Truth vs. proof
Gödel’s theorems traded on crucial distinctions such as truth versus proof, semantics versus syntax, and completeness versus formal consistency, distinctions that, though in the air, became fully clarified for the first time only after Gödel’s proofs had appeared. It was not that Hilbert, the founder of formalism, distinguished carefully between truth and proof and simply opted for the latter. Rather, as Gödel himself put the matter years later, “formalists considered formal demonstrability to be an analysis of the concept of mathematical truth and, therefore, were of course not in a position to distinguish the two.” In the realm of mathematics, proof, for the formalist, was indistinguishable from truth, and so any attempt to draw distinctions between them was simply incomprehensible.