Nicolas Bourbaki and the concept of mathematical structure

In the present article two possible meanings of the term mathematical structure are discussed: a formal and a nonformal one. It is claimed that contemporary mathematics is structural only in the nonformal sense of the term. Bourbaki's definition of structure is presented as one among several attempts to elucidate the meaning of that nonformal idea by developing a formal theory which allegedly accounts for it. It is shown that Bourbaki's concept of structure was, from a mathematical point of view, a superfluous undertaking. This is done by analyzing the role played by the concept, in the first place, within Bourbaki's own mathematical output. Likewise, the interaction between Bourbaki's work and the first stages of category theory is analyzed, on the basis of both published texts and personal documents.Several persons have read and criticized earlier drafts of this paper. I thank them all for forcing me to formulate my ideas more simply and convincingly. Special thanks are due to Professors Pierre Cartier (Bures-sur-Yvette), Giorgio Israel (Rome), and Andrée C. Ehresmann (Amiens) for stimulating and illuminating conversations.