Is mathematics mostly chaos or mostly order?

I’ve always considered math is something that is discovered, neither chaotic or orderly, it just… is. Really brilliant people make new discoveries, but they were there the whole time waiting to be found.

This article seems to kind of dance around yet agree with the discovery thing, but in an indirect way.

Math is just math. Music is just music. Even seemingly-random musical notes played in a “song” has a rational explanation relative to the instrument. It isn’t the fault of music that a song might sound chaotic, it’s just music. Bad music maybe. This analogy can break down quickly, but in my head it makes sense.

Disclaimer - the most advanced math classes I’ve taken: calc3/linear/diffeq.

What makes you consider it a "discovery" instead of a creation of us humans?

I am more on the side of seeing maths as a precision language we utilize and extend as needed, especially because it can describe physically non-existent things e.g. perfect circles.

It's not clear to me why people think perfect geometries do not exist, they occur all the time in physics.

Of composite matter, sure, because it's composite in a certain sort of way, you do not get perfect circles. But the structure of macroscopic material does not exhaust the physically.

Even here, one could define some process (eg., gravitational) which drives matter towards being a perfect circle, because perfect circularity is a property of that process. This is, as a matter of fact, true of gravity -- if it weren't we'd observe violations of lorentz invariance, which we do not.

When was the circle discovered? When it became essential to physics?

When it was essential to perception. Its necessary to have a model of a circle (, elipse...) in order to correctly parse (at least,) visual perception -- because space is inherently geometrical.