Here's a start of a different sort of approach to the question of what math is, taking off from a remark in Jordan Ellenberg's book "How Not To Be Wrong" about how a times b equals b times a, for all a and b, "because it couldn't be otherwise". Let's make this more modest and say that we can't IMAGINE how it could be otherwise. That is: I can just barely imagine (with a mental squint, and with an inner acknowledgments of my limitations as a reasoner) *that* ordinary multiplication of ordinary natural numbers might not be commutative. But I cannot imagine in any kind of detail *how* it might fail to be commutative. There are probably lots of things that humans aren't able to doubt (such as "I exist") that don't count as mathematics, so this definition will need to be modified before it comes close to drawing the line between math and non-math in approximately the right place. A variant of this approach would be to define pure mathematics as the study of fantasies that possess a certain kind of coherence. Jim Propp