The axiomatic method is at the heart of mathematics. The work of mathematicians is to derive the consequences of axioms. According to Euclid, axioms are evidently true, and deduction from them is a powerful method of learning new truths. The rise of non-Euclidean geometry disrupted the carefree connection between truth and proof and led many modern thinkers to adopt the formalistic attitude that the mathematician's sole endeavor is to work out the consequences of axioms, taking no professional interest in inquiring what, if anything, the axioms are true of.

In 1931 Kurt Gödel proved a deep theorem that showed that deduction from axioms cannot be all there is to mathematical understanding. Gödel showed that, for whatever system of truths of number theory we choose to regard as axiomatic, there will be statements of...