Is modern mathematics scholasticism?

Question

I have thought a lot a about mathematics and it's foundations. There have been several attempts to give it a solid foundation, and they all failed. Frege / Russell logical atomist approach failed, intuitionist logic failed, set theory did succeed to some extend, but it lead to extremely bizarre conclusions that many mathematicians think have nothing to do with reality (like infinite hierarchies of infinities and similar nonsense).

The more I think about it, the more it seems to me that all these projects failed because modern mathematics (with some exceptions like elementary number theory and classical geometry because they originated in ancient Greece) is nothing but medieval scholasticism in a different flavour. Real/complex analysis, set theory, modal logic and other related fields have nothing to do with reality, they are just hyper-logical nonsense abstraction for abstraction's sake.

Is mathematics, or a part of it scholasticism?

Answer 1

As I've studied set theory, I thought I was dying inside in trying to understand phrases like "zero sharp" or "premice." Then I saw the word "cobordism" in category theory and I died all over again. You say that mathematics has become nonsensical, and it's worth noting, then, that the phrase "generalized abstract nonsense" is current among category theorists, albeit not in a dismissive manner.

Now, too, then, it would be one thing to show that set/category theory are structurally/procedurally equivalent to scholasticism, another to show that this is to modern mathematics' demerit. I would offer that the equivalence goes through but that this rehabilitates scholasticism to that extent. One could try out adapting Trinity-talk to nontrivial-automorphism talk, for example. And Aquinas was not overly foolish; his labyrinthine deductions go awry (his premises on behalf of the Trinity, if faithfully tracked, would've led him to, "There are infinitely many divine persons," instead, but so he did pose that question but mangled his logic to avoid the unorthodox conclusion) but they were a valiant mental effort, if nothing else.

The main thing, though, is that whichever avenue we pursue our abstractions down will "look like" scholasticism in the limit, and there's nothing to be done, nor should there be done, about it. Not every question can be answered by observation; not every test is an empirical test. I might not take it on faith that premice are real, for example, but I take it on faith that model theorists know what they're trying to talk about when they go on about mice and weasels and so on. If I have a hard time following their words, that's on me more than on them (gatekeeping aside, though).

Answer 2

Is modern mathematics scholasticism?

Well, strictly speaking, not at all. They have similarities, but a medieval approach to theological education and contemporary mathematical research have a a lot of space between them conceptually. One can draw an analogy, to be sure, but the claim to strong similarity is a superficial one that would fall apart under scrutiny.

Frege / Russell logical atomist approach failed

Well, that's an black-and-white opinion; logical atomism was a hypothesis that helped philosophy progress. The correspondence theory of truth has a long, proud history and a certain air of pragmatic weight and is perhaps the simplest sense of truth to understand. It comes right out of naive realism and simple presumptions about representations.

intuitionist logic failed

Again, that's an opinion, not a fact. You seem unwilling to accept basic facts, such as the one Cork Screwdriver presented:

Intuitionistic logic is certainly not abandoned in math. Even if most people use classical logic, there is still a large community of people who work with intuitionistic foundations

Plus, you seem to harbor the notion that truth is a realist enterprise, and that there is a one-true-way to do philosophy and logic, instead of accepting the generalization of truth in terms of rational satisfiability and emprical adequacy and the inevitability of logical pluralism which is ironic, because such an ego-centric approach has a lot in common with the presumptions that undergird logical atomism.

Real/complex analysis, set theory, modal logic and other related fields have nothing to do with reality

This statement is the result of a superficial analysis. For starters, all of these formal systems are abstractions of reality. Real and complex analysis are mathematical formalisms to deal with "properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability". If you ever have used a fluid that flowed through a pipe, like in a car or a municipal water system, then real analysis is the mathematical conceptualization of fluid dynamics. Complex analysis and imaginary numbers are used in electrical engineering. And modal logic models epistemic modality. Maybe your reality doesn't include fluids, electrical current, and possibility, but I think I speak for most contributors here by saying it probably should.

There are two approaches to philosophy. One you romp through the field of language spouting this-good, that-bad, this-true, that-false. Or you could apply epoche when ask your questions consider the alternative points of view that may diverge from your intuitions.