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This is a by product of this recent question, where the concept of ultrafinitism came up. I was under the impression that finitism was just "some ancient philosophical movement" in mathematics, only followed by one or two nowadays, so It sounded like a joke to me.

But then I got curious and, after reading a bit, It seems to me that the only arguments against infinite mathematics that finitists seem to have are that "there are numbers so big that we couldn't computate in a lifetime" or the naive set theory paradoxes. The former doesn't seem like a serious argument, and the latter is not a problem now that mathematics relies on consistent axioms.

Are there some (maybe arguably) good mathematical reason to deny the existence of $\infty$ or is it just a philosophical attitude? The concept of unboundedness seems pretty natural to me, so what could be a reason to avoid it? Does this attitude even make any sense?

In short, why today-finitists have a problem with $\infty$?

Edit: First of all, thank you so much for your answers (and comments), they have been enormously illuminating. :)

I didn't know that "finitism vs. infinitism" was such a polemic topic. Now I myself agree this question might look as primarily opinion-based. However, It was not my intention to open a debate about "which posture is better"; I was just meaning to ask about what specific mathematical reason (argued and not-primarily opinion based) do finitists have to reject the "infinitists" use of infinity.

Based on the two excellent answers I've already had (thank you again :) It is my understanding that their main problem with the use of $\infty$ is that it leads to mathematical results (like the Banach-Tarski paradox) which they don't recognize as true when looked through the glasses of our real-world experience.

Final edit: After reading every answer and comment (specially Asaf Karagila's) I've came to the conclussion that there are not strictly mathematical reason to avoid the use of infinite. That my specified question on the last edit has no answer, and that the motivation to stick to finitists or infinitists view of mathematics relies on how much one expects mathematics to describe each one's "real world". As Wildcard's answer is the one that clarifies that matter best to me, I am accepting it. Thank you all again for your answers and comments!

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    $\begingroup$ Just a small caveat; we don't know that $\mathsf{ZFC}$ is consistent. We just haven't found a contradiction yet. $\endgroup$ – Hayden Oct 28 '16 at 22:54
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    $\begingroup$ Have you read about constrictivism? Finitism is a type of constructivism. en.m.wikipedia.org/wiki/Constructivism_(mathematics) $\endgroup$ – Paul Oct 29 '16 at 0:00
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    $\begingroup$ @Hugh I've just readed that post, pretty interesting. However I am not asking about finitism (or ultrafinitism) in a philosopical point of view. I am concerned with the mathematical point of view. Precisely if there are any mathematical reason (maybe arguable) to avoid infinity. I.e. why do they feel that infinite mathematics is a somewhat less reliable kind of mathematics? $\endgroup$ – user378947 Oct 29 '16 at 0:34
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    $\begingroup$ @Hayden: That is not entirely accurate. We can easily prove the consistency of ZFC, or any theory we haven't found to be inconsistent yet. The main question is what is the meta-theory from which you are trying to prove this. We know that ZFC cannot prove its own consistency, so the question is, do you assume a stronger theory, or not? It should be pointed that the stronger theory will also be subject to this issue, and will not be able to prove its own consistency. So you have an infinite descent. But you can argue the same about almost anything mathematical. $\endgroup$ – Asaf Karagila Oct 29 '16 at 14:38
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    $\begingroup$ [...] Ultimately, this is a matter of opinion, just like choosing your mathematical foundation to be ZFC or HOTT or category theory is a matter of personal preference: what mathematics you like to do, and how you think you'll have the easiest time arguing your mathematical argument as readably while being as formal as possible. Since there are no mathematical arguments as to why to prefer one foundation over the other (as long as all of them are not provably inconsistent, that is), the question becomes philosophical and a matter of personal and social choice. $\endgroup$ – Asaf Karagila Oct 29 '16 at 14:51
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To add to Reese's excellent answer, I will say first that I don't consider myself a finitist, but I can understand why finitists postulate as they do—even the ultrafinitists.

First, to quote the Wikipedia article already cited in a comment (emphasis added):

Even though most mathematicians do not accept the constructivist's thesis, that only mathematics done based on constructive methods is sound, constructive methods are increasingly of interest on non-ideological grounds. For example, constructive proofs in analysis may ensure witness extraction, in such a way that working within the constraints of the constructive methods may make finding witnesses to theories easier than using classical methods. Applications for constructive mathematics have also been found in typed lambda calculi, topos theory and categorical logic, which are notable subjects in foundational mathematics and computer science.

Now my own explanation of ultrafinitism has less to do with belief than practicality. But first, a discussion of ideas themselves.


Numbers, ultimately, are ideas, as are every other element in mathematics.

You say (emphasis added):

Are there some (maybe arguably) good mathematical reason to deny the existence of ∞ or is it just a philosophical attitude? The concept of unboundedness seems pretty natural to me, so what could be a reason to avoid it?

This is inherently a subjective position, and a perfectly valid one: "I can think of this idea, so how can someone say this idea doesn't exist?"

Of course it would be ludicrous to claim that the idea of infinity doesn't exist. You just thought of it (thought about it), didn't you?

I'll be an ultrafinitist for a moment and explain the position.

It's not that infinity doesn't exist as an idea, it's that you cannot point to an infinity anywhere in the real universe. Anything you point to is necessarily finite, or you couldn't point it out or demonstrate it.

Mathematics is all (all) based on working with symbolization of real or abstract data. You're dealing with ideas, fundamentally, and ways of representing those ideas to resolve, communicate about, or pose problems—again, either real or abstract.

Please don't be so attached to a single system for ideas and their symbolization that you fail to recognize that other ideas may exist outside of that scope.

You criticize ultrafinitists for failing to include the concept of an infinity in their abstractions and symbolizations. Very well, why is it that your own mathematics fail to include the concept of "certainty"? Or "knowledge"? Or "co-existence" (the same number having two different values at the same time)? Or how about "time" itself, since that is not included in mathematics?

If you can work with your mathematics and get results that work, or even just that you find interesting, fine. If I can work with my mathematics and get different results than you, but they work for me (produce a desired result when applied to the real world), great.

But this is all more general, covering the broad sweep of differences of mathematical ideas.


To answer your precise question, and provide the "arguably good" mathematical reason to omit consideration (not "deny the existence") of the infinite in a mathematical framework, it is:

If you omit everything that cannot be directly observed, and abstract only that which can be observed, your results will apply uniformly to the observable universe.

This conclusion itself can only be demonstrated by observation of the observable universe—it cannot be theoretically evolved. It itself is separate from the approach of theoretically evolving a set of ideas, so it cannot be measured by the yardstick of theoretical postulation of ideas.

Chew on that for a bit. :)


Even if I'm speaking as an ultrafinitist, I would still say there is one factual infinity:

The possible different ideas that can be conceived of and posed by the human mind is infinite.

But that doesn't make the idea of an infinity inherently superior to the idea of no infinity. ;)

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    $\begingroup$ I have to ask, though - are the possible different ideas that can be conceived of and posed by the human mind infinite? Or are they finite, since the configurations of the human brain are finite, and what's infinite is an abstraction of the human mind, like a Turing machine, failing to reflect it the same way circuit elements on a diagram don't reflect real-world components? $\endgroup$ – user361424 Oct 30 '16 at 9:52
  • $\begingroup$ @user361424, I don't understand which way you're posing your circuit elements analogy, so I don't quite get the question. But you posit a correspondence between the human brain and the human mind which is unjustified and undemonstrable. (And remember, I'm not generally an ultrafinitist, even though I'm capable of evaluating ideas from that perspective.) $\endgroup$ – Wildcard Oct 30 '16 at 14:51
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It's not known that modern set theory is consistent; in fact, by the Incompleteness Theorem, we can't ever have a system of axioms that we can prove is consistent. Which means that the only condition we can rely on for determining whether a set of axioms is "right" is whether or not it produces absurd results.

Under $ZFC$, we have different sizes of infinity - there are sets which are larger than the set of natural numbers in a precise sense. We also have a lot of weirdness involving the Axiom of Choice - for example, with the Axiom of Choice, a theorem of Banach and Tarski states that a hollow sphere can be disassembled into five pieces and then reassembled (without stretching, tearing, or otherwise deforming the pieces) into two spheres that are both identical to the first one in both size and shape. But the Axiom of Choice simply states that given a set of sets, we can "choose" one element from each set - which seems intuitively true.

A finitist's perspective on $ZFC$ is often that results like the hierarchy of infinite cardinals and the Banach-Tarski paradox are absurd - that they should count as contradictions, because they patently disagree with the intuitive picture of mathematics. The sensible conclusion is that one of the axioms of $ZFC$ is wrong. Most of them are intuitively obvious, because we can demonstrate them with finite sets - the only one we can't is Infinity, which states that there exists an infinite set. So a finitist's conclusion is to reject the Axiom of Infinity. Without that axiom, $ZFC$ becomes purely finitistic.

Now, many finitists are happy to stop here. But some are bothered by the fact that we still have an infinite collection of natural numbers; the infinite still "exists", in a sense, and gives the opportunity for the above weirdnesses to arise in the same way. So some people (including some mathematicians) subscribe to ultrafinitism and insist that there are only finitely many numbers at all. One ultrafinitist mathematician I know defines the largest integer to be the largest integer that will ever be referenced by humans.

Among mathematicians, ultrafinitists are much rarer than simple finitists. Finitists generally agree with you that "unboundedness" is a natural idea - it's essential, for example, in the definition of a limit. But they would go on to insist that this is just a formalism - that a limit, for example, is just a statement of eventual behavior, involving only finite numbers. So $\infty$ isn't an object, it's just a shorthand. This is (at least to my mind) more mathematically defensible than ultrafinitism.

EDIT: Since a lot of people seem to be having a hard time with my first sentence, I thought I'd clarify. The Incompleteness Theorem states that we cannot have a set of axioms powerful enough to express arithmetic and still be able to prove its consistency within the system. The reason I didn't include this phrase above is because it unnecessarily weakens the point. Any axiom system intended to codify all of mathematics defines the idea of "proof"; if, say, $T$ is intended to underlie all of mathematics, then by "proof" we must mean "proof in $T$". With such a $T$, we can say that $T$ cannot be proven consistent at all; because by Incompleteness, any proof of the consistency of $T$ would not be a proof from inside $T$, but $T$ is supposed to be powerful enough that all proofs are proofs from inside $T$. Thus: we can't ever have a system of axioms for mathematics that we can prove is consistent - full stop.

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    $\begingroup$ @Alephnull, to answer as an ultrafinitist would, you only referenced it abstractly. You can reference "infinity" also, but that's not actually an integer. How did you get this "largest integer" abstraction to which you refer? (Hint: You didn't construct it.) If you would argue with a mathematician, you must do so from the axioms with which he agrees. $\endgroup$ – Wildcard Oct 29 '16 at 3:09
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    $\begingroup$ I am a little bit confused, both incompleteness theorems start with the consistency assumption, thus the "incompleteness" in their names. So "by the Incompleteness Theorem, we can't ever have a system of axioms that we can prove is consistent", it IS important to add "from within the system" and this is the 2nd Incompleteness Theorem. Otherwise we are left with a sense of fatality ;) $\endgroup$ – rtybase Oct 29 '16 at 9:33
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    $\begingroup$ I stated it 4 comments ago <<it IS important to add "from within the system">>. That will be 100% accurate and pretty much all ;) $\endgroup$ – rtybase Oct 29 '16 at 10:33
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    $\begingroup$ Reese, I agree with your elaboration (mathematically, not quite philosophically), and this is why I didn't say that the first sentence is wrong, but rather a painful stretching of the incompleteness theorem. You're hiding essential details, and presenting it as if we can never prove that any theory is consistent. So you either question all the proof systems that we have in place, or you twist the arms of the incompleteness theorem behind the curtains to make it say what you want it to say. Either way, in a different discourse this might have been called out as dishonesty, if done on purpose. $\endgroup$ – Asaf Karagila Oct 30 '16 at 9:11
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    $\begingroup$ The statement by Reese that a hollow sphere can be duplicated with five pieces....should really be: "a hollow sphere can be duplicated with four pieces". Five pieces are needed for the solid piece, since the center of the ball causes a problem with the radial extension of the classic 4-piece solution. $\endgroup$ – stan wagon Dec 19 '16 at 18:53
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Leopold Kronecker was one of the leading mathematicians at the end of the 19th century. Kronecker disagreed sharply with contemporary trends toward abstraction in mathematics, as pursued by Cantor, Dedekind, Weierstrass, and others. Specifically, Kronecker rejected the notion of completed infinity. Many of today's mathematicians are so used to set theory being "the foundation" that they have difficulty relating to Kronecker's point to begin with. Kronecker's ideas were close to but not identical to the modern constructivist ideas; thus Bishop for one arguably did accept actual/completed infinity, since early on in his book he speaks matter-of-factly about functions $f$ from $\mathbb N$ to $\mathbb N$. The difficulty we have of relating to Kronecker's viewpoint has to do with our training. Recently Yvon Gauthier tried to explore Kronecker's position; see in particular his

Gauthier, Yvon. Towards an arithmetical logic. The arithmetical foundations of logic. Studies in Universal Logic. Birkhäuser/Springer, Cham, 2015

and

Gauthier, Yvon. Kronecker in contemporary mathematics, general arithmetic as a foundational programme. Rep. Math. Logic No. 48 (2013), 37–65.

Gauthier argues in particular that many applications Kronecker worked on can be handled without superfluous infinitary assumptions that merely clutter the picture.

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  • $\begingroup$ I think you could define functions in such a way that you dont have to think about the domain as a whole. You just specify what input is accepted, and what input would be rejected, like is done in static programming languages. $\endgroup$ – Kasper Jun 6 '18 at 17:16
  • $\begingroup$ @Kasper, sure, this is the Kronecker-Gauthier position and also that of many other mathematicians and I didn't say otherwise. In Reverse Mathematics they deal with numerous systems that are weaker than traditional ZFC fare, and show that often a rather minimalist system of axioms suffices for many purposes. $\endgroup$ – Mikhail Katz Jun 7 '18 at 8:53
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    $\begingroup$ I could add also that challenging realist beliefs about infinite sets tends to ruffle some people's feathers, as you can see from the breakdown of the voting on this question ;) @Kasper $\endgroup$ – Mikhail Katz Jun 8 '18 at 7:15
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If you believe that the natural world is all that exists, then that automatically makes you an ultra-finitist. There is simply no room in the physical world to store a monstrously large integer.

The axiom of infinity is obviously false when taken literally in the physical world, so if the physical world is all that exists, then the axiom of infinity is false.

This does not mean that one should reject ZFC or any of its theorems. While Con(ZFC) (ZFC is consistent) has not been proven in the usual meaning of proof, there are nevertheless very good reasons to believe Con(ZFC). It has been tested very thoroughly, both theoretically and practically, to the point that doubting Con(ZFC) is simply no longer a reasonable position to take.

Con(ZFC) means that all predictions of ZFC that are testable in the real world ought to be believed. There is no problem with using real numbers and infinite sets to prove theorems that are then used in weather forecasts and other scientific computations. Con(ZFC) means that in testable predictions, we can trust math.

That's why we should teach theorems derived from ZFC without any reservations. That also means: accept the Banach-Tarski theorem, different infinite cardinalities, etc. (not as "true" in the physical world, but as theorems of mathematics).

Ultra-finitism is true in the real world. However, none of the theorems based on ZFC ought to be controversial.

Scientific computations usually use approximate floating point numbers. It is hard to prove clean statements about such objects. It is much easier to first prove statements about infinite-precision "real" numbers (even though they don't actually correspond with anything in the "real" world). Infinite sets, and infinite-precision real numbers are so useful that it would be foolish not to work with them, and the highly probable Con(ZFC) means that math based on infinite objects can be trusted.

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  • $\begingroup$ I agree that we can only observe finite things. But surely one can believe that the universe is itself infinite. And that even if we don't understand its quantized nature, it might still be infinitely divisible via some other mechanism (which may or may not be useless for us, being so much larger than Planck length). $\endgroup$ – Asaf Karagila Feb 12 at 13:56
  • $\begingroup$ Whether infinite sets actually exist or not, I think this falls in the category of statements that physicists call "not even wrong" (a label reserved for statements that are not testable). When a question is impossible to settle with experiments, it is OK to believe it one way or the other. It is conceivable that the real world is continuous like the continuum in math, but I think that this is no longer the most common belief among physicists. But either way it is reasonable to doubt the existence of infinite objects (and still use them to write proofs!) $\endgroup$ – Mark Feb 13 at 16:03
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Many quality responses here, but ill add my perspective.

I agree that many defenses of finitism are silly or incomplete, but there is an underlying point to it that is quite valid and important.

The way I would put it, is that infinity as a concept should not be given the same epistemic status as other mathematical ideas. As has been noted elsewhere, the concept fundamentally cannot be defended in an empirical manner. A pure mathematician may shrug as to that argument; but even within the confines of mathematics, there are clear conceptual differences. Math involving infinity cannot be described in a system that is proven to be free of contradiction; incompleteness is a feature of math involving infinity; not of math or formal systems per se.

Personally, I am wholly unimpressed by the so called modern set theoretic 'solutions' to the properties of infinity known since ancient times. 'naively' speaking, youd think that 1-to-1 correspondence, and the ability to construct one set as a subset of the other, would be equally good axioms to decide on the relative size of sets. Both are observably true; and we should be willing to embrace all the conclusions that can be derived from either axiom, as truths about the real world. The 'modern' approach is to observe that you can just drop one of these axioms, thereby get rid of some obvious contradictions involving infinity, and be so enamoured with this to jump to the conclusion that therefore this must be a good idea, without much further consideration.

The Banach-Tarski 'paradox' can be seen as a rather direct result of this hasty philosophical choice. Its not a paradox of modern math of course; yet it puzzles a lot of people; because most people, physicists especially, do like to be able to assume their math has anything whatsoever to do with the real world.

The way physics should work, is that you try and identify what axioms must be true, and then try and reason about what other implications those givens have, and see if that actually holds in the real world. What infact we have is our best physicists (Feynman specifically) mocking the Banach-Tarski paradox as silly, rather than going out and trying to set up an experiment to reproduce it (and the latter goes for all physicists I know). Thats a big epistemological breakdown. Now I think Feynmans common sense is as usual spot-on; but if there are some conclusions from math that we should take less seriously than others, id like to know which one's those are; to codify that intuition, if you will.

Personally, I find the paradoxes of infinity hard to get around. If I divide a number by two an infinite number of times, is it zero? No, because non-zero-ness is conserved by halving; yet it has to be, since if it is nonzero, I divided a finite amount of times. Are there more natural numbers than even numbers? I could go on, and I find the arguments for and against usually equally compelling. Sure you can 'solve' these problems by 'clever' redefinition of terms and other trickery; but invariably this seems to involve driving a wedge between what you take to be axiomatically true, and any empirical obviousness of such axioms.

Which isn't only bad for the physical/empirical users of math; but generally lends a 'flavor' to modern mathematics that is off-putting to many. Bertrand Russel, who gave a big impetus to the idea that if we just boil down mathematics to as few and abstract axioms as possible, all will be great, actually disavowed the wisdom of doing so in his last years. And I imagine he would agree with the later ideas of Quine on epistemological holism. That is, the quality of an axiom should be decided on its interaction with the entirety of our web of beliefs; not on what pure mathematician or logicians thinks is 'elegant' or looks pretty on paper.

Personally, I am more in favor of a *para-consistent approach to infinity (*I guess thats what its called but it seems to mean many things to many people). That is, Id rather take an approach where I embrace the contradictions in infinity. Which doesn't mean you cannot do math at all; plenty of people knew how to do math throughout history while believing in contradictions of infinity, and it wasn't because they were 'naive', but because they had common sense. In fact, the principle of explosion is nothing but an artefact of so called 'classical' logic (specifically, the fact that it does not specify a deterministic rule for symbol-lookup). Its really not that hard in the 21th century to define your logical language to be robust to contradictions, while merrily continuing to perform deductions on your non-contradicting clauses.

What that means for the 'existence' of infinity, I dont know, but I am happy with that gray status until someone can show me any of these alleged paradoxes empirically. Personally, I am inclined to believe that the infinite divisibility of space is probably as much a figment of our imagination as the infinite divisibility of matter still was well into the 19th century.

This is mostly an academic debate with little practical implications in the end. I don't think I (or anyone with finitist inclination) should be expected to come to different conclusions on any key point. They just come to fewer conclusions. And I think research into say the real number continuum is quite valuable; even if we disagree as to the parity of the last digit of the square root of two. I say the answer is both even and non-even; you say that it's not a rational number and you think sidestepping the question like that makes you less 'naive'. Soit.

But if I was the world's science Tsar, I would definitely defund any research into the continuum hypothesis, or anything involving hierarchies of infinities. Id tell these people to stop chasing the implications of Cantors hasty assumptions, and start doing something comparatively useful for society, like solving Sudoku's for instance.

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  • $\begingroup$ Your description of "The way physics should work" would be better described as "The way physics would work if it were done by mathematicians". Fortunately for scientific advance, it's not mathematicians who do physics. What physicists actually do is rather different, even when they are applying mathematical methods to their work: they do not "try and identify which axioms might be true"; instead they try to formulate simple hypotheses that might accord with messy physical phenomena and that might predict new and testable phenomena. $\endgroup$ – Lee Mosher Mar 17 at 16:29
  • $\begingroup$ Well, I take the view that mathematics and physics are best viewed as spectrums of a single discipline, and that there is far too much specialization in these fields. Words like 'axiom' and 'theorem' play much, or ought to play much the same role on an epistemological level. If we are discussing the question of 'which are sensible mathematical axioms to adopt', id want that discussion to take place in a room with Gauss, Feynman, etc; and people like Cantor or Zermelo, well anyone is allowed to have their opinions but I have a-priori little confidence in their taste in axioms. $\endgroup$ – Eelco Hoogendoorn Mar 17 at 17:49

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