Bourbaki and Foundations

Bourbaki saw his project of the Elements as the culmination and synthesis of two trends leading to the state of mathematics in the thirties of last century: axiomatics and set theory. We argue that, in Bourbaki’s rendering, it was an unholy marriage. To stay together, set theory had to be reduced to a language and logic to the grammar of that language, and Hilbert to the caricature of a formalist. Both set theory and mathematical logic have suffered from this misrepresentation; they couldn’t force their entrance in the house, as other disciplines did, capitalizing on their necessity, since they were already, nominally and distorted, inside.