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.

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".

> Numbers don't inherently exist

How do we know that it is true?

They are symbols that we assign arbitrary meaning to. They are useful because of the axiomatic framework constructed to support them.

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.

> choose any axioms and then derive

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

> 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.

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

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

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

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.

Can you give some examples of axioms in pure math that run completely counter to our physical world? For example:

  It is NOT possible to draw a straight line from any point to any other point.
  It is NOT  possible to extend a line segment continuously in both directions.
  etc...

or

  Things which are equal to the same thing are NOT equal to one another.
  If equals are added to equals, the wholes are NOT equal.
  The whole is LESS than the part.

Note that the original forms of the above axioms "make sense" to us because everything in our physical experience agrees with them. So when you said that the "physical counterpart ... is immaterial", I was curious to see an example of a "physically impossible" axiom.

Most of large cardinal axioms.

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

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.

Well, then it sounds like your reasoning gets it backwards: the axioms that produce systems without significant consequences or connections outside of their own abstract realm end up being ignored.

Or in other words: the constraints on maths are imposed from outside of maths.

Doesn't this imply that, while you can invent all the axioms you like, you must discover which ones are consistent with each other and with experimental results.