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 mathematics.

Nicolas Bourbaki was influential in 20th-century mathematics, particularly during the middle of the century when volumes of the Éléments appeared frequently. The group is noted among mathematicians for its rigorous presentation and for introducing the notion of a mathematical structure, an idea related to the broader, interdisciplinary concept of structuralism.

Quotes

From Bourbaki, Nicholas (1950-04). "The Architecture of Mathematics". The American Mathematical Monthly 57 (4): 221–232. ISSN 0002-9890. DOI:10.1080/00029890.1950.11999523.:

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).
The "structures" are tools for the mathematician; as soon as he has recognized among the elements, which he is studying, relations which satisfy the axioms of a known type, he has at his disposal immediately the entire arsenal of general theorems which belong to the structures of that type. Previously, on the other hand, he was obliged to forge for himself the means of attack on his problems; their power depended on his personal talents and they were often loaded down with restrictive hypotheses, resulting from the peculiarities of the problem that was being studied... each structure carries with it its own language, freighted with special intuitive references derived from the theories from which the axiomatic analysis described above has derived the structure. And, for the research worker who suddenly discovers this structure in the phenomena which he is studying, it is like a sudden modulation which orients at one stroke in an unexpected direction the intuitive course of his thought and which illumines with a new light the mathematical landscape in which he is moving about.
... in a single view, it sweeps over immense domains, now unified by the axiomatic method, but which were formerly in a completely chaotic state... In place of the sharply bounded compartments of algebra, of analysis, of the theory of numbers, and of geometry, we shall see, for example, that the theory of prime numbers is a close neighbor of the theory of algebraic curves, or, that Euclidean geometry borders on the theory of integral equations.
At the center of our universe are found the great types of structures... they might be called the mother-structures... Beyond this first nucleus, appear the structures which might be called multiple structures. They involve two or more of the great mother-structures simultaneously not in simple juxtaposition (which would not produce anything new), but combined organically by one or more axioms which set up a connection between them... Farther along we come finally to the theories properly called particular. In these the elements of the sets under consideration, which, in the general structures have remained entirely indeterminate, obtain a more definitely characterized individuality. At this point we merge with the theories of classical mathematics, the analysis of functions of a real or complex variable, differential geometry, algebraic geometry, theory of numbers. But they have no longer their former autonomy; they have become crossroads, where several more general mathematical structures meet and react upon one another.
From the axiomatic point of view, mathematics appears thus as a storehouse of abstract forms -- the mathematical structures; and it so happens -- without our knowing why -- that certain aspects of empirical reality fit themselves into these forms, as if through a kind of preadaptation. Of course, it can not be denied that most of these forms had originally a very definite intuitive content; but, it is exactly by deliberately throwing out this content, that it has been possible to give these forms all the power which they were capable of displaying and to prepare them for new interpretations and for the development of their full power. The unity which it gives to mathematics is not the armor of formal logic, the unity of a lifeless skeleton; it is the nutritive fluid of an organism at the height of its development, the supple and fertile research instrument to which all the great mathematical thinkers since Gauss have contributed, all those who, in the words of Lejeune-Dirichlet, have always labored to "substitute ideas for calculations."

Quotes about Bourbaki

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 Bourbaki's bylaws.
- René Thom, "Modern" Mathematics: An Educational and Philosophic Error?, American Scientist 59.6 (1971): 695-699.
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.
- pages 23-24. Piaget, Jean; Maschler, Chaninah; Piaget, Jean (1971). Structuralism. London: Routlege and Kegan Paul. ISBN 978-0-7100-7044-9.