On Math and Language

I was thinking about the whole "Is math invented or discovered" question again.

I used to be, like I think a lot of STEM folk are, in the "math is discovered" camp.  But, over time, I have come to be firmly in the "invented" camp.  There are many reasons.  But here are a few:

-Math is a language and, like all languages, exists only in the minds of humans.

-Gödel's incompleteness theorems that basically show the limitations of formal proofs.

-Once, in grad school, a physics professor was dividing a term going to zero by another term going to zero and made some kind of hand-wavy argument and 'renormalized' the equation.  At the end of the derivation, out popped the mass of the electron (or something like that).  I raised my hand and asked "But that's not real math - can you do that?"  He grinned and said "Are you a mathematician or physicist?"  I said "I didn't know you had to chose." He said "Well, do you value internal consistency for its own sake or are you trying to reliably predict the results of experiments?"  It was a big deal to me when he said those words (thanks, Dr. Ling-Fong Li!).

-Lately, I've been thinking about basic mathematical ideas like infinity, geometric points, infinitely precise numbers, and continuous functions. These are all cool and important concepts, but they don't actually exist in the real world.  In a finite (at least observably/practically finite) Universe run on quantum mechanics, it simply isn't possibly to continually subdivide space with infinite precision nor can you just continue past the horizon forever.


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Isn't it interesting that we can hold and use these nonphysical concepts in our minds and they turn out to be so useful while simultaneously not being real?  We can't even come up with a wave function in quantum mechanics that doesn't use imaginary numbers!

Now that I think about it, all language can be described that way.  If I tell you a cautionary tale about walking on a lake with thin ice on it, it may never have happened and the story itself is certainly not a real thing out in the universe, but it could save your life in the future.