John Thacker:

I consider myself a finitist, but not an ultrafinitist; I believe in the existence of numbers expressed using Conway chained arrow notation. I am also willing to reject finitism iff a physical theory is constructed which requires me to believe in infinite quantities. I tentatively believe in real numbers and differential equations because physics requires (though I also hold out hope that e.g. holographic physics or some other discrete view may enable me to go digital again). However, I don't believe that the real numbers in physics are really made of Dedekind cuts, or any other sort of infinite set. I am willing to relinquish my skepticism if a high-energy supercollider breaks open a real number and we find an infinite number of rational numbers bopping around inside it.

I consider the Axiom of Choice to be a work of literary fiction, like "Lord of the Rings".

Bayesian probability theory works quite well on finite sets. Real-world problems are finite. Why should I need to accept infinity to use Bayes on real-world problems?

The two-envelopes problem shows the necessity of having a finite prior.

Godel's Completeness theorem shows that any first-order statement true in all models of a set of first-order axioms is provable from those axioms. Thus, the failure of Peano Arithmetic to prove itself consistent is because there are many "supernatural" models of PA in which PA itself is not consistent; that is, there exist supernatural numbers corresponding to proofs of P&~P. PA shouldn't prove itself consistent because that assertion does not in fact follow from the axioms of PA. (This view was suggested to me by Steve Omohundro.) Now, I don't believe in these supernatural numbers, but PA hasn't been given enough information to rule them out, and so it is behaving properly in refusing to assert its own consistency.

I have no desperate psychological need for absolute certainty or proof, which, even if PA proved itself sound, I couldn't have in any case, because I would have to believe in PA's soundness before I trusted its proof of soundness. Or maybe I'm in the grips of a Cartesian demon playing with my mathematical abilities.

Correspondence, not coherence, very easily justifies mathematics. Math can make successful predictions, ergo, it's probably true. No one has ever seen an infinite set, ergo, they probably don't exist, and at any rate I have no reason to believe in them.