 In this philosophical debate between the two mathematicians, we see a clash between two quite different ways of understanding the nature of mathematical theories and of their justification. The difference of opinion over the success of Hilbert’s consistency and independence proofs is, as detailed below, the result of significant differences of opinion over such fundamental issues as: how to understand the content of a mathematical theory, what a successful axiomatization consists in, what the “truths” of a mathematical theory really are, and finally, what one is really asking when one asks about the consistency of a set of axioms or the independence of a given mathematical statement from others.