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?

  • $\begingroup$ "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. $\endgroup$ – Arthur Dec 13 '18 at 7:20
  • $\begingroup$ 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. $\endgroup$ – aghostinthefigures Dec 13 '18 at 7:23
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    $\begingroup$ @aghostinthefigures That question is much more directly addressed by Reverse Mathematics. $\endgroup$ – Derek Elkins Dec 13 '18 at 7:25
  • $\begingroup$ @DerekElkins Agreed! $\endgroup$ – aghostinthefigures Dec 13 '18 at 7:27
  • $\begingroup$ @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. $\endgroup$ – JFox Dec 13 '18 at 7:37

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.

  • $\begingroup$ Think there is a lot of work to be done before that circle gets closed! But agree with the sentiment $\endgroup$ – Nadiels Dec 22 '18 at 22:39

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