Physicist: There’s a lot of math that doesn’t describe physical reality at all, and even some (few) mathematicians who feel that
“applicability” is just another word for “impurity”. The ability of math to describe reality is just a consequence of the fact that reality is nice and consistent.
The fact that the math we use (addition, subtraction, geometry, calculus, whathaveyou) works is no coincidence at all. Mathematics literally evolves in the sense that, if something doesn’t work, then people will ignore it. So if you have a theory that , great, but no one will use it because it’s patently, provably false. It doesn’t describe reality (in this case the reality that the ratio of the circumference to the diameter of a circle is ), so it goes the way of the Woolly Mammoth.
I assume that this question is about perceived reality (colors only exist in the brain, whereas in reality there is no “blueness” or “redness”), and not physical reality. The fact that we can only describe (mathematically and otherwise) the reality we perceive does guide the direction of mathematical research, and as we perceive more we find that the field of math expands accordingly. For example; number theory wasn’t much more than a hobby before digital communication and RSA encryption , and differential geometry was mostly a nuisance (and anal-retentive over-generalization) until general relativity cropped up. Now these are both thriving fields of research (in computer science and physics, respectively).
However, just because something works in your head has absolutely no bearing on whether or not it will work in reality (which you would expect if the physical world were created by our minds). Very good, very reasonable ideas get shot down by experiment every day, and we are constantly surprised.
Philosopher: If we assume the external world exists (independent of our minds), Math’s correspondence to reality is no more coincidental than the correspondence to reality of theories stated in any other language. This isn’t dependent on the existence of mathematical objects, and it’s not dependent on Mathematical truths existing independently of humans (though I think they do). If we assume the external world is merely an “invention of the human mind”, then the correspondence of Math to the world is even less coincidental, since the same thing is the author of both.