IIRC that is one of Zeilberger's opinions. For myself, I don't think "biggest number" makes sense per se in mathematics, but it is a fact that no matter what resources we muster we're left with some numbers equaling things other than their successor when incremented.

My sense is that infinitary axioms like the existence of N, successor, induction, etc., basically allow mathematicians to write shorter and more straightforward proofs. They are likely to be sound until the cows come home, and moreover they represent what mathematics is essentially all about - finding some shortcut through the calculations we need to make, understanding what part of some structure or procedure is irrelevant.

Within certain parameters, that's right.

And that probably makes as much sense (if not more) as the alternative for all I know. There is a way to think of mathematics as corresponding to computation in a way where anything that cannot be computed cannot be proven. If you cannot "construct" or "compute" infinity, then you cannot say that it exists.

I think what I am describing corresponds to the "constructivist" view of mathematics, but tbh I don't know enough about this to say whether the framework allows for infinities.

<sotto voce>These people are crazy

What do you mean?

At a very pop sci-fi level, think of writing down the biggest rational number that you can. As many 9s. Eventually, you'll run out of space in the universe to store 9s.

In a computer, this text is stored in binary format. That's a base2 number. Arbitrary concepts can be encoded into a form that can then be represented and worked with by mathematical operations.

If you want to get really cafe philosophical about it, the universe can be broken down into its smallest elements, quantified, and represented by numerical values. And thus the universe is a number. Of course, there's all sort of "gotchas" and "whatifs" and this is a wildly over-simplified version.

Yeah! Programmers are quite familiar with this universe. In many programming languages, you can't represent 2^64 with a built-in integral datatype. While it's certainly possible to implement data structures and algorithms to support arbitrary-size integer arithmetic, most programmers don't! They just pretend that numbers larger than 2^64-1 do not exist! And, that's a fairly safe assumption for many real-world applications.

But even with that arbitrary-size integer arithmetic, your mathematical universe is still bounded by your computational environment's resources.

Discussion with an ultrafinitist:

-Do you believe in 1?

-Yes.

-Do you believe in 2?

-Yes.

-Do you believe in 4?

-[slight pause] Yes.

-Do you believe in 8?

-[slightly longer pause] Yes.

Eventually you realize each question takes him twice as long to answer as the last.

That's funny. There is in fact a quasi-finitist number scheme called the counting numbers, closed under +1 and addition/multiplication generally but not exponentiation, so unbounded but not too "problematic" as they have limited ability to represent propositions (or machine states, etc) arithmetically.

I suppose a log-finitist takes O(1/n) to answer for 2*n? :-)

Yes. It's like writing code that's aware of MAX_INT. The code doesn't need to know the value of MAX_INT. It just has to handle special cases near that number.