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The title of this question is the exact title of one of the sections of a book written by Alexandre Borovik: Mathematics under the Microscope. Under the title, we read:

How should we approach the position of those mathematicians who refused to accept actual infinity? Henri Poincare is a prominent example. Weller et al. quote him on a par with modern day undergraduate students:

"There is no actual infinity; and when we speak of an infinite collection, we understand a collection to which we can add new elements unceasingly."

If I had in my class a student who made a similar statement and if the student's name happened to be Henri Poincare, I would not be in a hurry to correct him but would try to check whether Henri had a consistent vision of mathematics which was compatible with his thesis...Also, I have a gut feeling that Henri would still be able to solve every problem in the advanced calculus/analysis class. Indeed, within undergraduate calculus, every problem is solvable within the framework of the potential infinity of processes and sequences-you need only to be sufficiently attentive to detail.

Have you got the same gut feeling? Indeed, it is for years that a fundamental part of my course on the foundation of mathematics is to help my students to realize the actuality of actual infinity. That aside, next term I'll work with a group of mathematicians teaching Calculus, and today I have realized that the book they have chosen for the course is a kind of "soft analysis" that starts with real numbers and include different versions of the completeness of real numbers including the one that is based on decimal representation of numbers. That means, from very early on students are faced with the distinction between potential and actual infinity, and such a "famous" thing as $0.9999...$. For many, that thing just represents an infinite process. Shall we let them live with that conception providing that they have a consistent vision of what they do? (of course, this is a big assumption unless one of them be Henri!). Is there any problem in calculus that the distinction between potential and actual infinity indeed matters? That is to say, if one hasn't got the latter conception, the problem would become hard (or perhaps, impossible) to solve. If you think so, please give concrete examples.

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closed as off-topic by Andres Caicedo, Stefan Kohl, Tilman, Chris Godsil, Steven Sam Aug 12 at 4:49

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Andres Caicedo, Stefan Kohl, Steven Sam
If this question can be reworded to fit the rules in the help center, please edit the question.

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I always found the distinction between potential and actual infinity rather nebulous: what does "actual" mean in mathematics? Is the $2$ in $1+1=2$ a potential $2$ or an actual $2$ ? What about the $1$ or the $+$ or, gods forbid, the $=$ ? Doesn't the abstractness of mathematics preclude its actuality? The only good distinction I am able to make is that between considering infinite sets as sets and considering them as classes / types. Constructivism, as I understand it, does the latter. –  darij grinberg Aug 11 at 14:58
    
I think the distinction lies on the cognitive aspect rather than on the mathematical one. I understand the quote as expressing that mathematical concepts are necessarily finitistic and that looking for examples in the physical world of an actual infinity is doomed to fail; saying space or time are infinite is not the same as saying they are unlimited. The second is a finitistic notion, while the first simply does not make sense. –  godelian Aug 11 at 16:40
    
Well I would say what we can deal with in mathematical practice is potential rather than actual infinity while what we can perceive is actual rather than potential infinity. The first because we obviously only have finitely limited resources and the second since by nature of perception we only can perceive in leaps - either we don't, or we do. –  მამუკა ჯიბლაძე Aug 11 at 19:46
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I think this historical discussion hardly qualifies as "research-level mathematics". -- Voted to close as "off-topic". –  Stefan Kohl Aug 11 at 20:24
    
It would be inconvenient, akin to creating a translation of Brothers Karamazov into standard English, using only words that do not contain the letter `e'. –  Paul Fabel Aug 11 at 23:51

1 Answer 1

Okay, I'll take this one. We can't give a definite answer until we clarify exactly what kinds of reasoning belief in a potential, but not actual, infinity supports. However, a standard, and at least on its face, reasonable, answer is: reasoning in the formal system ${\rm RCA}_0$. (See this Wikipedia page for a brief description.)

The reverse mathematics school has produced a great deal of information about what can, and what provably cannot, be derived in formal systems of varying strength. Within ${\rm RCA}_0$, for example, we can prove

$\bullet$ the Baire category theorem

$\bullet$ the intermediate value theorem

$\bullet$ Urysohn's lemma for complete separable metric spaces

$\bullet$ Tietze's extension theorem for complete separable metric spaces

$\bullet$ existence of an algebraic closure of a countable field

$\bullet$ existence of a unique real closure of a countable ordered field

$\bullet$ the Banach-Steinhaus theorem

But we cannot prove

$\bullet$ every bounded, increasing sequence of real numbers has a least upper bound

$\bullet$ the Bolzano-Weierstrass theorem

$\bullet$ every covering of $[0,1]$ by a sequence of open intervals has a finite subcover

$\bullet$ every continuous function from $[0,1]$ to $\mathbb{R}$ is bounded

$\bullet$ every continuous function from $[0,1]$ to $\mathbb{R}$ is uniformly continuous

$\bullet$ every sequence of points in a compact metric space has a convergent subsequence

$\bullet$ every covering of a compact metric space by a sequence of open sets has a finite subcover

$\bullet$ every countable commutative ring has a maximal ideal

$\bullet$ every countable vector space over $\mathbb{Q}$ has a basis

$\bullet$ Konig's lemma

$\bullet$ local existence of solutions of ODEs

$\bullet$ Brouwer's fixed point theorem

$\bullet$ the Hahn-Banach theorem for separable Banach spaces

All of this and much more can be found in Stephen Simpson's excellent book Subsystems of Second Order Arithmetic (link to a PDF of the first chapter).

Edit: my answer is angrily challenged in the comments by Emil Jeřábek. He states "The main objects in ${\rm RCA}_0$ are (mostly infinite) sets of integers, so how is it not using actual infinity?" The answer is that a finitist can regard infinite sets of integers in roughly the same way that a set theorist regards proper classes. The concept "even number" is available to a finitist and he can make statements about all even numbers by treating the second order variables in ${\rm RCA}_0$ as class variables.

It is not the first time I've been subjected to abusive comments ("thoroughly confused and misleading", etc.) in a foundational discussion. For some reason the subject seems rife with this kind of thing. My policy in such cases is to clarify what I've said, if necessary, but not to get drawn into a debate.

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The main objects in RCA_0 are (mostly infinite) sets of integers, so how is it not using actual infinity? –  Emil Jeřábek Aug 11 at 21:33
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@EmilJeřábek: First of all, it is conservative over ${\rm PRA}$ for $\Pi^0_2$ sentences. But I guess the point is that the infinite sets of integers available in ${\rm RCA}_0$ are $\Pi^0_1$, so they are generated by finite computations. –  Nik Weaver Aug 11 at 22:03
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This is thoroughly confused and misleading. Yes, the only available means of constructing new sets in RCA_0 allow creation of sets $\Delta^0_1$-definable from other sets, but does not mean that the theory assumes all sets in the universe to be $\Delta^0_1$-definable, a priori they can be any sets of integers. This is not an oversight, but an essential property of the theory, which ensures that statements of the various results from analysis such as above with reals and continuous functions simulated by sets are sensible. Yes, the theory is a conservative extension of $I\Sigma_1$ (which is ... –  Emil Jeřábek Aug 12 at 11:31
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Let me make this simple. Present credible evidence that finitists consider RCA_0 a finitist theory, and stuff like the Baire category theorem finitist theorems, and I’ll shut up. –  Emil Jeřábek Aug 12 at 15:31
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The edit does not clarify anything. Of course you don’t need the actual set of all even integers in order to quantify over even integers or to state properties of even integers. You do need actual infinite sets of integers in order to quantify over infinite sets of integers (as in “for all sets”, rather than “for all integers in this set”) or to state properties of infinite sets of integers, which is what you are doing. I’m sorry if you see my comments as abusive, however if it happens repeatedly to you, maybe you should rather take that as a hint that you misunderstand something. –  Emil Jeřábek Aug 12 at 16:43

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