How to ask questions about the likelihood of "interesting" mathematical statements?

Question

I started reading about Gödel's theorems recently and found the idea of using the tools of mathematics to understand logic and what we can and cannot do with it. While doing my problem set for a basic real analysis class--in which we've been building the reals up from Peano's axioms--I got to wondering what the chances are that interesting results come out of some set of axioms and definitions. Is it at all surprising that the particular axioms we assume to be true yield so many amazing results when combined with the right definitions? If so, what's so special about the number system we find ourselves with. OTOH if it's not surprising, then what makes such fruitful axiomatic systems so prevalent?

I doubt the question as I've posed it has any sort of answer, but I'm curious what sorts of questions related to this have been asked within mathematics. Where would I look if I wanted to learn about the tools for asking these sorts of questions?

"the axioms we find to be true" That's a common misunderstanding. We assume axioms to be true. Our opinion on whether they really are true (if that even makes sense) is irrelevant. — Arthur, Dec 13 '18 at 07:20
A very related question to what you are asking is “What axioms/structure do I need in order to generate my favorite theorems/properties/results”, which is a question addressed by fields such as model theory and/or category theory. — aghostinthefigures, Dec 13 '18 at 07:23
@aghostinthefigures That question is much more directly addressed by Reverse Mathematics. — Derek Elkins left SE, Dec 13 '18 at 07:25
@Arthur, that's a good point. I think what I meant is "the axioms that yield a version of mathematics which is consistent with our intuitions/observations," but absolutely the correct word there is assume. — JFox, Dec 13 '18 at 07:37
@aghostinthefigures, thanks I'll look into those fields. Just doing a quick pass on the wiki articles now there seems to be quite a lot there..do you have any guidance on where to look within those disciplines for something like the questions at the end of my first paragraph? — JFox, Dec 13 '18 at 07:58
@KFox: I've some posts linked from my profile, including a brief sketch of foundational building blocks. Also a foundational look at PA and possible doubt of soundness of stronger theories are relevant. — user21820, Dec 31 '18 at 10:46

score 2 · Answer 1 · answered Dec 22 '18 at 22:14

What shapes mathematics? I'm rather certain that software could shape branches of mathematics that wouldn't be intuitive at all for human mathematicians. Like computers finding non intuitive continuations in chess.

I guess that mathematics from the origin starts with human perception and that the circle might soon be closed by disciplines as Mathematical Psychology. See for example

Journal of Mathematical Psychology

This is not a straight answer to your question, but maybe worth to keep in mind.

Think there is a lot of work to be done before that circle gets closed! But agree with the sentiment — Nadiels, Dec 22 '18 at 22:39

How to ask questions about the likelihood of "interesting" mathematical statements?

1 Answers1