Sorry for the purely philosophical post  I usually try to be more realworld than this.
Ok, so I'm most definitely not a Platonist in any woowoo, quasireligious sense; mathematical abstractions are not "real". On the other hand though, many of them do seem to be "natural". Let me be very concrete: I believe that if humans ever meet a technologically sophisticated alien race, they will have a number system that matches ours. I doubt they will have the Peano axioms of arithmetic, or the idea that the integers are polymorphic functions. They might not even care about prime numbers. But the aliens will have arrived at numbers as we understand them. I also believe they will understand the difference between spheres and donuts, but I doubt they'll express it in terms of pointset topology. In this sense certain math abstractions are the "right" way to describe the universe, and things that we prove about them can be taken to be "true".
Sorry for the purely philosophical post  I usually try to be more realworld than this. 
This blog features a series of miniessays about the history and nature of math, and how it fits into the spectrum of human activity. I'm considering putting them together into a book, so comments are encouraged! Archives
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