science

Ultrafinitism is pretty whacky since it requires tossing out the entire axiomatisation of mathematics. Ultrafinitism is inconsistent with axiomatisations even of fragments of Peano arithmetic (e.g. where you weaken the axiom scheme of induction to be restricted to a subset of formulae). You're left with something very cumbersome indeed. One thing the article fails to point out is that Gödel incompleteness still applies to such very weak theories. (It is actually misleading when talking about trying to prove the consistency of ZF - this too is not possible within ZF, provided ZF is consistent, due to incompleteness.)

There's a big difference between ultrafinitism and finitism - the latter allows a mathematical theory that allows you to talk indirectly about infinity but not directly. Peano arithmetic allows you to prove, for example, that, "for every prime number there is a larger one" - or as we might state it succinctly, "there are infinitely many prime numbers." But it doesn't allow you to talk about "the set of prime numbers" in the same direct way as you can talk about a single number or finite collection of numbers, and there is no such succinct way to say "there are infinitely many prime numbers" in the language of PA; you need the circumlocution.

These kinds of circumlocution are par for the course when dealing with weaker theories. But I think there is a huge practical issue with ultrafinitism which is exemplified thus: how do you prove a simple theorem in an ultrafinitist world, like for example, "the sum of any two even numbers is even" when it may be the case that the sum of two large even numbers is actually undefined? You have to write caveats all over every proof and I think for anything non-trivial it would swiftly become unmanageable.

On the philosophical side, I just don't think there is ever going to be a large number of people who think that "x+1 > x" is a statement that ought to be viewed with any level of suspicion. Mathematics is about creating abstractions from our real-world experience; numbers are such an abstraction. Real world objects are always finite, but that doesn't mean the abstraction has to be - to actually capture our intuition about how real world objects work, there can't be a limit.

There are two types of comments here lol

Mathematicians have gone too far

Clickbait. It's not mathematicians. It's a few fringe "mathematicians".

Two things occurred to me reading this:

1. Huge numbers are exceedingly common, but counting particles is the wrong way to find them. Combinatorics is where the real monster hunting lies. When you start calculating complex probabilities or numbers of possible arrangements of things, that's where the fuzzy boundary between "infinite" and "really, really, really finitely big" starts to blur.
2. I think looking to CompSci is the right move, but I still don't see many folks discussing computational complexity as a real, mathematical limit. We often treat two equal statements as though theres an immediate, single-step, jump between them. But discovering the equality requires computation/calculation. Shannon shows that information and entropy are the same thing. Computation is the process by which information is created. Ultrafinitist need to show that there is a finite quantity of information, which I don't think is true or possible.

First they came for Pluto, but I didn't speak up....

They should start with floating point numbers.