The Largest Number We Will Ever Need
True, numbers are infinite. The simplest, the bedrock as it were, of infinities is the natural numbers {1, 2, 3,...}
Now consider the fact that in a universe that's finite there's gotta be a number that is the upper limit of a counting processes that yields the largest number possible/required to describe this universe - this number will probably be the permutation of elementary particles of that universe.
For example in a universe with 2 elementary particles (X, Y) with the constraint that the permutations allowed involve only twos/pairs or less, the largest number needed to describe such a universe would be 4 (X, Y, XY, YX).
We can call such a number N[sub]max[/sub].
Question: What's the N[sub]max[/sub] for our universe?
Now consider the fact that in a universe that's finite there's gotta be a number that is the upper limit of a counting processes that yields the largest number possible/required to describe this universe - this number will probably be the permutation of elementary particles of that universe.
For example in a universe with 2 elementary particles (X, Y) with the constraint that the permutations allowed involve only twos/pairs or less, the largest number needed to describe such a universe would be 4 (X, Y, XY, YX).
We can call such a number N[sub]max[/sub].
Question: What's the N[sub]max[/sub] for our universe?
Comments (452)
Your supposition may not be valid. But entertaining idea.
Yeah.
Nevertheless, I wanna know, how shall I put it?, the largest finite number that would be required for science. Clearly we don't know how to do math with infinity and hence our inability to wrap our heads around phenomena such as black holes (re Michio Kaku). Ergo, we need to keep numbers finite in our science equations. For instance, in the equations for black holes, why not substitute a very large number (N[sub]max[/sub]) for infinity wherever, whenever it appears and then see what happens?
Good call!
If we substitute [math]\infty[/math] with 1000, we get 0.001 which to an accuracy to the hundredths is 0.00 or 0.
We go from undefined to defined (with caveats). Isn't that better than just staring blankly at a page with [math]\infty[/math] in an equation?
I was proposing a test-drive, you know, just to find out what happens. In all likelihood there's a number that would make us go "Yeah, this is it! It all makes sense now!" and that number is probably going to be unimaginably large but finite.
Note: We can't do (normal) math with [math]\infty[/math]. That's what I'm looking to find a workaround for .[math]\infty[/math], as per some well-known scientists, pops up in physics equations more often than we would've liked.
In other words, it is possible to count infinitely in an otherwise finite universe just in case we stipulate sortals whose extension is cardinally infinite.
With these suppositions in mind, it's important to note exactly the kind of environments that can possibly satisfy this finite N? project:
To answer your initial question directly:
Quoting Agent Smith
Unfortunately, our actual universe does not satisfy these conditions. But on the plus side, there are infinitely many possible universes which satisfy arbitrarily many N? values.
What about the observable universe? I read somewhere that the observable universe contains roughly 10[sup]80[/sup] atoms. That should be a good place to start at least when it comes to matter, oui?
By the way thanks for the detailed analysis of my query, much obliged!
Alright. Some extra suppositions, quarks are the simplest form of matter and are indivisible in principle. This is not currently known, of course, we have no idea if we'll discover something more basic in the future or if they really are fundamental. But for hypothetical's sake, we'll say so be it.
There are 22 quarks in an atom. So from simple multiplication N is 2.2e81 iff the only objects that exist are these quarks. Let's call this for now N? (and conceptually separate it from N?).
But what about things that the quarks make up, like apples? Should we count some trillion quarks that make up the apple, and also count the apple? If we can overcount like this, N? will be massively larger than N?. It can also be arbitrarily modified to be as large as we want in terms of approaching infinity (although it will never itself be infinite unless we deliberately define infinite objects).
This is because "apple" is not a joint-carving concept recognized by the universe but something imposed by us in our language; only the quarks that make the apple are recognized by the universe. Thankfully, one of our previous criteria already makes this qualification, so we can't "cheat" this number up using our massive array of sortals we can construct. That said, this notion of structure to the universe is itself controversial in the domains of metaphysics and the philosophy of science, but there simply is no N? if no concepts are joint-carving and purely reflective of our language because we defined N? in that structural way earlier (although this might not be true to your understanding of it in a literal sense, it's true to it in a spiritual sense)
There is also another thesis in mereology, "composition as identity," that holds that some whole and exactly the parts that make it up are identical. An apple and its quarks are identical, so it makes no sense to see "apples" as a problem because we won't be overcounting all the parts that make it up and the apple, as they're precisely the same thing (per this thesis). However, this itself is as controversial as some may find it intuitive (the controversy is to no surprise: this topic in general has very little consensus). One of the simple arguments against this view is that a whole is one and its parts are many, so this is at least one property true of the whole that's not true of the parts, so they can't be quite identical.
Well, the quarks also constitute everything else inasmuch as they compose everything else, so similarly as there is "composition is identity" there is the similar thesis of "constitution is identity." This runs into similar issues: consider a lump of clay and the statue it constitutes. We can intuitively say they are one thing, so this thesis has some intuitive force. But if somebody came to deform the statue completely, we will be no longer compelled to think the statue survived (since it is defined by its particular shape): however, we're fully inclined to think the lump of clay had survived, even if it now instantiates a distinct form. So this is another basic argument against the mirror thesis to say that they can't be quite identical. To connect this back to the apple and the quarks: the quarks will survive if someone ate the apple, even if they may now instantiate a different form altogether. But we'd no longer say the apple exists, perhaps some human waste and other chemicals now exist, so clearly the things that the quarks constitute and compose are not quite identical to them.
This motivates the idea that including our natural language concepts, referred to by our ordinary nouns and proper names, forces us to overcount. We can restrict the overcounting to be finite, but since we can make our language whatever we want it to be, we can still count to any arbitrary finite number, giving us no clear answer. While we can simply discard those, per our earlier criterion, it goes back to something else you said:
Quoting Agent Smith
When it comes to us describing the universe, we usually don't care about joint-carving the structural properties of the universe itself whatsoever. When we ask "how many apples are on the table?", quite literally no one means "how many independent apple-forming quarks are on the table-forming quarks?", returning some massive number. We just mean our own language, instrumental to our utility, regardless of whether it reflects the universe's structural properties.
However, we still think we're meaningfully describing the universe: at least in the way it appears to us.
In omission of our ordinary concepts, we remove the "cheating" problem of overcounting to any finite number, but this also removes otherwise true descriptions of reality that exceed N? (it's not exaggerating to say, per the post's title, that we need these descriptions, sometimes for our own survival). But, when we include those ordinary language descriptions, N? itself loses its intended meaning and becomes whatever we suppose it to be per the intensions of our non joint-carving concepts. Either we can't describe everything we need to describe (in a practical way) or there is no principled N?, its value is only a function of our choice! We're at a dilemma here :(
Quoting Agent Smith
Cheers,
This was a fun write!
That said, I was hoping to find a number such that
1. No calculation ever would exceed that number
2. if in an equation with [math]\infty[/math] in it, replacing [math]\infty[/math] with the number (N[sub]max[/sub]) allows us to, at the very least, approximate the state of affairs, mathematically.
Physicists tend to throw their hands up in the air with disgust mixed with utter frustration when they see [math]\infty[/math] when number crunching. My post is an attempt to, as you rightly pointed out, cheat the system if possible.
Danke again for your helpful analysis although I must admit some of it is above my pay grade.
On what principles would you decide how to count all the dark matter?
Like I said, most of this is really speculative. A candidate particle is the d-star hexaquark d*(2380) which is hypothesized to account for the universe's dark matter. This particle is composed of six quarks.
Then again, there's also the debate about ontic structural realism and other types of scientific realism versus instrumentalism: whether our best physical categories really carve the universe or whether they're just useful to us. I'm obviously assuming the former is true so that this question is in principle answerable.
Quoting Agent Smith
Why? Are you planning on investing in real estate?
Why is it I think you are not serious? :smile:
Sure, just fix whatever large number you wish and round off to that number. But you might make a mess of computations that follow. In physics renormalizations work in various settings.
Wiki
:blush: I'm serious but looks like my idea is ridiculous.
Quoting Cuthbert
Muchas gracia señor! You took the sting out of jgill's remark.
Quoting jgill
Well, I was hoping that a seasoned mathematician like yourself could bring some precision to N[sub]max[/sub]. For the last coupla years I've been thinking of very large numbers and the way I do it is pick a number at random and raise it to a power that's large and also random e.g. [math]23^{2017^{3138934}}[/math]. Much to my amazement I found a youtube video on the topic of thinking of a number which no human has ever thought of on the channel Numberphile. It seems that if one is systematic and methodical one can do better than just guesswork.
There's an infinite number of numbers which no human has ever thought of. What's the point in trying to name a random one of these? Here's one for you though, which might be worthwhile. Try naming pi to its final decimal place. That's a meaningful number which no one has ever thought of.
[quote=Ms. Marple]Most interesting.[/quote]
'Describe the universe' is a notion not defined by you.
Quoting Agent Smith
Clearly that is false. Infinite sets are basic for calculus.
Quoting Agent Smith
Let that number be M. Then M+1. Poof.
Quoting Agent Smith
Example please.
Quoting Metaphysician Undercover
There is no last decimal place of pi.
Good points, worth pondering upon.
Let's back up a little for my sake.
Is there a finite number (N[sub]max[/sub]) such that no calculations ever in physics will exceed that number?
From special relativity, the Lorentz factor is unbounded as v approaches c.
[math display="block"]\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}[/math]
[quote=Ms. Marple]Most interesting.[/quote]
Infinity is then some kinda test for impossibility, at least in this case. We need infinity to demarcate the boundary of what's doable and what's not.
In a sense then [math]\infty[/math] in science has a job description similar to contradictions - separates the possible from the impossible.
Is infinity a contradiction? It does lead to some rather odd conclusions: a part is equal to the whole and all that. No wonder many mathematicians (recall Kronecker's vitriol against Cantor) were dead against it.
No, set theory does not say that there is a proper subset of a set such that the proper subset is the set. Set theory does say that there are sets such that there is a 1-1 correspondence between a proper subset of the set and the set.
This is another example of you running your mouth off on this technical subject of which you know nothing because you would rather just make stuff up about it rather than reading a textbook to properly understand it.
I know so little about physics or cosmology that I can't answer that.
:lol:
Quoting TonesInDeepFreeze
Too bad! I had high hopes, expected more (from you)!
You should not be disappointed. You should be encouraged. I gave you an outstanding example: when one does not have sufficient knowledge then one should defer from making wild claims.
The problem isn't me, mon ami - if a particular topic is conducive to wild claims then something's wrong with the topic. I have no truck with people who make claims that are far out.
You regularly and willfully spread ignorant disinformation about mathematics throughout many threads, over the course of at least several months, even repeating yourself after you've been given detailed explanations why you are incorrect. It's disgusting.
Gives TonesInDeepFreeze 8 mg of Zofran.
From your personal stash. Side effects include confusion and disorientation. Explains a lot.
Perhaps! :snicker:
:chin:
Quoting Agent Smith
TIDF can be very entertaining that way, and also quite enlightening. But eventually it gets boring when TDIF refuses to divulge the secrets of the mathemagician's smoke and mirrors.
Quoting Agent Smith
This is actually a very good point. Every time infinity is employed in the application of mathematics, it's like employing a contradiction. This becomes very clear in an analysis of the common mathematician's claim to have resolved Zeno's paradoxes.
Which means [math]\infty[/math] is impossible, squaring with Aristotle's decision to make the distinction potential vs. actual (infinity).
If all the energy of the universe were somehow marshalled to accelerate a single particle, what would its Lorentz factor be? I have no clue, but I suspect it wouldn't come near 23^2017^3138934, let alone a real monster like 10^10^10^10
What is the size of the set of possible states of the universe? I suspect this is the true nmax. Would this be a number best expressed as x^y^z?
There is no magic. Very much to the contrary. At a bare minimum, it is algorithmically verifiable whether a given formal expression is well formed and then whether a given sequence of formulas is a formal proof. That is a courtesy given by formal logic that is not hinted at in various handwavings and posturings by cranks as often found in a forum such as this. And I have given extensive explanation of many of the formulations I have mentioned.
Quoting Metaphysician Undercover
Yet no one who says things like the above has ever demonstrated that Zermelo set theoretic infinitistic mathematics implies a contradiction.
Quoting Metaphysician Undercover
Zeno's paradox is not a formal mathematical problem. Saying that calculus provides a problem solving tool in which paradox does not occur does not imply a contradiction.
Yet, again, we remind that a contradiction is a statement of the form "P and not-P". No such statement has ever been derived from Zermelo set theoretic infinitistic mathematics, no matter that, perpetually, cranks groundlessly and ignorantly claim otherwise.
.
The statement that there exists sets that are infinite is not a logical impossibility.
The idea of allowing 'there exist potentially infinite sets' but not 'there exits infinite sets' is fine as a motivation for an alternative mathematics. But, if one cares about mathematics being formal (in the sense that it is utterly objective by algorithmic checking whether a sequence of formulas is indeed a proof from axioms), then the notion of 'potential infinity' requires formal definition and axioms to generate the desired theorems about it.
Saying, "I don't like the notion of infinity so I'll use potential infinity instead" but without even hinting at how that alternative would be formulated is no better than saying "I don't like that human life requires breathing oxygen so we should breathe hydrogen instead" but not giving a hint as to what technology would allow hydrogen to do the job of oxygen.
Whether an expression is well formed or not is irrelevant to whether it is self-contradictory, because to determine contradiction we must analyze the meaning, and this is the content, not the form.
Quoting TonesInDeepFreeze
Have you forgotten the conversations we've had earlier? The empty set for instance, involves contradiction.
Quoting TonesInDeepFreeze
Oh, now I remember, you have a very odd notion of what constitutes contradiction, and this is how you insist that there is no contradiction even after contradiction is demonstrated to you. If the statement doesn't explicitly say "P and not-P", then there is no contradiction in your interpretation, regardless of what the statement means.
The law of noncontradiction states that the same object cannot both have and not have, the same property, at the same time, in the same respect. So consider the empty set for example. For simplicity, let's say that a set is a collection of objects. Therefore a set necessarily has objects. The empty set has no objects. Therefore "empty set is self-contradicting. The empty set is said to have objects (necessary to being a "set", or collection of objects) and also to not have objects (necessary to being empty), at the same time, and in the same respect
You are entirely ignorant of what contradiction is in mathematics.
Moreover, even if contradiction were, in some sense, couched semantically, then no contradiction, even in some sense of a semantic evaluation, has been shown from ZFC.
Moreover, if an expression is not grammatical, then it does not admit of semantic evaluation.
Quoting Metaphysician Undercover
"Involves contradiction" has not been given meaning by you. Either the theorem that there exists an empty set implies a contradiction or it does not. No contradiction has been shown to be derived from the theorem that there exists an empty set.
If you mean that the notion of 'set' is not compatible with a set being empty, then that just entails that your conception differs from a different conception in which there is an empty set.
Moreover, the phrase "the empty set" is not in the formal theory. Rather, there is a theorem:
E!xAy ~yex
and definition:
x=0 <-> Ay ~yex.
Moreover, even if you persisted to object to mathematicians using the informal locution 'the empty set', then mathematicians could say, "Okay, we won't say 'empty set' anymore. Instead we talk about sets and one particular object, whether it is a set or not, such that that object is the only urelement (an object that has no members), and then all these things - the sets and the urelement - are called 'zets'. So there is an empty zet." That would not alter the mathematics of set theory one bit, especially formally, and even informally except that the mouth pronounces a 'z' instead of an 's' for that one word.
So you are terribly ignorant and self-misguided in every aspect of this matter.
Quoting Metaphysician Undercover
No, I have the standard logical and mathematical notion.
Quoting Metaphysician Undercover
That is one informal formulation. It is equivalent though to the standard formulation. That is:
~Ex(Fx & ~Fx)
is equivalent to
~(P & ~P).
Quoting Metaphysician Undercover
First, 'set' is not a primitive of set theory. An actual definition can be:
x is a class <-> (x=0 or Ey yex)
x is a proper class <-> (x is a class & ~Ey xey)
x is a urelement <-> (~x=0 & ~Ey yex)
x is a set <-> (x is a class & ~x is a proper class)
Second, even informally, you mention a certain definition of 'set'. Mathematicians are not then obliged to refrain from having an understanding in which "collection of objects" does not preclude that it is an empty collection of objects, notwithstanding that that seems odd to people who have not studied mathematics, and so more explicitly we say, "a set is a collection, possibly empty, of objects". You are merely arrogating by fiat that your own notion and definition must the only one used by anyone else lest people with other notions and definitions are wrong. That is an intellectual error: not recognizing that definitions are provisional upon agreement of the discussants and that one is allowed to use different definitions in different contexts among different discussants. It's like someone saying "a baseball is only one such that is used in major league baseball" and not granting that someone in a different context may say, "By 'baseball' I include also balls such as used in softball". It is intellectually obnoxious not to allow that. And it is one in the deck of calling cards of cranks.
OK, Mathematics is allowed its own special definition of "contradiction", so that statements which would qualify as contradictory in a rational field of discipline do not qualify as contradiction in mathematics.
Quoting TonesInDeepFreeze
When you explain to me how a set which has no objects also has a collection of objects, and this is not contradictory, then I'll start to listen to you.
So please explain to me, your understanding of "collection of objects" in which there is no objects. As far as I can tell, either you have a collection of objects, or you have no objects, but to have both is clearly contradictory. What if you had one object? It is neither a collection of objects nor is it no objects. Do you agree? Or do you just abandon rationality for the sake of mathematics?
It is a formal definition. But it still captures the ordinary sense of "To claim a contradiction is to claim a statement and its negation." For example, in ordinary conversation we may say, "'Mike is a car mechanic and Mike is not a car mechanic' is a contradiction."
Quoting Metaphysician Undercover
You switched form "is a collection of objects" to "has a collection of objects".
I said nothing about "has a collection of objects". Rather, I said
Quoting TonesInDeepFreeze
Quoting Metaphysician Undercover
An object that has no members is either the empty set or an urelement. And of course, an object that has in it only one object is a non-empty set.
So my explanation stands and all the rest of my remarks demolishing your ignorant and self-misleading remarks stand.
Empty generalization and bluster.
I quite understand that human thinking, including about mathematics, involves intuition. Indeed I'm interested in the relation between formal theories and intuitions. And I know vastly more about the school of intuitionism compared with your lack of knowledge about it.
Henri Poincaré!
I dunno! That's what I'm trying to find out.
Well that truly settles the question!
:ok:
And your emoticon doubly seals it! Who could ever defeat an emoticon?
You must free yourself (of mathematics).
:ok: Please edify me then. How does intuition work in math? How is it related to so-called mathematical/logical rigor? Talking to you is like conversing with a computer. DOES NOT COMPUTE! DOES NOT COMPUTE! From start to finish, that's all you say! I should call tech support! :snicker:
Quoting Agent Smith
Calculations in physics. The Lorentz factor is unbounded.
I'll havta take your word for it.
Suppose there was an upper bound to the Lorentz factor, [math]\gamma < M[/math].
Then the free variable v would also be bounded below c, which, in theory and thus computation, it is not:
[math]0< v < c \sqrt{1-\frac{1}{{{M}^{2}}}} < c [/math]
Quoting Agent Smith
As it does in everyday life. But that is not the meaning of intuitionism in the philosophy of math.
An "object" is one. To allow that an object has members requires a special definition. We can say that this object is a set, and define a "set" as a collection of objects. But to say that there is a collection of objects with no objects is contradictory, whether you admit to this fact or not.
It seems that you do not recognize the fundamental distinction between one and many, and you are now trying to reduce many to one, by saying that a set is an object. One is not a plurality, and a plurality is not one. That is a basic self-evident truth. If you want to talk about a category, or class of objects, this is something completely different. We cannot name the category "an object", and also name the members "objects", without equivocation. Do you apprehend the category mistake, and consequent fallacy of equivocation which occurs if we call both the category, and the members within that category, by the name "object"?
Quoting Agent Smith
I thought for a while, that Tones was a bot, endlessly repeating the same thing over and over without grasping the logical problems with what was repeated. Then I realized that this is simply the way mathematics is. The students are taught very specific principles, and their minds are funneled down a very narrow path, You could say that they are programmed, like a computer is programmed, and the possibility that the program is not a very good program, so that they are being misled, is excluded from the program. The students are discouraged from looking outside the program, and seeking the truth, like philosophers do, because truth is not important to mathematics.
It hasta make sense and (rigorous) logic is all about that - hats off to @TonesInDeepFreeze - making sense (of the cosmos).
I seek more and more, with the idea that more will eventually result in completely full. More knowledge, more experience, more understanding.
But more doesn't lead to 100% full, or so I've been told
The more that is had, the bigger becomes the comparison of measurement
So that even if had the whole earth, I would end up comparing my little earth to the universe and feel quite poor.
Instead, aescetics have told me through writings, that I have to become 100% empty. Only a 100% empty mind has room for infinity.
As motivational coach Tyler Durden of 'Fight Club' put it, "It's only when we lose everything that we are free to do anything."
And:
"When the doors of perception are cleansed, we see things as they really are, infinite" - One of those classic poets, William Blake, I think. Also what the band, The Doors, based their band name on, or so I've heard.
Reality ultimately can't be limited, because limitation requires two. One thing limiting another.
I also reminded of an interpretation of the symbolism of the tower of Babel.
They tried to build a tower tall enough to reach God.
Is that we are are doing with the intellect? Trying to attain Infinity through building conceptual realities?
Fun to think about. Infinity literally is the one thing that can't ever become boring, since things only become boring once we understand everything about them!
The universe is physical. Numbers are not. They are abstract objects. Whether the universe is finite or infinite has nothing to do with numbers being finite or infinite. Their nature is totally different and their existence is of a different kind, too. The only relation between numbers and universe that I can see is the following: numbers are representations of things in the universe, considered individually or in groups, categories etc. When I say "5 (five) apples" I refer to "5 (five) individual objects" or "a group of 5 (five) objects". Both the numeral "5" and the word "five" are symbols that represent a "quantity", which is also another abstract word, a concept.
So, if the universe is finite, it means that it contains a finite number of elements, such as atoms (microcosm), stars (macrocosm), etc. If it is infinite, then it means that it also contains an infinite number of elements.
Now, we also have irrational numbers, like PI, which have a sequence of decimals that looks infinite. But this is theoretical and has nothing to do with whether the universe is finite or infinite.
Finally --but not the last-- we talk about the magnitude of physical objects, which is also an abstract idea. For example Plank's constants refer to the smallest and largest number calculable magnitudes, which range from 5.4 x 10^-44 ("Planck time in seconds, the shortest meaningful interval of time in seconds, and the earliest time the known universe can be measured from" and 5.1 x 10^96 (Planck density, the density in kg/m^3 of the universe at one unit of Planck time after the Big Bang). (Re: https://www.physicsoftheuniverse.com/numbers.html)
But I don't think that these can be used as a proof that the universe is finite. Well, my knowledge of Physics is quite limited to be able to judge.
Nice angle, the psychologist's view on [math]\infty[/math]. Speaking for myself, I find the expression "I love you infinity" very thought-provoking and awe-inspirng. Also in the same category is the answer "11" to the question "how's the pain on a scale of 1 to 10?"
@Alkis Piskas
:ok: It's just that I wanted to know if scientific calculations invovling infinities could be tamed in a manner of speaking.
Renormalizations :roll:
Whazzat?
:scream:
Renormalizations & Regularizations
https://en.wikipedia.org/wiki/Graham%27s_number
Don't go messing' with mathematicians with your "c'mon, be reasonable" attitude; they'll have none of it.
Graham's number appears in a mathematical proof but what about calculations that use actual measurements such as mass, velocity, charge and so on? Infinity appears in black hole calculations and once that happens, the results evoke the response nec caput nec pedes. I only wanted to explore the possibility of some kinda workaround to this rather lamentable state of affairs in the math of physics.
Initially I was of the opinion that infinity had to be replaced by an extreme number like Graham's number but the result [math]\sum_{n=1}^{\infty} = -\frac{1}{12}[/math] (used in string theory) is evidence the number that we swap infinity with in a calculation, surprise, surprise, doesn't have to be large, a wannabe infinity; a small, nevertheless most special number like [math]-\frac{1}{12}[/math] will do just fine.
I guess that's why TIDF split the scene. But we were just playing, it's nothing harmful.
Quoting Agent Smith
Fractions are actually very tricky. The common approach is to assume that any object can be divided in any way, so there is an infinity of possible divisions for each thing to be divided. In reality though, the way an object can be divided is highly dependent on the composition of the object. So producing ratios in theory, and applying them without proper standards as to the true way that things can be divided, can be very misleading.
This becomes quite evident in wave theory, as the ancient Pythagoreans who studied harmonies and the properties of musical tones found out. There is a very real problem with the assumption that time (hence frequency) can be divided arbitrarily (in any way that one wants). This produces the uncertainty principle of the Fourier transform.
That last part... holy shit!
:strong: :wink:
The equation is unbounded. But is it unbounded in this universe? We cannot accelerate an object arbitrarily near to c, since that requires arbitrarily high amounts of energy, which is not available.
Quoting Agent Smith
False actually. I cannot think of a single number that is, let alone all of them.
That aside ...
Quoting Agent SmithIf that number causes it to all make sense, it probably isn't unimaginably large.
Quoting Agent Smith
Perhaps a reasonable place to start when talking about large useful numbers.
Now consider all the possible ways to arrange all those atoms in a given visible universe volume. This gives all the possible states of a visible universe, which is quite useful for positing such navel-gazing concepts as "How far is it to the nearest exact copy of me?". That's a really huge but still useful number, but it starts with the number you gave above.
Tegmark did such a calculation and got a number something like 10^10^28 meters away (quick google search) which is smaller than your random useless number above, but still much bigger than the 10^80 one.
Tegmark doesn't read his own book. A better answer is somewhat closer by if the whole book is taken into consideration.
PS: I couldn't figure out how to do the superscripts like you have in your posts. Maybe it's buried in the terse menu above, but I can't find it.
What I think, is that we allow "infinite" so that we will always be able to measure anything. If our numbers were limited to the biggest thing we've come across as of yet, or largest quantity we've come across, then if we came a cross a bigger one we would not be able to measure it. So we always allow that our numbers can go higher, to ensure that we will always be able to measure anything that we ever come across. In that way, "infinite" is a very practical principle.
I wonder what Max Tegmark would've said about my query.
I'm neither a mathematician nor a physicist so you'll have to cut me some slack here. Imsgine you're doing a calculation on black holes and you end with [math]\infty[/math] in your result.
Last I checked there are many infinities, each larger than the other. the "smallest" of them being the infinity of natural numbers viz. {1, 2, 3,...} aka [math]\aleph_0[/math]. Are all the infinities that appear in physics calculations [math]\aleph_0[/math]?
Infinities can appear at singularities. This is the notion of infinity as unboundedness and/or non-infinite but bizarre behavior at such points. What you are talking about are ways of categorizing infinite sets in set theory. The terms of the series S=1+2+3+... form a set having cardinality aleph naught, but the series is unbounded. If Tones tunes in he could explain the details. In my research I never needed anything more.
Why don't you send Tegmark a question about you OP? Then report back. :cool:
Yea, that much I know; fell in love with that word since I was, what?, 17!
Anyway is the [math]\infty[/math] in a singularity physics calculation [math]\aleph_0[/math]?
Now we're talking!
They're not 'counts' of things, so the question doesn't apply. Those infinities just mean that the equation fails to describe the physical situtation. So for instance, it take infinite coordinate time for a rock to fall through the event horizon. That just means that this choice of coordinate time is singular there, so it cannot meaningfully describe the rock falling through. It doesn't mean the rock doesn't fall through, or that anything even particularly different happens to it there.
It means that the concept of a rock falling through the event horizon, is incoherent. In other words, the theories applied, mathematics applied, or both, are faulty, because they produce an incoherent scenario.
It's as if some phenomena that are said to be real are off-limits, no-go red zones; our most powerful tool - math - is rendered useless. Intriguing possibilities emerge, one of which is is Max Tegmark who believes the universe is mathematical right?
Good one! You seem to be on the right track - take the smallest physically meaningful quantity and check how many of 'em fit in the (observable) universe! You get points for being systematic.
:lol:
Isn't what I said implied by finitism?
Finitism simply avoids infinity. There is no such thing as the "largest number" as far as I know.
Here's where you are going (Wiki):
As per some sources, the Greeks kept [math]\infty[/math] at arm's length because of paradoxes that it generates (vide Zeno's paradoxes) and other such as x + x = x [math]\to[/math] 2x = x [math]\to[/math] 2 = 1 (:brow:)
The number that's to be swapped with [math]\infty[/math] doesn't have to be "very large" as the video on [math]-\frac{1}{12}[/math] I posted demonstrates.
But, but, but... that number arises from calculating infinite sums, and explicitly not from setting a finite limit. You cannot get to it if you set a finite limit.
Quoting Metaphysician Undercover
I would rather say:
The mathematical approach is to assume that any object can be divided in any way, so there is an infinity of possible divisions for each thing to be divided. In physics though, the way an object can be divided is highly dependent on the composition of the object.
But the problem with setting a largest number is that it rules out irrational numbers such as pi, sq-root 2 etc because they cannot continue to infinity as decimals and therefore become expressible as ratios. Of course physicists don't care about such things, and always just fudge their calculations to get roughly the right result any way, Blah blah, experimental error, uncertainty, whatever. But mathematicians have no mercy, and maths is full of irrationality ever since Pythagoras. Irrational numbers are the devil in the detail that he proved the existence of geometrically, and the fact that mathematicians (and others) are still trying to insist that maths should be fitted within the limits of their thinking is more to do with psychology than mathematics.
You could, though, go for taxicab geometry, as long as you like your circles roughly square.
Yea! Notice however that the [math]\infty[/math] sum has a finite value ([math]-\frac{1}{12}[/math]). That's the killer move!
When one tries to do analytic continuation of the Riemann Zeta function where it is not warranted this kind of nonsense results. What makes it useful in physics is beyond me.
Yes, in physics the way an object "can be divided" is highly dependent on the composition of the object, but sometimes this fact is ignored by physicists. That's the problem I referred to with the way that frequencies and wavelengths are treated. The problem appears to be that there is no identified medium within which the electromagnetic waves exist, so the real properties of the waves cannot be determined or described, and physicists are left with theoretical mathematics which assumes an infinity of possible divisions.
Quoting unenlightened
I would say that if the geometry can prove the existence of irrational numbers, then this is an indication that the geometry is faulty. The issue with pi and the square root of two appears to involve the traditional geometer's use of distinct dimensions. So the irrational nature of the square root of two shows that any two distinct dimensions, produced by the right angle, are fundamentally incompatible. Set two new points, equidistant from a starting point, on two traditional dimensions (making a right angle), and the distance between those two points will always be indefinite (an irrational number).
The issue with pi is very similar. A straight line has one dimension, and a curved line has two dimensions. These two types of lines are fundamentally incompatible. Some might say that an infinitely large circle, has an arc so gradual that it's actually a straight line, but how would you ever get two dimensions, a curve, or a circle then? Likewise, some might say that a polygon with infinite sides is actually a circle, but this doesn't really reconcile the incompatibility, because each side is still a one dimensional line, at an angle to the next side, it's sides are distinct straight lines and that's not a curved line.
I like your example of the taxicab geometry. I'm not sure, but it appears to deny the reality of the one dimensional straight line. But it appears to me, like if we deny the straight line, then we'll need an infinite number of dimensions, because there would be an infinite number of possible ways to get from one point to another point. So this doesn't really get rid of the irrationality.
This [math](\sum_{n=1}^{\infty} = -\frac{1}{12})[/math]"nonsense result" has applications in science, in string theory to be precise. I thought mathematicians like yourself would know. Oh, but I repeat myself. Apologies.
You seem fascinated by this anomaly. It's a kind of summation result for series that don't converge in the mathematically acceptable manner. Here's a link that should keep you occupied until bedtime:
1 + 2 + 3 + ...
Well, it is fascinating is it not? If we could somehow prove that [math]-\frac{1}{12}[/math] can take the place of [math]\infty[/math] in every calculation [math]\infty[/math] pops up in, we've struck gold, oui monsieur?
Fool's gold.
Why? Are you saying string theory and all the other physics topic in which [math]\infty[/math] is swapped for [math]-\frac{1}{12}[/math] is nonsense?
[math](\sum_{n=1}^{\infty} = \infty) \land (\sum_{n=1}^{\infty} = -\frac{1}{12})[/math]
The expression on the right is a way of summing a divergent series. (assuming we know what you mean with the notation)
You're not the spoon feeding kind are you? :up:
You got me wondering if anyone does this line of research anymore. My conclusion, rarely. Here are a couple of papers, the second being more a survey.
Summing Series
Summing Series
For finite cardinals, if ~x = 0, then ~x+x = x.
For infinite cardinals, x+x = x and 2x = x.
But you cannot infer 2 =1 from 2x = x when x is an infinite cardinal. Cardinal addition where one of the multipliers is an infinite cardinal does not have the cancellation property.
Once again, your ignorance and intellectual dishonesty have enabled you to post a false claim.
No derivation of a contradiction has been shown in ZFC.
And you write (I'm using plain text):
Sum[n = 1 to inf] = inf
Sum[n = 1 to inf] = -(1/12)
As far as I can tell, those are not even well formed.
Sum[n=0 to inf] requires a term on its right*, otherwise it's just a dangling variable binding operator.
* E.g., Sum[n=0 to inf] 1/(2^n) is well formed and meaningful.
There's lot to unpack there.
(1) (a minor point) You were responding to a quote of mine about intuitionism. That's the wrong quote, since the subjects of intuition and intuitionism are related but very different subjects. Instead the pertinent quote is "I quite understand that human thinking, including about mathematics, involves intuition. Indeed I'm interested in the relation between formal theories and intuitions."
(2) "How does intuition work in math?" The subject of intuition and mathematics is a big one. I could not even summarize my thoughts about it in an ad hoc post. And I do not have conclusive things to say about it. I said that I think about the subject a lot; but I do not claim to have arrived at firm conclusions about it.
(3) The subject of the relation between intuition and rigor is narrower than just mentioned, but still a big one. It's not clear to me where the best place to start would be. But perhaps one area to begin is the notion that formalization in its most basic intuitive sense can, in principle, be reduced to series of discrete observations about discrete objects conceptualized as being indivisible, such as, witnessed in physical form, tally marks on paper or 0s and 1s on paper. My own personal imagination doesn't provide that there could be any form of participation in mathematical reasoning more basic.
(4) Even if it seems to you that my postings read as mechanical or computer-like, that is not a refutation of anything I've posted. Moreover, it would be a non sequitur to infer that I don't think about the subject of intuition in mathematics from your premise that my writing is computer-like. Moreover, as to writing style, one should take into account that my purpose is not to entertain you, nor to present to you as loosely gabbing. When you post plainly incorrect claims about mathematics, then usually my main purpose is to clearly point out that you are incorrect and often to explain why; and often that is best achieved by explanations that use uniform terminology and parallel forms.
(5) You are lying. It is not even remotely true that I only say that you are incorrect. Over many months, I have also given you quite generous and detailed explanations why you are incorrect. It is remarkably dishonest and boorish of you to say otherwise.
You skipped my previous remarks about that.
Quoting Metaphysician Undercover
If I recall, we discussed this at length many months ago. Again, I don't mean a physical object. I mean a set is an abstract object.
If you don't accept the notion of abstract objects, then I admit that there's not much for us to discuss. I do not feign to be able to explicate the notions of 'abstract' and 'object' beyond ordinary understanding of such basic rubrics of thought as acquired ostensively or by whatever means people ordinarily understand them.
If you do accept the notion of abstract objects, then I point out that a set theoretical intuition may begin with the notion of a thing being a member of another thing: The notion of membership. First, that obviates the need even to use the word or notion 'set', as instead we merely discuss the membership relation. Second, even though the word or notion 'set' is dispensable, we can still go on to define it. Also, in the formal theory itself, the word or notion 'object' does not occur, though when we informally talk about informal theories it would be cumbersome to eschew the word and notion 'object'. In that context, mathematicians readily understand that a set is an abstract object. There is nothing in the definition of the word 'object' that precludes an abstract object nor that precludes that certain abstract objects, viz. sets, are in relation to others, viz. membership.
Quoting Metaphysician Undercover
That offers at least these prongs of refutation:
(1) I am mostly (but not exclusively) self-taught from textbooks; and textbooks in mathematics don't indoctrinate. Rather they put forth the way the mathematics works in a context such as presented in the book. A framework is presented and then developed. There is no exhortation for one to believe that the framework is the only one acceptable.
(2) Indeed, mathematics, especially mathematical logic, offers a vast array of alternative frameworks, not just the classical framework, including constructivism, intuitionism, finitism, paraconsistency, relevance logic, intensional logic .... And mathematics itself does not assert any particular philosophy about itself, as one is free to study mathematics with whatever philosophy or lack of philosophy one wants to bring to it.
(3) It is actually cranks who are narrowminded and dogmatic. The crank insists that only his philosophy and notions about mathematics are correct and that all the mathematicians meanwhile are incorrect. The crank doesn't even know anything about the mathematics yet the crank is full of sweeping denunciations of it. The crank makes wildly false claims about mathematics, and then doesn't understand that when he is corrected about those claims, the corrections are not an insistence that the crank agree with the mathematics but rather that the corrections merely point out and explain why what the cranks says about mathematics is untrue. It's as if the crank says, "classical music is all wrong because classical music never has regular meter" and then when it is pointed out that most of classical music does have regular meter, the crank takes that as narrow minded demand that he like classical music. And the crank is not even aware that mathematics, especially mathematical logic, offers a vast array of alternatives. Meanwhile, the crank's usual modus operandi is to either skip, misconstrue, or strawman the refutations and explanations given to him, thus an unending loop with the crank clinging to ignorance, confusion, and sophistry.
Quoting Metaphysician Undercover
Does Metaphysician Undercover have actual incidents to cite? Is there a particular incident to which he is witness, or widespread reports of them that would justify such a sweeping generalization or even a more modestly limited claim?
Wrong again.
I cannot agree that abstractions are objects, unless we restrict "object" to refer only to abstractions. But then we could not use "object" to refer to anything else, or we'd have equivocation. And we would have to create a special form of the law of identity, such that when 'the same' abstraction exists in the minds of different people, we can still refer to it as "the same" abstraction, despite accidental differences between one person and another, due to different interpretations. The current law of identity requires that accidental differences would constitute distinct 'objects' which are therefore not the same, so we'd need a different law of identity.
Quoting TonesInDeepFreeze
If the law of identity describes what an "object" is, then I cannot accept the notion of abstract objects, for the reasons explained above. I might accept the notion of abstract objects, so long as you agree with me that the law of identity does not apply to this type of object, and we proceed with caution, so as not to equivocate between these two distinct types of objects, physical objects being described by the law of identity, and abstract objects being a different type of object to which the law of identity does not apply. Can you agree to that?
Quoting TonesInDeepFreeze
Sure, "a thing" in this context, I assume is an abstract object, with no possibility for identity, so you can say whatever you want about "a thing", or "object". You can even say contradictory things about an object, because the object has no identity as a physical object does. You could say one thing about "the object" and I could say a contradictory thing about it, because there's nothing to ensure that we maintain consistency between the object in your mind, and the object in my mind, which both bear the same name. How would we even know that we're talking about the same object, except that we are using the same name? Do you propose that the name is the object? Then where is the abstraction?
So there are no restrictions in the sense of truth or falsity by correspondence, and to say contradictory things about the same object might be completely acceptable. The "thing" is a purely imaginary fiction, and we can use contradiction in fiction without a problem, though it might make the imaginary "thing" seem counterintuitive.
Quoting TonesInDeepFreeze
All right then, considering the above conditions, I'm ready to try and understand what "membership" means. What does it mean for one thing to be a member of another thing? Is it necessary that the thing which is a member be a different type of thing from the thing which it is a member of? If not, how would I distinguish between which things are the partakers, and which are partaken of?
Quoting TonesInDeepFreeze
I disagree on a number of key premises here, so I do not consider any of this to be acceptable refutation. But that's all beside the point, just a difference of opinion on trivial matters.
Quoting TonesInDeepFreeze
I'm glad to be wrong, of course. I actually enjoy discourse with you TIDF, you're generally well behaved and intelligent. I just don't see that you know any strong principles.
My Thoughts on Intuition & Logical rigor in Math
Intuition is more a feeling than a thought. I've experienced it in doing high school math. There was this time when I was solving a problem in an exam and I soon realized the numbers mid-calculation were just too large & awkward, they didn't feel right to me; I went back to check my work and found out I had made a boo-boo. This is what I mean by intuition in general and mathematical intuition in particular.
Too, last I checked, mathematicians love to guess, formally termed conjectures. As far as I can tell, mathematical conjectures are intuitions and not just wild guesses - the former tend to be hard to prove/disprove while the latter are easily tackled by even amateurs.
That's all she wrote!
So those who're learning are guilty of intellectual dishonesty? Gimme a break!
:chin:
Yep. The maximum entropy or information content of the visible universe is 10^123 k. So if you wanted to number every individual degree of freedom that exists for all practical purposes, theres your number.
At least it is the current best stab. See https://mdpi-res.com/d_attachment/proceedings/proceedings-46-00011/article_deploy/proceedings-46-00011-v2.pdf?version=1612356633
:fire:
I said nothing that can be construed as "learning is intellectual dishonesty".
I don't know what that emoticon you keep posting means.
/
I found out more about the -(1/12) thing. It requires taking the infinite sum in a different sense from the usual sense. It doesn't imply that there is a contradiction in mathematics.
Okey dokey! Muchas gracias kind person! Please cut me some (more) slack. I'm a only a beginner as you already know.
For many months you have continued to post disinformation, even repeating items on which you were already corrected or refuted. That is a pattern not merely of a beginner, but of a crank.
And you recently lied about me personally.
This has to be exciting news for theoretical physics!
I'm sorry you feel this way! Good day.
Quoting jgill
Indeed there's potential there to revolutionize science! Glad that you see it.
It's not a matter of feelings. They're facts.
Come the revolution everyone will like strawberries and cream.
A dialogue between Georg and Ernst:
G: How's your rock collection these days?
E: I sold all my rocks. Now my collection is empty.
Ha ha, nice try Tones. Ernst's proper reply would be: "I have no collection of rocks. I sold all my rocks".
OED, collection: 1. the act or process of collecting or being collected. 2. a group of things collected together, esp. systematically. 3. an accumulation; a mass or pile (a collection of dust)."
I really don't know how you can conceive of a group of thing, or a mass or accumulation, without anything there.
Sure, you can't conceive of an empty set. But lots of people do.
But the problem is more fundamental with you. You can't conceive of abstract objects.
Here's a difference between you and me: You're a dogmatist. I am not.
In these kinds of matters, you cannot be bothered to give fair consideration to frameworks other than your own. Not only do you know nothing of the mathematics involved, you know hardly anything (if anything) of the many views of modern philosophers of mathematics. And you willfully avoid knowing anything about them. So you are brutally stunted in your capability to view from different perspectives, to intelligently compare different frameworks, to conceive. No matter that there is a rich, intensive, and intellectually competitive cornucopia of thinking in and about mathematics, you insist that all of those very smart and dedicated people must be wrong all the way to the core, while you alone stand above.
On the other hand, I can see the attractions of various points of view - from among even physicalist, materialist, idealist, phenomenologist, platonistic, nominalist, structuralist, finitist, constructivist, intuitionist, etc. Unlike you, I don't demand that my own personal framework for understanding mathematics is the only reasonable framework.
As I explained, I readily conceive of abstract objects, but I maintain a difference between abstract objects and physical objects, as necessitated by sound ontological principles. You it appears, do not seem to be ready to accept the dualism required for a true understanding of "abstract objects". That's the real difference between you and me, I understand abstract objects from sound and consistent metaphysics, while you simply assume "abstract objects" for the purpose of mathematics, without any kind of understanding of what an abstract object might consist of.
Quoting TonesInDeepFreeze
This is untrue. I've given you adequate opportunity to explain the principles that you adhere to, which I find contradictory. I'm very willing to proceed with you, but not until we resolve apparent contradictions in your primary principles. I refuse to proceed from faulty principles. That would be nothing but unsound logic and a waste of time. So I give you fair consideration, but it is you who has not given fair consideration to the ontological problems I have raised. Instead of addressing my concerns, you now insist that I ought to just drop them, and take up some "different perspectives", even though I still apprehend your perspective as based in contradiction because you have done nothing to resolve this problem.
So, as I proposed, we can have an empty set, so long as "set" refers to the category, or type of thing which is going to belong to the collection, not the group of things itself. If the group had no members it could not be a "set" if "set" referred to the group itself. No members, no group.
This proposal allows that we could have a type, but no things of that type. Would you agree with this as a compromise principle? Then the "set", which is a category or type, is not a group, so that it can be empty, and the members of the set have a distinct kind of existence from the set itself, being the things which are categorized as being members of the set. So we have sets, and we have members, such that there are these two aspects of any set, the set (category), and the members (things classed as within that category, and there might not be any. And, we might even classify sets as things of a sort, such that we could have a set where the members are sets. But the set which has sets as members, would be a special type of category. So for example, "colour" could be a set which would have sets as members. The members would be "green", "red", etc., each being a category, or set on its own.
Quoting Metaphysician Undercover
Now:
Quoting Metaphysician Undercover
We should stop right there.
But you also put words in my mouth:
Quoting Metaphysician Undercover
I haven't said anything about duality. This is yet another instance of you putting words in my mouth (except weaseling with "do not seem").
Quoting Metaphysician Undercover
I don't advocate any particular philosophy. But I have explained to you crucial notes about the mathematics itself, and I have touched on certain aspects of frameworks in which mathematics is discussed. You have made it a point to either ignore, evade, misconstrue or strawman all of it.
Quoting Metaphysician Undercover
I don't insist that you do or don't do anything (other than that you don't put words in my mouth or lie about me).
And I did not even hint that you have to "drop" your philosophy. I only said that you are not capable of also giving fair consideration, let alone study, to other points of view in the philosophy of mathematics, not to even the basics of mathematics on which you have such vacuously dogmatic opinions. It's characteristic of childish mentality to think that you can't look at things from other people's points of view without giving up your own.
And putting "different perspectives" is scare quotes is also jejune. Even if you disdain other perspectives, it should not be at issue that the notion of a different perspective is common and doesn't need scare quotes. That's not even a big point, but it is an emblematic detail.
Two posts now I've attempted compromise, but you still haven't addressed my proposal. It appears very much like you are the one incapable of giving fair consideration.
You've been addressed:
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Friend, I think it's time to back away. You're debating with circle-squarers. It's like attending a flat-earthers' convention and joining a debate on what causes the phases of the moon.
Notice that most TPF folks are giving this topic a wide berth. Don't keep propping up their soapbox.
When v = c, [math]\gamma = -\frac{1}{12}[/math]
But the Lorentz factor is always positive, so how can that be? :chin:
Nevertheless, you're on an amazing roll. :clap:
You, as a mathematician, are better equipped to answer that most intriguing question.
Quoting jgill
You jest!
This gets more and more interesting by the minute.
Take the irrational number [math]\sqrt 2[/math]. To what decimal place of accuracy must we calculate it to construct the dome of the Hagia Sophia (google for details)? I read a book that relates how the engineer used a rational approximation instead of the actual value of [math]\sqrt 2[/math].
Remember high school when [math]\pi[/math] was [math]\frac{22}{7}[/math]? We could use rational approximations that are accurate to an arbitrary number of decimal places for [math]\pi[/math] and other irrational numbers and I think modern calculators do exactly that.
The same logic is true in this case too - replace the infinite decimal part with a finite one using a rational approximation. My guesstimate is we won't be needing the actual value of [math]\pi[/math] ever; a rational approximation accurate to the required number of decimals will be just fine.
Likewise, infinity, no need! A very large but finite number will suffice. If memory serves the Greeks did just that - they used arbitrarily large numbers in place of infinity.
Friend, my admonition to you is the same as that for TDIF : when you see certain names pop up on math topics, run the other way. Otherwise, you're wasting your time.
Without an audience, they will go away.
(My understanding is that banishment can be initiated given a low quality of posts. So I caution the posters trying to tear down 3,000 years of well-established mathematics and invent their own. If any of the moderators have a knowledge of math, they may frown upon such repeated nonsense as "infinity = -1/12". These ideas are not up for debate in math. Such ridiculous pronouncements would not be tolerated on a "purely" philosophical thread.)
You quote the first line of a post and you ignore the rest. I see no point.
Quoting Real Gone Cat
Sorry to disillusion you Real Gone Cat, but this is a philosophy forum, not a math forum.
The rest doesn't mitigate your contradiction. I've been through this before with you where you shift your position making coherent discussion impossible.
Quoting Metaphysician Undercover
Philosophy of mathematics requires knowing the mathematics being philosophized about.
Sorry Tones, You have not pointed out any contradiction. If I remember correctly, you define contradiction as saying 'is' and 'is not' of the exact same proposition. "I cannot agree to abstractions as objects, without specific restrictions", does not contradict with "I can readily conceive of abstract objects". All conceptions require restrictions, that's what conception is, understanding the specified restrictions.
What you wrote:
Quoting Metaphysician Undercover
So I take it that restricting 'object' to refer only to abstractions is not acceptable to you. Thus, indeed you do not agree that abstractions are objects. Thus, indeed you contradict yourself when you also said:
Quoting Metaphysician Undercover
To put it starkly:
"I cannot agree that abstractions are objects" is tantamount to "abstractions are not objects".
"We restrict 'object' to refer only to abstractions" is tantamount to "only abstractions are objects".
So what you said is tantamount to: Abstractions are not objects unless only abstractions are objects. But you also deny that only abstractions are objects. Thus you affirm that abstractions are not objects.
Or:
Let 'P' stand for 'abstractions are not objects'.
Let 'Q' stand for 'only abstractions are objects'.
You say 'P unless Q'. But you deny Q. So you affirm P.
You host a continually silly shell game. I shouldn't indulge in responding indefinitely.
If part of one's philosophizing about mathematics includes criticisms of certain actual mathematics, then one should know enough about that actual mathematics that one doesn't misconstrue and misrepresent it. And if one proposes a certain alternative philosophy of mathematics, then it is natural to ask "To what actual alternative mathematics does your alternative philosophy pertain?"
It's kind of funny in retrospect how arrogant you came off in accusing Agent Smith, while, unknowingly, you are completely wrong! Part and whole have nothing to do with set and subset: one is mereological, the other is set-theoretic, yes, they can overlap, but no, they're not the same thing.
To interpret his statement set-theoretically, when "part and whole" are specifically the technical terms used in formal mereology, is either (1) delibarately uncharitable or (2) you not being aware of what he's referring to. But whatever the case is, (1) or (2), it does not excuse your hostility.
In any case, (non-proper) parthood is a transitive, antisymmetric, and most importantly reflexive relation such that ?x P(x, x): every whole is a (non-proper) part of itself. This axiom is true in virtually all of contemporary formal mereologies (X, M, AX, GEM, AEM, AMM, EM, etc.) and is perhaps the least controversial mereological axiom: even the transitivity of parthood is sometimes disputed!
So AgentSmith was correct, and your "correction" of him is a result of conflation of mereology with set theory on your part, so before telling him to do his homework, do your own.
Quoting TonesInDeepFreeze
Quoting Kuro
I didn't say that subset and set align with the mereological notions of part and whole.
Agent Smith said that said set theory allows that a part can be equal to a whole. I correctly pointed out that that is not true. (For that matter, 'part and whole' are not even terms of set theory). And I correctly pointed out that what set theory does say is that in some cases a proper subset is equinumerous with its superset.
And Agent Smith made no reference to 'part' and 'whole' as technical terms of mereology. And it doesn't matter anyway. Whatever mereology has to say, set theory does not say that a part can be equal to a whole.
Quoting Kuro
I am hostile to him only in a broad sense that includes that I decry his continual (over many posts and many months) ignorant and willful falsehoods, misrepresentations and confusions of the subject.
Quoting Kuro
You are either confused about the context of the posts or you are willfully fabricating about it.
What I responded to:
Quoting Agent Smith
He's not talking about infinity per mereology. He's talking about the Cantorian notion. The mathematicians who were against Cantor's notion of infinity were not taking Cantor to have presented a mereology but rather indeed a mathematical notion of sets.
Agent Smith is claiming that the notion of infinity, as in set theory (for example, as he mentioned, set theory engendered by Cantor), leads to the conclusion that a part is equal to the whole. And I correctly replied that that is not true, though it is true in set theory that in some cases a proper subset S or T is equinumerous with T.
/
Bringing a mereological perspective to the subject is fine. But it is a red herring to put my exchange with Agent Smith in context of mereology when the context was clearly set theory.
I responded based on the quotation of him where he says "part is equal to whole" - this is what you've included in your post in whole with no further passage/text in the quote. Perhaps you could've made a larger quotation for context, but otherwise I do not think my presumption was particularly irrational (since, reading "part is equal to whole" at face-value, just is a mereological truism.)
You are entirely correct in that if he refers to proper subset and set by "part and whole", which, per context seems to be the case, then it is completely inaccurate.
Quoting TonesInDeepFreeze
I made clear that I was responding regarding set theory. Granted, I didn't include in his quote the part - that makes even more explicit that the context is set theory - where he specifically mentioned 'infinity' and 'Cantor'. Also, I didn't belabor that the previous context was infinity especially as Agent Smith faults the notions of infinity and set theory (with your comments about mereology running alongside but separate from the particular exchanges between Agent Smith and me). In all that context, if one paid attention to the conversation, rather than just knee-jerking to one quote in it, it was clear that set theory was what was being discussed at that juncture.
Quoting Kuro
(1) It's my very point that Agent Smith seems to have conflated the set theoretic notion that an infinite set T has proper subsets equinumerous with T with the mereological notion of part/whole.
(2) Agent Smith was rejecting the notion that a part is equal to a whole. Obviously, that is not about a non-proper part being equal to a whole, but rather a proper part being equal to a whole. So that he's not adducing the truism you mentioned.
/
I would have made it easier for you if I had quoted him more fully. But even then, I did not distort him by quoting more narrowly. I was correct in my reply to Agent Smith. It was not arrogant of me. You were wrong to claim I was not correct or that I was arrogant about it.
No need to hammer something down when my previous post already agreed that, per set-theoretic context, you were correct. You can relax. I simply made the point that my misinterpretation was not unreasonable.
I reiterated the point that I was correct to support the additional point, which you did not mention, that I was also not arrogant about it.
Sure. You were not arrogant about it. Extra brownie points.
Your defensive sarcasm is misplaced.
Why do you conclude that? If I said, I'll only agree to call that type of thing which we sit on "a chair", if that's the only type of thing we will call "chair", would you conclude that it is not acceptable to me to call that type of thing a chair? What I said, is that we can call an abstraction an "object" if that's the only type of thing we call by that name. It's not uncommon to restrict definitions in this way, it makes deductive logic more productive by avoiding equivocation.
Quoting TonesInDeepFreeze
This is a complete misunderstanding. Abstractions are a very specific type of thing. I have no objection to saying that abstractions are objects, if I reject other senses of "object" which I am familiar with, thereby naming a category "objects", and placing abstractions within this category. Just like in my example, we can name a category "chairs", and place the things we sit on in that category, and we can name a category "objects" and place abstractions there. However, since abstraction are such a unique, and very specific type of thing, I do not see how anything else could be placed in this category.
Or, we might realize that just like chairs are only one type of thing that we sit on, objects are just one type of abstraction. But if this is the case, then all objects are abstractions, but not all abstractions are objects. And we still don't have anything other than abstractions as objects.
I will though, on reconsideration allow the possibility that we could have a category "objects" and there might be things other than abstractions, which may be similar to abstractions in some way, which might also be placed in that category, "objects". So if you believe that there are other things similar to abstractions, which you believe ought to be placed in that category of "objects", you might demonstrate to me the reasons why you believe this.
But what is there left to move when the Cosmos arrives at its de Sitter heat death condition where all its degrees of freedom are embedded in its event horizon.
Time and change as scaled by the Lorentz factor - have effectively ground to a halt. The only action is the quantum sizzle of the radiation attributed to the event horizon. We have reached the edge of the conformal disk in a finite fashion. The universe beyond may be supraluminally infinite. But it too is most likely to be in the same generalised condition.
So this may be the surprising thing. The reciprocality of the Planckscale start to the Comos means that a Big Bang with a minimum spatiotemporal scale and maximum energy density just simply turns itself inside out to become an equally finite Heat Death of maximum spatiotemporal scale and minimum energy density.
Lorentzian invariance would thus be emergent as particles formed with local momentum and inertial mass. And then it would dissolve from relevance as those particles get swept up and their energy radiated over the cosmic event horizon.
Of course, there is the little issue of the dark energy sustaining the whole show after the Heat Death. From that perspective, the Comos does expand forever and so "something" is always being added in terms of an infinite metric.
But once everything we physically could care about has reached its max ent state, do all those extra degrees of freedom actually count for anything? Even if there is potentially an infinite amount of them as dark energy repulsion just keeps stacking up in its supraluminal "biggest picture" way?
And it is still a remarkable thing if the cosmos had to find a way to close itself in this fashion so as to achieve concrete existence. It had to have a cut off at the beginning of the reciprocal kind that could once again be its cut off at the end.
Out of curiosity, do you have citations on this point? I would argue something somewhat different. But I'm interested if there are discussions that support your view here.
:up: Aren't odd numbers a part of natural numbers? Is it not true that the cardinality of the former equals that of the latter? :chin:
Stop it. Nothing makes me doubt what I just wrote more than someone's apparent agreement. :grin:
But I was thinking of you and rock climbing mathematicians the other day when I caught up with this sad story https://profmattstrassler.com/2019/08/06/a-catastrophic-weekend-for-theoretical-high-energy-physics/
And also read this NYT story on how physicists (or at least mathematical physicists) are keen on mountaineering and bouldering - https://www.nytimes.com/2001/02/20/science/a-passion-for-physical-realms-minute-and-massive.html
I always thought from my experience that maths types were invariably into classical music. It was the physicists who scaled peaks.
For what its worth, I don't like a landscape that presents a technical challenge one that has to be solved like a riddle. I like running fast and dangerously on twisty goat trails in the hills in a way that becomes quite unconscious. While listening to punk rock. A flow experience.
I take it that you don't take it that the only objects are abstractions, because you went on to say why you don't take it that the only objects are abstractions. You wrote:
"I cannot agree that abstractions are objects, unless we restrict "object" to refer only to abstractions. But then we could not use "object" to refer to anything else, or we'd have equivocation. And we would have to create a special form of the law of identity, such that when 'the same' abstraction exists in the minds of different people, we can still refer to it as "the same" abstraction, despite accidental differences between one person and another, due to different interpretations. The current law of identity requires that accidental differences would constitute distinct 'objects' which are therefore not the same, so we'd need a different law of identity." [Bold added]
The bold part is your argument why you don't restrict 'object' to refer only to abstractions.
Or, are you now saying that the only objects are abstractions?
Yes, I knew you would reply by shifting back around again from your own words but pretending that you haven't. This will go on indefinitely with you, as you play a silly game that is the forum equivalent to a child's peek a boo.
Yes. That has never been in question here. Indeed I reiterated just what you said in my post that you are replying to now! What is in question here is your your ignorant misunderstanding that equality of cardinalities in set theory implies that set theory says that a (proper) part can be equal to the whole. Please pay attention to exactly what you have said and what I have said.
Again, in set theory it is the case that infinite sets are such that they have proper subsets of the same cardinality as the set. But that is not at all to say that there are sets S and T such S is a proper subset of T and S is equal to T.
If T is infinite, then there exists a proper subset S of T such that there is a bijection between S and T (thus card(S) = card(T)). But there are no sets S and T such that S is proper subset of T and S = T.
By the way. In set theory we have these definitions:
T is infinite <-> T is not finite
T is Dedekind infinite <-> there is a proper subset S of T such that there is a bijection between S and T
In set theory, even without choice, we prove:
there is a proper subset S of T such that there is a bijection between S and T -> T is infinite
In set theory, with choice, we prove:
T is infinite -> there is a proper subset S of T such that there is a bijection between S and T
So, in set theory with choice, the definitions of 'infinite' and 'Dedekind infinite' are equivalent anyway:
T is infinite <-> T is Dedekind infinite.
Also, even without choice, we can prove that there do exist Dedekind infinite sets, such as your example of the odds and the naturals.
[quote=Ms. Marple]Most interesting.[/quote]
Please continue posting but I suggest you engage with those who have the same level of understanding or higher as/than you. I'll read your posts as and when I can. Bonam fortunam TonesInDeepFreeze.
I will not impose upon myself a restriction from commenting on your posts.
No problemo! It only means I should improve the quality of me posts! Have an awesome day monsieur!
And the way for you to do that is to read a book on the subject.
Thanks for the advice! On it!
Oh really? What book?
Recommend one, a simple one, to me. I have about 1.5 Terabytes of books, includes those on math.
First:
Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar
That is to get a solid understanding of the first order predicate calculus, which is the ground level for formal mathematics and formal philosophy.
Supplement:
Chapter 8 of Introduction To Logic - Suppes
That is for the best explanation of formal definitions I have found.
Supplement:
The introductory chapter of Introduction To Mathematical Logic - Church
That is for the very best overview of the subject of modern logic one would ever find.
Then:
Elements Of Set Theory - Enderton
De nada.
:up:
Sorry, no reference at all It just seemed obvious that if there is a largest number, decimal iterations must end at it - one cannot have the largest-number-plus-one-th decimal place. Thus no irrational numbers, Or to put it another way, the number of points on the number line between 0 and 1 will be finite.
But indeed so much would have to change that I cannot see how the numbers game would survive - most numbers would have no square for example, and calculations would keep ending in ERROR like on early calculators.
So, in my humble opinion, there's got to be a finite number such that it's, for all practical purposes, [math]\infty[/math] to us. For example, did I mention this already?, the speed of light (186000 miles/sec, a finite number) behaves like [math]\infty[/math] - we can never attain a speed of 186000 m/sec and that, in a sense, is infinityish. Wouldn't you agree?
Sounds like you can remember Texas calculators and Polish notation too. :up:
Choose the highest number you will allow
You would admit that someone else might choose a higher number.
And whenever someone chooses their highest number, someone else can choose one plus that number.
So which should we take to be the highest number we allow?
Mathematics should be limited to only the numbers you will personally allow?
/
Suppose there is a highest number any living human can now conceive in the manner you require. It is not ruled out that future humans may conceive higher numbers. So when that happens, we will reformulate mathematics to accommodate the new highest number? And if the person who conceives the highest number dies, then we'll reformulate mathematics to bring down the highest number? Then every day, we'll check the newspaper to see what the new highest number is?
/
Consider a music CD, with about 850 mb. Consider all the possible combinations of those 850 mb as zeros and ones. Big number. I can conceive it though, since I can conceive of changing every one of the bits on the disc. But I can also conceive of 8 billion discs - one for each person on the planet - and each of those dics having capability of changing every zero and one. A bigger number. But I can also conceive of each person on the planet having a 100 terrabyte computer to store lots and lots of 850 mb music albums, and then to switch the ones and zeros around on each of them as many ways as combinatorically possible. A bigger number. But I have to check with you each time I proceed to a bigger number whether you also can conceive it?
Don't you see that mathematics is general - not confined to only numbers that particular human beings can visualize in the way that you require at a particular time in the history of the universe.
For any natural number n, there is the natural number n+1. So there are infinitely many natural numbers. That is not thwarted by your limitations of what you can personally conceive at this particular moment.
No, I don't agree.
The the number of states my lamp can be in is 2 - on or off. There is no counting past the number 2 when counting the number of states my lamp can be in. You can never attain more than 2 possible states for my lamp. So 2 is infinityish?
/
If 186000 is infinityish is then so is 186000 x 1000, since 186000x1000 m/millisecond is the speed of light. Etc.
First time in my life I was able to count from one to infinity and then back. Thanks.
And it did not take forever, either.
However, the speed of light (186000 miles/sec) isn't personal; it's not something I wish and nor did I order it to be so. In fact I would like it not to be true, but that's a different story altogether.
My point is, from what I know of physics, no calculations on velocities at least will ever exceed 186000 miles/sec. In other words if there's a claculator dedicated to calculation velocities, any result for speed that exceeds 1860000 miles/sec should return ERROR.
So what? We agree that 186000 is not the greatest number. Nor is 186000 x 1000, which is the speed of light in milliseconds. Etc.
In some sense 2 is infinityish.
You seem to be an expert on math so what I'm gonna say is going to sound familiar. There's the Pirahã tribe in the Amazon whose mathematics is limited to 1 and 2, anything greater is many which, to me, is [math]\infty[/math].
We would have to restrict oursleves to specific units of measurement.
I am most definitely not an expert on mathematics.
You're claiming that there is a greatest number. It's not 186000. And there's no law of thought that says I can't use different units of measurement. And there's no law of thought that says I can't add 1 to whatever number you claim is the greatest number.
Your view is dogmatic.
Let's play a little, simple game. For money, $5 a turn. Who can name the higher number (finite positive integer).
You go first in each turn.
Sorry, I don't mean to be disrespectful. At all. It is just that this thread put me in a giddy mood.
(Reminds me of a scene in an animated TV show. Two robots sit down to a chess set, set up on the board. They both think very hard, for a long time. Finally, one says, without either of them having moved a piece, "White checkmates black in 325 moves." The guy who plays black, pipes up, "Awww... you always win!")
/
Yes, chess is determined. Either there is a winning strategy for white, or winning strategy for black, or a strategy for both to draw. von Neumann made that observation. But we don't know which it is.
I also proved it for myself before I heard that von Neumann already had. It's kinda trivial really. Induction on the number of moves, if I recall.
Automatic, dogmatic, same difference.
Another whiff from the past. It was a button in the era when people wore large buttons on their shirts, attached by a pin in the back of the button. This one showed a car speeding away from the carcass of a dog, with tire marks across him. The inscription said, "My Karma just ran over your Dogma."
automatic is to dogmatic as karma is to dogma. Cute.
Granted.
I changed my reply. I didn't notice that it was you who posted and not the other poster. Then I realized that you were making the same point as I was.
Well, I'm not in any way trying to say my way or the highway. Anyway, nice talking to you.
No, you're not. But you hold to your position even though it can't withstand easy objections.
(I know that irrelevant flaming is reason enough to delete posts. Irrelevant to the topic. But are irrelevant compliments also sufficient reason for deletion?)
Duly noted.
I don't know what kind of person he is away from posting, but I find him to be flippantly dismissive in my interactions with him as a poster.
Right, but as I said, I was willing to restrict my definition of "objects", to mean abstractions exclusively, for the purpose of proceeding with this conversation if that's what you desire.
There is a problem with using ambiguous terms in deductive logic. It leads to equivocation. So I wanted to make sure that this problem was avoided.
Quoting TonesInDeepFreeze
I told you, I can go either way, you seem to be having great difficulty with that idea. So long as we adhere to one definition or the other, I'm ready to proceed. But equivocation is a waste of time. My inclination, through my habit of common use, and therefore preference, is to say that abstractions are not objects, and define "object" in a way which is consistent with the law of identity, as I explained. But an abstraction, as an "object" is not consistent with the law of identity, because what we call "the same" concept exists in many different minds, with accidental differences. So if we proceed by defining "object" so that abstractions are objects, we must forfeit the use of the law of identity as a defining feature of an object.
You seem to be having difficulty with this proposal. Could you explain the problem you are having?
Quoting TonesInDeepFreeze
Yes, so long as we reach no agreement as to how to define "object" to ensure that neither of us will equivocate in our arguments, this situation could go on indefinitely without any progress.
Yes, your obfuscation seems to be inexhaustible.
Oh, a new angle... AS did not argue that there is a greatest number... he argued that there is a greatest number we'd ever need.
That is also deterministic. The number itself is beyond our capacity to find out, but I daresay if we find a way to define "we", then yes, AS is right on.
Please note: AS means the greatest number we'd ever NEED, and not the greatest number we'd ever be able to THINK OF.
Apolgies, I'm lazy, plus I lack the wherewithal. I hope you understand.
I am going by these:
Quoting Agent Smith
Quoting Agent Smith
Probably those are not exact enough to say whether he means "there is a greatest number" or "there is a greatest number that will ever be needed". He can say for himself. But I would be very interested if he said "No, there is no greatest number. But there is a greatest number we'll ever need." If he said that, then it would be hard for me to dispute him, since I have no idea whether there is a greatest number that anyone (and let's include any conscious being in the universe now or ever) would ever need.
But still, again, as far as mathematics is concerned, there is no compelling reason that mathematics should limit to only numbers that are not greater than the one we guess to be the greatest ever needed for practical question. When I have a hankering to add one to that number, what's going to happen? The universe will tell me I can't do that?
Good point.
But I still would be interested to know whether he does recognize that there is no greatest number. If he does, then I would need to retract some of my previous comments about this.
I think it follows. Everyone needs numbers. One of the numbers one needs in a lifetime is the greatest number. This applies to everyone.
One of the complete lot of people will have a number that he or she needs that is greater or equal number compared to the numbers needed by everyone else in the lot. That is the greatest number we'd ever need.
Although it may not be a positive integer, a rational, an irrational, or even a real number.
This can be further analyzed. We, not a person individually. So the greatest number we'd ever need, is the greatest number which at least two people will ever need. And therefore it may not be necessarily the greatest number ever needed by a human being, because one or more individuals may have other, non-equal-to-each-other numbers, that are greater than the greatest number at least two people will need.
I don't know whether there is a limit on how long there will be conscious beings.
True. But we are talking "we". Supposedly humans. That's what I'm sticking with. Computers are not humans. Salient, conscious beings, wherever they are and however they are made, that are smart like humans or smarter, but are not humans are also not part of "we".
Sure, but your refusal to define and adhere to a definition of "object" is inexcusable. It can mean nothing other than that you rely on the game of equivocation as your mode of persuasion.
Quoting TonesInDeepFreeze
This is relevant to our discussion of how to define "object". Let's assume that mathematicians might define "object" in any way that one likes. This means that there is no necessary consistency between one mathematician's "object" and another's. Therefore, we could have contradiction, incompatibility, and incommensurability between one mathematicians idea of "object", and another's.
Now suppose that we agree that there is an advantage to providing consistency in what qualifies as an object, such that measuring quantity could proceed in a standard way, by some agreed upon convention. Do you not agree that this would be advantageous? And do you not agree also that we need a definition of "object", and we need to adhere to that definition, to ensure that inconsistency between one mathematician and another, in the measuring of quantity is avoided?
Of course that is a plausible idea. But I don't know that there is a limit on how long there will be conscious beings.
I actually can't see the UNIVERSAL advantage of unifying objects and terms of measurement. Sailors still measure velocity in knots; Americans still measure weight and volume in ounces and gallons and pounds. If there were an advantage to unification, everyone would be using the same system, and yet we don't.
We could stick to humans. But there may be a time when humanity becomes some other kind of being. Also, for philosophical purposes, it seems arbitrary to confine this question to a particular species at a particular point in history.
I wasn't even talking about computers. I would take it though that any number needed by a computer that is used by a human (or whatever) is number needed by humans (or whatever).
Human beings are predicted to die out at the latest when the sun becomes a Red Giant. Whether we would escape the confines of the Solar system by then, is questionable.
However, if we are able to escape the confines of the solar system, then our lifetime as a species is still limited, even under the best of circumstances, because heat entropy will make all habitable environments inhabitable. That is to come in a trillion, trillion, trillion Earth years. (1 followed by 36 zeros Earth years.) There may have been some rounding error.
I did not make this up. I read it in a newspaper (before the advent of the rampant use of the Internet)
You're mixed up again and lost your place in the exchange. The question is not how I define 'object'. And it's not even how you define it. Rather, no matter how you define it, you contradicted yourself about it, as I quoted you doing that.
I predict that now you'll write yet more rambling, obfuscatory paragraphs in which you elide your own posted words.
if we become something else, we are something else but humans.
And like I said, I go by the title. The title clearly states the set of salient things who will ever need a number that is larger than all other numbers they will need. And the named set is "we". Since it was written by a human (supposedly), it is humans we talk about.
I'm just saying that, in principle, I don't know whether consciousness, even if no longer human or was never human, will end.
The question of what this particular species needs seems (at least to me personally) too quotidian to be much of a really philosophical question about mathematics.
I have already granted that if Agent Smith does agree that there is no greatest number (despite his persistent anti-infinitstic lobbying in other threads), then I misconstrued him, forgot to refer back to the title post, and a retraction by me would thereby be due.
Yes, as I had gone already many posts ago:
Quoting TonesInDeepFreeze
It would free us from the need for conversion, translation, in communicating with each other, which would of course be an advantage.
Quoting TonesInDeepFreeze
Oh, if that's how you interpreted what I said in all those posts, it's very apparent that you did not read what I said. I conclude that I was wasting my time.
I proved that I read what you posted, as I quoted it more than once and gave exact and detailed comments about it. And though my comments were clear as day and formatted quite perspicaciously, you couldn't intellectually cope with them.
Quoting Metaphysician Undercover
Indeed your confused, ignorant and intellectually dishonest postings are a waste of your time, as you'd do so much better for yourself by reading a book on the subject.
What sayest thou?
How about, if I don't know what degree of precision my friend needs to complete his task, I just tell him that the value he need is pi, then he can use whatever approximation is suitable.
Ok, but there's no need for [math]\infty[/math] oui?
I have explained over and over that in rigorous mathematics:
There are points of infinity on the extended real line
And there is the notation "goes to infinity" such that that notation is only a convenience as we can dispense with it by writing the formulas with greater rigor.
And there is the adjective 'is infinite' but not the noun 'infinity'
But when you use the leminscate, you do so without rigorous grounding to explain what you mean by it as actual mathematics.
/
And I have explained over and over the importance of infinite sets for axiomatizing mathematics.
/
And you keep pointing to the -1/12 thing. You brandy it around like a chimpanzee beating his chest, as if you've brought in some kind of killer refutation of infinitistic mathematics. Yet you ignore jgill when he explains that it's really just a gimmick that plays on purporting that a divergent series converges.
I really do not understand why someone would keep spouting the same mistakes over and over, even when explained to him, and while not bothering to actually learn something about the subject. Well, I can guess actually: vanity - the fun of fancying oneself as a thinker on a subject one thinks one doesn't even have to study.
How would it be if there were a Biology discussion group, and I serially posted that ordinary academic biology should be replaced with my own personal notions about biology, and yet I didn't even know what a cell is, didn't even know what a carbon atom is. That is essentially what you do.
/
And, by the way, in ordinary axiomatized mathematics, there is no pi if there are no infinite sets. You would see that easily if you only took a bit of time to learn something about this subject.
Oops, looks like there already exists such a Theory :sad:
BIG NUMBER THEORY! Yeah that sounds about right - if there's a finite number that exceeds a supercomputer's ability to grok it (returns an error), then a fortiori humans can't handle it. We could of course always program a computer to display an arbitrarily large number instead. Imagine a calculator that can't deal with numbers larger than 5. If I ask it to do 2 + 3, it does so and gives the result 5, but if I ask it to calculate 3 + 9, it should still give 5 as the sum (it doesn't "understand" 12, a number > 5).
I knew Lester Germer, a multi-dimensional person. He was a fighter pilot in WWI.
My oldest and best climbing friend ,Dave Rearick, was a math prof at the U of Colorado.
Quoting TonesInDeepFreeze
But you are on a philosophy site. And one that is far from rigorous in its willlingness to allow all types of speculative thought. So no need to continually clutch your pearls in shock at the rude habits of the nasty natives.
More productive would be to engage in the deeper issue being expressed - the fact that physics may indeed be in ontological conflict with maths on the issue of whether infinity is usefully thought of as potential or actual. The result of a generating process or the existence of an actual value.
The idea of a largest number makes no sense to either of these positions. But the idea of largeness being arbitrary does.
On the physicalist side of the debate, the ontological argument is that reality is relational. It is born of a structure of constraints. Stability cant be taken for granted as existence is a process - the process of instability being stabilised.
In that light, we can well wonder what the numerical value of pi is in a real world where we are not even sure if the metric is flat - just flat to some currently measured degree of precision.
Likewise, if the values of numbers are being understood as the limits of terminating processes, it becomes legitimate to ask if all the irrationals have generating algorithms, or just some of them, like pi? So what is going on there.
If you come on to a philosophy site, you ought to expect a traditional rivalry - such as that between potential and actual infinity - to be treated as an open and interesting question, even if the level of discussion is lay.
I've said much of this before:
I do not object to broad and speculative philosophy, even regarding mathematics. And, of course, I appreciate that there are alternative mathematical systems and motivating philosophies - constructivist, predicativist, finitist, dialtetheist, etc. - and that it is philosophically relevant to compare mathematics with the sciences. But when a poster also brings the actual mathematics into consideration, especially to critique it, then the poster should not mangle or misrepresent that mathematics. Certain posters do this serially (over months and years) and willfully. To ignore multiple corrections and explanations while posting, over and over again, confusions and falsehoods about the actual mathematics is intellectual abuse. Crankery corrodes knowledge, understanding, clarity and communication.
You mention what is constructive to post. Crankery is dishonesty and it is destructive. It is honest and constructive to flag that dishonesty and destructiveness. (Anyway, a poster's intent is very often not to be constructive but rather merely to be expressive.)
To reiterate: I have no principled objection to people philosophizing as wildly as they want, but when they cite actual mathematics as part of the subject of that philosophizing, then they should not, from ignorance and self-misinformation, be posting falsehoods and confusions about that mathematics. And while I don't begrudge them the prerogative to do that, I don't begrudge myself the prerogative to refute it and denounce it.
Sure. But again, this is you now setting your self-appointed standards for the site. And there are moderators who actually are responsible for deciding the limits of tolerance.
So I understand your point of view. But pragmatically, have you achieved a measure of success? By your own admission, hammering on @Agent Smith doesn't seem to have the desired effect. You can set him as much reading homework as you like, but that becomes performative if you have no real expectation anything will change.
Quoting TonesInDeepFreeze
This speaks of a joyless rigidity to life. And you are mistaken if you think "actual maths" trumps "actual philosophy" on a philosophy website. It's a simple category error.
The site could crack down harder on folk's critical thinking skills, of course. :grin:
I don't set any site-wide or general standards. Rather, I comment based on my own standards.
Quoting apokrisis
Moderators don't censor cranks. And I don't advocate that moderators censor cranks. But that doesn't entail that I must refrain from my own comments about cranks.
Quoting apokrisis
Cranks never* come around to reason and knowledge. They go on for years and decades. They are, by nature, deeply stubborn and are narcissistic in criticizing the profound intellectual developments in mathematics over the last 200 years without feeling the slightest need to study to know anything about those developments.
But I find it worthwhile to post the corrections to the record. I don't have an inflated sense that this makes any "hill of beans" difference in the outcome of humankind or the world. It's just, for me, satisfying, even if only in principle, to articulate and enter my comments. And I believe it is constructive to do that.
Quoting apokrisis
Absolutely the contrary. I find real joy and solace in the fact that people are free to post what they like and I am free to reply as I like. Rather than the rigidity of disallowing people from posting as they like, I celebrate the freedom that they may post as they like and that I may reply as I like.
Moreover, it is presumptuous to extrapolate to life in general. Posting refutations to cranks doesn't nullify the joys of life such as friendship, art and study. Moreover, I find posting as I do to be a mildly gratifying pastime.
Quoting apokrisis
It's good that you phrased that as a conditional. Because indeed I don't think mathematics trumps philosophy. I said nothing like that. What I said, if you would be so kind as to reread it, is that when we bring in the actual mathematics as a subject of the philosophizing, then we should not misrepresent the subject we are talking about; we shouldn't misrepresent or cause confusions about how that mathematics is actually formulated.
/
* I've witnessed just one exception. Years ago, in a different discussion group, there was a poster who was abysmally confused about set theory, yet he kept posting ignoran, and blatantly incorrect purported proofs that set theory is inconsistent. Later, he kept popping up alternative formal proposals, but they were ill-formed nonsense. For a long time, I offered him suggestions on how he could fix his formalizations so they made sense. Gradually, he started to get the hang of it. Then it became clear that he is brilliant. Once he got his feet on the ground, he was able to propose a number of meaningful and interesting alternative foundational systems, to the point that he proved a result that was published, as a prominent logician took him under wing. Years ago he went way past my own meager knowledge, so that he talks about advanced subjects I can't even begin to keep up with. (Yet, he never seemed to let go of the habit of making large claims without adequate support or rigor; but he would retract and admit that he did have more work to do on it when it was pointed out to him.)
You're not. Instead of thinking for even a moment about the corrections and explanations given you, you keep popping back to prorogate your misunderstandings and flat out errors about the mathematics you comment on.
I'm questioning your definition of "constructive".
Quoting TonesInDeepFreeze
Now reading further back in the thread, your excuses seem even thinner. Don't you realise that the more dogmatic you become, the more flippant will be the reply in this situation. That is how the dialectics of real life social interaction works. So you are producing the very thing you are complaining of in the end.
Rather than stamping out crackpottery, you are fanning its flames. :clap:
This is similar to my diagnosis.
I find myself as frequently frustrated as Mr. Tones with respect to the mathematical, logical or other formal/technical errors that are somewhat frequent on this forum. However, a rude attitude seldom yields anything productive: there is the option of politely leaving a discussion, perhaps at no fault of your own but the inadequacy of your interlocutor, or explaining their mistake at a reasonable level.
"Fanning the flame", so to speak, is unnecessary in whole. As a wise Greek philosopher said:
I suggested the sense I have in mind.
Quoting apokrisis
I don't make excuses. My posting doesn't require excuses.
Quoting apokrisis
In this thread, I might have made a certain major error about the other poster's position. As mentioned, I would retract that error if needed clarification shows that indeed I was in error.
Quoting apokrisis
What dogmatism do you refer to?
Quoting apokrisis
That seems likely. It is a Catch-22. If one refrains from commenting on crank posts, then cranks more greatly dominate. But if one comments on crank posts, then that also causes cranks to post even more.
My approach has usually been to first simply state the corrections and sometimes to add explanation. But then the crank replies with even more ignorant, confused and intellectually dishonest posts, and usually skipping even recognition of the most substantive parts of the rebuttals (form of strawman). An indefinitely long back and forth ensues. And at a certain point, what becomes most glaringly pertinent is no longer the subject itself but the cranks' inability (or refusal) to reason.
I accept that. Again, I don't claim that such engagement is constructive in the sense that it leads to the crank desisting from posting disinformation. The crank will continue to post disinformation whether he is left alone to do it or whether he does it in response to being rebutted about it. Rather, I find that it is constructive at least to have on record that the crank was rebutted. That is the best that can be achieved.
The very first time I post to a crank, I don't impart attitude*. I simply post the correction and sometimes added explanation. But after a while, the crank gets even more dishonest, especially by countering as if he has rebutted the corrections while skipping the most important substance of them. Or, often, the crank becomes snide or condescending. It's often a key element of the crank's bag of tricks to impart that dismissiveness, condescension and putdown so subtly that the crank is ostensibly on high ground while still getting his digs in.
At a certain point, what becomes most glaringly pertinent is no longer the subject itself but the cranks' inability (or refusal) to reason and his abject intellectual dishonesty.
AtQuoting Kuro
Keep in mind that these are not mere technical lapses, but basic and ongoing critical systematic disinformation.
/
* Possibly there have been an exception or two?
I agree with that.
But I don't agree with the utilitarian framework you apply here*. First, I don't think utilitarian result is the only consideration. Second, for utility it doesn't matter anyway: The crank will continue to spew disinformation no matter whether left unresponded to or responded to with correction.
* I don't claim you adhere to utilitarianism. I am just saying that in this particular context your framework is utilitarian.
I think you can even be rude if you are funny with it. And hammering a crackpot could be constructive if you explain yourself well enough that others are engaged and learning something.
I think being too nice can be unproductive as well. Discussions have to have energy.
But this particular interaction seemed just an attempt to whack the stubborn pupil over the head with the textbook.
I mean it is not as though "actual mathematics" doesn't have textbooks by Norman J Wildberger as well. :rofl:
[No slight on Wildberger who I really enjoy.]
I know of him. Watched his History of Mathematics on youtube. If memory serves, he's a finitist. His reason, again if memory serves, was that the close bracket "}" doesn't square with the ellipsis " ... " preceding it in {1, 2, 3, ... }. I feel him in a way, but that's about it!
Formalization, it's a big deal! TonesInDeepFreeze seems well-versed in that department. Hence, I suppose, his annoyance at my rather informal approach to math.
Nope. Over many months (a year or even more?) I have offered the poster the copious explanations. My purpose is not to merely rhetorically bludgeon, but rather it's as I described in previous posts. But, as I mentioned, after a certain point, it emerges that the poster is unwilling to desist from posting disinformation. At that point, I am not obligated to just give up posting the corrections and also to point out that the source of that disinformation is crankery.
Quoting apokrisis
Wilderbeger in a particular aside, you skipped that I already said that I am enthused that there are alternative mathematics. I have never, in any way, advocated that the only reasonable context is ordinary classical mathematics. Indeed, I very much appreciate that there are reasonable intuitive and philosophical quandaries about classical mathematics. On the other hand, self-malinformed cranks dogmatically claim, with terrible ill-reasoned arguments, that ordinary classical mathematics is not acceptable by any means or has certain problems (that it actually does not have).
So, rather than me be burdened by a strawman, I'll reiterate:
(1) I do not urge that all philosophical and mathematical discussions be anchored with respect to classical mathematics.
(2) I have never said anything that can even remotely be fairly construed as a claim that mathematics trumps philosophy.
(3) I am enthused that there are alternative philosophies and mathematics.
(4) I'll add that I admit my own personal framework for understanding mathematics is not itself a developed philosophy of mathematics and that there are quandaries that I don't claim to explain.
(5) So I am quite the opposite of dogmatic concerning mathematics and philosophy of mathematics.
(6) The crank, though, dogmatically, and from a position of ignorance on the subject, claims that classical mathematics is wrong.
(7) When the crank proposes a philosophical discussion in which classical mathematics is to undergo examination, then when the crank (as invariably he does) misstates what that classical mathematics is or how it works, then it is not improper to correct the crank about that, no matter how long he persists. And it is not mere formal nitpicking to do that.
:lol: I can vouch for that!
That is blatant strawman. Especially after I have over and over stated and explained that I don't object to informal rumination in and of itself. What I object to is making false claims about the actual mathematics you presume to criticize without understanding even its basics..
Please do not persist to so egregiously mischaracterize my point of view, as your mischaracterization is blatantly refuted by the actual posting record.
[quote=Ranjeet]A thousand apologies.[/quote]
You deploy that "aw shucks" shtick repeatedly. It's so transparent.
Emoticons are as eloquent as you ever get.
That is quite missing the point. Wilderberger, as I glean, is an ultrafinitist.
Finitism has a broad range. Hilbert was a finitist, yet Hilbert famously celebrated infinitistic mathematics. And that is not a contradiction, as one would see upon learning the specific sense of Hilbert's finitism.
So what is salient about Wilderberger is not just that he's a finitist but that he's an ultrafinitist.
What's that?
I'm emotional.
So am I. That's why I usually find emoticons inadequate and ugly.
Sometimes they're useful, but yours directed to me strike me as mere insouciance.
You're joking, right?
It's pretty much at the basis of this thread you started.
And you ask me without even bothering to type it into a search?
You're trolling.
Tones! Your audacity is unbelievable. You took two months to reply to my post. After two months, you called me back to defend what I wrote. After two months! Then, you never once addressed my defense, which I wrote in the last couple days, only referring to it as confused, ignorant, dishonest, rambling, obfuscations. What the fuck is the point? If I'm wrong, show me where!
You practice nothing but bull shit! Please, for the love of all who participate in this forum, shut the fuck up!
A point of logic. How is the statement that Wildberger may be a finitist rendered untrue by him being also some subset of that set?
Im sure that is a basic error that needs pointing out to stop the internet descending into crackpottery and ignorance.
What was the salient point in the discussion that demanded a need for the further distinction?
There is no expiration on a poster's prerogative to reply.
Quoting Metaphysician Undercover
That is false. I did not ask nor suggest that you reply. I stated no preference whether you replied or not. Indeed, it would be just as well if you hadn't replied, since your replies are invariably even more obfuscation and sophistry.
You're putting words in my mouth again.
And now you're flat out lying about my posting.
I gave specific analysis, quoting you, and arranging my argument about it in conspicuously clear formatting.
I didn't say it was untrue. I said the generality of him being a finitist misses the most salient point about him that he's an ultrafinitist. And I didn't even say the poster is personally remiss in that regard. I merely pointed out that we can address the reference to Wilderberger more specifically than generally.
Please do not continue to misconstrue my plain words to the point of essentially strawmanning me.
Quoting apokrisis
Now you have become egregious. I didn't claim that the poster's comment was crankery. I merely added that his comment admits of additional specificity. That I have crossed with the poster in other posts and said that he posts crankery does not entail that every reply to him includes a suggestion that the particular post is crankery.
Please stop essentially putting words in my mouth.
Quoting apokrisis
It wasn't demanded, but it is helpful.
The topic of this thread is whether there is a greatest number for practical purposes. That goes to utrafinitism. So it's quite pertinent to mention that alluding to Wilderberger in this discussion suggests highlighting that he's not just a finitist but an ultrafinitist.
No, i never said you "asked", or "suggested" that I reply. It's actually you putting words in my mouth, just like in your false accusation that I contradicted myself
Quoting TonesInDeepFreeze
This is what you always end up saying to me. You're a liar! Let me remind you of your own words:
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
And my words:Quoting Metaphysician Undercover
Looks like I quoted your words correctly, unlike your lame reply.
Quoting TonesInDeepFreeze
No, all you did was falsely accuse me of contradiction through a stupid strawman. You never even considered what I actually wrote.
You wrote:
Quoting Metaphysician Undercover
But I did not call on you, or call you back, to defend what you wrote not did I call on you, or call you back, to do anything at all.
But maybe what you meant is that merely you felt, upon your own sense, that you should defend what you wrote. Not that, as you literally wrote that I called on you to do that. In that case, fair enough. If I should not have taken you literally, then you may consider my previous comments about that now scratched.
/
Quoting Metaphysician Undercover
Talks about audacity!
You don't speak for all who participate.
You are just repeating that falsehood, ignoring that there are actual posts in which I discussed your argument in detail, quoting it and analyzing it explicitly.
So you made a pointless point as if you were adding some significant and necessary correction to the discussion.
Of course I do get why you felt it worth saying. This is how you troll @Agent Smith. If he says something true, you have to make it seem untrue by pretending some even more sophisticated textbook distinction - one beyond his ken - was at stake.
He trolls you his way, and you troll him your way. And so it will continue. :ok:
I respect your judgment although I really haven't the foggiest how to troll.
Ok, so just out of interest then, If you were not calling me back to defend my statement, then what were you doing with that false accusation (two months later) that I contradicted myself two months earlier?
Quoting TonesInDeepFreeze
When someone publicly accuses me of contradicting myself, I take it that I am being called on to defend myself. If you were not calling on me to defend myself, then what was the purpose in your false accusation, simple defamation?
Quoting TonesInDeepFreeze
There was no contradiction. Go back and reread please. And only reply when you can demonstrate that you have understood what I wrote. No more of your whiny behaviour, please. You poor pathetic baby, my writing is just too confused and "obfuscatory" for you to understand, so you'd better just dismiss it as contradictory. Come on, grow out of the diapers.
And if you cannot understand philosophy don't bother replying please. Go read some books first!
It had a good point. This thread is about a greatest number. Wilderberger is an ultrafinitist. By pointing that out, we see that he's especially relevant to this thread.
Quoting apokrisis
I made no such pretense that it was necessary. You keep essentially putting words in my mouth. It's beneath you.
Quoting apokrisis
I decidedly do not troll. I do not post to provoke reactions, and especially not for gratuitous reasons. I do not post to confuse or gaslight people.
By merely commenting that we gain by a more specific designation, I am not at all trolling.
Meanwhile, it seems to me that I my postings have rankled you to the degree that you've lost your balance so that you read into my posts the worst interpretation even though that interpretation is false, and your imbalance is causing you to now strawman me as a pattern.
Quoting apokrisis
I did not suggest at all that it was untrue or even that it seems to be untrue. I addressed that in a post a few back. Please stop claiming that I've said or suggested things I have not.
Stating my own argument about it.
Whether you reply to my statement of my argument about your contradiction is entirely your own prerogative.
And, since you seem now to be leaning toward a claim that I called you back to defend your argument, I withdraw my comment that I might have taken you too literally.
Um, you can google ultrafinitism if you don't know what it is. It is clear the OP is an example of ultrafinitism. I suspect the original poster knew that when they started this (and has been laughing all along).
It was apokrisis who first mentioned Wilderberger in this context.
Then Agent Smith replied that Wilderberger is a finitist.
Then I mentioned that, more particularly, he's an ultrafinitist.
And for my doing that, apokrisis is ludicrously reading into my comment.
There was no argument made by you, just a blatantly false accusation, that my previous post was contradictory.
Quoting TonesInDeepFreeze
You clearly demonstrated that you never even understood what I said, calling it obfuscation. Now you confess to not knowing whether you took it literally or not. Admit the facts, you have no understanding whatsoever, concerning the ontological principles I stated. So you decided your best course of action was just to publicly state that what I said is contradictory, (even though it clearly is not), instead of asking me to clarify. That is bull shit!
Yet again you repeat your false claim that is refuted by content of the posts.
Quoting TonesInDeepFreeze
Your notion of mentioning is as disingenuous as your definition of being constructive. We can leave it there.
Granted, I didn't just say that Wilderberger is an ultrafinitist but that it is quite missing the point to refer to him merely as a finitist. In my post just quoted, I didn't intend that my saying that I mentioned that Wilderberger is an ultrafinitist was meant to elide that I also said that it is quite missing the point not to more specfically say he's an ultrafinitist.
That you read into such details that I am dishonest for referring to myself has having mentioned something is nutty.
Quoting apokrisis
I gave no definition. I just said that there is a sense in which I think posting replies to cranks is constructive. And I as much as happily granted that that is limited.
Yep. Just trying to help you counter the three-headed anti-modern-math beast.
The OP is too clearly an example of ultrafinitism to be an accident. All protestations aside, I think someone is trolling.
What do you mean by "ultrafinitism to be an accident"?
Quoting Real Gone Cat
I think Agent Smith is sincere that he thinks there should be an alternative to infinitistic mathematics. And he thinks he's contributing to that goal when he broaches certain considerations and (what are to him) juicy tidbits about mathematics.
Where he is not sincere is in not bothering to learn very much about the subject while he serially misrepresents the infinitistic mathematics he objects to.
I sense that where he's actually trolling is in his cutesy, self-effacing, would-be disarming, [parody:] "Aw shucks, I'm just a poor country boy educated in an old one room country school" shtick.
Quoting Real Gone Cat
That is nuts. You think @Agent Smith is pretending to be a finitist when he is really an ultrafinitist at heart?
This little drama sounds more and more like that scene in Life of Brian
That's addressed to Real Gone Cat, but for myself, I don't think he's pretending that. He doesn't even know what the scope of the rubric 'finitism' includes, and he didn't even know what ultrafinitism is until (maybe) he looked it up an hour or so ago.
Anyway, it's not clear to me what his view is: Does he grant that there is no greatest number, while suggesting that there is a greatest practical number? Or does he hold that there is a greatest number period?
:snicker:
I find that really amusing! I'm afraid my acting skills aren't up to the mark, I wouldn't be able to pull off such a stunt.
All of the above. :grin:
His position is vague. But one can develop it in a fruitful direction - like ultrafinitism.
The enquiry can be fruitful. But poorly so if the vines are mangled and torn apart by misinformation and unnecessary confusion.
Muchas gracias for understanding.
I recall @andrewk (hope s/he's well) said something to the effect that we could be like the finite calculator I hypothesized in my post to @jgill [a calculator that can't calculate beyond 5 would display 5 (the arbitrarily large number) for both the queries 2 + 3 = ? and 3 + 29 = ?]; he asked, paraphrasing, how can a finite brain grasp infinity? What follows is there hasta be a finite number to think of which would require every single neuron in the brain to be activated, let's call it N[sub]max[/sub]. Any number larger would be, in a way, truncated to N[sub]max[/sub] whatever that is (a cognitive numerical ceiling).
[math]\infty[/math] = ERROR!
All this hullabaloo about an error message!
Depends on what 'grasps' means. If it means mentally seeing an infinite number of concrete or abstract objects all at once, then probably it can't be done.
But if it means grasping that a certain property is held by more than a finite number of objects, such as we describe that situation as there being a set of all and only those objects, then I grasp that easily.
In that regard, I have little difficulty in grasping, for example, that there is the set of all and only the natural numbers. And I am not dissuaded by such mere facts that, for example, there is a bijection between the set of even natural numbers and the set of natural numbers. I don't find that baffling. Especially I don't think I need to reject that fact only because at an earlier stage in my life, as a child, I didn't think about such things so that I might have mistakenly jumped to the conclusion that a proper subset of a set T cannot be in correspondence with T (though that incorrect intuition wouldn't have been stated in those terms when I was a child).
[quote=Ms. Marple]Most interesting.[/quote]
I did? I'm old. Maybe I was closing up shop for the night. :chin:
Then there is the heuristic question: What practical advantage would there be in throwing out the methods of infinitistic calculus for an ultrafinitist approach? Again, how is it any better for me to have to know what degree of accuracy my friend needs when instead I could just say, "pi is the value" and let him use whatever accuracy he needs in its practical application?
And, from a formalizing perspective, we can pretty much anticipate that an ultrafinitist system would have a much more complicated axiomatization.
From an aesthetic point of view: At least for me, it is unbeautiful to pick some particular number, either arbitrarily or on the basis of some physicalist constraints, to be the greatest number. I lean toward being more sympathetic with a system of greatest possible generality, not particularized by whatever physical science determines at any given point in the developments of the sciences.
But there are puzzles I find with infinite sets in certain aspects of mathematical logic. I admit that my framework of understanding hits a wall in those particular aspects.
:lol: :up:
Is [math]\infty[/math] like God as Cantor believed? Lt. Worf said of Klingon gods, "We killed them. They were more trouble than they were worth."
I'm no mathematician but I believe it's a simple rule of thumb that if a mathematician wants to propose a new idea, s/he'll use [math]\infty[/math] only if absolutely necessary and that too with much reservation.
When I said "too clearly an example of ultrafinitism to be an accident", l was implying that these ideas most likely originated from an ultrafinitist source. I suppose you could arrive at such a conclusion on your own, but not if you've taken math at university, or even high school. The opposite is taught in math today. (Isn't this why you keep admonishing AS to get some learnin' on the subject?)
Not telling you anything you don't know, but adopting ultrafinitism leads to some odd results
Why am I nuts?
Ultrafinitism is usually defined as the belief that really large FINITE numbers do not exist because of constructive limits - what is physically realizable in the universe. And what does the OP posit? That there is a maximum number needed to describe the universe. Anything bigger doesn't need to exist. Did I get something wrong?
First, as I've said ad nauseum, (with exceptions of figures of speech) mathematics doesn't have a noun 'infinity' but rather an adjective 'is infinite'.
You keep using 'infinity' as a noun. Okay, then what is your definition? What object do you think is named by the noun 'infinity'?
And as I've said ad nauseum, infinite sets are necessary for the ordinary system of the mathematics for the sciences. There are alternative systems, for many approaches, but I don't know of an actual ultrafinitist systemization. (Maybe there is one? Someone can adduce one? Quite some time ago I did read through a lot of Lavine's book, but I don't recall how formally satisfying his proposal is.)
Quoting Agent Smith
You skipped what I wrote about that. That is your M.O., in true crank style: Skip responding to points that don't support your own position and instead just keep reiterating your position.
Quoting Agent Smith
Oh come on, why do you keep harping on something that died over a hundred years ago? Mathematicians, at least in their actual formal mathematics, don't relate to the part of Cantor in which he equated an Absolute Infinity with God. That's just not on the table. Forget about it. It's a huge red herring.
Quoting Agent Smith
When you make claims like that, I ask you to provide your basis. By what actual evidence, from what actual sources, do you make that claim?
It's patently false anyway. Many mathematicians freely make use of infinite sets. All over in analysis and other other branches. Not to mention set theory itself which is a branch of mathematics. Day one of even a high school class: The real number line. That's an infinite set of points ordered by the less than relation. Day one of freshman calculus: A sequence converging to a limit. That's an infinite sequence. Infinite sets are as basic to ordinary mathematics as wheels are to automobiles.
You're just making claims straight from your smoke blowing orifices, with no regard, no sense of responsibility, for supporting those claims or even thinking for one moment whether you are right about them. Pure crank.
I say he should learn about the subject so that he'll desist from spewing misinformation about it. He fancies himself a critic of classical mathematics but he persistently makes false claims about it. One might as well go into a biology discussion group and say, "Academic biology is all wrong. Just look at the concept of a cell. They say a cell is an organ of the body. That makes no sense. Look at their concept of carbon. They say carbon is the fluid that runs through veins. How much more wrong could they be?"
If you know a thing or two about that quote you agree with, you'd know that I'm not employing a utilitarian framework here* (though I'm delighted that utilitarians and Kantian deontologists alike would also agree with me): this is simply the framework of Hellenic virtue ethic theories, and I'm saying that, in a less blunt way, that in not acting rightly one retracts from the virtue of their character, which is less so a property of any particular fault or flaw in action rather than those habits which become second nature to them and come to form their decisions & methodology (which is what the discussion came about, the generalized implications versus the particular one, hence the relevance of the quote).
*I'm not saying this to imply that what I said disagrees with what utilitarians or deontologists have to say, just that I'm not myself using their theories.
If you believe @Agent Smith was being that cunning and conspiratorial in framing his OP.
What would be the point?
My main point is that I don't accept your argument of posting in terms of what is "productive" or outcomes ("fanned flames"). Evaluation per what is productive or outcomes, as I would think in that framework, productive outcomes are good and unproductive outcomes are at least less good. That is what I meant by a utilitarian point of view. My comment about the quote was that I agreed with that part of it that I quoted (no comment on the rest of it), but also I don't accept the "utilitarian" (the productivity and outcomes of posing) part of your argument; I did not all intend to suggest the quote itself is from a utilitarian point of view. Indeed, I would have no reason to think it is.
Is that clear? There's the part of the quote that I quote, which I agree with. And there was another part of your argument that I don't agree with. I don't take the quote to be from a utilitarian point of view. But I find an productivity and outcomes argument to be, at least generally put, utilitarian.
But I accept that you don't consider 'utilitarian' as a correct description of your productivity and outcomes argument. Indeed, my point doesn't rely on the particular rubric 'utilitarian' but rather that I reject your productivity and outcomes argument, whatever rubric it correctly falls under.
Also, I don't claim that evaluation of the merits or advisability of posts cannot include productivity and outcomes as a factor. Just that I don't think it is in and of itself determinative and may be trumped by other factors including even a poster's own prerogative to express himself and as a post may offer others access to expressive and cogent and/or well articulated thoughts even if they are not productive in the immediate context of a particular posting exchange. Even further, sometimes it is simply fitting that a poster gets stuff off his chest.
You're correct, of course. I apologize to AS.
I now realize that AS is asking for something even more restrictive than the ultrafinitists. Correct me if you disagree :
Finitism rejects the existence of an actual infinite set (sometimes, as in the case of Hilbert, allowing for a potential infinity). Ultrafinitism rejects the existence of very large integers (sometimes positing that only what is constructable in this universe, or by humans, exists) - and by extension, irrationals. AS rejects any number that is not useful to humans, such as Pi beyond a few decimal digits. Ultra-ultrafinitism?
My bad.
That does have some sense if you wanted to play with it as a conjecture. What would be the best rebuttal? Would we start by saying we couldnt limit the scope of future human ambitions and the numbers that would involve?
There is a line of thought if you wanted to be serious about dismissing it.
Likewise he made the point that animals and even tribal human cultures dont require a number sense beyond one, two, a few and many. So where does that leave our love affair with an unbounded capacity for assigning distinct names to an infinity of values?
Again a perfectly fair start to a discussion that can draw on plenty of scholarship.
How often do random or confused starting points lead to entertaining discussions on PF? Far more often than the dogmatic exposition of something anyone could read up as the received view in an undergrad textbook I would say.
So yes, give AS a break. :smile:
The claim was not false, as I showed with a number of explanations. You would not even address the content of my replies, insisting they're too confused, obfuscatory, rambling, dishonest and ignorant.
In other words, you could not understand what I wrote, and instead of asking for clarification you decided that it must be contradictory, so you just kept repeating that it is contradiction. That's not only disingenuous, it's bull shit!
Quoting TonesInDeepFreeze
How familiar. Now I can call you a "crank", by your own description.
Quoting Real Gone Cat
There's a very simple ontological principle to employ. Potentialities do not "exist". To "exist" requires actuality. So when I say tomorrow I might build myself a new chair, that chair does not exist because it's only an idea, a potentiality, or possibility, without actual existence.
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
OK, let's see if you can clarify what is meant here to make sure you are not contradicting yourself. "Infinite" is an adjective. There is something which is said to be "infinite" and that is a "set". Now, a set is not an object, otherwise the set would be an infinite object, and this would be an infinity. But "infinity" as a noun, is not a word used in mathematics, so a set is not an object.
So what type of existence does a set have? It's not an object, or else it would be an infinite object (referenced by the noun "infinity"). What is this 'thing'?, called a "set", which we cannot say is a thing, referenced with a noun, otherwise it would be an infinite thing, i.e. an infinity. How is it that "infinite" can be predicated without a thing to be the subject of that predication?
Now the nature of infinity is an interesting topic to explore!
I have much empathy for your position. Since our lives are finite, it seems impossible to experience an infinite set. But it may surprise you to know that many mathematicians today believe that actual infinite sets exist in math!
I stated earlier in the thread, what I believe to be the reason for the concept of infinite:
Quoting Metaphysician Undercover
Quoting Real Gone Cat
That doesn't surprise me at all. I've had numerous discussions in this forum with mathematicians, and I've already been well exposed to the absurd ontology which seems to be exclusive to that cult.
It correctly falls under 'contemporary analytic virtue ethics', and is not consequentialist in the substantive sense of the word but the trivial one (the trivial one where all ethical theories are 'consequential' in that Kantians would care about that consequence of violating the imperative, or Aristotelians about virtue, and so on). The substantial sense of 'consequentialism' and 'utilitarianism' (which falls under the former) is not as broad, and the argument translates to the fact that your actions fall short of moral excellence (in that indignancy fail prudence and justice): indeed, no virtue ethicist would ever grant you that your personal frustrations offset this.
Yes, we're a wicked bunch intent on the corruption of the intellects of youth in order to bring them to the alter of our Satan, Paul Erdos RIP. All bow.
[math]0 + 0 = 0[/math]
[math]\infty + \infty = \infty[/math]
[math]\infty[/math], rather [math]0[/math]ish!
Supertask (James F. Thomson of Thomson's lamp fame); What o'clock is it after [math]\infty[/math] time has elapsed?
Not all mathematicians are the same, just like not all theists are the same. It's just that some are fanatics with a cultlike attitude, who are inclined toward professing absurd ontologies to support their beliefs.
I had to look up Paul Erdos, to see that he is famous for his work on Ramsey theory. Seems like Erdos was very socially active. Is he responsible for the famous notion "six degrees of separation"? Or was he just paranoid about aliens? I see you can still earn money by solving Erdos' problems. Have you ever managed to get any reward?
I gotcha!
[math]\frac{5}{5}[/math]=1
and
[math]\frac{8}{8}[/math]=1.
What about [math]\frac{0}{0}[/math] or [math]\frac{\infty}{\infty}[/math] ? Gotta be 1, right?
0 and [math]\infty[/math], rather 1ish!
A little advice, friend : get ya some math learnin'
What surprises me about our math-phobic friends on TPF, is that philosophy majors usually love the esoteric. You would think they would revel in knowing more about mathematics than the Great Unwashed. Instead, they make up their own notions and denigrate 5,000 years of developments by some of the greatest minds that have ever lived. Weird.
I like the patterns you found. They're natural extensions of numerical operations on numbers and they are kinda sorta right. Kudos to you for that.
P. S. Does that mean 0 and [math]\infty[/math] aren't numbers like 1, 3, 2938, 10[sup]100[/sup]?
0 is a number.
I don't understand why you keep skipping my point that using the leminscate as if it stands for an object makes no sense the way you do.
Unless you mean the point of infinity in the extended reals, though I don't think you are even that specific. But even as a member of the extended reals, the point of infinity is not necessarily itself an infinite set. It can be any object that is not a real number. Then we define an ordering using the standard ordering on the reals but setting the point of infinity as greater than all reals.
Means "Whatever, dude", eh?
Agreed that many of our TPF worthies misunderstand what is meant by [math]\infty[/math]. But as a minor defense, we often do get sloppy with our use of this symbol. In calculus classes, for example, when a limit evaluates as [math]\frac{\infty}{\infty}[/math], we tell students to try applying L'Hopital's rule. I doubt AS has this usage in mind.
It's a conceptual issue. He seems to think that, because mathematics is infinitistic, it has a thing that is called 'infinity'. As if the leminscate stands for that thing like the golden arches stand for a hamburger place.
I don't know about the six degrees or aliens. I met him once and talked with him briefly at a meeting in Hungary years ago, but I am not in that select group of mathematicians who have "Erdos numbers". He stayed with my advisor for a week long ago as he traveled around the world, giving talks and working with colleagues. Universities supported him on his visits to their campuses. He lived out of a suitcase and shopping bag and, while his mother was alive, stayed at her apartment in Budapest on and off.
Their is a curious parallel in the world of climbing. Fred Beckey, who died in 2017, was the most prolific climber in American history. Like Erdos he lived a vagabond life, sleeping on the sofa of whoever he was visiting. I knew him slightly and we bouldered together occasionally. I suppose there could be something like a "Beckey number" earned by doing notable climbs with him. :cool:
:lol:
Yes, they certainly like to talk about what Kant meant or how Aristotle would know, but math is more precise, with less room to wiggle. :nerd:
Another interesting discussion topic (and perhaps what the OP was alluding to) would be the distinction between pure math and applied math. And the usefulness of the former.
And now in support of the much-maligned OP - a quote from Marc Rayman, a chief engineer at NASA (with the coolest name any employee of NASA ever had) :
He's "The Crank With The Friendly Face".
If we agree that there's a largest number we'd ever need to use, then still, what's the harm in having all the rest of the larger numbers in the attic, just in case we ever feel like looking at some larger ones, even if only to play around with them?
I wonder whether the debate lines up with certain kinds of personalities.
One personality type just cannot stand that there are infinite sets. It terribly rankles them that we would allow such a sweeping abstraction. It goes against their way of looking at mathematics as expression of our experiences as a stream of immediate and concrete based perceptions.
The other personality type just cannot stand limiting mathematics to particulars. It bugs them that we would cut off the numbers at some arbitrary point (or a point merely estimated by current cosmological theories) in the succession of numbers. It goes against their way of looking at mathematics as being of greatest abstract generality.
For me, ultrafinitism is ugly that way. I'd rather study mathematics that is not embroiled with a bunch of messy physical stuff like how small an atom is and what is it's rounded-off length, given in a bunch of base ten digits. It's ugly to me to say, "The greatest number is 492^(3327989025)" or whatever. It's so ... choppy.
Oh, I'm no finitist. As I've pointed out, finitism (or worse, ultrafinitism) leads to some odd results : you have to truncate [math]\pi[/math] (which turns circles into polygons), you have to deny irrationals, you destroy the foundations for calculus, lines no longer consist of an uncountably infinite set of points, etc.
You may be onto something about personalities. I think it's also possible that one of the sources for the OP may have been some half-remembered quote such as the one from the NASA engineer, above. But perhaps I'm being too generous.
NASA telling me what's the largest number I can use is like the Department Of Agriculture telling me how many taste buds I can have.
That's a list of features, not bugs. It's long past time to drive the Platonists out of maths. Pragmatism makes for a sounder metaphysics when it comes to how one would model reality. :smile:
I don't believe that infinitistic mathematics requires a platonist commitment.
Of course not. It's not a formal claim.
Anyway, I think the burden of argument is on the side saying that infinitistic mathematics does require a platonist commitment. There are powerful arguments for that side. I very much respect that. But there are two different questions:
(1) Can one have a cogent philosophy of mathematics with infinite sets but that is not platonist? I don't have a firm opinion on that.
(2) Can one work in ZFC without committing oneself to platonism? That's more an empirical question. Such mathematicians as Abraham Robinson do*. And Robinson's own explanation might be pretty good. (but If I recall, I found it not to be entirely glitch-free). For myself, even though I am not a mathematician, I happily study ZFC without having the platonist commitment that the abstract objects of mathematics exist independent of mind.
* A fair number of set theorists do, as does Robinson, say that the notion itself (not even just the existence) of infinite sets is literally nonsense, yet they work in ZFC and recognize its fruitfulness.
I was teasing. If you gatecrash a comment, you could at least have the courtesy to set out your reasons for your assertions.
Quoting TonesInDeepFreeze
So you reveal yourself as a pragmatist. Infinity is a useful idea as far as it goes in the real world of doing things like deciding whether some circle is in fact a very fine construction of flat lines, or a polygon is in fact a very fine construction of flattish curves.
Truncating pi is practical. One can be a finitist and it looks exactly the same as being an infiinitist. Outside the culture wars of philosophy of maths, who could tell the difference? It becomes a difference that doesn't make a difference.
Okay, you were joking with the 'formal' part. Maybe because you perceive me as asking posters to back up with formal proofs? Or you think I can be charactured that way? I don't know. Anyway, of course I don't ask people to provide formal proofs of informal assertions.
Quoting apokrisis
Oh come on. I didn't "gatecrash" anything. You posted essentially a one-liner on the subject, itself not an argument. That's fine. And it should be allowed that one may reply in kind. And even if a poster replies tersely to a longer argument, that's not "gatecrashing" or necessarily even rude or whatever.
Quoting apokrisis
I don't have a philosophy of mathematics; and not one that could be called anything, including 'pragmatism'.
Quoting apokrisis
I am sympathetic to that idea. But I don't personally stake my own understanding of infinitistic mathematics primarily to it.
Quoting apokrisis
Of course no one expects engineers to write an infinite sequence of digits.
Oh come on. You yourself have said you have no philosophy to defend on this forum, just a self-appointed need to police it for its mathematical thoughtcrimes and disinformation campaigns.
Having no philosophy is not a disqualifier. Posting is not paintball where you can't participate unless you you are on one of the teams. Not having a philosophy doesn't entail that one doesn't have meaningful things to say. And I find it refreshing when a person doesn't have a philosophical ax to grind.
There are no thought crimes. On the contrary for me. As I don't hew to a particular philosophy, I don't have strong oppositions to other philosophies; and I relish that there are so many tantalizingly different philosophies of mathematics and formal systems; and I believe that freedom to imagine is to be cherished. Spewing of disinformation though is abundant. Moreover, much of my posting is not just making corrections. Your categorical reduction is false. And beneath you, just as your multiple strawmans earlier
I've seen that cartoon, and it is funny.
Have you noticed how much you assert the negative so as to avoid having to support the positive?
What is life without some form of ontological commitment?
I don't usually support the affirmative because I'm humble enough to admit that I don't have the vision, education, confidence and constancy to arrive at a fixed philosophy. If I were a philosopher, I'd be nowhere in that way. But I'm not a philosopher.
Quoting apokrisis
That is a great line.
Hey. Now you are on a site where you get the chance to learn! Don't waste it.
Your strength is deductive rigour. Pragmatic philosophy stresses that rational thought involves the three steps of abduction, deduction and induction.
A scientific mindset means making the creative leap of forming a hypothesis, properly deducing the general constraints of that hypothesis, then inductively confirming the truth or otherwise of that hypothesis in terms of the observed particulars or practical consequences.
So think of this site as presenting you a true intellectual challenge. There is an arc to thought that transcends all forms of human rationality. If you are strong in one of the three aspects of reasoning, why would you be content with leaving the other two weak?
Can you just make an inspired guess, a creative leap in the dark. Can you follow through a formal model to its practical and measurable consequences?
The deduction of a theory is indeed a formal exercise. But pragmatism explicitly recognises that its inductive confirmation is necessarily informal. And that is OK. Theories make predictions and we find confirmation in the messy real world business of making measurements. We say the facts fit, even if we only have the first two or three significant digits. Three sigma could be good enough if not a lot is at stake.
So when you hammer on posters, some are indeed just fools or cranks. But also, they might at least be discovering something about how to abductively form hypotheses, or inductively confirm their theories.
You clearly are confident in your logical rigour. But Pragmatism tells you that that is only one third of what you need to be a "reasonable person" in this world we share. Time to be properly humble and get involved with PF in ways that challenge your weak spots.
I glean a thing or two here and there. But posting is only a side hobby. I don't have ambitions for philosophy. Sometimes, though, I see things in discussions that I can't resist finding out about more, then I look them up or grab a book.
Quoting apokrisis
Of course.
Quoting apokrisis
I don't know that I am so terribly relatively weak. And I'm never content. But there's so much else I also need to be doing. Forumcombing is itself a distraction that I probably shouldn't allow myself as it eats so terribly into my time needed for my main pursuits.
Quoting apokrisis
That's disastrously overgenerous. I've studied cranks for over 20 years, in forums from here to Timbuktu, and (speaking of inductive inference) one thing is clear: They never learn. They are dogmatic, irrational, intellectually dishonest, and narcissistically self-sure to the core. They persist in their favorite forum for years spewing confusion and disinformation.
Quoting apokrisis
I am reasonable in forums. Almost always, my first posts to a crank are without attitude. Merely a statement of the correction. Then, over time, the crank entrenches with even more dishonesty and often with passive put downs and things like that. Eventually, what becomes salient is the crank himself. And eventually I frankly say what is up with them. Believe me, I have so many times practiced restraint hoping that a crank might, miraculously, come to reason. Never happens. And that is not a function of my style. Hundreds and hundreds of other posters in many forums have tried with cranks, and they always fail to get anywhere with it. Not even a millimeter. Always.*
In this thread, you're seeing only recent interchanges. But there is a context with this poster going back over a year(?) or two years(?). I don't know whether you've read much of those threads. If not, then I would understand that you think my approach is arbitrarily harsh. (By the way, this poster is not as overtly dogmatic as usual cranks. Indeed his skill is to deflect by feigning that he is considering the corrections, which I perceive to be disingenuous.)
/
Your comments in your above post are well taken though. Even if I am in countering mode in this post, probably soon later I'll reflect more and benefit more from your point of view.
But now that you've given me advice, I will return the favor:
Your annoyance with me should not permit you to read into my plain words things that are not in them, not even plausibly, not to willfully misconstrue what I say in the worst way, and not to strawman me over and over as you did yesterday. That is beneath you.
/
* Except that one fellow I wrote about recently. The sole counterexample.
That is my hypothesis. The evidence continues to mount.
Quoting TonesInDeepFreeze
Of course. We are not worthy of you. That makes your pointless asterisking of "dangerous disinformation" even more pointless.
Quoting TonesInDeepFreeze
So someone is different in lacking a prime characteristic of the true crank. And yet you only hammer harder when that someone finds you being overly hostile and wants to laugh you off?
Quoting TonesInDeepFreeze
Great.
Quoting TonesInDeepFreeze
You wildly exaggerate. But it is true I hadn't read any of your posts before this thread.
You did it again! You falsely twisted what I said to reflect it in the worst possible way, little doubt as a spite you're exercising. You are incorrigible.
To say that I have other pursuits, closer to my heart, that I sometimes neglect for posting, does not at all entail that I think that there are not posters worth reading. I could even say that the ghost of Kurt Godel himself visits me each night and wants to give me free lessons, but I can't take him up on it, because mathematics is far from my main pursuit. That wouldn't entail that Kurt Godel is not worthy of me! Get a grip, man.
Quoting apokrisis
He's a variation.
Quoting apokrisis
I have recently.
Quoting apokrisis
Not this time.
:lol:
Quoting TonesInDeepFreeze
Too bad I don't have an emoticon to express that for you.
Just to let you know that I haven't disregarded your post. I wish to give it more thought. I hope eventually to reply.
What, no leminscate to go with that?
Your lack of training in philosophy really shows. And, it is very annoying for a philosopher, when a person without philosophy comes to a philosophy forum, and enters into a philosophy of math discussion, insisting that philosophers ought not discuss the metaphysical principles upon which mathematical axioms stand, if they have not first studied mathematics. Clearly, it is philosophy which is being discussed in the philosophy of mathematics, not mathematics.
Quoting TonesInDeepFreeze
The above is unreasonable behaviour. And, you personally increase the degree of unreasonableness with the use of insult. When you do not understand the philosophical principles being discussed, because you have no philosophy, you simply hurl insults at the philosopher. Try some introspection, to reveal to yourself, your unreasonableness. You may find the way toward respect.
For instance take the Goldbach conjecture. If the N[sub]max[/sub] were 10, I would do the following
4 = 2 + 2. Check.
6 = 3 + 3. Check.
8 = 3 + 5. Check.
10 = 5 + 5. Check.
There, I just proved the Goldbach conjecture to/for kindergarten kids! :cool:
It might be the case that an ultrafinitist system would not be subject to the incompleteness theorem? I don't know. We'd have to see the system or at least a sufficient statement of its relevant properties.
/
Nothing can be understood to stand in for what you think the leminscate stands for, unless you say what you think the leminscate stands for.
1. The real numbers no longer form a continuum
If [math]N_{max}[/math] is the largest possible number, then [math]\frac{1}{N_{max}}[/math] must be the smallest (Otherwise, if there exists some k such that [math]k<\frac{1}{N_{max}}[/math], then [math]\frac{1}{k}[/math]>[math]N_{max}[/math], a contradiction). This smallest number in turn implies that the set of reals is discrete (i.e., no more irrationals). Thus, the diagonal of the square is now commensurable with its side, which we've known to be untrue since the time of the Pythagoreans! (Also, points have size!)
2. Series can be expressed that have no (acceptable) sum
Consider the series 1 + 2 + 3 + 4 + ... If the last term is [math]N_{max}[/math] (which should be allowable), then the series sums to a quantity greater than [math]N_{max}[/math], a contradiction.
For these reasons (and others) ultrafinitism is inadequate to describe modern mathematics.
Is it a coincidence that the word "irrational" means illogical/makes zero sense? I recall starting a thread on how irrational numbers could be the smoking gun that there's something seriously wrong with mathematics and the universe itself.
I remember watching a video about how math teachers generate problems with numbers such that the answer is a nice whole number. The point? If you're ever find that your calculations lead you to an answer that has a decimal expansion then you've made a mistake. As you can see even ordinary fractions (rationals) are red flags, forget about an irrational number as an answer.
As for series sums, what about modulus arithmetic?There is no 13 o'clock on an analog watch; there's 1 o'clock though.
You have got to be kidding! Please tell me you are kidding!
Consider the root of "rational" is "ratio". Now think about an irrational ratio such as that expressed as pi, and you'll get a glimpse at the problems which pervade mathematics.
pi is the ratio of the circumference of any circle and its diameter. But if the diameter is rational then the circumference is not. So still pi is not the ratio of two rational numbers.
So there is not a contradiction.
I don't know what "problems" in mathematics are supposed to have been caused by pi.
https://www.jpl.nasa.gov/news/on-pi-day-how-scientists-use-this-number
So I guess people that don't like pi being used in mathematics have not need for such things as satellite technology, etc. Fair enough.
Irrational numbers can't be expressed as a ratio of two whole numbers. That didn't jibe with the way math was supposed to be in the eyes of the Pythagoreans. If memory serves Pythagoras discovered that harmonious musical notes were rational i.e. the note combinations were pleasant to the ear when the length of string producing one was a whole number multiple of the length of string producing another. Since music is numinous in nature, it being somewhat of a bridge between us and the universe, the Pythagoreans probably extrapolated the math found therein to the universe itself.
The discovery of irrationals, kind courtesy of Hippasus of Metapontum who was thrown overboard to prevent word of this getting out, threatened to overturn what was up to that point a perfect world. A simple and yet magnificent way mathematics could serve as the foundation of the universe had to be abandoned. I wonder what Max Tegmark has to say about this?
Will we, somewhere in the future, come across a kind of number that would do to us (mathematical universe hypothesis) what [math]\sqrt 2[/math] did to the Pythagoreans (mathematical universe hypothesis)?
There's nothing wrong with math, so it must be the universe. I wouldn't put anything past that sucker!!!
Quoting Metaphysician Undercover
It'll take more than a glimpse for me. I know, I know, pay no attention to the man behind the curtain. :cool:
:smile: Couldn't there be something wrong with both?
No
If that is the case, then it's a mistaken premise that we can decide on a greatest practical number, or greatest number of places to approximate pi. Science and engineering doesn't work with just specific numbers but also rather with general formulas. I take engineers at their word that they use trigonometry with pi all over the place. Not that they plod along with just individual values.
Couldn't there be something wrong with you?
Sadly, it seems math discussions on TPF are doomed to descend to the level of farce. Notice that most folks on TPF avoid these topics like the plague. This is most likely because they've seen what happens far too often before.
Consider the latest comments regarding irrationals. Let's draw an analogy. Imagine that every time you saw a topic where consciousness was under discussion, you stepped in and declared that consciousness obviously did not exist because the first three letters of the word spelled "con". How long before you got canned for the low quality of your posts? That's the equivalent of what's happening here.
Some TPF worthies are convinced they know more about math than some of the greatest minds that humanity has produced over the past 5000 years. They know what they know, and there's no arguments that will convince them otherwise. How foolish of us to question the folk wisdom of anonymous forum posters!
Some months ago, on another thread, I implored one of these commenters to write up their math musings and send them off to prestigious math journals. The world languishes without access to these amazing insights. (I don't think he took my advice.)
Why? Well, given how math is supposedly an axiomatized system, there can be no issues with the conclusions that follow among which number the irrationals. However, that means we could play around with the axioms to disallow irrationals, oui? Is this possible/no?
My favorites are cranks who say that mathematical logic, even just sentential logic, is all wrong, as they are typing and reading on computers that are packed to the panels with Boolean programming and would not even exist if not for the invention of modern digital computers in the very crucible of mathematical logic.
Si.
I've told you about a million times already, you can have axioms for whatever you want*, even inconsistency if that's your thing.
* 'ceptin sometimes you can't have all of what you want at the same time, as we found out from Godel.
:rofl:
To be fair she could've been talking about our interplanetary ambitions, moon landing and all that jazz. Frankly speaking, she has a point - all that money earmarked for space ventures could be spent on more pressing matters of which, if the news is to be believed, there are many. Perhaps it's an escape plan. :chin:
Gödel? If it is consistent, it is incomplete; if it is complete, it is inconsistent.
@javi2541997 [math]\uparrow[/math]
If it is recursively axiomatized and has enough of arithmetic, then if it is consistent then it is incomplete (which is to say that if it is complete then it is inconsistent).
What could possibly be more pressing than people getting their satellite dishes on the roof to watch reality TV shows of people eating bugs?
Good quote and reference! :up:
It remembers me about shûnya (empty) on Buddhist metaphysics.
Relative Existence or No Self Nature: Nothing has a essence, nature, or character by itself. Things in isolation are , shûnya, "empty." The nature of things only exists in relation to everything else that exists. Existence as we know it is thus completely relative and conditioned by everything else.
Just like you can't do very much math without relations. But with just one relation* you can do it all.
* membership
Yet, in line with the Buddha's thoughts, we're all alone. The relations we build are, some say, fragile to the point of being mere illusions (the imagery of rats fleeing a sinking ship is enough to send shivers down me spine). That, perhaps, is what Buddha meant - the ties that bind us are maya, no self (anatta).
:lol:
The measurement of one is incommensurable with the measurement of the other, therefore the relation between the two measurements is an irrational ratio. So I wouldn't really call it a contradiction, it's just an attempt to do the impossible, to establish a relation where one cannot be properly established due to that incommensurability. A straight, one dimensional line is incommensurable with a curved two dimensional line. Likewise, an attempt to give the cardinality of an infinite set is an attempt to do the impossible.
Quoting Agent Smith
This was primitive wave theory.
Quoting Agent Smith
The temporal nature of waves always seems to throw a wrench into the cogs of the application of mathematics toward understanding the foundation of the universe.
Quoting jgill
Step out from behind that curtain. On with the show this is it!
Quoting TonesInDeepFreeze
Yes sir! But what happens when understanding the foundation of the universe is "the task at hand"?
Good luck with that. :roll:
Absolutely correct: the level of crank on this site is ridiculous
Quoting Real Gone Cat
Interestingly enough, Avicenna's argument against (mereological) atomism was that applying the Pythagorean theorem is empirically successful, and it could not have been had our physical space been akin to a taxicab geometry, something like what the atomists suggested (in fact, in a taxicab geometry, using the Pythogrean theorem would not even approximate our values!). This later on came to be known as the distance function argument
Thanks, but unlike the undisciplined mathematicians who make willy nilly axioms however they please, we don't rely on luck here.
Luck's role in your life is inversely proportional to the quality of your plans.
:snicker:
Oh Great Anonymous Forum Poster - who clearly knows better than some of the greatest minds humanity has produced for the past 5000 years - enlighten us : please give an example of these wicked axioms that mislead our youth that we might know them and revile them.
(This should be good.)
I didn't say the axioms are wicked, I said they are wlly nilly. And it was Tones who stated that idea. I just went on to draw the conclusion that when you create your principles in such an undisciplined way, you need luck in the application of them. Here:
Quoting TonesInDeepFreeze
That one can do X, or is permitted to do X, doesn't entail that one does X.
Quoting Kuro
If I understand, your view is that by doing things that are not right, one become habitual in doing things that are not right, thus harming one's character.
My point is that I don't agree that it is not right to decry the egregiousness of cranks.
Here you apply an outcomes/productivity argument:
Quoting Kuro
So, if I understand, you base the claim that the act is not right, and thus harms character, on the claim that the act causes bad outcomes and is unproductive. Also, you think the act is wrong because it is acting from indignancy which is not prudent and just.
(1) I think that the act has a better outcome than not decrying the egregiousness of cranks, and that it is productive. Not productive toward having bonhomie with the crank, but productive in another way.
(2) I don't think it is wrong for posters to post for reasons other than certain outcomes or productivity, and I don't limit my own reasons that way.
(3) I don't view virtue ethics as determining, not even the main determinant. I see that virtue ethics, in my limited understanding, offers a lot, but I don't see that it should be the sole, or even main, framework to be used alongside a balance among other frameworks.
(4) I don't think it's incumbent to so seriously apply ethics, let alone pretty demanding ethical frameworks, to all aspects of posting. There is a lot of posting that I don't think has good outcomes or is productive that I wouldn't think of cudgeling with application of an ethical theory or even think of objecting to it at all. Posters don't ordinarily think "Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. People usually post at their pleasure, to express themselves, to advocate for their ideas*. And that expression may include voicing of indignation.
* Though there are other motives too, such as learning from other posters, enjoying pleasant agreements (rare, in this forum), etc.
(5) Perhaps I don't know your context, but I don't think that expressing indignation is necessarily imprudent or unjust.
Fine, willy nilly then. You miss the point : somehow the truths of math have been revealed to no one else but you.
I ask that you stop being so selfish and share this revealed folk wisdom with the world. Begin here by detailing some misbegotten axioms you have encountered. Then write up your math musings and send them off to prestigious math journals. I am sure they will fight to be first to publish. Your fame and fortune await.
(1) I did not say I have not studied philosophy. In philosophy, I am not a scholar, and I have not retained many of the particulars I learned a long time ago, but I have taken courses in philosophy (and not just philosophy of mathematics) and read books. In the philosophy of mathematics, I am not a scholar, but I have read many books and articles. In mathematical logic and set theory, I'm not a scholar, but I have a good handle on the basics through taking courses and careful study of several textbooks, and I have also compiled a rigorous log of formalizations, definitions and proofs in set theory and the first stages of some other branches of mathematics. Meanwhile, I don't opine in threads on various philosophical discussions where the level of conversation would require me to be more adequately versed.
And whatever particulars I've forgotten about Western philosophy through history, I still retain much of the understanding and appreciation of philosophical methods themselves.
Philosophy of mathematics may be discussed at a general level that does not reference particular mathematical developments, but usually, as in this forum, discussions in the philosophy of mathematics do turn the mathematics itself. In that regard, it is crucial that that mathematics not be mistaken or misrepresented.
Indeed, threads in this forum are often premised as critiques of set theory and classical mathematics. Critiquing set theory and classical mathematics is great and vital. But the critiques need to be based on actually knowing what is being critiqued. A critique from ignorance, confusion, and misrepresentation is intellectual and philosophical garbage.
So what about the crank himself? There is not a hint in what he writes that he knows even the least thing about such basic contexts of philosophy of mathematics as Frege, logicism, Hilbert's formalism, Godel's 'What Is The Continuum Hypothesis?' constructivism, intuitionism, reverse mathematics or anything else really. The crank himself knows not a thing about the first order predicate calculus, second order logic, formal axiomatics, set theory or, as far as I can tell, any mathematics beyond everyday arithmetic. But that doesn't stop the crank from claiming it's all nonsense. Indeed, the crank insults mathematics itself and mathematicians themselves.
Of course, the crank does not fault himself for lack of training in philosophy of mathematics, though he spews his ignorance, illogic and literal nonsense about it regularly, for years.
And what would the crank say about other posters who we can tell are sorely "lacking in training" in the philosophy of mathematics? What is the training in philosophy of mathematics of the original poster of this thread? Let alone what are his philosophical commitments? Does the crank (and another poster) hold that everyone must have a philosophical commitment to qualify for posting? I don't believe posters must.
* The original poster started with a presumably empirical question, then went on top claim or insinuate that infinitistic set theory is wrong, and it is still not clear whether the original poster has clear position as to whether he's an ultrafinitist, even that he is a particular kind of finitist.
The crank pounced on my humble statement that I don't hold to a particular philosophy, saying that I "lack training" in philosophy. The crank shows his pettiness, illogic, and factual incorrectness. And lies about me.
(2) It is not necessarily a fault not to have a philosophy. One can be open minded about many philosophies and employ their virtues.
/
$ The crank says it is annoying for a philosopher to have to see a non-philosopher post in a philosophy of mathematics discussion and insist that philosophers shouldn't discuss the metaphysical bases on which mathematical axioms stand, if they have not first studied mathematics.
(3) The crank shouldn't presume to speak for others.
(4) The crank is lying about me. I have never said that one must study mathematics to talk philosophically about mathematics. What I have said is that one should not spew disinformation and confusion about the mathematics without knowing anything about it. The posts by cranks are not just philosophy but they include claims about the mathematics that is the subject of the philosophy. I don't even say that one shouldn't talk about that mathematics without first studying it; rather that one should not misrepresent it, and that the first step to not misrepresenting it is to learn at least a little bit about it.
$ The crank says that it is philosophy that is being discussed not mathematics.
(5) The cranks lies again. The profuse record of actual posts show that the cranks make many claims about the mathematics and that the discussions reference the mathematics. This very thread is one of them.
The crank himself, at nearly the start of this thread wrote: "Try naming pi to its final decimal place."
The crank lies, directly belying his own posting, when he says the discussions are about philosophy but not also the mathematics itself. How can that be topped?
Hey, I'm a fan. You have my respect. :cool:
I used to pick up bits of modern math knowledge from @fishfry before he departed.
Really, where did you get that idea? I am in the habit of dismissing what are commonly touted as the "truths of math", for being in some way faulty. How do you get from this to the point of saying that the real "truths of math" have been revealed to me.
That was quite the rant Tones. I hope you're feeling better now, to have gotten that off your chest. And if so, I'm very happy to have been able to assist you, in feeling better about yourself: if that was even possible. If it wasn't possible, for you to feel better about yourself, as I fear is the case, then I'm sorry, for you.
So there are no truths of math? What, in your wisdom, is math after all?
I assume that - other than simple arithmetic with positive integers (basically, what you can do with your fingers and toes) - you believe mathematics to be made up gobbledy-gook. To live in the modern world and be such a willful know-nothing is breath-taking.
I begin to discern what happened : someone received a low grade in calculus at university and has had it out for those idiot mathematicians ever since.
Math is based on made up axioms, like Tones described. It's imagination, fiction, not truth. The majority of the axioms which get accepted into the mainstream do so because they prove to be useful. Some though, may be accepted simply for beauty or eloquence. Usefulness is quite distinct from truthfulness.
(What is the) The Largest Number We Will Ever Need (?)
I'd say one, if all other axioms in the mainstream system are applied. 1. It can be added, the sums can be multiplied or divided or subtracted from each other, and the whole shkebam can be developed just form one number, which is one.
This is a specific example of the largest number. It could be a half, a million, any number, really, real or imaginary, and rational or irrational. Any one number could satisfy the question, "What is the largest number we shall ever need?" Provided, of course, that the generation of other values, expressed in numbers, in an infinite variety, is possible from the axioms used.
That's wonderful. :up:
All we need to get the ball rolling is 1.
[quote=Charles Darwin]Thus, from the war of nature, from famine and death, the most exalted object which we are capable of conceiving, namely, the production of the higher animals, directly follows. There is grandeur in this view of life, with its several powers, having been originally breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved.[/quote]
Le Monad.
Can infinity even be considered as possible or impossible? it's a concept. If an axiom calls it to be true/in need then I see no reason to assume otherwise.
I haven't read the original argument made by Aristotle - Wikipedia offers only a rough sketch. It seems as though Aristotle considered real/actual entities as those that had an end; consider the process of constructing a chair. It begins (wood, nails, glue, etc.) and ends (a chair). If one is unable to complete the task, we have a potential chair and not an actual one. The same goes for [math]\infty[/math], it, by definition is endless.
I don't agree that it's a valid application of Aristotle's rule of the difference between actual and potential.
The infinity is there. The only reason it can't be counted is because counting is a process which is always finite. However, it is the tool of the test, the counting, that is the culprit here, so to speak; it is the weakness of the tool that stops us from realizing the actuality of infinity.
If there were an instrument that meausred infinity, then the actuality would immediately show through.
Quoting god must be atheist
I think this is a good point. It would be harmful to assume that infinities don't exist in the real world simply because we can't process them in a tangible manner.
Yeah, we need a different tool!
Quoting Agent Smith
What about a cyclical process?
I proposed that; no takers!
Why? First Cause?
Some mathematical systems require an infinite set to function. I think it's just a question of where your looking at; assuming infinity isn't intrinsically false, or bad, it's just another way to look at something, which happens to have some pretty interesting conclusions.
https://en.wikipedia.org/wiki/Axiom_of_choice
Dunno!
I will say that the benefit of virtue ethics is that you'll no longer have to reconsider this in straightforward situations that are not ethical dilemmas. In acting virtuously, virtuous action becomes habit
That makes it appear that I said that we can't expect that posts have good outcomes, etc.
But what I posted:
Quoting TonesInDeepFreeze
That is to say that I don't expect that ordinarily posters ask those questions before posting.
Quoting Kuro
I understand that view.
Well to begin with i would calculate all the countable space in the universe in the smallest units possible (Planck lengths). Then i would calculate the total lifespan of the universe in Planck time units. After that i multiply the last two results to get the total amount of countable space in the universe times the amount of countable time in the whole lifespan of the universe.
Total space in universe calculation:
------------------------------------
Planck length in meters:
1.6 x 10^-35 =
0.000000000000000000000000000000000016
Diameter of the universe in meters:
8.8 x 10^26 =
880000000000000000000000000
Diameter of universe in Planck lengths:
5.4453761 x 10^61 =
54453761000000000000000000000000000000000000000000000000000000
Volume of universe in Planck cubes:
1.61467 x 10^185 =
161467000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Total time of universe calculation:
-----------------------------------
Current age of universe till now in years:
13.7 x 10^9 =
13700000000
Time from now till universe heat death in years:
1.7 x 10^106 =
17000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Total life (past + future) of universe in years: (insignificant difference)
1.7 x 10^106 =
17000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Total life of universe in seconds:
5.3618 x 10^113 =
536180000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
A few more calculations...
Planck lengths per light second (number of plank lengths light travels in 1 second):
1.855 x 10^43 =
18550000000000000000000000000000000000000000
Universe Planck volumes per second over total life of universe:
9.946139 x 10^156 =
9946139000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Universe Planck volume * Planck time instances over life of universe:
1.605973225913 x 10^342 =
1605973225913000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
After that i can add matter into the mix by calculating the number of subatomic and atomic particles in the universe at any one plank time instance. Then we multiply that number into our last result.
Number of subatomic and atomic particles in the universe:
3.28 x 10^80 =
328000000000000000000000000000000000000000000000000000000000000000000000000000000
Result for all countable things in the universe so far:
5.267592180994640 x 10^422 =
526759218099464000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
We can add a few more things like stars planets and galaxies:
-------------------------------------------------------------
Stars in the universe:
10^24 =
1000000000000000000000000
number of planets in the universe (star orbiting and rogue planets):
10^51=
1000000000000000000000000000000000000000000000000000
number of galaxies in the universe:
13 x 10^15 =
13000000000000000
All stars, planets, and galaxies in the universe:
1.3 x 10^91 =
13000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
My final result:
6.847869835293032 x 10^513 =
6847869835293032000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
This number can be significantly larger if we multiply in the number of molecules in the universe along with the number of space dust particles, asteroids, comets, black holes, and whatever else i'm missing.
Surely you jest.
I jest you not!
And don't call me Surely.
Here comes . . . . . . Zepto !
:nerd:
Very easy: aleph-1. The the infinite cardinal of the real numbers
Because I think there's still something for us to understand with infinity, it isn't so easy that to use finite logic. And more interestingly, bigger infinities seem not to be usefull for example in physics, computing, etc.
card(reals) = aleph_1 is the continuum hypothesis. It is not provable in ZFC. It is thought to be true by some mathematicians and false by other mathematicians - an unsettled question.
What did I say? There's still something for us to understand with infinity.
Actually, I think the continuum hypothesis is that from aleph_0 the next is aleph_1.
It is correct that the next aleph after aleph_0 is aleph 1. That follows trivially from the definition of the alephs. Since the alephs are indexed by the ordinals, and 1 is the next ordinal after 0, It is trivially the case that the next aleph after aleph_0 is aleph_1, regardless the continuum hypothesis. Meanwhile, actually, it is incorrect that "the next aleph after aleph_0 is aleph 1" is the continuum hypothesis.
The continuum hypothesis is that the cardinality of the set of real numbers is aleph_1. Or, equivalently, that 2^aleph_0 = alpeh_1.
Quoting ssu
You said there's more for us to understand regarding infinitude. Indeed, whether the cardinality of the set of real numbers is aleph_1 is something more that is not clearly understood. Again, settling that question is to settle the continuum hypothesis.
It's unsettled because there's a problem with what constitutes a "countable" cardinality. As soon as we define "countable" such that an infinite set might be "countable", we create incoherency.
Yes, the continuum hypothesis is about the first two infinite cardinals. Meanwhile, what I said stands:
Quoting ssu
That is the continuum hypothesis.
In other words,
The cardinality of the set of real numbers is aleph_1
is the continuum hypothesis.
Quoting ssu
That is not the continuum hypothesis.
In other words
aleph_1 is the next aleph after aleph_0
is not the continuum hypothesis.
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Well, it's about what can be put into one-to-one correspondence with the set of natural numbers and the reductio ad absurdum proof that this cannot be done with the set of reals. Here 'countable' has it's problems, when ordinarily everything that we can map into one-to-one correspondence is countable (a+b=c).
When ZF was meant to do away with the paradoxes, it's obvious that it has problems with infinity. After all, it's just taken as an useful axiom.
Quoting ssu
is the continuum hypothesis
and that
Quoting ssu
is incorrect, since "from aleph_0 the next aleph is alelp_1" is not the continuum hypothesis.
/
Quoting ssu
Cantor's proof was not by reductio ad absurdum.
it's easy to give a name to an infinite cardinality (aleph_1 for example), just like we might name it "an infinity", but naming it in no way demonstrates that it is countable.
By the way, I did not say that the cardinality of the set of real numbers is aleph_1. I said that "The cardinality of the set of real numbers is aleph_1" is the continuum hypothesis.
Do notice that Cantors system is the sequence of cardinal numbers: aleph_0. aleph_1, aleph_2, aleph_3 and so on. The question is if this hierarchial system holds and if there is a cardinality or not between the naturals or the reals. The continuum hypothesis is that the reals is the next aleph, that there isn't anything else.
Quoting TonesInDeepFreeze
Your somewhat correct. The first proof in 1874 wasn't. But he did give this in 1891 with the diagonal argument, which I find more simple.
The incoherency is quite clear, and I'll explain it to you. You can deny that it exists, and call me whatever name you like, but that doesn't address the problem.
In simple terms, counting is a task. To be "counted" implies that the task is completed. To be "countable" implies that the task may be completed. In such common terms, no infinite number is countable, because the task to count an infinite number can never be completed.
Now, it may be possible to define "countable" in a way such that completion of the task would not be required as a criterion for being countable. If mathematicians have successfully done this, then an infinite number would be countable.
However, mathematicians have not successfully done this. They have defined "countable" in relation to another task, bijection, and proper bijection would also require completion of the task, just like counting. Since bijection is a task which cannot be completed in the case of an infinite number, they have not defined "countable" in a way such that an infinite number can be truthfully said to be countable.
So mathematicians pretend that an infinite number, the cardinality of the set of natural numbers, is countable when it really is not. "Countable" as defined by mathematicians is not consistent with "infinite" as commonly used in reference to the natural numbers, so the idea that an infinite set is countable is a pretense.
This pretense produces a new meaning for "infinite", one which is consistent with the pretense. However, this meaning of "infinite" is not consistent with how "infinite" is commonly used by mathematicians, hence the word "transfinite" is sometimes employed. I will call this concept of "infinite" a phantom infinite because it's a completely imaginary concept, totally distinct from actual usage, created solely for the purpose of making it appear like it is possible to do the impossible task, count the infinite natural numbers. The phantom infinite concept is a product of that pretense.
Now we have a concept, the phantom infinite, which hides the inconsistency between "countable" and "the natural numbers". The natural numbers are not really countable, (being infinite as implying a task which cannot be completed), but the phantom infinite makes it appear like they are by changing the meaning of "infinite". The phantom infinite is a false concept because this sense of "infinite" is not consistent with how "infinite" is actually used in relation to the natural numbers. We do not allow that one can actually complete the task of counting the natural numbers. The result being a false representation of the natural numbers, having been created by the phantom concept of infinite. The pretense requires a false representation of the natural numbers, for its support. So the phantom infinite is imposed onto the natural numbers, as if this is the real way that "infinite" is used in relation to the natural numbers, but this is not a true representation of how the natural numbers are actually used, and how they are said to be "infinite".
Keeping all that in mind, the problem with the continuum hypothesis ought to become crystal clear to you. The idea that an infinite set is countable, or has a specifiable cardinality, is a product of the phantom infinite concept. This is not consistent with "infinite" as used by mathematicians. Therefore there is an inconsistency between the concepts of "infinite set", and "countable" inherent within the continuum hypothesis. The "infinite set" of natural numbers takes the traditional meaning of "infinite" (implying not countable as the task cannot be completed), and the designation that such a set is "countable" uses the phantom infinite concept which is inconsistent with the traditional concept.
The inconsistency between the traditional concept "infinite", and the phantom concept of infinite makes the two completely incompatible. Because of this, any attempt to establish commensurability between the two fails.
Not merely a lack of rigor. Rather, your statement "the continuum hypothesis is that from aleph_0 the next is aleph_1" is plainly false.
Quoting ssu
There is no question of whether it "holds". It doesn't even make sense to say that a sequence holds or not. What hold or not are statements.
Quoting ssu
Yes, the continuum hypothesis is the claim that there is no cardinality between the set of naturals and the cardinality of the set of reals. That's another way of saying what I've been telling you.
Quoting ssu
The continuum hypothesis is that the cardinality of the set of reals is aleph_1. That is equivalent to saying that there is no uncountable subset of the set of reals that is not 1-1 with the set of reals. Of course, no matter the continuum hypothesis, there are cardinals greater than aleph_1.
Quoting ssu
The diagonal argument given by Cantor was not a reductio ad absurdum.
Cantor proved that you cannot make a bijection between the natural number to the reals, hence the reals aren't aleph_0 like for example rational numbers.
Quoting TonesInDeepFreeze
When you first assume that there is a bijection between the natural numbers and reals, then show that there is a real that cannot be in this bijection, that is a reductio ad absurdum.
Correct.
Quoting ssu
Cantor didn't make that assumption.
Here's the argument, which is not reductio ad absurdum:
Let f be a function from the set of natural numbers to the set of denumerable binary sequences.
Construct a denumerable binary sequence not in the range of f.
Conclude there is no function from the set of natural numbers onto the set of denumerable binary sequences.
/
He also could have used reductio ad absurdum, but he didn't:
Assume there is function from the set of natural numbers onto the set of denumerable binary sequences.
Derive a contradiction.
Conclude there is no function from the set of natural numbers onto the set of denumerable binary sequences.
When you construct "a denumerable binary sequence not in the range of f", aren't you deriving that contradiction? There's the negative reference to f.
After all, when let's say one asks if the set of natural numbers and rational numbers have the same cardinality, there is a direct proof (the set of rational numbers can be well ordered and Cantor showed this).
When you use reductio ad absurdum, you construct a denumerable binary sequence not in the range of f, which contradicts the assumption that f is a bijection between the set of natural numbers and the set of denumerable binary sequences. But Cantor didn't do it that way.
Quoting ssu
The proof doesn't rest on showing a well ordering. Showing a well ordering of the set of rational numbers is not adequate for showing that the set of rational numbers is countable. (Even uncountable ordinals are well ordered.)
Countable, right. Thanks for the correction.
You've picked up the scent mon ami but que sais-je.
What's [math]\aleph_1[/math]?
Are we talkin' about the reals?
aleph_1 is the least cardinal greater than aleph_0.
That is the case by definition.
"aleph-1. The the infinite cardinal of the real numbers"
That is the continuum hypothesis. The continuum hypothesis is "aleph_1 is the cardinality of the set of real numbers" (or equivalently, "aleph_1 = 2^aleph_0") and it is not a theorem of ZFC nor is its negation a theorem of ZFC.
So is ssu right?
And even if this will irritate the mathematicians here, we do use infinity quite a lot, we just don't talk about it as infinity. Just like with limit sequences and limit points etc.
As to the topic at hand, I was in search of a finite number N[sub]max[/sub] such that no physical calculation ever exceeds N[sub]max[/sub].
Nonetheless, it seems you're on the mark, [math]\aleph_1[/math] (the reals?) is the only level of [math]\infty[/math] required for our universe.
Yes, x is a natural number iff x is a finite cardinal.
And aleph_1 is not a finite cardinal.
And the poster is asking about finding a certain natural number, so aleph_1 is not an answer.
I addressed that about half a dozen times in posts above.
The statement "aleph_1 is the cardinality of the set of real numbers" is the continuum hypothesis. It is not derivable in ZFC and its negation is not derivable in ZFC.
Cardinality as in the cardinality of the set {a, &} is 2?
Quoting TonesInDeepFreeze
Apologies ... patience is a virtue. :smile:
thm: n is a natural number <-> (n is finite & n is an ordinal)
dfn: card(x) = the least ordinal k such that x is 1-1 with k
dfn: c is a cardinal <-> there exists an x such that c = card(x)
thm: n is a natural number <-> (n is finite & n is a cardinal)
thm: x is finite <-> card(x) is a natural number
thm: x is infinite <-> card (x) is infinite
/
So, yes, if x is finite, then its cardinality is a natural number.
But if x if infinite, then its cardinality is not a natural number but rather is an infinite cardinal.
In either case, every set is 1-1 with its cardinality.
/
dfn: aleph_0 = the set of natural numbers
dfn: aleph_1 = the least infinite cardinal that is strictly greater than aleph_0
dfn: R = the set of real numbers
dfn: x is denumerable <-> (x is 1-1 with the set of natural numbers & x is infinite)
thm: card(R) = 2^aleph_0 (in other words, the cardinality of the set of real numbers is the cardinality of the set of denumerable binary sequences)
The great question of set theory: Is the cardinality of the set of real numbers aleph_1? Put another way: Is it the case that there is no set that has a cardinality strictly greater than the set of natural numbers but strictly less than the cardinality of the set of real numbers? Put another way: Is aleph_1 = 2^aleph_0?
The assertion "the cardinality of the set of real numbers is aleph_1" is called 'the continuum hypothesis' ('CH'). Cantor thought CH was true, but he couldn't prove it. Hilbert proposed that finding a proof, one way or the other, was a priority of mathematics. Later, Godel proved that ZFC does not prove the negation of CH, then later Cohen proved that ZFC does not prove CH.
So the great question of set theory does not have an answer per merely ZFC. So mathematicians have been proposing and studying axioms that they consider to be intuitively true and could be added to ZFC to settle CH. Some mathematicians propose axioms that prove CH, and other mathematicians propose axioms that prove the negation of CH. There is no consensus.
So it is silly just to say "the cardinality of the set of real numbers is aleph_1", without citing a context for belief in the assertion, when it is a profound open question.
/
PS: The generalized continuum hypothesis (GCH) is that for any x, aleph_(x+1) = 2^aleph_x.
Of course, GCH implies CH, and the negation of CH implies the negation of GCH. Godel actually proved that ZFC does not prove the negation of GCH. And Cohen, by proving that ZFC does not prove CH perforce proved that ZFC does not prove GCH.
I thought the Continuum Hypothesis was about the existence/nonexistence of an infinity (I) such that cardinality-wise, set of naturals ([math]\aleph_0[/math]) < I < set of reals ([math]\aleph_1[/math]).
What would be the implications of there being an I?
No. The way you wrote it is wrong. The continuum hypothesis is that the cardinality of the set of reals is aleph_1. This point keeps getting lost. Don't just take it for granted that the cardinality of the set of reals is aleph_1, when that is the very point that is in question with the continuum hypothesis.
But this part that is suggested (though mangled) by you is correct: The continuum hypothesis is equivalent to the assertion that there is no set that has cardinality strictly greater than the cardinality of the set of naturals and strictly less than the cardinality of the set of reals.
One more time. Here are three equivalent ways of saying the continuum hypothesis [here 'N' stands for the set of naturals]:
(1) aleph_1 = 2^aleph_0
(2) card(R) = aleph_1
(3) There is no set that has cardinality strictly greater than card(N) and strictly less than card(R)
Those are three ways of saying the continuum hypothesis.
If you say card(R) = aleph_1, then you are asserting the continuum hypothesis, which is an assertion that can't be proven in ZFC nor disproven in ZFC.
Or, since this point seems not to be getting through, I'll say it this way:
Don't just assume that card(R) = aleph_1. "Is card(R) = aleph1 ?" is the QUESTION. Don't just assume its answer is 'yes'.
Thank god! :pray:
So we're looking for an infinity bigger than N and smaller than R. What are the ramifications of the existence/nonexistence of such an infinity? You said or someone else did that Hilbert wanted the CH proven/disproven ASAP.
If you think the continuum hypothesis is false, then you think there is a set with cardinality strictly greater than the cardinality of N and strictly less than the cardinality of R.
For more about the ramifications, you should read more about set theory, to gather the whole context. But perhaps the more pertinent matter is what are the ramifications of added set theoretic principles that would prove CH and of added set theoretic principles that would disprove CH. This is at the heart of contemporary set theory research. I couldn't do the subject justice in posts (and I'm not expert enough anyway).
Yeah, Hilbert enjoined mathematicians to either prove or disprove CH. Godel did half by proving that ZFC doesn't disprove CH and Cohen did the other half by proving that ZFC doesn't prove CH. So a kind of "stalemate": Hilbert's challenge can't be answered on the terms Hilbert had in mind. This is a fascinating, profound and remarkable intellectual turn of events.
So it's like I can't prove whether Jesus existed or not from the fact that green light has a wavelength of 555 nm.
No, it's not like that. You say it is only because you think being a wiseracre suits you.
Insightful comments. A veteran ... with scars to prove it!
Back to main topic. I guess I was wrong to think that there could be a finite number N[sub]max[/sub] such that all physical calculations [math]\leq[/math] N[sub]max[/sub].