Mathematicians who are finitists.

[A Toy Windmill]

Do finistists accept the existence of irrational numbers?

I don't think so, at least not in the sense that everyone else does. The classic proof that a square's diagonal is incommensurable with its side looks finitistic to me, but that's not what a modern mathematician means by "the square root of 2 exists."

Still, it's not too shameful if the finitist position on the nature of incommensurable magnitudes is on par with the pre-moderns.

Just curious. I know that you may be playing the DA here.

I'm not. I'm not a finitist, but I see value in research in finitistic mathematics.

Modern finitism was started by David Hilbert who, as far as mathematicians goes, wasn't exactly a wally. So I don't think it is fairly compared to creationism. Hilbert regarded Cantor's mathematics as a "paradise", and insisted we would not be expelled from it, but he was also aware that it contained serious contradictions, as have the many other attempts to naively treat of infinite sets and higher order logic. Some might say that this business is inherently precarious and we should aim to build our mathematics on much more secure footing. Some say we should start with a world that is logically finite, and if some finitist shows that huge swathes of existing mathematics can be done this way, that's an awesome thing to know.