Early reviewers such as Mostowski wrote that Bourbaki's chosen foundations were "cumbersome"; I had not realised to what extent till I read a footnote in Bourbaki, reproduced in Godement, saying that the term for the number 1 would take some tens of thousands of signs to write out in full. I thought, "That must be false, surely only a couple of hundred;" and then the truth emerged.
I see in the hopeless unwieldiness of their system of logic, with its remarkable explosion in the length of formulae, a possible explanation of the psychological stress suffered by some readers of Bourbaki. What will happen to a young innocent who decides to learn mathematics by reading Bourbaki, and to start with VolumeI? It will tie him in knots. Either he will shut the book in disgust, or he will persevere and then he will be paralysed by the mental effort required to disentangle the formalism.
Bourbaki themselves took the first course: as remarked by Corry, they shied away from their own foundations. I expect that they came to the conclusion that logic is crazy — they had to conclude that to protect their sanity; but were they aware that the picture of logic they were giving to their disciples is merely a grotesque distortion and diminution of that subject? Is it too fanciful to see here, in this choice of formalism, with its unintuitive treatment of quantifiers, the reason for the phenomenon (which many mathematicians in various European countries have drawn to my attention whilst beseeching me not to betray their identity, lest the all-powerful Bourbachistes take revenge by depriving them progressively of research grants, office facilities and ultimately of employment) that where the influence of Bourbaki is strong, support for logic is weak? How does one get the message across, to those who have accepted the Bourbachiste gospel, that logicians are actually not interested in a formal system of such purposeless prolixity, still less do they advocate it as the proper intellectual framework for doing mathematics?
