I find the problem of induction interesting. I’ve long had this strange doubt whether, for example, the natural numbers actually correspond to a natural phenomenon—or whether classical logic is anything more than an approximate model of whatever logic actually underlies the universe.
Humans mostly understand counting intuitively, logic perhaps less so. But how can we make inductive statements about natural numbers if they’re merely a by-product of how our brains work? How could we use the “pure” language of mathematics to communicate with extraterrestrials if our understanding of mathematics is based on entirely different principles?
As a pragmatist, I mostly just keep my head down and say “numbers and Booleans may not be anything, but we can use them to do things, and that’s good enough”.
http://en.wikipedia.org/wiki/David_Hume#Hume.27s_.22Science_...
http://en.wikipedia.org/wiki/Problem_of_induction