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.