I Found It on Sale at Walmart

An interesting and natural question arises about mathematics. Is mathematics created or discovered? One does not need to build a lot of philosophical machinery to start thinking about the answer.

When a mathematician says she discovered a mathematical concept, theorem, or proof, what does she mean? Did she find it lying on the sidewalk while strolling on the boulevard? Did she find it growing in the forest while hiking? Did she see it in aisle six at Walmart while shopping?

None of the answers to those sorts of questions will do. Mathematical objects are not that kind of thing. It seems as if mathematical objects inhabit some supernatural realm if they are to be discovered. That puts us on familiar ground. Everyone has thought about supernatural worlds even if some have found them uninhabited.

If mathematicians create mathematics, then it seems an object for the cognitive sciences to investigate. In fact, the cognitive sciences do investigate it. How much research needs to be done before there is a theory is an open question.

I recommend reading Rebecca Goldstein's excellent book, Incompleteness, about Kurt Godel, his incompleteness theorems in mathematical logic, and their impact across philosophy, science, mathematics, and culture. No prior background in the subject is required. It's that good.

Lakoff and Nunez's Where Mathematics Comes From? is a very interesting attempt at a theory based on research coming out of the cognitive sciences. You can read a long way into the book if you know arithmetic.