>>12929609Thus doubting set theory in general is not the same as doubting the infinitary mathematics done with it (which I believe is meaningful).
A great argument in favor of infinitary mathematics is how closely it's linked to computational mathematics (i.e. arithmetic) and how beautiful and simple the arguments concerning statements in number theory become if you allow infinitary tools. Think about the proof of Dirichlet's theorem using zeta functions, or proofs of prime number theorem using complex analysis and so many others.
At the end of the day, the finitist is right to say that the infinitary definitions are much too vague to be called "true definitions", they are not computational. He is also partly right in his critique of set theory. But then it is the job of the finitist to propose his own system in a clear and rigorous way, and demonstrate how it can accommodate the beautiful body of mathematics already discovered in his system. And there is a need for such a system, as evidenced by the vast disagreement among different finitists themselves as to what counts as a finitistic system of reasoning (e.g. is PA finitistic?).
The finitist also seems to be overemphasizing rigor, and in a sense putting the cart before the horse. Rigor is a good thing, but it is not be all end all of mathematics. Plenty of great mathematics has been done nonrigorously, and only later accomodated into a rigorous framework. Remember complex numbers, Heaviside stuff with distributions, italian algebraic geometry, Euler's manipulation of infinite series, Ramanujan's work and so on. Mathematics is about exploration using the faculty of intuition and reasoning. The greats go in, mess about and discover things, the lesser ones come after them to clean things up and make it "rigorous", sometimes in the process lessening the aesthetic appeal of it and making it harder to understand. It's clear where the real work and ideas are.
