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I know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist. Could someone make this a little bit more precise? Are there good reasons for taking this point of view? Can you actually get math done from that perspective?

Noah Snyder
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    How is this question not community wiki? – Asaf Karagila Oct 03 '10 at 23:00
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    It is fun to also consider this question: What is "ZFC" and why do people beleive in it? –  Oct 03 '10 at 23:38
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    @Asaf: It seems OK - it's asking for an explanation of an established position in the philosophy of maths. The 2nd subquestion, "are there good reasons", is maybe a bit open-ended, but the question has hardly generated a deluge of opinionated posts. – Charles Stewart Oct 04 '10 at 09:07
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    Hm... does induction only work up to the amount of things I can prove before the death of the universe? =P – Simply Beautiful Art Dec 28 '17 at 01:55
  • Ultrafinitsm always seemed counter-intuitive because I do not see how can you build the Natural numbers when there's a limit. And what stops us from adding 1 to that limit and getting a bigger number? I guess you need to dismiss Induction if we're talking in terms of Peano axioms, but without induction, how do you prove something for all the Natural numbers. Are there finite constructions of the Natural numbers even? – Everstudent Oct 03 '20 at 21:18

3 Answers3

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Ultrafinitism is basically resource-bounded constructivism: proofs have constructive content, and what you get out of these constructions isn't much more than you put in.

Looking at the universal and existential quantifier should help clarify things. Constructively, a universally quantified sentence means that if I am given a parameter, I can construct something that satisfies the quantified predicate. Ultrafinitistically, the thing you give back won't be much bigger: typically there will be a polynomial bound on the size of what you get back.

For existentially quantified statements, the constructive content is a pair of the value of the parameter, and the construction that satisfies the predicate. Here the resource is the size of the proof: the size of the parameter and construction will be related to the size of the proof of the existential.

Typically, addition and multiplication are total functions, but exponentiation is not. Self-verifying theories are more extreme: addition is total in the strongest of these theories, but multiplication cannot be. So the resource bound is linear for these theories, not polynomial.

A foundational problem with ultrafinitism is that there aren't nice ultrafinitist logics that support an attractive formulae-as-types correspondence in the way that intuitionistic logic does. This makes ultrafinitism a less comfy kind of constructivism than intuitionism.

Why do people believe it? For the same kinds of reasons people believe in constructivism: they want mathematical claims to be backed up by something they can regard as concrete. Just as an intuitionist might be bothered by the idea of cutting a ball into non-measurable pieces and putting them back together into two balls, so too an ultrafinitist might be concerned about the idea that towers of exponentials are meaningful ways of constructing numbers. Wittgenstein argued this point in his "Lectures on the Foundations of Mathematics".

Can you actually get math done from that perspective? Yes. If intuitionism is the mathematics of the computable, ultrafinitism is the mathematics of the feasibly computable. But the difference in ease of working with between ultrafinitism and intuitionism is much bigger than that between intuitionism and classical mathematics.

Ben Millwood
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Charles Stewart
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    To make your final statement more explicit: the math one can actually get done under ultrafinitist constraints is essentially a sub-discipline of theoretical computer science — i.e. deterministic polynomial time algorithms. (Not that all theoretical computer scientists are ultrafinitists, of course.) The difference between ultrafinitism and constructivism can in a sense be represented by complexity classes, and studying the vast chasm between ultrafinitism and constructivism is then a central activity of the field! – Niel de Beaudrap Oct 04 '10 at 07:31
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    Just to nitpick: you don't need 10^50 non measurable pieces for the Banach-Tarski paradox. 5 are enough. – LIE Feb 09 '11 at 09:23
  • @LIE: Indeed. I shall fix the text. – Charles Stewart Feb 09 '11 at 09:45
  • It is not the case that whatever you do with ultrafinitism is a subset of math, rather, whatever you can't do with ultrafinitism, is nothing but literature. The issue is not that choice theory based infinities have unknown unknown complexities, rather that it's inappropriate to question anyone in mathematics. It's a positivist field and deeply corrupted. You say easier, I but I know what problems floats cause in practice. – Ashnur Feb 27 '20 at 23:27
  • Isn't it easier to use infinity as an abstract concept even if you deny its existence? I mean, often you need tools which are more abstract to prove something concrete? Like for example you may use Gaussian integers to prove things for integers? So in the end - if something is useful, what's the point in thinking about whether it exists or not? After all people dismissed negative and complex numbers for similar reasons historically. But now that they are useful(accounting, electric circuits, etc.), no one does that. – Everstudent Oct 03 '20 at 21:10
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The philosophy is explained in Doron Zeilberger's article. Basically, it's the belief that there is a largest natural number!

I've heard a funny story (on Scott Aaronson's blog) about someone who was an ultrafinitist.

-Do you believe in 1?

-Yes, he responded immediately

-Do you believe in 2?

-Yes, he responded after a brief pause

-Do you believe in 3? -Yes, he responded after a slightly longer pause -Do you believe in 4? -Yes, after several seconds

It soon become clear that he would take twice as long to answer the next question as the previous one. (I believe Alexander Esesin-Volpin was the person.)

Akhil Mathew
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  • This is a good one :) – BBischof Jul 23 '10 at 03:32
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    Zeilberger's article is wrong though, the world isn't a digital computer, because of quantum mechanics... – Noah Snyder Jul 23 '10 at 03:36
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    or a digital computer simulate 'quantum mechanics'. If you are in the matrix, they can make you believe there is no gravity. – Chao Xu Jul 23 '10 at 04:01
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    There are recent theories that say the universe is effectively a quantum computer though, where (qu)bits represent the superposition of states. – Noldorin Jul 23 '10 at 08:15
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    This is a very nice answer. A quibble: *it's the belief that there is a largest natural number* - Not quite. If you don't believe in the principle of induction, it doesn't follow from accepting the existence of zero and the successor function, that numbers get very big. At least, not if big means much, much bigger than the amount of time you are prepared to spend constructing them. – Charles Stewart Jul 23 '10 at 09:24
  • Indeed it was Alexander Yessenin-Volpin. The blog post of Scott Aaronson is [here](http://scottaaronson.com/blog/?p=103). (Also [some context here](http://dialinf.wordpress.com/2009/02/16/achilles-tortoise-and-yessenin-volpin/).) – ShreevatsaR Oct 02 '10 at 14:07
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    I think it should be $2^1$, $2^2$, $2^3$, $2^4$, _etc._ His interlocutor was trying to get him to agree that $2^{100}$ exists. – Trevor Wilson Nov 25 '13 at 03:28
  • @TrevorWilson: Yes, that fits the original (and more sensible) quote on Wikipedia: https://en.wikipedia.org/wiki/Alexander_Esenin-Volpin#Mathematical_work – Blaisorblade Feb 19 '16 at 23:27
  • @Akhil Mathew Just to clarify : Do Ultrafinists really believe that there is a largest natural number ? If yes, do they make any effort to concrete, which it is , or at least which approximate magnitude it has ? – Peter Jun 10 '16 at 10:31
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    @Peter The bound is on complexity not on the magnitude of numbers. In an ultrafinitist system representation *really* matters. There are about $10^{80}$ atoms in the (observable) universe. Assuming we can use each and every one to represent a bit, then in a Peano representation of numbers the largest number is ~$10^{80}$. In a binary representation it would be ~$2^{10^{80}}$. (1/2) – Derek Elkins left SE Dec 18 '17 at 01:40
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    However, in a representation that lets us stack exponentials, you can make vastly larger numbers but the complexity is no longer monotonically related to the magnitude. So while there is still a "largest" number, it's less relevant. Instead, the natural numbers in that representation have a fractal structure with huge gaps of numbers too complex to represent that are nowhere near the "largest" number. For example, $2^{2^{2^{2^{1000}}}}$ is compactly represented, but there are huge swathes of numbers between it and $2^{2^{2^{2^{999}}}}$ that can't physically be represented in this way. (2/2) – Derek Elkins left SE Dec 18 '17 at 01:40
  • Do you think it actually was the case that they would take twice as long after each question? Maybe after a really long time, they would decide it is a matter of deciding what you want to assert and you can assert what ever you want to for what ever reason you want which will generally come from seeing what results you get after performing calculations by a different method. That's why they will eventually say yes to the question of whether they believe in 7. Sometimes you can seek a definition of something to fit the desired properties. I believe we could invent our own definition of a formal – Timothy Mar 12 '20 at 00:35
  • proof for a certain system where you start with the conclusion and then for each of the two statements the conclusion derives from, you write a full formal proof of one and then a full formal proof of the other. In this case, we need a QED symbol at the end of a formal proof in order to have each string of statements uniquely represent a proof. This method of writing a formal proof is similar to Polish notation. Now this method of writing a proof might actually make it so that each natural number takes twice as long to prove exists than the previous natural number. – Timothy Mar 12 '20 at 00:40
  • Ultrafinitism isn't a monolith. Zeilberger has said that there is a largest natural number, but other ultrafinitists generally don't. Some have said that $10^{10^100}$ (that's even the title of one paper), but here we have a commenter who says that it does (only some of the numbers in between don't). – Toby Bartels Jun 23 '21 at 14:38
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Greg Egan has some fun with this idea in one of his best short stories, "Luminous" (published in the collection of the same name). A pair of researchers are exploring an apparent "defect" in mathematics:

"You still don't get it, do you, Bruno? You're still thinking like a Platonist. The universe has only been around for fifteen billion years. It hasn't had time to create infinities. The far side can't go on forever-because somewhere beyond the defect, there are theorems that don't belong to any system. Theorems that have never been touched, never been tested, never been rendered true or false."

Terrific stuff!

timday
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    It is not known whether the universe "started out" at the big bang being finite or infinite. The observed fact that space appears flat suggests that it is infinite in extent, in which case it would have been infinite already at the big bang. Not that it detracts much from a nice fiction. – Tommy R. Jensen Jul 17 '18 at 11:11
  • @Tommy: To be fair the text does say "It hasn't had *time* to create infinities". Even if it started spatially infinite, no theorems at all might have been "tested" across that space initially. Not sure it's worth prodding too much at the cosmology of a scifi short story though. – timday Sep 17 '18 at 22:43
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    I think you're *supposed* to prod at the cosmology of this story. There's a plot with humans, aliens, and an evil corporation; but the cosmology is what it's really about. – Toby Bartels Jun 23 '21 at 15:21