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Is there any formal foundation to ultrafinitism?

Very recently I have come across the skeptic opinions of a school of mathematicians(ultrafinitist) over a physically perceptible existence of "infinity". As of me, although I neither subscribe to the "infinitist" school of thought, nor the "ultrafinitist", I find it wiser to consider both the opinions with equal seriousness before forming one of my own. Considering that even after some 400 years of our conventional mathematics of infinity we have not been able to describe physical reality to the desirable extent, it seems unwise to eradicate the possibility that our mathematics has been fundamentally flawed as far as its goal has been to aid describing the physical reality, due to introduction of the "infinity" whose physical existence cannot be verified, possibly ever.

But it cannot be denied that due to introduction of infinity, we have other allied concepts like irrational numbers, real numbers, convergence etc. which has been extremely useful in describing the physical reality. If we are to replace "infinity" with "an extraordinarily large but albeit finite, number" then we also need to amend or replace these important notions of analysis to derive a working model that would aim to describe the reality. I have some queries regarding such a model, and I would list them in the following sections

1) The concept of infinity, in my understanding, is introduced in the axioms "There is no natural number that has zero as its successor" and "all natural numbers except zero, has a unique successor" which are essential parts of the Peano's axioms to describe natural numbers. To do away with infinity, we need a set of axioms to replace the Peano's axioms. Is there any such set of axioms? In case, there is, what are they?"

2) If the previous question has a positive answer, then how to define addition, multiplication and a total order on these numbers that emulate our natural understandings of the concepts. (Like, a>b implies a+c>b+c, or a>b and c>0 implies ac>bc)? In case we cannot do so, then is there a way to make this model work with our intuitive understandings?

3) From these numbers, once we can define multiplication, it is fairly easy to define rational numbers. But, for defining irrational numbers, we needed Cauchy sequences in our conventional calculus. Since in absence of infinity there is nothing called "convergence" (or is there?), how to define an analogue of irrational numbers, and consequently an analogue of real numbers? In case, it is not possible to do so, can we search for reality in mathematics without them? (Can computers do calculus from the first principle? It does not understand infinity or irrational numbers, I guess. Then, how does it do it?)

4) What would be the geometric interpretation of this discrete system?

I guess it is implied here, that I am not searching for a model that is a discrete approximation to the continuous mathematics (or vice versa) we are accustomed with. It should rather be an replacement of the conventional analytical model, with its own complete structure of axioms and definitions. Only and only then it can be used as an alternative mathematical model to describe reality.

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I fixed the formatting, which had made some parts v hard to read. –  David Roberts Jun 20 '11 at 9:20
    
You should perhaps look at Nelson's arithmetic, where the natural numbers are restricted to an initial segment closed under the $+$ and $\times$. However, exponentiation isn't defined in this system. I seriously don't think that one can think about defining real numbers (as opposed to some real numbers) if you want a finite universe. Perhaps interval arithmetic is what you want... –  David Roberts Jun 20 '11 at 9:25
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This seems to be a duplicate of: mathoverflow.net/questions/44208/… –  S. Carnahan Jun 20 '11 at 9:27
    
It is also related to mathoverflow.net/questions/66776/alternative-arithmetics/66780#66780. –  Ali Enayat Jun 20 '11 at 11:32
    
I will try again: mathoverflow.net/questions/66776/alternative-arithmetics/… –  Ali Enayat Jun 20 '11 at 11:34
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marked as duplicate by André Henriques, Gerald Edgar, Andres Caicedo, S. Carnahan Jun 20 '11 at 15:41

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1 Answer

There is a simple answer to your question, which I hope is not too trivial, and that is to do everything in modular arithmetic modular some huge prime. This can quite easily be axiomatised. Calculus would then be reduced to the study of step functions. With regards to modeling the physical reality, we do this already when we use computers. Some of the laws of nature may turn out to be rather ugly but I do not see in principle why one cannot work in such a way.

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Presumably the modular rationals you have in mind are not the quotients that already exist in the integers modulo $p$. For example, for any odd prime $p$, the integer $(p+1)/2$ when multiplied by 2 gives 1 (mod $p$), but I'm guessing this integer is not what you want as the modular rational 1/2. –  Andreas Blass Jun 20 '11 at 13:37
    
Problem is, in an integer modulo p structure, i am not being able to define a total order (actually, one can prove that there is no ordered finite field). also, we also do not have any corresponding concept of convergence. but without these two crucial concepts, it is difficult to emulate nature with this structure, unless we find a way to replace these concepts altogether with newer discrete structures that also satisfies our intuitive grasp of the real world satisfactorily. –  Mandal Jun 20 '11 at 13:51
    
if we assume the universe is finite we do not need rationals to describe it... there is a smallest unit which we cannot divide... there is no way to decide whether or not the physical universe is finite... we will never know that we have found the smallest unit of the universe until the largest unit of time has passed, by which time we will not exist... (or maybe we shall exist modulo p ;) which amounts to the same thing)... thus ultrafinitism is a philosophical position that is tenable if somewhat horrific to our sensibilities... –  George Lazou Jun 20 '11 at 17:28
    
an interesting question is whether such a system can be axiomatised in fewer than p - 1 characters... note a negative would still not rule out the universe being finite, but rather that we cannot fully describe it in the finite time allowed... –  George Lazou Jun 20 '11 at 17:37
    
@Mandal the naturals modulo p are well ordered... sequences are finite and converge to their last term... integrals are simply the area under step functions... differentials are simply the step function of the ratio at each point if it exists... in the case the ratio doesn't exist for each point then the function is not differentiable at those points... etc. etc. –  George Lazou Jun 20 '11 at 17:47
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