Metaphysician Undercover
Like the two ways I showed of representing the cyclic group of four elements? So are you talking about structuralism? Two things can be the "same" if they have the same structure, even if they are very different as things.
Quoting Metaphysician Undercover
Well ... no. There is only one Platonic 5. We may all think of it different ways, but we all have the same concept of fiveness. So yes and no to your point.
Quoting Metaphysician Undercover
Particulars versus instances. I take your point there too. I think. Fruit and apples, colors and red. Two subtly different concepts. Which is which?
Quoting Metaphysician Undercover
Ok. I accept that.
Quoting Metaphysician Undercover\
Hmmm. Ok.
Quoting Metaphysician Undercover
Ok. Like apple is a subclass of fruit, but a specific apple in my hand is an instance of the class of apples. Is that right?
Quoting Metaphysician Undercover
A type of number. No, don't agree. Real numbers and complex numbers and quaternions are types of numbers. The real number 5 is an instance of a real number hence an instance of a number. It must be so, mustn't it?
Quoting Metaphysician Undercover
I do not agree that five apples is an instance of the number 5. The collection of five apples has cardinality 5. That is, 5 is an attribute of the group of five apples. Numbers are pure, they are not apples. A group of apples is not an instance of 5. It might be a representation of the number 5, I could live with that, in the sense that xxxxx is a representation of the number 5.
Quoting Metaphysician Undercover
I already addressed this point.
I said in the context of doing math blah blah. When I'm not doing math, the subtleties of sameness matter. Of course the subtleties matter in math too, as I just showed. Two different sets can represent the same group. These days structuralism is how we think about things in math. Two things are the same when they have all the same relationships to all the other things.
The symbols are standard. The words are ordinary for logic and mathematics, or if personal, they're defined.
So maybe it's something else.
Most glaringly of all, what accounts for you recently claiming that I hadn't specified 'identity theory' when I had specified it multiple times in this thread, including multiple times addressed to you, and even twice quoted by you? Your claim is bizarre.
Identity. I don't know how to make it more clear than I already have.
x is x.
With models:
'=' is interpreted as {
Quoting javi2541997
5 = 5
5 dollars not= 5 pounds
If you have 5 dollars and I have 5 pounds, then the number of your dollars is the number of my pounds. 5 is 5. But that does not entail that 5 pounds is 5 dollars.
The number of things is not the things.
'x = x' doesn't equal 'is'. Rather, '=' stands for 'is', so 'x = x' stands for 'x is x'.
None of that is a contradiction or problem.
Quoting TonesInDeepFreeze
:up:
This is exactly the problem. Notice you refer to "any textbook in mathematical logic", rather than any textbook in mathematics. If you look at a textbook in mathematics, you'll find "=" defined in the way of my reference, "equals"... "indicates two values are the same". So there is a discrepancy between what "=" means in mathematics, and what "=" means in mathematical logic. And, since mathematical logic is supposed to provide a representation of the logic used in mathematics, we can conclude that mathematical logic has a false premise. The proposition of mathematical logic, which states that "=" indicates identity, or that it means "is" or "is the same as" in mathematics, is a false premise. This is a false representation of how "=" is used in mathematics. As I explained in other threads, if "=" indicated "is the same as", then an equation would be absolutely useless, because the left side would say the exact same thing as the right side. (Incidentally, this is what many philosophers have been known to say about the law of identity, it is a useless tautology). In ontology though we see the law of identity as a useful tool against sophistry like yours.
Quoting fishfry
I agreed with you about "pure math", for the sake of discussion, so that we could obtain some understanding of each other. But I will tell you now, as came up one other time when we had this discussion, I do not agree that there is such a thing as "pure math" by your understanding of this term. So I agree that if there was such a thing as pure math, that's what it would be like. However, I think your idea of "pure math" is just a Platonist/formalist fantasy, which is a misrepresentation of what mathematics is. In reality, all math is corrupted by pragmatics to some degree, and none reaches the goal of "pure math". You criticize me to say, it's not a goal, it's what pure math is, but I say that's false, it is a goal, an ideal, which cannot be obtained. Therefore "pure math" as you understand it, is not real, it's an ideal.
I think the issue being exposed here is a difference of opinion as to what mathematics is. Since this is a question of "what something is", the type of existence it has, I think it is an ontological issue. Would you agree with this assessment? For example, the head sophist refers to "mathematical logic", and I find this defined in Wikipedia as the study of the formal logic within mathematics. So we have a distinction here between the use of mathematics (applied mathematics), and the study of the logic used by mathematicians (mathematical logic). "Mathematical logic" would be a sort of representation, or description, of the logic used in mathematics. What you call "pure mathematics", I believe would be something distinct from both, applied math and mathematical logic, as the creative process whereby mathematical principles are developed. But I think that this process is not really "pure", it's always tainted by pragmatics and therefore empirical principles.
The issue I have with the head sophist is with the way that mathematical logic represents the use of the = symbol as an identity symbol. In applied mathematics, it is impossible that "=" is an identity symbol because if both sides of an equation represented the exact same thing, the equation would be absolutely useless. This I've explained in a number of different threads. In reality, as any mathematics textbook will show, "=" means "has the same value as". Therefore we can conclude that any mathematical logic which represents "=" as an identity symbol is simply using a false proposition. When a "textbook in mathematical logic" states that "=" is an identity symbol, this can be taken as the false premise of mathematical logic.
Quoting fishfry
I have conceded the point regarding what you call "pure math". However, I am now qualifying this concession to say that "pure math" is just an unreal ideal. There is no such thing as pure math. It's a term which people like to use in an attempt to validate their ideals. In reality though, such ideals are fiction, so all that I have really conceded, is that within the fictitious conception which you call "pure math", this is the way things are. Of course, I'm not going to argue about the way things are in your work of fiction, but I will argue about the way that your fiction bears on the real world.
Quoting fishfry
Sure, there are thousands of people who might call themselves "pure mathematicians". In reality though, these people are not engaged in "pure mathematics", as I believe you understand this to mean. As I said above, all mathematics is tainted by pragmatics (applications), and there is no such thing as "pure" mathematics.
This is very evident in our discussion of the meaning of "=". In what you call "pure mathematics", we might say that "=" signifies "is the same as". This would remove the basic fact that what mathematicians work with are values. To make the mathematics "pure" we must remove this content, what the mathematicians work with, values. We remove the inherent nature of the thing represented by the symbols (i.e. that the symbols represent values) to allow simply that the symbols represent things without any inherent nature, no inherent content. Then we might claim the left side of the equation represents the exact same thing as the right. However, this type of equation would be totally useless. We could do nothing with an equation, solve no problems.
Furthermore, there would be a disconnect, an inconsistency between the mathematicians practising "applied" math, who use "=" to represent "is the same value as", and those "pure" mathematicians creating mathematical principles which were inconsistent with the applied mathematics. Since the supposed "pure mathematicians" actually produce principles which are compatible with, and are actually used in applied mathematics, we can conclude that the supposed "pure" mathematics is not really pure, and the principles they are using and developing do not really treat "=" as meaning "is the same as". That's just a misrepresentation, supported by the misrepresentation that these people are doing "pure" mathematics.
Quoting fishfry
I can't say I understand everything you wrote following this, but it mostly makes sense to me. I'll have to work on these ideas of "mod 4", and "cyclical group".
Quoting fishfry
What I mean, is that if you recognize that two things are different from each other, then that difference has already made a difference to you (in the subconscious for example) by the very fact that you are recognizing them as different. So for example, if you see two chairs across the room, and they appear to be identical, yet you see them as distinct, then the difference between them must have already made a difference to you, by the fact that you see them as distinct. So to say that the difference between them is a difference which doesn't make a difference must be a falsity from the outset. We might even say that they are identical in every way except that they are in different locations, but this very difference is the difference which makes them two distinct chairs instead of one and the same chair.
Quoting fishfry
I knew you wouldn't agree, but i wouldn't agree that the real number 5 is an instance of a real number. The problem I think has to do with the statement "a real number". "The real numbers" is a conceptual construct in itself. This conception dictates the the meaning of "a real number". So in reality any supposed instance of "a real number" is just a logical conclusion drawn from the dictates of "the real numbers". In other words its not a distinct or individuated thing, which would be required for "an instance", it's just a specific part of "the real numbers". Can we agree that the real number 5 is a specific real number?
The crank says that we may look in a textbook in mathematics to see that the definition of '=' differs from mathematical logic. What specific textbook does the crank refer to?
"two values are the same"
Indeed, in both mathematical logic and in other mathematics:
'1+1 = 2' means that the value of the expression '1+1' is the same as the value of the expression '2'.
The crank claims that we may look in a textbook in mathematics to see that mathematics doesn't agree. What textbooks are those?
And notice that the crank has shifted his argument. Previously he recognized that mathematics regards '=' as 'is the same as' and that mathematics is wrong to do that, but now he's claiming mathematics doesn't but that it is only mathematical logic that does, modulo his latest tact of blaming only pure mathematics. The crank is always greased for easy shifting.
Most simply, when we say "1+1 = 2" we mean that '1+1' and '2' name the same number. Or we are to believe they name different numbers? What numbers are those?
Clerk: Okay, we have one plus one plus one plus one. Okay, that is four. At $2 each, that's $8.
The Crank: No, one plus one plus one plus one is not four. I only pay for one plus one plus one plus one, not for four
Clerk [into mic]: We have a problem at register ten.
My deficiency, I'm sure.
Quoting TonesInDeepFreeze
I'll retract it then, as an alternative to arguing the point. Or if you consider the second clause as adding fuel to the fire, I'll retract it.
Those corrupt math professors. Something must be done. Pure math is math done without any eye towards contemporary applications. That's a decent enough working definition. Mathematicians know the difference.
Quoting Metaphysician Undercover
Not really. Nailing down a definition is unimportant. Mathematics is whatever mathematicians do in their professional capacity. Circular, but as good a definition as any. What difference does the definition of mathematics make?
Quoting Metaphysician Undercover
No. Mathematics is a historically contingent human activity. It's different every day, every time someone publishes a new paper.
Quoting Metaphysician Undercover
You must think I'm sticking to some pure/applied distinction. I'm not.
Quoting Metaphysician Undercover
Repeatedly calling a fellow forum member that makes you look like an asshole.
Quoting Metaphysician Undercover
This is a standard complaint. If math follows from axioms, then all the theorems are tautologies hence no new information is added once we write down the theorems. But that's like saying the sculptor should save himself the trouble and just leave the statue in the block of clay. Or that once elements exist, chemists are doing trivial work in combining them. It's a specious and disingenuous argument.
Quoting Metaphysician Undercover
Repeating a mistake is no virtue.
Quoting Metaphysician Undercover
We agreed long ago that 1 + 1 and 2 are not the same string; and many people have explained the difference between the intensional and extensional meanings of a string. Morning star and evening star and all that.
Quoting Metaphysician Undercover
What math teacher hurt your feelings, man? Was it Mrs. Screechy in third grade? I had Mrs. Screechy for trig, and she all but wrecked me. It's over half a century later and I can still hear her screechy voice. I hated that woman, still do. When I'm in charge, I'm sending all the math teachers to Gitmo first thing.
Quoting Metaphysician Undercover
You are the one making a big deal out of the distinction. Besides, in my last post I ended up talking myself out of my entire thesis due to the structuralist turn in mathematics.
Quoting Metaphysician Undercover
I really haven't made a big deal of the distinction; and if I did, I shouldn't have.
Quoting Metaphysician Undercover
A piece of math without any known application is pure math. A hundred years from now it could be applied. The most striking case is number theory, which was totally useless for 2000 years then became the basis of public key cryptography in the 1980s, the basis of online commerce.
Quoting Metaphysician Undercover
I just think you're working yourself up over nothing. I'm losing interest. Can you write less? This is tedious, I find nothing of interest here.
Quoting Metaphysician Undercover
Whatevs. I can't follow you. And I've already noted that the difference between pure and applied math is often a century or two, or a millennium or two.
Quoting Metaphysician Undercover
Sorry. Nevermind all that. Point being that two groups are essentially the same even though their presentation as sets is entirely different. Just like the real number 5 and the integer 5 are essentially the same even though they are different sets. Now what do I mean by "essentially the same?" Well now we're into structuralism and category theory. Sameness in math is a deep subject. I'll take your point on that.
Quoting Metaphysician Undercover
Ok. They're two instances of chair-ness.
Quoting Metaphysician Undercover
I can't even imagine how it isn't.
Quoting Metaphysician Undercover
Yes.
Quoting Metaphysician Undercover
Even so, 5 is one of the real numbers. What do you call it if not an instance? What WOULD be an instance of a real number?
Quoting Metaphysician Undercover
I agree that 5 is a specific real number. But ok, instance means something else. Still a bit fuzzy but sort of seeing your point.
Why does it take so long to understand an axiom that appears so simple?
I once read Bertrand Russell's works, and one of them was Principia Mathematica. Well, it took him and another colleague of his more than 300 pages to prove that 1+1 = 2. I acknowledge that I struggled to understand some of their pages and axioms due to my lack of familiarity with logical language. There was even some criticism of the work of Russell and Whitehead because it seems the work was based on finding 'truth logic'
I asked myself then. Is 1 + 1 = 2 a logical truth? And I found on the Internet big debates among mathematicians and logicians about whether it is a tautology, a logical truth, or a theorem.
In the following link (1 + 1 = 2) you will see similar answers to yours: 1 + 1 = 2 is a 'definition'.
2 is another way of defining '1 + 1' if I am not mistaken...
Frege proposed a way that it would be a logical truth. But his way was inconsistent.
Quoting javi2541997
That's often the case, and per the definition, '1+1 = 2' means that '1+1' names the name number as is named by '2'. 1+1 is 2.
It is bizarre to suggest there's any arguing the point, when the point has been so profusely documented. Your retraction and your offer to retract the bizarre qualifier in the retraction are a self-serving and sneaky way to put the ball back in my court where it doesn't belong.
Isn't that a bit too much to put on the Basic Law V?
If we have problems with infinite sets, why would you throw away also everything finite?
How about Peano axioms or Peano Arithmetic?
Are they inconsistent also according to you?
If you desire to avoid the long posts, I think, by the end of my reply here, that I have isolated the primary point of disagreement between us. It is exposed in how you and I each relate to what is referred to by "the real numbers", and what is referred to with "5" in the context of "the real numbers". And further, how this relates to the extension/intension distinction.
Therefore, I think you might just read through my post and reply to the aspects which are related to this issue. However, the issue of what mathematics is, how you and I would each describe "what mathematicians do", might also be important and relevant.
Quoting fishfry
The issue though, is that even supposedly "pure" mathematicians work toward resolving problems, and problems always have a real world source or else they are really not problems, but more like amusements. A mathematician working in pure abstractions works with abstractions already produced, and may not even know how real world problems have shaped the already exist abstract structure. Even if we attempt to step aside from existing conceptions, and 'start from scratch' as philosophers often do, we are guided by our intuitions which have been shaped and formed by life in the world. And intuition comes from the subconscious into the mind, so we cannot get our minds beneath it, to free ourselves from that real world base. And since it is from the subconscious, we have no idea of how the real world effects it.
Quoting fishfry
I agree, but the description of what mathematicians do, is very difficult to get an agreement on. It's not a circular definition, but a proposal of how to produce a definition. So to actually provide the definition of mathematics, we need that description. It will be very difficult for you and I to agree on such description. You will probably place as the primary defining feature, (the essential aspect), of what mathematicians do, as working with abstractions. I will say, that description is problematic because then we need some understanding of what an abstraction is, and what it means to "work" with this type of thing. This almost certainly will lead to Platonism because we've already assumed as a premise, the existence of things called "abstractions".
Therefore I look at what mathematicians are doing as "solving problems". That's what they do, and there is a specific type of problem which they deal with. You are most likely not going to like this proposal for a description of what mathematicians are doing, because it eliminates the distinction between "pure" mathematics and "applied" mathematics. In the way described above, there is no such thing as "pure" mathematics. However, my starting point has the advantage of applying equally to all mathematicians, by applying the initial assumption of pragmaticism. Instead of saying "mathematicians are working with abstractions", we say "mathematicians are working with symbols (language), to solve problems. This way we avoid the messy ontological problem of "abstractions" It is only when we start sorting out the different types of problems which mathematicians work on, do we get the divisions within mathematics.
Quoting fishfry
This is not the point at all, and you are not paying respect to the difference between the two distinct fields, mathematics, and mathematical logic, so your analogy is not well formed. If the field of mathematics is represented by the sculptor, then the field of mathematical logic is represented by the critic. Whenever the critic mistakenly represents what the sculptor is doing, then the critic is wrong. When mathematical logic represents mathematicians as using = to symbolize identity, the logic is wrong.
Quoting fishfry
Fishfry, wake up! Was it getting late there or something? There is no physical object involved! There is no star! I think we've been through this before. The intensional/extensional distinction is completely irrelevant in this case because everything referred to is meaning (intensional). There is nothing extensional, no objects referred to by "1+1", or "2". That is the heart of the sophistic ruse. This intensional/extensional rhetoric falsely persuades mathematicians. It wrongly misleads them due to their tendency to be Platonist, and to think of mathematical abstractions as objects. As soon as meaning is replaced by objects, then "extensional" is validated, the sophist has succeeded in misleading you, and down the misguided route you go. In reality, there is only meaning referred to by "1+1", and by "2", everything here is intensional, and there is nothing extensional.
This is why I was very steadfast on the previous issue, to explain that "5" is not "an instance of a real number". It is that type of nomenclature, that type of understanding, which leads one into allowing that there is a place for extensional definitions in mathematics. Really, "5" in that example is just a part of that conception called "the real numbers". It receives it's meaning as part of that conception. there are no extensional objects referred to by "the real numbers", and "5" is just an intensional aspect of that conception. When you apprehend "the real numbers" as referring to a collection of things, instead of as referring to a conception, then you understand "5" as referring to an instance of a real number, instead of understanding it as a specific part of that conception. Then you may be misled into the "extensionality" of real numbers, instead of understanding "the real numbers" as completely intensional.
Quoting fishfry
Again, you are not distinguishing between "mathematics", and the "mathematical logic" which the head sophist preaches. One is the artist, the other the critic. My beef is not with mathematics (the art), it is with mathematical logic (the critic). I see mathematical logic as sophistry intended to deceive. And I will explain the reason why i say there is an intent to deceive.
Mathematics has a long history of exposing us to problems which we just cannot seem to solve. These are issues such as Zeno's paradoxes, and other apparent paradoxes discussed at TPF, which generally amount to problems with the conception of infinity, the continuity of space and time, etc.. What mathematical logic does, is create the illusion that such problems have been solved. So, the intent to deceive is inherent within the conceptual structure, which makes these problems solvable. It deceives mathematicians into thinking that they have solved various problems, by allowing them to work within a structure which makes them solvable. The problem though is that the basic axioms (extensionality for example) are blatantly wrong, and designed specifically so as to make a bunch of problems solvable, regardless of the fact that incorrect axioms are required to make the problems solvable.
Quoting fishfry
Future application is not the issue here. The issue is that mathematicians work toward problem solving, by the very nature of what mathematics is. The problems are preexistent. Therefore mathematics by its very nature is fundamentally "applied". If you remove problem solving from the essence of mathematics, then it would be random fictions. But mathematics is not random fictions, the mathematicians always follow at least some principles of "number", already produced.
Quoting fishfry
What I think, is that there is really no such things as sameness in math, and this is better described as a misleading subject. Mathematics actual deals with difference, and ways of making difference intelligible through number. Similarity is not sameness, but difference which can be quantified. To me, "essential the same" just means similar, which is different.
Quoting fishfry
This appears to be the substance of our difference, or disagreement. If you do not like long posts, we could just focus on this specific issue. The issue is whether "the real numbers" refers to a conceptual structure, or whether it refers to a group of things, numbers. I believe the former, and the fact that "numbers" is plural is just a relic of ancient tradition. From my perspective, "5", in the context of "a real number" is just a specific part of that conception. Then the relations are purely intensional, and there is nothing extensional here. If however, you apprehend "the real numbers" as referring to a group of things called "numbers", then "5" refers to one of those things, and there is the premises required for extensionality.
If you reject my retraction and apology, that is your right.
What do you mean by "put on"? I only said that Frege's system is one attempt to derive mathematics solely from logic, and the system is inconsistent.
/
I don't know what you have in mind about "throwing away everything finite"?
Frege's approach to even defining the number 0 from logic alone requires an infinite class.
/
Frege's system was proven to be inconsistent.
First order PA has not been proven to be inconsistent. I don't see a reason to believe that first order PA is inconsistent. And accepting the premises of Gentzen's proof, first order PA is proven not to be inconsistent.
So I don't know why you would ask.
It is true that some mathematicians are "problem solvers", perhaps the majority. But for the others, myself included, a mathematician is an explorer trying to find a path extending knowledge in a particular direction or discovering new directions. Creation and discovering are two sides of the same coin: we create, for instance, simply by virtue of defining and we discover where those creations lead.
And the evening star (which is the morning star) is not a star, it's a planet, and exists as a physical object.
And the crank adds to displaying his lack of intellectual capability by showing that he cannot comprehend intensionality and extensionality, which is not surprising since he is incapable of comprehending use and mention.
Looking ahead, you wrote a long post. To which, in my own verbose style, I will reply to at length para by para, increasing the overall length of the thread.
I see an out. In this para you have stated your aim about the real numbers and the number 5. I don't think I have any interest in this topic. I know it's important and meaningful to you, but it isn't to me. Perhaps I'm to dim to grasp all these philosophical subtleties such as you raise. If so, so be it.
But secondly, and I'd be remiss if I didn't add, that I have formally studied the real numbers and the number 5. That doesn't make me right and you wrong, by any means. What it does mean is that I'm not likely to ever defer to your opinions about the real numbers or the number 5.
And if, as you say, that's all you want me to know, then now I know it. We disagree on the real numbers and the number 5. Ok. I am a pluralist. It doesn't bother me when people have different opinions than I do. I don't have to convert you nor you me. We can let the matter rest. I'm for that.
Quoting Metaphysician Undercover
I have no strong or absolute opinion about "what math is," nor do I feel any need to argue for or against any particular interpretation of that question. The history of math is the evolution of the answer to that question! "What math is," is always changing. It literally is what mathematicians do.
Quoting Metaphysician Undercover
If your claim is that by definition, what pure mathematicians do amounts to amusements, in the sense that they solve problems that were only inspired by their own meaningless work; and not by anything that we currently know about in the world.
If so, it's perfectly and trivially true, if that's your definition. What of it? Doesn't mean anything.
Read Hardy's A Mathematician's Apology. He'd have been insulted if you told him his work was useful. The irony is that he did number theory, which had been a beautiful but utterly useless branch of math for over 2000 years. And then finally in the 1980s, people invented public key cryptography, and Hardy's work was at the heart of grubby world commerce.
So you just never know.
But still. Amusement? Ok. Whatever. Like Picasso, he made amusements too.
Quoting Metaphysician Undercover
Such a triviality. Everyone knows that math started when some caveman put a mark in the ground when he killed a wooly mammoth, and then put another mark next to it when he killed another one. That was the first mathematical abstraction, and it is the paradigm for all others.
Everybody knows this. You think you just discovered it. Just like someone could say that abstract art is an evolution of representational art, or is influence by it or is a reaction to it.
It's just the process of abstraction. And abstraction is always based on our own experience of the world. Nobody is denying that. It's your strawman.
Quoting Metaphysician Undercover
Correct. Mathematics is a historically contingent human activity that changes every day as new papers are published. And every few decades new ideas come out that change our very conception of what math is.
Even in contemporary practice, there is professional disagreement about what is mathematics. I refer to the amazing dispute over the work of Shinichi Mochizuki, about which mathematicians have been arguing for over a dozen years, and illustrating the fact that mathematical truth is subject to social agreement.
In view of the history of math, we see that this has always been so.
Quoting Metaphysician Undercover
Trying to nail down a definition of mathematics is like a cat chasing its own tail. Not as cute though. Why do you persist? Why do you even think it matters? It changes throughout history, and it's not even agreed on by all professional practitioners today!
But who said I'm not a Platonist? I am? When it suits my argument. I'm a formalist as well at times. Mathematical philosophies are tools, nothing more. Conceptual tools, frameworks for thinking about the development and structure of math. They aren't "true" or "false," they're just models, if you will.
Quoting Metaphysician Undercover
I don't like it because it's factually false. There's a famous essay on that, about the kind of mathematicians who solve problems, and the kind of mathematicians who build theories. To the Internet! Yes here it is, The Two Cultures of Mathematics by Timothy Gowers.
Problem solvers and theory builders. The theory builders don't solve problems at all. They create conceptual frameworks in which others can solve problems.
Quoting Metaphysician Undercover
Ok, there's no such thing as pure mathematics. So what? Why do you care? Why should I? I've already explained that not only is math historically contingent, there's not even universal agreement today about what math is.
You are arguing against a strawman of your own imagination.
Quoting Metaphysician Undercover
Ok you didn't like my sculptor analogy.
Quoting Metaphysician Undercover
LOL. 1 + 1 and 2 are each representations of the same set in ZF, with "1" and "2" interpreted as defined symbols in the inductive set given by the axiom of infinity; and likewise "+" is formally defined.
But we've been having this conversation for years, and I don't think today is the day for any more.
Quoting Metaphysician Undercover
Well.
I will stipulate that the real number 5 has a relatively shake ontological status. The mathematical real numbers are a very strange gadget, and I genuinely doubt that they are instantiated or exemplified by anything in the physical world. That is, the real world is not a continuum in the sense of being isomorphic to the mathematical real numbers. The real numbers are far too weird to be real.
BUT! Are you telling me that you don't believe in the physical instantiation of the natural number 5? Just look at the fingers on your hand. I rest my case.
Quoting Metaphysician Undercover
Mathematical logic. Anyone in particular? Do you go back to Aristotle, or are you annoyed with Russell, or Godel, or what? What is the specific nature of your beef, as you put it.
Quoting Metaphysician Undercover
Your level of mathematical knowledge is so naive, that you actually think that Zeno's paradox is a mathematical problem; and that the extremely naive conceptions of mathematics exhibited by innumerate philosophers that drive these inane discussions of supertasks, has any relationship whatsoever to the professional activities of mathematicians. You just have no idea. In argument, you wield your ignorance like a club. You have no idea what you are talking about.
Quoting Metaphysician Undercover
Instead of addressing these bitter complaints to me, have you thought about going down and picketing the math department at your local university? "Lying corrupt sophists all! Should not drink and derive!" Get some press for sure.
Why me?
Quoting Metaphysician Undercover
Yeah ok, all math is applied. I have no strong opinion. It's all an abstraction of the first caveman who made a bijection between marks on the ground and wooly mammoths he killed. I have no problem with that. But actually you have zero idea what pure mathematicians do, and why they are so regarded by their peers. You're just making stuff up that you don't know anything about. You keep saying mathematicians do this and mathematicians do that, and you have repeatedly demonstrated to me that you have no understanding of mathematics nor mathematicians.
Quoting Metaphysician Undercover
If I'm understanding you, I agree. I don't think the mathematical real numbers refer to anything in the world at all. They describe the idealized continuum, something that we have no evidence can exist.
I would say that this is a type of problem solving, wouldn't you? The problem being worked on is not necessarily a practical issue. Philosophy is like this too, as well as speculative theorizing, there is a wide range to the types of problems. Sometimes, problems are being worked on without any obvious practical implications.
Quoting fishfry
Well, "the real numbers", and "5" being an instance of a real number, was your example. I agree that by some accepted principles of mathematics, the axioms of set theory, etc., 5 is an instance of a real number. This I believe to be the influence of Platonism which assumes that a number is an object. I disagree with this, and think that a number is a concept, and conceptions are quite different from objects. The way that one concept relates to another for example is completely different from the way that one object relates to another.
You might think that it doesn't matter whether a number is an object or not. You might think that within the confines of the logical system of "the real numbers", a number can be whatever the mathematician who states the axiom wants it to be. My argument is that numbers are used billions of times a day by human beings, and according to that usage there is some truth and falsity about what a number is. Therefore when an axiom makes a statement about what a number is, and it's not consistent with how numbers are actually used, the axiom can be judged as false.
Quoting fishfry
Like I explained earlier, formulism is just a specific type of Platonism. It takes Platonist principles much deeper in an attempt to realize the ideal within the work of human beings, while other Platonists allow the ideal to be separate from human beings.
Quoting fishfry
Do you not look at mathematics, and mathematicians as real human beings, carrying out activities in the real world? If so, then don't you think that there is such a thing as true and false propositions about what those mathematicians are doing? If you follow, and agree so far, then why wouldn't you also agree that mathematical philosophies, as tools, or models, ought to be judged for truth and falsity? If a mathematical philosophy provides false propositions about what mathematicians are doing, offering this philosophy as a tool for understanding the structure and development of math, it is likely to mislead.
Quoting fishfry
As I explained to jgill above, theory building is a form of problem solving, it just involves a different type of problem. There are many different types of problems which can be categorized in different ways.
Quoting fishfry
Yes, this is the problem, axioms of set theory are false, in the way described above.
Quoting fishfry
I said that 5 is not an instance of a real number. Also, I would say that the fingers on my hand are not an instance of the number 5, they are an instance of a quantity of five. You see, this is the problem of mixing up the ideal with the physical. "The natural number 5" is an ideal, a type of Platonic object called "a number". There is no physical instantiation of numbers, they are by definition ideal. So we need to refer to the use of "5" to see its meaning, and then we can find a physical representation for its meaning. In the context of usage of the natural numbers my understanding is that 5 represents a specific quantity, and the fingers on my hand provide an example of this specific quantity.
If we say that the numeral 5 represents a number, which goes by that name, 5, we have no meaning indicated to assist us in finding a physical example of the number five. All we have is that there is a type of thing called a number, and one of them is named 5. In order for numbers such as 5 to be used in practise, we need to provide something more, otherwise we're stuck with the interaction problem of idealism, these ideal things have no bearing on the real world. But if we give the number 5 further meaning, such as "a specific quantity", to allow it to be useful in the world, then the ideal, the number 5 becomes redundant, and completely useless. Why not just say that the numeral "5" means a specific quantity, and be done with it. Well I'll tell you why not. The numeral "5" is assumed to represent a number, 5, which is an abstract, Platonic object, for another purpose. The other purpose is mathematical philosophy, building structures and frameworks to be used as tools for understanding the development of math. However, as explained above, rather than assisting understanding, it misleads.
Quoting fishfry
You are free to abandon me anytime you want.
Quoting fishfry
If you truly believe this, then how would you validate your claim that the number 5 is an instance of a real number. Do you see that when you talk about "a real number", and "the real numbers", you validate the claim that "the real numbers" refers to a collection of individual objects? And that is contrary to what you say here. And do you see that in set theory, "numbers" also must refer to individual things, and this is contrary to being a description of "the idealized continuum".
Are the two really mutually exclusive?
Quoting Lionino
So you are asking "couldn't a formalist not be a nominalist?"
I'll try to check it out.
Right, I caught that a moment later, and edited mine.
That, and also "Couldn't a logicist not be a platonist?".
Right.
I wonder about the categories. The schools could be something like:
realist
logicist
formalist
structuralist
constructivist
Why not? Maybe if 'logical truth' was regarded as a property of formal semantics? I mean, can't we regard 'logical axiom' as merely a logical notion without ontological commitment?
I think the first and fourth are about the metaphysics of mathematics, while the second, third, and fifth are about the foundations of mathematics, so then they would be grouped in two separate sets. Some of the questions in my thread are also about what arrows we are supposed to draw between those two supposed sets.
Quoting TonesInDeepFreeze
I would imagine so, there is nothing about logical statements to me that imply an ontological commitment. On the other hand, my problem is that I am very unpersuaded by platonism, so I am not keen on spotting flaws in arguments against it.
Perhaps. The matter of mathematics being invented or discovered is absolutely derivative from the ontology. It is possible that the foundations are also derivative from ontology or vice versa.
Yes. Well put.
Quoting Metaphysician Undercover
Meaningless word games. The fingers on your hand are a physical instantiation of the number 5. Positive integers have the property that the smaller among them may be physically instantiated. 12 as in a dozen eggs, 9 as in the planets unless an astronomical bureaucracy demotes Pluto. That's one for the philosophers, don't you agree? The number of planets turns out to be a matter of politics, not math or astrophysics.
Quoting Metaphysician Undercover
Actually some numbers happen to have physical instantiations and some don't. 5 does.
But it's a distinction without a difference. It's something that consumes you but nobody else. Least of all me.
Quoting Metaphysician Undercover
Nah. Not buying any of it.
Quoting Metaphysician Undercover
Why do you think I take a position on any of this?
Quoting Metaphysician Undercover
The swine.
Quoting Metaphysician Undercover
Judged by who? Politicians? Academic administrators? Philosophers? How about by their fellow mathematicians? That's the standard of what counts as math.
Quoting Metaphysician Undercover
Just change the meanings of words to suit your argument. Pigs fly, if I redefine pigs as birds.
Quoting Metaphysician Undercover
False. Not just "not true," or "lacking a truth value," but literally false. If they can be false then they can also be true.
Quoting Metaphysician Undercover
Meaningless word games. The fingers on your hand are a physical instantiation of the number 5. Positive integers have the property that the smaller among them may be physically instantiated. 12 as in a dozen eggs, 9 as in the planets unless an astronomical bureaucracy demotes Pluto. That's one for the philosophers, don't you agree? The number of planets turns out to be a matter of politics, not math or astrophysics.
Quoting Metaphysician Undercover
I can establish a bijection between the fingers on one hand, and the elements of the set {0, 1, 2, 3, 4}. Done.
Quoting Metaphysician Undercover
Because sometimes it's useful to study things in the abstract; just as one guy goes fishing, and another studies ichthyology
Quoting Metaphysician Undercover
Thank you. I should avail myself of that option at the end of this post. I think I will.
Quoting Metaphysician Undercover
It depends on how you look at them.
Quoting Metaphysician Undercover
Suppose math is entirely fraudulent. Would it then be any less useful in the world? Would you fire all the professors? What is your suggested remedy for all these academic crimes?
Quoting Metaphysician Undercover
You are free to judge things as you wish.
Quoting Metaphysician Undercover
They're meta-false, as I understand you. They're not literally false. If the powerset axiom is false, you get set theory without powersets. You don't get some kind of philosophical contradiction. You are equivocating levels.
Quoting Metaphysician Undercover
A model, not a description. Is that better?
That's funny. @Metaphysician Undercover just told me that formalism is just a deeper kind of Platonism. Which actually makes sense in this context. It explains how one can be both.
I don't see what this all has to do with your claim that a concept like a number, 5, could have a physical instantiation . Fingers are fingers, and are therefore physical instantiations of fingers, not of numbers, not matter how many of them you have. Wittgenstein took up this issue in the Philosophical Investigations, showing why there is a lot more involved with learning a language than simple ostensive definition. Abstraction is very complex, and with complex concepts like number, an explanation of what it is about the thing which is being shown, which is being referred to with the word, is a requirement.
A person cannot simply look at the fingers on a hand and apprehend the concept 5. An explanation about quantity, or counting is required. The concept 5 is learned from the explanation, not from the ostensive hand, therefore the hand is not a physical instantiation of the number.
Quoting fishfry
It can be judged by anyone. The issue though, is that many, like yourself refuse to make such a judgement. You say that there is no truth or falsity to mathematical axioms, they are simply tools which cannot be judged for truth. Since mathematicians tend to think this way, they are not well suited for judging truth or falsity of their axioms. But I've shown how axioms can be judged for truth. If an axiom defines a word or symbol in a way which is inconsistent with the way that the symbol is used, then it is a false axiom.
So for example, if a mathematical axiom defines "=" as meaning "the same as", yet in applied mathematics the mathematicians use "=" to mean "has the same value as", then the axiom makes a false definition. This axiom will be misleading to any "pure mathematician" who uses it to produce a further conceptual structure with that axiom at the base, just like if anyone else working in speculative theories in other fields of science starts from a false premise. False propositions are fascinating, sometimes leading to theories which are extremely useful, because they are designed for the purpose at hand.
Quoting fishfry
Sorry, I don't understand what you mean by "meta-false". I am talking about "literally false". False to me, means not corresponding with reality. For example, if someone says that in the use of mathematics, "=" indicates "the same as", but in reality, when mathematicians use equations, "=" means "has the same value as", then the person who said that "=" indicates "the same as" has spoken a falsity. Do you agree that this would be an instance of "literally false"?
Quoting fishfry
That doesn't help. Numbers form discrete units, and discrete units cannot model an idealized continuum. There is an inconsistency between these two, demonstrated by those philosophers who argue that no matter how many non-dimensional points you put together, you'll never get a line. The real numbers mark non-dimensional points, the continuum is a line. The two are incompatible.
Ontological assumptions are what foundations are made of, and Platonism provides the assumptions required for formalism, the idea of pure form.
Some would disagree. But it is quite possible.
Quoting Metaphysician Undercover
That is not true for every formalist. If you want to know why, look it up.
I believe it is required to validate any formalist approach. If you think otherwise maybe you could explain.
Then I would say that they misunderstand the foundations of the principles they believe in.
I was only relating what @Metaphysician Undercover said. Of course it doesn't make sense :-)
I'm not really too expert on formalism versus Platonism, I have a very lay-person understanding of those philosophical terms.
5 is an attribute of the fingers on your hand, would you grant me at least that?
I think of fingers as a physical instantiation of the concept of 5. But if you disagree, then we must be using the word differently. I'm ok with that. How about representation, in the same sense that the first cave man to kill five mastodons and make five marks in the ground to keep track.
Quoting Metaphysician Undercover
As far as I know, Wittgy utterly failed to understand Cantor's diagonal argument; therefore his mathematical judgment is deficient in my opinion.
Perhaps abstraction is difficult to define or pin down with words. But we all know it when we see it, and with practice we become good at using it.
Quoting Metaphysician Undercover
This is manifestly false. Not a matter of opinion or interpretation or language. Flat out false. On the contrary, it is exactly through the experience of looking at one's hand that one at first does apprehend the number 5; and only later, by analogy and induction, all the other natural numbers. Most of the others are far too big to have any such convenient physical representation. Our introduction to the natural numbers, our first concept of them, is by counting the things around us when we are babies. I"m no expert on child development, but there must come a time that a small human looks at his or her hand, and says, "Five." I just did it myself as an experiment. I looked at my hand and saw five. I believe you when you say you don't. You just lack the abstraction and math genes.
Quoting Metaphysician Undercover
Oh no. 5 is learned by bijection with the fingers, not with counting. Counting is a higher function. Bijection is more primitive or intuitive. If you've seen a mother cat missing a kitten from her litter, she is not going "One, two, three ..." She's comprehending the total number instinctively and knowing when she's one short.
Quoting Metaphysician Undercover
There is a modern trend of misspelling judgment, and I can't let it go by. No middle 'e' in judgment.
Quoting Metaphysician Undercover
There's none in the theory. When you're thinking ABOUT the theory, you can have truth if you like. There is no "truth" in the axioms of group theory, but they are true about the symmetries of a triangle.
Quoting Metaphysician Undercover
There is no such axioms. You make stuff up then tilt your lance at strawmen.
Quoting Metaphysician Undercover
You are the only one making up these definitions so that you can disagree with them. Your characterizations of what mathematicians do is in your imagination.
Quoting Metaphysician Undercover
You're off on your own thing here.
Quoting Metaphysician Undercover
You said that the axioms are false.
But you misunderstand what that means. Take the powerset axiom. Suppose as you claim, it's false. Then we would be studying the class of set theories that lack powersets. It would be interesting math. It's actually done. The powerset axiom is one that is often assumed to be false, so that we can work out the set theory that doesn't depend on it.
But you think that by an axiom being false, it's committing some kind of metaphysical no-no or faux pas. That's what I mean by meta-false. You think that axioms aren't properly defined or conceptualized or something. NOT that they are literally propositions that are to be taken as false. That's an entirely other thing.
Quoting Metaphysician Undercover
If 2 + 2 is 5, then I am the Pope.
That is a true statement that does not correspond with reality.
Ahab is captain of the Pequod. That is a true statement that does not correspond with reality, since both Ahab and the Pequod are fictional entities.
So you are wrong. But isn't the knight move a truth of chess that does not correspond to reality? Axioms in math are like that. Statements assumed true in a fictional context so as to work out the consequences.
Quoting Metaphysician Undercover
I can't sort it out. That paragraph broke my parser. An instance of literally false? I have no idea how to approach this question; nor would I be inclined to do so even if I did.
Quoting Metaphysician Undercover
Not sure how the discrete/continuous thing got into this convo. Do you mean the set of real numbers doesn't contain all the real numbers? What's a discrete unit? Starting from the natural numbers we can logically construct the real numbers, even there you're wrong.
Quoting Metaphysician Undercover
If you didn't get the number line in high school, there is not much I can say. The modern mathematical theory of the real numbers is logically unimpeachable. We can indeed start from the empty set and the axioms of ZF, and construct a model of a continuum; that is, an infinite totally-ordered Archimedean set. And we can show that all models of such a thing are isomorphic. So we can indeed construct the real numbers out of discrete units.
It's call the arithmetization of analysis. It's a thing in late 19th century math. Basically founding math, including calculus and continuous processes, on set theory.
https://en.wikipedia.org/wiki/Arithmetization_of_analysis
I am aware.
No, I'd say "it has five fingers" is an attribute of your hand. An easy way to think of attributes, is as what something has, a property. So ask yourself, do the fingers on your hand have 5. It doesn't make any sense to say that your fingers have the number 5 as an attribute. Number is a value, and values are proper to the subject, not the object. 5 is not an attribute in the way you propose it's a value.
Quoting fishfry
Using what word differently, instantiation, or 5? As I said before, I don't believe that numbers have any physical instantiations. Numbers are values and values do not have physical instantiations. So I don't understand what you're asking.
Quoting fishfry
OK, so we have a difference of opinion, and you are extremely convinced that you know the truth, and my opinion is false. This indicates to me that unless you can prove to me the truth of your opinion, then discussion is pointless. Maybe you can explain it to me. Imagine a person with no understanding of number, a young child just learning to speak for example. You believe that this person can stare at one's own fingers and abstract the concept 5, without any explanation. Please explain how this would be done.
Quoting fishfry
Come on fishfry, say something reasonable. This is ridiculous. You are asserting that the number 5 is the first number that a person learns.
Quoting fishfry
Now it's time for me to say that I think you are wrong. I never learned bijection with my fingers, I learned how to count. We learned how to count to ten. Then we were given examples of the quantities which each name signified, but that was only after we learned how to count. Learning how to count was first because that's how we memorized the names, and their order. Once the names were memorized we could learn the quantity signified by the name. We did not learn bijection, that's a much more complex skill then simply memorizing the order of some words. All simple arithmetic was a matter of memorizing. Did you not use flash cards?
Cats don't do bijections, nor do young children learning about numbers. The mother cat knows each kitten intimately, and knows when one is missing because she misses it. She does not count them in any way.
Quoting fishfry
Sorry, the devil made me do it. For some reason, out of all the words that have multiple spellings British/American mainly, people on this forum complain about judgement/judgment. Why is this worthy of a correction? You didn't correct me when I spelled color colour.
Quoting fishfry
That's nonsense. There is nothing to relate "2+2=5" to you being the pope. So this conditional is clearly false, not true as you claim. If 2+2 is 5, how could that make you the Pope, there's no logical connection to support your claim of truth.
Quoting fishfry
Uh huh, fictional statements which are assumed to be true. That's contradiction. Do you mean a counterfactual? Obviously they are not assumed to be true. You and I seem to have a completely different idea as to what constitutes truth, so I think we'd better leave that alone.
Quoting fishfry
"Literally false" was your terminology. Why pretend not to understand it?
"They're meta-false, as I understand you. They're not literally false. If the powerset axiom is false, you get set theory without powersets. You don't get some kind of philosophical contradiction. You are equivocating levels."
This discussion has degenerated. Let's evacuate.
Oh that's interesting. Is judgement with the extra 'e' a Britishism? I was not aware of that.
Quoting Metaphysician Undercover
Ok.
I don't know, but there are lots of US/Brit differences, the common one being the "o/ou", which most are familiar with. I'm Canadian so I'm stuck in between, getting it from both sides. For us, the 'proper' way is the Brit way, which my spellcheck hates. I have the keyboard option for Canadian English, but it seems to default to US. There are some interesting nuances, such as the practice/practise difference. We would use "practise" as a verb, an activity, but if a professional like a doctor, or lawyer, sets up a practice, we have the other form as a noun. It's not a very useful distinction, and difficult to figure out when you're writing, so screw it! What's the point in such formalities?
Back to the question of formalism... How does a formalist typically account for the ontology of rules? What kind of existence do rules have? Consider the rule of how to spell "judgement" for example, how does that rule exist?
You managed to entirely misunderstand my answer.
Since judgement is apparently a Britishism, I accept that and have no trouble with it; no more than I have with colour. I speak perfect English. I left my spanner on the bonnet of the lorry!
So I did NOT realize that judgement with an e is a Britishism. I just thought it was the same common misspellings that's on my personal list of stuff that annoys me.
And now that I know it's a Britishism, the next time I'm about to chide someone for misspelling judgment, I'll first ask them if they're British; and if they are, I'll belay that chide.
Is that any more clear?
I explored this question somewhat in my Grundlagenkrise thread, specially in my chat with Banno, but there was no interest in the topic died after 3 days folks prefer to go around circles about ethics instead and keep it shallow. The ontology of rules are ultimately derived from logic, be it first-order or second-order and logical terms can be taken as primitives defined from their truth tables and the usage of undefined terms, such as "line", "+", or, in the case of ZF, membership ?.
5 views per day. The title doesn't resonate with many apparently (including me). Nevertheless, an important movement.
If logic is following rules, as formalists seem to think, then to say that rules are derived from logic is circular. That's the issue with formalism to avoid the vicious circle, rules must exist as Platonic Forms. So formalism really cannot avoid Platonism, because the only ontologically coherent formalism is Platonism.
Virtually every professional mathematician lives in the world created by this movement. Nobody notices because it's like fish not noticing water. It's just the familiar founding of analysis in set theory.
Logic is not "following rules". Your argument is failed as it relies on a nonsensical definition.
[quote=Wikipedia: Formulism (Philosophy of Mathematics)] In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. [/quote]
However you frame it, rules are an essential aspect of formalism. So the ontology of rules needs to be addressed if we want to determine whether formalism can actually avoid Platonism, or whether it is as I say, just a deeper form of Platonism.
Thus, were set theory removed mathematicians would perish. I think not. But mathematics would not be nearly as robust as it is today. My humble opinion.
Back in the late 1960s my advisor remarked on the separation of the nitty gritty at ground level and the efforts to fly high and look down on mathematics, an abstract perspective to see how the various parts fitted together and document how parts from one branch were like parts form another. He gave me a choice and I felt far more comfortable working in the lowlands, (particularly after learning a bit about algebraic topology). I came into the profession exploring convergence and divergence of analytic continued fractions and related material. Pretty much an extension of the efforts during the 1700s and 1800s to solidify those properties of series. Grubby stuff, but I still enjoy grovelling in it. :cool:
I said no such thing. If the freeway didn't exist, I'd take the old dirt road that people used before they built the freeway. I'd still get where I'm going. Set theory is the modern general framework for most math. That doesn't mean math couldn't get along without it. But if you want to do math these days, you have to use the language of set theory simply because everyone else does.
Quoting jgill
Yes ok. But people are into alternatives these days. Category theory and homotopy type theory are two big alternative foundational frameworks. It's all a matter of historical development.
Quoting jgill
Sounds like fun.
I don't opine as to what that other poster has in mind. But:
Rules themselves may be mathematical objects. Languages, axioms, rules, systems, theories, and proofs can be defined and named in set theory. Even informally, when, for example, we say "by the rule of modus ponens", the rule of modus ponens is a thing named by 'the rule of modus ponens'.
The ontological status of rules. If rules are real, then they have some form of existence. Ontology is the study of what there is, and the possibility of different forms of existence. An ontological study of rules will determine if there is such a thing as rules, and if so what type of existence they have.
Consider it is the same sort of issue as the ontological status of numbers, for comparison. We can ask, is there such a thing as rules, just like we can ask is there such a thing as numbers. If we answer yes, there is such a thing as rules, then we may proceed to ask questions like are they objective, and if the answer to this is yes, then we are in Platonism. If we answer no, rule are only subjective, then we are faced with a whole lot of problems as to how such a thing as a rule could actually exist.
For example:
https://ontology.buffalo.edu/smith/articles/Hart_Rawls_Searle.pdf
Not quite the same.
Quoting Metaphysician Undercover
It doesn't work like that for numbers. In any case, one has to ask what kind of rules you are talking about. If any rules at all, the idea that every rule we may come up with is a platonic object is silly, especially when so many rules are absolutely dependant on us being around. If you are talking about rules of logic and mathematics, then wonder why it is only such rules that get a special status. As I said before, if you want to break it down to rules of logic, what is wrong with them being defined as:
Quoting Lionino
? Those are set up by convention.
Quoting TonesInDeepFreeze
I should add that the above does not opine that those things are platonic things. Moreover, there is not a particular sense in which I am saying they are things. Moreover, I'm not opining that saying "things" or "objects" requires anything more than an "operational" sense: we use 'thing' or 'object' in order to talk about mathematics, as those notions are inherent in communication; it would be extraordinarily unwieldy to talk about, say, numbers without speaking, at least, as if they are things of some sort. But, it is not inappropriate to discuss the ways such things as rules are or are not mathematical things of some kind.
My point is that there may be many views as to what mathematical objects are or are not, including realism, fictionalism, nominalism... But that, in that inquiry, not just things like sets, numbers, algebras may be considered, but also rules.
It's not clear to me what you're claiming. Example?
I am asking him if there is anything wrong with taking logical operators, such as & ? ~, to be defined from their truth tables, instead of being mysterious platonic objects that float around in another dimensions.
Like A?B being defined (convention) exactly by what it gives in a truth table according to each value of A and B, and A&B, etc.
Yes, I think that is what Metaphysician Undercover is talking about.
Do you disagree with the point that inference rules may themselves be a mathematical object?
Quoting Lionino
The symbol '->' may be a primitive or defined from primitive symbols.
The truth or falsehood, in a model M, of a sentence of the form 'P -> Q' is determined by the definition of 'S is true in model M'.
Meanwhile, 'P -> Q' is a formula (if 'P' and 'Q' are formulas) or it stands for a set of formulas (if 'P' and 'Q' are meta-variables ranging over formulas). It is not something that is "defined". Rather, it is shown to be a formula from the defintion of 'is a formula'.
Yes, I think this is the issue, why would some rules get special status, and if they do, how could we know which ones deserve that special status. For example if we say some rules are objective, and other rules are subjective, what would distinguish the two?
Quoting Lionino
So it appears like you want to start with the basic premise that rules are fundamentally arbitrary. Why should we agree to some rules and not to others then? Why would we want to start with something like "truth tables" as the primary rule?
It seems to me, that rather than jump right into the process of deciding which rules to accept, and which not to accept, we ought to first determine precisely what a rule is, When we have a complete understanding of what a rule is, then we will be much better prepared for making such a choice, by having some understanding of what the consequences of that choice might be. So rather than start from a truth table, as the basis for which rules to accept, we should start with the definition of a rule, as the basis for which rules to accept.
The two key words you used. Social rules are (inter-)subjective because, as soon as we die, they are not carried out, the "rules" of physics are carried out independently of an observer.
Quoting Metaphysician Undercover
For 2000 years at very least, people thought that the LNC was fundamental. Then came dialethism.
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
Dictionary.
I haven't thought about it deeply, so no. The matter of [s]mathematical[/s]abstract objects naturally goes back to Plato. If numbers and sets and so on are mathematical objects, rules are, in some way, the relationships betwen those numbers. I am not sure and can't imagine how the relationship between universals has been tackled by platonists, if at all, so I can't give a strong judgement on the matter.
Inference rules may be rigorously defined as relations on the power set of the finite set of formulas cross the set of formulas. So, if sets are mathematical objects, then, as rules themselves are sets, rules also are mathematical objects.
Let S be the set of formulas. Let T be the set of finite subsets of S. Let PT be the power set of T. Let x be the Cartesian product. Then:
An inference rule is a subset of PT x S.
Every rule is a set of ordered pairs, such that for each pair
For example, with that definition, the rule of modus ponens is:
{
All the rules of natural deduction can be written in that manner.
And then 'proof' may be defined as a sequence of formulas such that latter entries are conclusions from previous premises per the rules. So, proofs also are mathematical objects.
In general, languages, syntaxes, axiom sets, inference rules, systems, theories, and interpretations are also formalizable as mathematical objects.
I have no comment about the other poster in this context.
But I am glad that I made my quite relevant point that rules also may be regarded as mathematical objects.
It appears like you are confusing descriptive rules with prescriptive rules. This is why we need a good definition of what a rule is. The laws of physics describe the way things behave. Social laws prescribe the way we ought to behave. The latter requires an agent who understands, and intentionally conforms one's activity to follow the law, the former is an inductive conclusion derived from observations of how things behave.
Some philosophers mix the two, so that a social rule is just a descriptive principle of how people behave in general. I think this is done to avoid the fact that people choose to follow rules. But this is problematic, because many people step outside the bounds of social rules, so it would be faulty induction.
Quoting Lionino
Tones is arguing that rules are Platonic objects just like numbers are. If that is the case, then formalism does not escape Platonism, it is a deeper form of Platonism, just like I said.
To pull this structure out of TIDF"s Platonic cesspool, and give it a nominalist foundation, you need to address the problems which I stated above. How do we get beyond arbitrariness? What makes some rules more acceptable than others. This commonly leads to pragmaticism
[quote=https://plato.stanford.edu/entries/truth-pragmatic/]As these references to pragmatic theories (in the plural) would suggest, over the years a number of different approaches have been classified as pragmatic. This points to a degree of ambiguity that has been present since the earliest formulations of the pragmatic theory of truth: for example, the difference between Peirces (1878 [1986: 273]) claim that truth is the opinion which is fated to be ultimately agreed to by all who investigate and James (1907 [1975: 106]) claim that truth is only the expedient in the way of our thinking. Since then the situation has arguably gotten worse, not better. The often-significant differences between various pragmatic theories of truth can make it difficult to determine their shared commitments (if any), while also making it difficult to critique these theories overall. Issues with one version may not apply to other versions, which means that pragmatic theories of truth may well present more of a moving target than do other theories of truth. While few today would equate truth with expediency or utility (as James often seems to do) there remains the question of what the pragmatic theory of truth stands for and how it is related to other theories. Still, pragmatic theories of truth continue to be put forward and defended, often as serious alternatives to more widely accepted theories of truth.[/quote]
No, I am giving examples of subjective rules and objective rules because those are the keywords you used, not the new two keywords. A subjective rule may be descriptive or prescriptive, an objective rule also may be either otherwise prescriptive grammar wouldn't exist.
Quoting Metaphysician Undercover
Application, just like 2000 years ago. During ancient times, mathematics was an empirical endeavor. Many mathematicians of today in fact take pride in their research being useless meaning having no application.
Quoting Metaphysician Undercover
He said they are mathematical objects, not platonic objects.
That's yet another of the crank's lies about me. The crank needs to stop lying about me.
Yes, it was a good explanation.
I explicitly said that I do not claim platonism. And I explicitly said that I am not advocating any particular sense of the notion of object. And I even said that we may discuss ways in which rules are or are not mathematical things of some kind. And I said that even if we don't commit to mathematics as talking about objects, communication about mathematics would be extraordinarily difficult if we did not at least talk as if we are talking about objects.
I wrote it explicitly. Yet the liar crank flat out lies that I said the opposite. The crank has no shame.
More exactly, I said they may be regarded as objects, and that we may discuss in what sense they are or are not objects. But the crank runs all over that like a loose lawn mower. The crank lies at will.
Thank you.
So truth for you is pragmatic then?
Quoting Lionino
I don't see how one could ever distinguish between these two. When an idea is said to be an "object" this is Platonism, by definition. Platonism is the ontology which holds that abstractions are objects.
Nominalists agree that if mathematical objects exist, they would be platonic objects. But nominalists deny that mathematical objects are real, some think they are useful fictions. There are other kinds of realism besides platonic, including psychological and physicalist.
Note: it is platonism with lower case, we are not talking about Plato when the discussion is modern mathematics.
Quoting Metaphysician Undercover
That is a deep topic in itself and, though related, distinct from the metaphysics of mathematics.
That's not a credible definition of 'platonism'.
https://plato.stanford.edu/entries/platonism-mathematics/ :
"Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. [emphasis added]"
"Mathematical platonism can be defined as the conjunction of the following three theses:
Existence.
There are mathematical objects.
Abstractness.
Mathematical objects are abstract.
[b]Independence.
Mathematical objects are independent of intelligent agents and their language, thought, and practices[/b] [emphasis added]"
https://plato.stanford.edu/entries/platonism/ :
"Platonism is the view that there exist such things as abstract objects where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental [emphasis added]."
The crank doesn't know jack about logic, mathematics or philosophy of mathematics. But still he's good for serially shooting his ignorant mouth off about them.
OK, I was unaware of that convention. We are definitely not talking about Plato, but modern day platonism (my spell check does not like the lower case).
Quoting Lionino
If the so-called mathematical objects are fictions then they are not really objects, but fictions. Therefore we cannot correctly refer to the tools of mathematics as "objects". If we're nominalist we'd say that anyone who speaks about mathematical objects is speaking fiction; fiction meaning untruth, because abstractions are simply not objects for the nominalist. And to call them objects would be false by the principles of that ontology.
Quoting Lionino
The point though, is that if we reject platonism, then we need some other ontology to support the reality of rules. We cannot defer to "intersubjectivity", because that makes a rule something between subjects, therefore outside the subject, and this external existence of rules is just a disguised platonism. Therefore we need to properly locate "rules" as properties of subjects, internal to thinking minds. I have my rules and you have your rules. Then the reality of agreement, convention, needs to be accounted for, and pragmaticism is designed for this purpose.
'Sam Spade is a detective'.
'The Maltese falcon is a statuette'.
'Sam Spade' and 'The Maltese falcon' are nouns. We regard a noun as referring to something, whether it ('it' is a pronoun that refers to something) is a material piece, a collection of material pieces, such things as a sports team, an experience, event, series of events, physical experience, mental experience, mental state, idea, concept, number, set, vector space, word, sentence, literary piece, story, narrative, etc. - now existing, once existing, predicted to exist in the future, conceived to possibly exist now, conceived to possibly to have existed in the past, or possibly conceived to exist in the future, or hypothetically now existing, once existing, Or, fictional instances ('instance' refers to something) of any of those ('those' is a pronoun to refer to some things) hypothetically.
But to accommodate a position that fictional things are not really things, the fictionalist could say, "When I say 'thing' or 'object' regarding mathematics, 'fictional' is tacit. I may be taken as saying 'fictional thing' or 'fictional object'".
The Maltese falcon is fictional, but I can still refer to the Maltese falcon without having to say each time "the fictional Maltese falcon". And it makes perfect sense for someone to ask, "What kind of thing is the Maltese falcon?" If the crank says I must answer, "It is a fictional thing. It is fictional statuette" then okay. But in context, it might be understood that we're talking about fictional things.
The number 1 is not a concrete object, but I can still refer to the number 1 without having to say each time "the fictional number 1". And it makes perfect sense for someone to ask, "What kind of thing is the number 1?" If the crank says I must answer, "It is a fictional thing. It is a number, which is a fictional thing" then okay. But, as mentioned, the fictionalist may take it as understood that we're talking about fictional things.
For anyone who doesn't understand what the number 1 is, it makes perfect sense to ask: "What kind of thing is the number 1. The answer, for a platonist is: "It is a mathematical object". The answer, for a nominalist is: "It's not a thing".
Pick your choice. Or, you can maintain your position as someone who doesn't understand what the number 1 is, and keep asking "What kind of thing is the number 1?"
To say, "the number 1 is a fictional thing," provides nothing by way of explanation, because we are left with essentially the same question; "What kind of thing is a fictional thing?". That's just an evasion; an indecisive move of avoidance practised by a slippery sophist. The sophist then slides off and starts talking about the number 1 as if it is a platonic object, hypocritically insisting that it is not.
Quoting Metaphysician Undercover
The fictionalist position is that mathematical truths are fictional (that was already explained further by me above), and, since they are nominalists, that there is no such thing as abstract objects. That is the position, engaging in a debate about "object this object that" is pointless as you will choose to tailor the meaning of "object" to suit your ends.
Quoting Metaphysician Undercover
Ok. Psychologism, I choose you!
Quoting Metaphysician Undercover
Ok, am I supposed to disagree? Formalism is still not disguised platonism, much less nominalism.
I think we've gotten beyond this talk of "objects". We've moved on to "rules", because rules are what formalism takes for granted. Platonism takes mathematical objects for granted, formalism takes rules for granted. These seem very similar to me, though you seem to have a desire to drive a wedge between platonism and formalism. That is why I said we need to look into the ontology of rules. If rules are supposed to exist in the same way that platonic objects are supposed to exist, then there is no real difference between the two.
Your reference to fictionalism inclines me to think that you want to portray rules as fictions. But this does not suit the nature of rules. Rules are guidelines for behaviour, principles of what one ought to do. Rules fall into a completely different category from fact and fiction and cannot be classed as either.
Quoting Lionino
We still haven't determined that yet, because we haven't determined how formalists account for the existence of rules. I think formalism generally takes rules for granted. Then the rules simply "are", just like platonic objects simply "are", and formalism is a form of platonism.
It has already been discussed that rules may be considered objects.
Quoting Metaphysician Undercover
"Taking something for granted" means nothing in this context.
Quoting Metaphysician Undercover
That is just projection from you. I am correctly stating these two are distinct views, one may be a formalist without being a platonist and vice-versa.
Quoting Metaphysician Undercover
That is just going back to square one but talking about rules instead of numbers, see my first line in this post.
Quoting Metaphysician Undercover
Idc.
Quoting Metaphysician Undercover
If you had a single reference to support this nonsensical claim, you would have given it already. You talk about "rules" without specifying what kind of rule you are talking about. If it is a mathematical rule, see the first line in this post and then this post.
Huston Lover: I saw 'The Maltese Falcon' again last night. What a great film.
Muddlefizzle Undergarment: Never heard of it.
Huston Lover: It's terrific. Humphrey Bogart plays this detective Sam Spade whose partner was murdered.
Muddlefizzle Undergarment: What do you mean "he plays"?
Huston Lover: What do you mean?
Muddlefizzle Undergarment: You said, "he plays". What is that?
Huston Lover: Humphrey Bogart was an actor. He plays the character Sam Spade.
Muddlefizzle Undergarment: A character? What's that?
Huston Lover: A character. A fictional person.
Muddlefizzle Undergarment: Fictional people don't exist.
Huston Lover: Right. They don't exist like you and I. But they exist in the stories. Like the statuette, the Maltese Falcon in the story. It doesn't exist like the Stanley Cup exists, but it is a fictional object.
Muddlefizzle Undergarment: So you're a Platonist!
Huston Lover: I don't even know what that is. I was just trying to tell you about the movie.
Muddlefizzle Undergarment: As I've told the Platonists at The Philosophy Forum, there are no fictional objects; there are only actual objects. It is nonsense to talk about fictional objects. Like it is nonsense when mathematicians try to make you believe that numbers and sets are objects. That breaks the law of identity! So please don't try to make me believe that there are fictional people and fictional objects.
Huston Lover: Then how can I tell you who Sam Spade and Brigid O'Shaughnessy and Kasper Gutman are? I mean, they're not living people or dead people, so what else can I call them except 'fictional people'? What can I call the Maltese Falcon if not 'a fictional object'?
Muddlefizzle Undergarment: You may call them 'fictions'. Otherwise, I'll post that you're a Platonist.
Huston Lover: Would that be bad?
Muddlefizzle Undergarment: Very bad. Because then I can say that you're a slippery sophist dragging us all into the Platonic sewer. But good for me, because then I'd have the satisfaction of exposing you as a slippery sophist sewer bound Platonist.
Huston Lover: Okay, but if Sam Spade and the Maltese Falcon can only be called 'fictions' and not 'a fictional person' and 'a fictional object', then how do I refer to them so that you know what they are?
Muddlefizzle Undergarment: Okay, you can call him 'a fictional detective' and you can call it a 'fictional statuette'. As long as you never say they are a fictional person and fictional object. Because, just like at The Philosophy Forum, I've explained that there are no fictional objects or people.
Huston Lover: Thanks for that, Muddlefizzle. But isn't a detective a person, and a statuette an object? So a fictional detective is a fictional person and a fictional statuette is a fictional object.
Muddlefizzle Undergarment: No! Not unless you want to be dirty slippery sophist sewer seeking Platonist rat!
Huston Lover: Okay, Muddlefizzle. But you should check out the movie.
Muddlefizzle Undergarment: I will not be doing that. I only watch documentaries. They're about real things.
...
I am correctly stating these two are distinct views, one may be a formalist without being a platonist and vice-versa.[/quote]
The question then, is how do you think that rules can be taken as objects, in a way which is not platonist? I already explained how the concept of "intersubjective objects" is inherently platonist.
Quoting Lionino
I am talking about prescriptive rules, as I said, ones which people follow when they are doing what they should do, such as when they are doing mathematics. All prescriptive rules can be classed together as the same type, and this distinguishes them from descriptive rules which are commonly understood as inductive principles. Any further division of type is not necessary at this time.
Quoting Lionino
I don't deny that rules are subject to the platonist/nominalist debate. In fact that's what I am claiming, and that's what we are debating. I am saying that if you state that rules are understood as objects (whether or not those objects are claimed to be fictional is irrelevant), then you have taken a platonist stance in this debate. Further, I am claiming that all common formalist positions on this matter are platonist as well. I am not saying that it is impossible to produce a nominalist ontology of rules, I am saying that formlism relies on platonist ontology. And I've supported my claim with explanations. I am waiting for you to produce evidence which is contrary to what I am claiming, but you do not seem capable of finding any.
And I have no idea what Tones has been talking about.
Quoting TonesInDeepFreeze
Our language is very much platonist. And this is the reason why Plato's writings are still available to us after more than two thousand years. He has had great influence over our culture, and those human beings in authoritative positions have had great respect for platonism, so platonism is extensive through our institutions. Therefore our culture and language have extensive platonist features. If you had education in the history of philosophy, you'd know this. Furthermore, the Catholic church was very much platonist influenced, and the Church, for an extended period of time, controlled the use of language through the use of force. This is known as The Inquisition.
I do like when occasionally the crank speaks truthfully.
And characteristic of him to not know what is being talked about even when what is being talked about is what he's been talking about.
For starters, the crank throws around the word 'platonism' but he doesn't even know what it is.
Then the crank says, "Our language is very much platonist."
Right, it was Plato, Platonists, and platonists who installed the words 'it', 'thing' and 'object' and the need for them.
And notice no response to:
Quoting TonesInDeepFreeze
or
Quoting TonesInDeepFreeze
Keeping my promise. In any case, at the very least, the introductory SEP articles on platonism and nominalism should read. Clearly no one here has done that.
What are you saying, that it's just by random chance that the words "platonism" and "Platonism" are extremely similar? Are you saying that it's wrong to assume some sort of relationship between the meanings of these two words, even though they don't both mean the exact same thing?
https://thephilosophyforum.com/discussion/comment/920596