Are there ways to build mathematics without axiomatizing? [duplicate]

Solution 1:

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.)

Solution 2:

Gödels incompleteness theorems are widely interpreted to mean that it is impossible to construct a reasoning system that can prove itself to be consistent. As such, it is almost universally believed that it is impossible to construct mathemathics without starting with some "self evident" axioms.

The Wikipedia article explains the details of the theorems better than I can hope to do here and also includes sketches for various methods of proving the theorems.