Mathematical Structures from Hilbert to Bourbaki: The Evolution of an Image of Mathematics

The notion of a mathematical structure is a most pervasive and central one in twentieth-century mathematics. Over several decades of this century, many mathematical disciplines, particularly those belonging to the “pure ” realm, developed under the conception that the aim of research is the elucidation of certain structures associated with them. At certain periods of time, especially from the 1950s to the 1970s, it was not uncommon in universities around the world to conceive the ideal professional formation of young mathematicians in terms of a gradual mastering of the various structures that were considered to embody the hard-core of mathematics. Two important points of reference for understanding the development of twentieth century mathematics in general pertain the ideas of David Hilbert, on the one hand, and of the Bourbaki group, on the other hand. This is also true, in particular, for the development of the conception of mathematics as a science of structures. In the present article I explain how, and to what extent, the idea of a mathematical structure appears in the works of Hilbert and of Bourbaki. My presentation will be based on defining the idea of a mathematical structure as