Nicolas Bourbaki
Nicolas Bourbaki
Nicolas Bourbaki is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathe
"Nicolas Bourbaki is the nom-de-plume adopted in the 1930s by a group of out standing young French mathematicians who undertook the monumental task of reorganizing mathematics in terms of basic structural components. This enterprise is an ongoing effort, whose members must resign at age fifty according to Bourbakis bylaws."
"It can now be made clear what is to be understood, in general, by a mathematical structure. The common character of the different concepts designated by this generic name, is that they can be applied to sets of elements whose nature has not been specified; to define a structure, one takes as given one or several relations, into which these elements enter (in the case of groups, this was the relation z = xτy between three arbitrary elements); then one postulates that the given relation, or relations, satisfy certain conditions (which are explicitly stated and which are the axioms of the structure under consideration.) To set up the axiomatic theory of a given structure, amounts to the deduction of the logical consequences of the axioms of the structure, excluding every other hypothesis on the elements under consideration (in particular, every hypotheses as to their own nature)."
"But in the eyes of contemporary structuralist mathematicians, like the Bourbaki, the Erlanger Program amounts to only a partial victory for structuralism, since they want to subordinate all mathematics, not just geometry, to the idea of structure. Classical mathematics is a quite heterogeneous collection of algebra, theory of numbers, analysis, geometry, probability calculus, and so on. Each of these has its own delimited subject matter; that is, each is thought to deal with a certain “species” of objects... Transformations may be disengaged from the objects subject to such transformation and the group defined solely in terms of the set of formations. The Bourbaki program consists essentially in extending this procedure by subjecting mathematical elements of every variety, regardless of the standard mathematical domain to which they belong, to this sort of “reflective abstraction” so as to arrive at structures of maximum generality., since they want to subordinate all mathematics, not just geometry, to the idea of structure."
