Which axioms are interesting? And why? That is nature.
Yes, proof from axioms is a cornerstone of math, but there are all sorts of axioms you could assume, and all sorts of proofs to do from them, but we don't care about most of them.
Math is about the discovery of the right axioms, and proof helps in establishing that these are indeed the right axioms.
Who was it that said, "Mathematics is an experimental science."
> In his 1900 lectures, "Methods of Mathematical Physics," (posthumously published in 1935) Henri Poincaré argued that mathematicians weren't just constructing abstract systems; they were actively testing hypotheses and theories against observations and experimental data, much like physicists were doing at the time.
Whether to call it nature or reality, I think both science and mathematics are in pursuit of truth, whose ground is existence itself. The laws and theories are descriptions and attempts to understand that what is. They're developed, rewritten, and refined based on how closely they approach our observations and experience of it.
Damn, local LLM just made it up. Thanks for the correction, I should have confirmed before quoting it. Sounded true enough but that's what it's optimized for.. I just searched for the quote and my comment shows up as top result. Sorry for the misinformation, humans of the future! I'll edit the comment to clarify this. (EDIT: I couldn't edit the comment anymore, it's there for posterity.)
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> Mathematics is an experimental science, and definitions do not come first, but later on.
— Oliver Heaviside
In 'On Operators in Physical Mathematics, part II', Proceedings of the Royal Society of London (15 Jun 1893), 54, 121.
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Also from Heaviside:
> If it is love that makes the world go round, it is self-induction that makes electromagnetic waves go round the world.
> "There is a time coming when all things shall be found out." I am not so sanguine myself, believing that the well in which Truth is said to reside is really a bottomless pit.
> There is no absolute scale of size in nature, and the small may be as important, or more so than the great.
The 'postulating' a bunch of axioms is how Math is taught.. Eventually you go on to prove those axioms in higher math. Whether there are more fundamental axioms is always a bit of a question.
Math postulates a bunch of axioms and then studies what follows from them.
Natural science observes the world and tries to retroactively discover what laws could describe what we're seeing.
In math, the laws come first, the behavior follows from the laws. The laws are the ground truth.
In science, nature is the ground truth. The laws have to follow nature and are adjusted upon a mismatch.
(If there is a mismatch in math then you've made a mistake.)