[danck]

If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered

The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.

[red75prime]

Arithmetic existed long before its axiomatization. Arithmetic was useful and no one stumbled upon contradictions in it. So it was natural to suppose that it can be described by some axiomatic system. Peano found it.

[kevin_thibedeau]

It is a system for modeling concepts invented by man. Everything that falls out of such a system is a product of the invention. Numbers don't inherently exist. Everything derived from that concept can't be a "discovery".

[mmmBacon]

This is an excellent point. When I took algebra as an undergraduate I was blown away by the fact that you can choose any axioms and then derive an algebra based on those axioms. I was blown away because prior to that course I just assumed that our “standard” axioms were immutable.

[Koshkin]

> choose any axioms and then derive

Sound almost like "jump off the roof and see what happens."

[naasking]

> If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered

How would you revise this statement if we lived in a "Mathematical Universe", like Max Tegmark's hypothesis.

> The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.

It's actually hard to avoid Turing completeness, and once you have that, any recursively enumerable function is calculable. All you need is addition and multiplication on numbers.

[pfortuny]

Oh no: the axioms come much much later. The order is exactly the reverse one.

[danck]

Well, I have to agree. From a practical perspective.

[symplee]

Isn't it the other way around? Axioms are chosen because there are no observable counter examples in the real world.

[karmakaze]

Not at all, pure math is in part about exploring axiomatic systems that may or may not have a physical counterpart. The latter is immaterial.

[vanderZwan]

Isn't that simply because axioms that don't lead to consistent conclusions are rejected?

[danck]

You can invent and pick axioms in many ways that (probably) won't lead to inconsistencies. But they won't all be powerful enough or relevant in the real world.