I've heard about mathematicians who defend a strictly finite conception of mathematics, with no room for infinity. I wonder, how is it possible for these people to do this? Are there any concepts that they use to replace results that have to do with the infinite, or are these results not taken into consideration? Is there a name for this kind of mathematics? I would like to read and learn more about this, especially the arguments on why infinity is absurd or even superfluous.

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    $\begingroup$ You mean ultrafinitists? See this for instance. $\endgroup$ – J. M. is a poor mathematician Sep 22 '11 at 0:32
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    $\begingroup$ Holy everloving... why was this downvoted? $\endgroup$ – J. M. is a poor mathematician Sep 22 '11 at 0:34
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    $\begingroup$ @J.M., I'd venture the rather less severe finitism also fits the bill. Jeroen, there have been a few questions about finitism and ultrafinitism on math.SE which you may find interesting, e.g. math.stackexchange.com/q/531/856 and math.stackexchange.com/q/501/856. $\endgroup$ – Rahul Sep 22 '11 at 1:08
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    $\begingroup$ @J.M. I agree - the downvote is puzzling. It seems there is some sort of higher-order rejection in action. Hopefully there'll be no thread on mathematicians who reject mathematicians who reject $\infty$! $\endgroup$ – Bill Dubuque Sep 22 '11 at 1:14
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    $\begingroup$ To add to the list of links Rahul gave, this MathOverflow question also contains a lot of references. $\endgroup$ – Willie Wong Sep 22 '11 at 1:22

Few people these days argue that the concept of infinity is absurd per se, but there are still those who hold that it is epistemically intractable, or unnecessary, or that infinite objects simply do not exist.

Finitistic systems are generally very weak. Steve Simpson claims that primitive recursive arithmetic (PRA) is finitistic, since it involves reference to only potential infinities—that is, indefinite iteration—not completed infinities such as the set of natural numbers $\mathbb{N}$. Some finitists accept the existence of countably infinite sets: $\mathbb{N}$ is finitistically acceptable but the set of real numbers $\mathbb{R}$ is not.

One might think that this is the end of the road for finitist mathematics, but various clever dodges have been thought up by philosophers to avoid this trap. Shaughan Lavine, for example, has developed a finitary version of ZFC which is not mathematically revisionary but nonetheless allows one to maintain that there are only finitely many mathematical objects.

There are a number of considerations which motivate people to take finitist positions. The first is the desire to repudiate abstract objects. If there are no true infinities then we can be realists about mathematics without having to accept the existence of a platonic heaven or otherwise account for the existence of infinitary mathematical objects. Mathematics can be understood purely in terms of the combinatory properties of physical objects.

Alternatively, one might think that mathematical objects are merely ideas in our minds: they do not exist independently of their construction by us. Since we are finite, we cannot construct all of the natural numbers, even though in principle we can keep going forever. Because of this, there is a strong tendency among constructivists to reject the completed infinite.

Epistemology provides a different reason. We only have finite computational power available with which to derive mathematical truths, and thus we cannot grasp any infinite objects (if such things exist) in their entirety. Mathematical knowledge must thus be arrived at via finitary means.

Some concepts, like that of the natural numbers and recursive functions on them, seem basic: they are primitive ideas which do not admit of further justification. Higher mathematics, however, is incredibly useful, even if it does not admit of such straightforward justification. We might therefore want to treat all mathematics beyond arithmetic as merely a game with symbols that has only instrumental value. We can see just such an approach in Hilbert's programme, with its restriction that all proofs must be finitary.

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    $\begingroup$ Your final two paragraphs are, I think, the most important arguments for finitism, and also show that finitism (in Hilbert's sense) isn't all that restrictive. $\endgroup$ – Zhen Lin Sep 22 '11 at 6:53

Since you have some computer science background, a good and quick introduction to some of the (ultra)finitism ideas and objections can be found in Ed Nelson's talk "Warning signs of a possible collapse of contemporary mathematics".

  • $\begingroup$ "Warning signs of a possible collapse of contemporary mathematics" - that sounds awful :) Math shouldn't be so much dependent on current "fashion" paradigms, or am I wrong? $\endgroup$ – Tomas Sep 22 '11 at 1:24
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    $\begingroup$ Titles to talks tend to be more provocative than titles to actual papers or books. For example, Nelson's technical work which gives the whole shebang behind the stuff in that talk is called Predictive Arithmetic. Much less rousing, don't you think? $\endgroup$ – Willie Wong Sep 22 '11 at 1:28
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    $\begingroup$ @Tomas: The standard of rigour demanded of mathematics is entirely a function of sociology, so actually, yes, it does depend on the current fashions. $\endgroup$ – Zhen Lin Sep 22 '11 at 1:28
  • $\begingroup$ @WillieWong Great article! Sorry for the late reaction but I only saw the article now that I was going over the thread again :) $\endgroup$ – user12205 Dec 27 '11 at 17:02
  • $\begingroup$ @WillieWong - Of course you meant Predicative Arithmetic :) $\endgroup$ – r.e.s. Apr 26 at 13:47

Yes, in set theory there was an axiom of infinite set that could have been skipped ... but why you are so much surprised by this? We don't use the infinity very often and I would say - in fact it somehow improves our imagination, but we don't really need it. We usualy use $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$ numbers and none of these contains infinity! You might argue that these sets themselves are infinite, but do we need to treat them as sets in order to work with these numbers? Limits? We can also define them without infinity. So for most of the applied math and statistics I think the infinity would not really be needed, it's just some "added value" :-)

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    $\begingroup$ $\mathbb{N}$ is not a set unless we have the axiom of infinity. Conversely, once we have the set $\mathbb{N}$, the axiom of infinity is satisfied. $\endgroup$ – Zhen Lin Sep 22 '11 at 1:17
  • $\begingroup$ @Zhen, yes, but do we need to say that $\mathbb{N}$ is a set in order to work with natural numbers? $\endgroup$ – Tomas Sep 22 '11 at 1:21
  • $\begingroup$ You can avoid saying $\mathbb{N}$ is a set if you work only in first-order Peano arithmetic. But then you can't prove things like the strengthened finite Ramsey theorem. $\endgroup$ – Zhen Lin Sep 22 '11 at 1:25
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    $\begingroup$ For more on necessary uses of infinity in finite mathematics see here. $\endgroup$ – Bill Dubuque Sep 22 '11 at 2:30
  • $\begingroup$ Since real numbers have infinitely many digits, you need infinite sets to construct real numbers. Limits are defined in terms of sequences with infinitely many members. $\endgroup$ – Mark Feb 12 at 18:45

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