Are there ways to build mathematics without axiomatizing?

I remember reading the abstract of an article (or description of a book perhaps) that claimed to answer this using the principles of evolutionary biology; essentially, the author performed various simulations suggesting that organisms that take, as their fundamental logic, anything other than $2$-valued boolean logic tend to die off in the long run. I think if you Google around, you'll probably be able to dig something up in that vein.

One might object: ah, but you're using classical logic to build computer simulations and interpret the result of those simulations. That's circular! My gut feeling is that actually, this isn't circular (but my thoughts on this aren't sufficiently well-developed that its worth me trying to write them here.)