Can we reset at this point?
I was particularly interested in the post questioning whether 0.999...=1. Normally, of course, one would simply append a reply, but at my last viewing the post already had 539 replies, and the thread has long since decomposed into a river-delta of irrelevant issues and private conversations. So, if the moderators are friendly, I'm hoping we can reset the discussion within a more rigorous, immediate, and relevant context.
I would say that whether 0.999...=1 is crucially dependent upon which number line is presupposed.
Briefly, the position appears to be that in the (classical) real number line, 0.999... is the largest real number which is less than 1; Cantor's Diagonal Argument certainly seems to support this interpretation, and natural intuition concurs: however many decimal places you add to the manifold, you can never close the remaining gap by more than 90%.
But if we consider the question in the context of the hyperreal number line - that is, the real number line augmented by adding infinite values at each end, namely Aleph-null and h, the goalposts move.
Abraham Robinson's definition of h revolutionised mathematics in the 1960's. Briefly, he defined the infinitesimal as a number which, for all values of a, is <a and >-a. Thus the infinitesimal may have a range of values, including 0. Within THIS number line, it appears to be undeniable that 0.999... meets the limit of 1, and thus 0.999...=1.
I would say that whether 0.999...=1 is crucially dependent upon which number line is presupposed.
Briefly, the position appears to be that in the (classical) real number line, 0.999... is the largest real number which is less than 1; Cantor's Diagonal Argument certainly seems to support this interpretation, and natural intuition concurs: however many decimal places you add to the manifold, you can never close the remaining gap by more than 90%.
But if we consider the question in the context of the hyperreal number line - that is, the real number line augmented by adding infinite values at each end, namely Aleph-null and h, the goalposts move.
Abraham Robinson's definition of h revolutionised mathematics in the 1960's. Briefly, he defined the infinitesimal as a number which, for all values of a, is <a and >-a. Thus the infinitesimal may have a range of values, including 0. Within THIS number line, it appears to be undeniable that 0.999... meets the limit of 1, and thus 0.999...=1.
Comments (48)
The question is predicated on a faulty assumption.
Number lines do not exist (or, at least, cannot be described).
Definitions
All definitions are of the form: X is not(Everything Else)
The real numbers as intrinsic values are a phantom; an illusion.
When we describe numbers we actually describe the relationships of numbers.
'1' is understood by its relationships with '2', '3', '-4.8776', 'apples', ...
When we talk about X we are actually talking about X's relationships.
Infinitesimals
While it is worth considering what the smallest possible relationship is...
The difference between 0.999 recurring and 1 is (in part) their relationship(s) to each other.
The relationship between 0.999... and 1 defines a difference between them.
Indistinguishable
A relationship and a difference are, for all practical purposes, the same thing.
The presence of a relationship between two perceptions demonstrates a difference between those perceptions. We can distinguish between them.
0.999... is different from 1 because we perceive a difference.
The nature of difference
'Difference' is defined as not(not Difference).
0.999... is different to 1 - but we can only understand that difference by comparing it to other differences.
Sorry to be weird at you
I know this isn't the sort of answer you were looking for - but the question is only troublesome because of the mistaken belief that number lines exist.
It is flat impossible to objectively describe number lines (or anything else).
All descriptions are of relationships. We can describe the relationships of numbers. And we can describe relationships in comparison to other relationships.
This is the structure of all knowledge. This is the mechanism of understanding.
The real number line, as an objective entity has never been described; can never be described.
We can, and do, describe the relationships of numbers. This dense network of relationships is our concept of what numbers are.
.9999... = x
9.9999... = 10x
10x-x = 9.999... - .999...
9x = 9
x = 1
But you made it the other way around.
Quoting alan1000
0.999... is equal to 1 here, not lesser than 1.
Quoting alan1000
Here, 0.999... can indeed be less than 1, because 0.999... is ambiguous.
Number lines do not exist (or, at least, cannot be described).
Wow, I did not see that coming. More than 2500 years of mathematical development flushed down the toilet in a few seconds!
The real numbers as intrinsic values are a phantom; an illusion.
What are your supporting arguments? Give me some help here, I'm trying to understand your position.
All definitions are of the form: X is not(Everything Else)
The assertion that X has no identifying properties in its own right is certainly a courageous approach. To my knowledge, nobody in the previous history of mathematics, from Euclid to Penrose, has ever adopted such a definition. By this method, how would you define "prime number", for example?
When we describe numbers we actually describe the relationships of numbers.
"Relationships of numbers" is a defining property of the relational number line (the line of negative and positive integers). But you deny the existence of number lines. Can you develop this point?
The example provided by Flannel Jesus conforms with the parameters of the hyperreal number line, as set out in my original post. Of course, it does not call on any infinite or infinitesimal values, but if he wishes to argue therefrom that it is consistent with the logical properties of the real number line, he will need to provide clarifying arguments.
TC: I'm sorry, but "We'll have to trust Chat GPT" is not a philosophical argument. I'm not looking for someone to tell me the "right" answer; I am posing a question in the (apparently vain) hope that someone out there actually understands mathematical philosophy.
I think it is you who doesn't have any familiarity with high school math:
Quoting alan1000
For the real numbers, 0.999... is exactly equal to 1.
@flannel jesus gave a clear and obviously correct answer using simple arithmetic operations on real numbers. What does number theory have to do with it?
This is an exaggeration. There are probably universities around where this is taught regularly, but it has not caught on to any significant degree in general. A colleague of mine who taught at the U of Colorado told me they made an attempt to start a course in the subject, but it flopped. I don't see any course in their curriculum now that focuses on non-standard analysis. But there are courses in foundations where it may crop up.
So, rather than drift off into systems that depart from the standard material on the real numbers, its best to stick with the widely accepted ideas. Just my opinion.
Even though it is right, its authority cannot be assumed. It confabulates.
Nicely put. I've been searching for the right word and that's it.
Quoting alan1000
Let's hope they are not.
Folks are abducted by the philosophy of mathematics during summertime. The effect of being burned down by the sun...
You have it backwards.
[math]1 - 0.999 = x[/math]
With the standard reals this equation can only be:
[math]1 - 0.999 = 0.000 = 0[/math]
There is no standard real number greater than 0 that can satisfy the equation.
But with the hyperreals this can be:
[math]1 - 0.\underbrace{999 }_H = {1\over{10^H}}[/math]
Where H is an infinite hyperinteger.
What is worth discussing is why the laymans mathematical intuitions favour nonstandard analysis, and why standard analysis is standard.
The same intuition that stands under Zeno's paradox.
Quoting Michael
Calculus and its wide applications, I would imagine.
Yup. Sort of. Descartes, Gödel, Alfred North Whitehead and The Foundational Crisis In Mathematics (among many others) have covered much of this territory before. It gets ignored because it is inconvenient - but this isn't entirely new.
Evidence
Try creating a non-circular definition.
Different approaches
Gödel's incompleteness theorems talk about the limitations of a system referring to itself.
Descartes observes that nothing can be proven outside one's own existence.
Alfred North Whitehead formalised process philosophy.
The foundational Crisis in Mathematics is a number of different people pointing out that axiomatic mathematics cannot establish a firm foundation from which to proceed.
General Relativity demonstrates that the assumptions of Newtonian Mechanics do not apply to the universe.
Context matters: the meaning of a thing depends (entirely) on the context.
Subjective experience exists.
Count Timothy von Icarus provides several relevant quotes in response to NOS4A2
Precedent
This really isn't out of the blue.
Mathematicians have tried really hard to establish a set of definite, unambiguous axioms. It can't be done.
Every statement within a (closed) system is one part of the system describing other parts.
All definitions are necessarily circular.
More pertinently: every definition is by reference to other things.
Bold, Underline & Italics
We can describe the relationships of X. We cannot describe X.
There is nothing complicated here. Whatever X is - we cannot access it. We cannot experience X. We cannot describe X.
In this respect I am simply reiterating an observation that is over two millennia old.
Quoting alan1000
We cannot describe X. We can describe the relationships of X.
The majority of the time this distinction doesn't matter. When sitting at the dining table it would be redundantly pedantic to note that we are experiencing the dining table's relationships rather than the dining table itself.
However, in pure mathematics, philosophy and metalanguage discussions the distinction becomes crucial. Such as when we are discussing whether 0.999... = 1.
Relative vs Objective: Change vs Static
The Law of Identity states that objects (like the number line) are static, unchanging.
A defining characteristic of relationships is that they change.
As you change, grow and learn, your relationships with concepts changes. Your understanding of numbers now is significantly different than when you were first being taught to add and subtract.
Each person has (slightly) different ideas of what numbers are, and the significance of them.
There are, of course, similarities. Common experiences create similar networks of relationships. An ordered series of numbers (1<2<3<4<5<...) is a near universal experience. It is easy to confuse many apparently similar subjective experiences for objective truth.
Your understanding of the number line is dynamic. Your sense of knowledge and meaning changes.
The concept of "number line" that you posses is constantly evolving, developing, changing.
In closing
That all concepts (such as number lines) are dynamic is in direct opposition to The Law of Identity.
It is a big step to swallow in one go.
However, the individual components are simple enough observations:
These aren't shocking, groundbreaking revelations.
It just so happens that a static number line with fixed (true) rules is a direct contradiction of these observations.
My 2 cents
No, it depends on what is meant by '...'. In ordinary mathematics, '...' in that context refers to the limit of a certain sequence, and we prove that that limit is 1.
Quoting alan1000
Wrong. In classical mathematics, '.9...' is notation for the limit of a certain sequence, and that limit is proven to be 1.
Quoting alan1000
Cantor's argument has nothing to do with it. They are different matters.
Quoting alan1000
Robinson came up with non-standard analysis. That is a different context.
That's not a proof. It's handwaving by using an undefined operation of subtraction involving infinite sequences. Actual proofs are available though.
It's handwaving. The argument invokes an utterly undefined notion. It's a garbage argument as far as mathematics goes. And it doesn't even have explanatory value, since it merely defers the question of what '...' means to the question of what subtraction on infinite sequences means.
Definition: .999... = lim(k = 1 to inf) SUM(j = 1 to k) 9/(10^j).
Let f(k) = SUM(j = 1 to k) 9/(10^j).
Show that lim(k = 1 to inf) f(k) = 1.
That is, show that, for all e > 0, there exists n such that, for all k > n, |f(k) - 1| < e.
First, by induction on k, we show that, for all k, 1 - f(k) = 1/(10^k).
Base step: If k = 1, then 1 - f(k) = 1/10 = 1(10^k).
Inductive hypothesis: 1 - f(k) = 1/(10^k).
Show that 1 - f(k+1) = 1/(10^(k+1)).
1 - f(k+1) = 1 - (f(k) + 9/(10^(k+1)) = 1 - f(k) - 9/(10^(k+1)).
By the inductive hypothesis, 1 - f(k) - 9/(10^(k+1)) = 1/(10^k) - 9/(10^(k+1)).
Since 1/(10^k) - 9/(10^(k+1)) = 1/(10^(k+1)), we have 1 - f(k+1) = 1/(10^(k+1)).
So by induction, for all k, 1 - f(k) = 1/(10^k).
Let e > 0. Then there exists n such that, 1/(10^n) < e.
For all k > n, 1/(10^k) < 1/(10^n).
So, |1 - f(k)| = 1 - f(k) = 1/(10^k) < 1/(10^n).
"Applying the rules consistently breaks down. Therefore we do not apply the rules consistently".
If you apply the rules of naive set theory - those rules lead to a contradiction. Therefore the rules of set theory cannot possibly be correct (The Principle of Explosion).
When you have to make an exception to the rules in order for things to work - your rules don't work.
As such - I don't think your comparison to an inconsistent set theory and its ad hoc fix is very helpful.
More pertinently - Godel was specifically working within the rules of axiomatic mathematics and exploring the limits of those rules. If he had to step outside those rules then the whole point of the exercise collapses.
The whole point of Godel's incompetenesses is: Given premise; what can we say?
For the incompleteness theorems to exclude themselves would be to disregard the whole point of the exercise in the first place.
Quoting Gregory
Yes-ish.
You have to exist in order to have a concept of mathematics. The universe has to exist for you to exist for you to conceive of mathematics.
So - yes - absolutely - mathematics is founded on the existence of the universe.
The trouble is that a full understanding of mathematics requires a full understanding of the universe as a pre-requisite.
Your concept of numbers derives from your experience of the world around you.
But no-one can define the universe in a fixed, objective manner.
Axioms
Given a set of axioms we can create an axiomatic system.
But...
In order to uniquely define a set of axioms we need a set of instructions that describes how axioms should be interpreted: axioms^2
In order to accurately interpret axioms^2 we need a set of instructions the describe how axioms^2 should be interpreted: axioms^3.
...
Etc, etc and also etc.
So - yes - we do in fact take the universe as a foundation and explore that foundation. But we can't say anything definitive about that foundation. Consequently we cannot say anything definitive about anything derived from that foundation.
So - we are free to propose the existence of the real number line - but we cannot say anything meaningful about it. Any definition faces the problem of axioms - infinite regression (or a closed loop of A defines B and B defines A).
No matter how hard we try - we are only ever able to describe the relationships of X, never X itself.
Quoting Gregory
In mathematics - a paradox (inconsistency) demonstrates a faulty set of axioms.
Zeno's paradox demonstrates that some assumption (such as the continuous nature of space) is mistaken.
I would argue that zeno's paradox is a demonstration that space is not, in fact, continuous. That space cannot be infinitely divided - just as we currently believe matter cannot be infinitely divided (c.f. electron is a fundamental particle).
Not necessarily. The Diagonal paradox can be extended to a sequence of smooth curves that converges to a limit curve in the complex plane in which the disparity of lengths is infinite. There is no argument I have heard of that implies fundamental axioms of the real (and complex) numbers is at fault. I seem to recall Aristotle was aware of this discrepancy of lengths.
Why can't we just choose to say the set of all sets that do not contains themselves is the highest in order and so is not included in itself? What in math or language requires that include itself in itself?
And i also am curious why Godel thought self reference a necessary step in mathematics instead of being contingent on our will
Maybe i am an intuitionalist
First: I like the example. I do enjoy coming across these sorts of things.
Second: Mathematicians have a long career of coming across inconsistencies and hurriedly changing the rules so that this particular inconsistency no longer counts.
You may remember that I don't think anything other than relationships exist. I don't think Euclidean space, points, lines or triangles exist.
{Exist = can be described}
The relationships of a triangle do exist. We can describe those relationships.
In the staircase paradox there is a presumption of a continuous manifold within which exist infinitely divisible lines of arbitrary length.
And the definitions of all those terms are circular.
As such, inconsistency in mathematics occurs when people try to describe things that are indescribable.
So long as we only ever try to describe describable things, there is never any inconsistency. But trying to do something impossible always leads to some kind of system error.
We can, then, describe the relationships of our hands. We cannot describe our hands.
We can describe how (the relationships of) our hands pick up (the relationships of) a ball.
The distinction between X and Relationships-of-X is usually irrelevant since we can only ever describe "Relationships of X".
Mathematics tries to describe X. To the extent that descriptions of X and descriptions of X's relationships share common ground this appears to work.
But:
This means that the staircase paradox cannot actually define a continuous space or a continuous line. There are no demonstrably continuous relationships for a continuous space to be similar too.
There is no definable limit for a staircase to approach.
P.S.
That isn't very clear.
Simplified:
We can only describe the relationships of X.
Number lines, continuous spaces, points and lines are defined to be static.
It is not possible to describe static (or continuous) objects using dynamic and discrete relationships.
Quoting Gregory
I think Axiomatic Mathematics is wholly mistaken. At the same time, I'm trying not to mislead you about Godel's incompleteness theorems and Axiomatic Mathematics in general.
I think Intuitionism(sp?) is giving up too easily - but it is arguably at least as well founded as axiomatic mathematics (which isn't saying a lot, but...).
Godel is pulling on a piece of string and seeing where it leads: Given a set of assumptions; what conclusions can we draw.
Godel's specific arguments are about what an axiomatic system can say about itself.
As such, the self-referential nature of Godel's arguments is baked into the premise
Godel could have asked different questions - but the questions he asked were specifically and explicitly self-referential. Removing the self-referential bit doesn't leave anything behind - it is all about self-reference.
Axiomatic Mathematics is wrong but the question is right
Axiomatic Mathematics cannot definitively define axioms. Without axioms, Axiomatic mathematics is nothing.
Godel's theorems are part of axiomatic mathematics and fatally flawed even before we reach the self-referential stage.
However, the question of what we are able to say about the universe we inhabit remains a legitimate question.
For example, if we describe electrons as having wavelike properties we are describing one part of the universe in terms of other parts.
Our sense of waves and wavelike properties come from our experience of the universe. We then try to describe the universe by saying that electrons behave (somewhat) like waves.
We are describing the universe as being similar to the universe.
This isn't wrong, of course. We can compare and measure similarities and differences between parts of the universe.
But there is no explanation. We can't step outside the universe and objectively describe it independent of our experience within it.
We are here
All of our experience, understanding and knowledge derives from our existence within the universe.
When we try to describe the universe itself (physics) we find that our measuring sticks are part of the thing we are trying to measure.
We can still make measurements. The Earth's circumference is roughly 40,000km. And a metre is one ten millionth of the distance from the north pole to the equator through Paris.
{The metre is now defined as the distance travelled by the speed of light in vacuum in a particular time.}
As a species, we are getting pretty good at measuring things. But we can't explain what we are measuring.
We don't know what distance is. From small scale to large scale, we can measure distance - but we have no clue how to create it from first principles. We can describe what we observe; but not why we observe in the first place.
Not nihilism
We cannot explain the universe. All of our explanations have the universe as a given.
All we can do is describe/measure one piece of universe using other pieces of universe.
This is a limit on knowledge - if you expect omniscience. We can't explain the universe.
We can describe what the universe is, though.
We cannot say what an electron is. The intrinsic properties of an electron are permanently outside our ability to speculate upon.
We can measure electric fields (whatever they are).
In the case of the staircase paradox mathematicians simply accept the fact that the sequence of arc lengths does not converge to the length of the arc that the sequence converges to under the supremum metric on a space of contours. No changing of the rules.
Not hurriedly, sometimes it takes centuries.
As a historically contingent activity of humans, math evolves. We extend our concepts to incorporate new situations and paradigms. Negative numbers, complex numbers, transfinite numbers. Set theory, category theory, homotopy type theory.
This is a natural process. You seem to take a pejorative view of mathematical evolution.
Quoting Treatid
Drat that Euclid, and Ernst Zermelo too.
Quoting alan1000
Alan1000, did you get your money's worth from this thread?
.999... = 1 is a theorem of the hyperreals. It must be so, since both the standard reals and the hyperreals are models of the same first-order axioms, therefore they must satisfy the same first-order theorems. This is a simplified statement of a technical fact in model theory.
https://en.wikipedia.org/wiki/Transfer_principle
Aleph-null is not a hyperreal, by the way. You can add symbolic points at plus and minus infinity to the standard real line to get the extended real numbers, but their only use is to make some notation simpler, such as infinite limits and limits at infinity.
https://en.wikipedia.org/wiki/Extended_real_number_line
There is mathematics - and there is the justification/explanation of mathematics.
Applied mathematics is concerned with whether the results are useful. I'm more than fine with this.
It is where pure mathematics tries to establish a foundation of knowledge that I am disgruntled. The effort is laudable - but mathematicians have gotten themselves stuck in a dead end and appear unwilling to extricate themselves.
Axiomatic Mathematics is the show piece of mathematics within which reside logic, formal languages and the majority of mathematical proofs.
But it doesn't work. Axiomatic Mathematics can't define axioms. As deal breakers go - this is one.
There is an argument that a flawed system is better than no system. "We know axiomatic mathematics is flawed - but it is better than nothing".
Except that axiomatic mathematics without axioms isn't merely flawed - it doesn't exist. The axiom bit is not negotiable. You can define axioms or you can't.
As it stands, axiomatic mathematics strives to find the essence of meaning by stripping away extraneous fluff like relationships.
In fact, meaning resides entirely in those relationships.
All progress in modern thought is emphatically despite axiomatic mathematics. The presumption of objective truth has been a catastrophic mistake in modern thought.
There is nothing to be lost by discarding axiomatic mathematics.
As it happens, we can describe relationships. The thing that axiomatic mathematics is trying to dispose of is exactly where knowledge, understanding, meaning and... everything is.
Mathematics' insistence that the path to truth is in defining inherent properties is holding back human progress.
To be fair - mathematics is merely making explicit general societal assumptions. By making implicit assumptions explicit, mathematics makes it much easier to understand what our assumptions are and consider them critically.
I do think that the idea of an objective universe is a dead end and mathematicians should have examined their failures more critically. And we still need the rigour and pedantry of the mathematical process.
Quoting jgill
That is debatable.
The central problem is that the rules aren't, and cannot be, defined.
When nobody knows exactly what the rules are, it is hard to determine whether the rules are being followed consistently.
We could spend decades arguing back and forth over whether mathematicians are applying rules consistently to the staircase paradox. And we will never come to a conclusion without a definite and unambiguous statement of what the rules are.
It is impossible to have a definite and unambiguous statement of what the rules are. Definitions are relational. X is not(Everything else). Roughly: A defines B and B defines A.
Any definition of rules (or axioms or anything else) results in infinite regression or circular definitions.
This is where mathematics is stuck. In order to make clear, unambiguous mathematical statements, we first need clear, unambiguous (mathematical) statements.
There is no starting point to jump start mathematics.
..................
I didn't mean to sucker you into a debate on the foundation of mathematics...
But mathematics doesn't have a foundation (c.f. foundational crisis in mathematics).
As much as mathematicians really, really want to make categorical statements - they can't. Anymore than philosophers have managed to establish a fixed point to build philosophy on.
To be clear - I'm not a nihilist. We can and do describe the relationships things have.
It just sp happens that you can either describe X, or the relationships of X but not both.
In this universe - we can describe the relationships of X.
I don't think you would find a mathematician who would spend more than hour on it.
Ok. You don't like formal math, as you envision it to be. You might be interested to know that there are other approaches to 20th century set theory, such as category theory and homotopy type theory. Some of these areas involve non-traditional intuitionist logic. So you should not think of math foundations as static. Quite a lot of work is being done.
Now I'm not sure if you'e unhappy with axiomatic math, or pure math. As in math for the sake of math, the kind of math that might never be useful. Or it might turn out to be useful in a hundred years or longer.
But you're unhappy with pure math, or math done from axioms. That's ok with me. I am not the Lord High Defender of Math. If you don't like pure math that's ok with me.
But having said that I'm perfectly ok with whatever your feelings are in this matter, let me try to address some of your concerns the best I can.
Quoting Treatid
Better than nothing for what? You know, if you ask a thousand pure mathematicians what axiom system they're using, or if they can state the ZFC axioms, they'd look at you funny. They don't study axioms. They study the math of quantum field theory, or exotic topological spaces, or deep properties of the natural numbers. They're never thinking about foundations. They have no opinions on foundations and they wouldn't even understand the question. It doesn't come up.
Just ask @jgill, who has had a career as a professional mathematician without having much contact with the foundational side of things.
It's like living in a house. How often do you climb underneath the house to play around with the foundation? Most people just hire specialists for that.
So nobody is making any compromises with anything or using a flawed system. They're just doing math.
Quoting Treatid
Aha. I can answer that.
It's like the game of chess. It has formal rules. Within the game there are moves and positions that are legal; and moves and positions that are not legal, according to the rules. The rules are more or less arbitrary. None of it "means" anything; but people themselves invest the game with varying levels of meaning, from the occasional player to the tournament grandmaster. People devote their lives to the game, yet the game has no intrinsic meaning at all.
If it helps, you can think of math that way. It's a formal game. It means nothing. Some people enjoy it. Some make it their life's work.
If some physicist, or architect, or bank teller, comes along and has some use for math; all the better. It makes no difference to the pure mathematicians; except insofar as it motivates their universities to employ them to teach some low-level math skills to the physicists and bank tellers; and then leave them alone to do their obscure, pure research.
So what's wrong with that?
Quoting Treatid
Of all the intellectual disciplines, math is the least concerned with objective truth; and makes no claims to it whatsoever. A mathematician will tell you that proposition P implies proposition Q within some particular axiomatic system; but they never claim it means anything about the real world.
You should take up your complaint with the physicists and bank tellers. They're the ones trying to apply all of this fictitious math.
Quoting Treatid
You want to make all those math professors go out and get real jobs? That's cruel.
But what do you mean? There would be nothing to be lost by throwing out music, or mountain climbing, or chess, or sports. None of those are necessary to life, they just make life worth living. Entertainments. Fields of study.
People study the 15th century British kings and queens. They study the great wars. They read literature or pulp novels. Why on earth should't those who are so inclined, study abstract math?
And besides -- once in a while, a previously useless area of math becomes useful. Mathematicians invented non-Euclidean geometry in the 1840's. It was regarded as a curiosity. Then in 1915 Einstein found that it was just the thing for him to express his new theory of relativity.
So math is often about reality, but in the future. And if you ban all the pure math, you'll lose all those tools that might be needed.
Quoting Treatid
If you like the math of relationships, you should chec out category theory. It's the "abstract theory of arrows." You have a bunch of things and a bunch of arrows between pairs of things. Relationships if you like. You can express quite a lot of math in that framework.
https://en.wikipedia.org/wiki/Category_theory
Quoting Treatid
You are just making that up. It's your own straw man. There are no mathematicians asserting any such thing. Mostly they're just trying to crank out the next paper to justify their latest grant; and teaching calculus to the budding physicists and engineers.
Where are you getting such ideas that mathematicians are insisting on any such thing?
Quoting Treatid
Math doesn't make any assumptions about society. Are you thinking about quantitative sociology perhaps? Or epidemiology, where we apply statistical methods to see how diseases spread
Mathematicians don't do the things you think they do.
If I may be so bold as to tell you: All of this is entirely in your head. Your ideas do not refer to anything real about mathematics. You are tilting at windmills of your own imagination.
Quoting Treatid
By the time I got to the end of this I did not think you had an argument or thesis at all. Speculations regarding the nature of the universe belong to the philosophers and sometimes the cosmologists and quantum physicists.
Definitely not mathematicians. When mathematicians study quantum field theory, they do the math. They don't do the metaphysics. And if they do, at that moment they are acting as philosophers and not mathematicians.
You have your basic facts all wrong.
:clap: :cool:
You put too much effort in a post towards someone who won't learn from disagreement. In another thread, he said that he is quite sure about he was talking about, and in the same post he said that the twin paradox is a paradox in Newtonian physics only and not in relativity.
Quoting fishfry
Study who?
Sorry did I get my history wrong? Not sure what you meant. I was making the point that people find value in studying all kinds of stuff so why not math. Was my analogy off the mark?
Quoting Lionino
Quoting fishfry
.
Quoting fishfry
No, it was a contrarian joke implying that people (me) don't study them.
Speaking of category theory, I came in contact with it (again) to explore the subject of vector spaces with irrational dimensions. Naturally, vector spaces traditionally defined have a dimension n, n E N, naturally because the set of its basis can't have ? elements, but something like that is the case of fusion categories, if a mathoverflow user is to be trusted.
Sorry, must be quoting issues.
Quoting Lionino
Oh I get it.
Quoting Lionino
Wow that's a new one on me. The Wiki page on the subject is brief and unhelpful.
How would a vector space dimension work that wasn't a positive integer? Have a link?
ps ... found this.
https://math.stackexchange.com/questions/1466820/vector-spaces-with-fractional-dimension
There is this link https://ncatlab.org/nlab/show/fusion+category
It mostly goes over my headQuoting fishfry
That is the stack user I was referring to.
Pretty weird stuff ...
It is not a popular function of "pure" mathematics to delve into these issues. For example, arXiv.org lists the number of math papers submitted in this last week: Total-783, Logic-5, History&Overview-1. To compare: Category Theory-18, Complex variables-18. Others-751 (29 additional categories).