Not being facetious: I suggest you read the book. It discusses precisely these objections. I should say also the only reason I read this book was because I lik

Not being facetious: I suggest you read the book. It discusses precisely these objections. I should say also the only reason I read this book was because I liked the author's other books which have nothing to do with mathematics. Before reading it I thought "man who gives a shit about philosophy of mathematics? can we just do some real math please?", but this book changed my mind about that. Also: I agree that traditional mathematics is useful. I trust that FLT true (not to mention it's an awesome achievement). And I even trust that proofs which make heavy use of concepts I find philosophically dubious will almost always turn out to still hold when placed under the "right" sort of foundations. There are some exceptions to this though. For example, I absolutely don't trust stuff like this: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox By "don't trust", I mean I seriously doubt there is anything at all out there in the universe (or in the metaphysically possible universe) that behaves like this. Instead, it's just a logical result which leans heavily on some problematic mathematical assumptions which were never actually justified. It should be taken as a reductio ad absurdem that something went wrong with the inputs (or at least evidence as such). 100+ years ago, people spent time wrestling with the foundations of math, but now it's looked down upon as a waste of time/not "serious" math, etc.