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I know Nicolas Bourbaki "is the pseudonym of a group of (mainly) French mathematicians who publish an authoritative account of contemporary mathematics." But what characterizes "Bourbaki's style in mathematics"?

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    $\begingroup$ A lot of what characterizes the "Bourbaki" style is their stringent adherence to definition/theorem/proof/definition/theorem/proof ... writing. Very few if any examples. Everything as general as possible. Very little or no motivation for why you're doing what you're doing. $\endgroup$ – Bill Cook May 12 '14 at 18:58
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    $\begingroup$ It comes from an assumption (wrongheaded IMHO ) that knowledge and understanding build up/develop naturally ( or can be developed naturally) in a strictly bottom-up format. $\endgroup$ – user99680 May 12 '14 at 19:00
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    $\begingroup$ Probably also avoiding pictures and geometrical understanding. $\endgroup$ – Johntra Volta May 12 '14 at 19:03
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    $\begingroup$ They invented the terms injective, surjective, and bijective? interesting $\endgroup$ – Geremia May 12 '14 at 19:16
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    $\begingroup$ You can see two papers from Leo Corry : Writing the Ultimate Mathematical Textbook : Nicolas Bourbaki’s Éléments de mathématique and Nicolas Bourbaki and the Concept of Mathematical Structure. $\endgroup$ – Mauro ALLEGRANZA May 13 '14 at 6:25
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From Leo Corry, Writing the Ultimate Mathematical Textbook : Nicolas Bourbaki’s Éléments de mathématique :

In the decades following the founding of the group, Bourbaki’s books became classic in many areas of pure mathematics in which the concepts and main problems, the nomenclature and the peculiar style introduced by Bourbaki were adopted as standard.

The branches upon which Bourbaki exerted the deepest influence were algebra, topology and functional analysis and they became the backbone of mathematical curricula and research activity in many places around the world.

Notations such as the symbol ∅ for the empty set, and terms like injective, surjective, and bijective owe their widespread use to their adoption in the Éléments de mathématique.

And :

Bourbaki’s extremely austere and idiosyncratic presentation of the topics discussed in each of the chapters – from which diagrams and external motivations were expressly excluded – became a hallmark of the group’s style and a main manifestation of its thorough influence.

Also the widespread adoption of approaches to specific question, concepts, and nomenclature promoted in the books of the series indicate the breadth of this influence.

Concepts and theories were presented in a thoroughly axiomatic way and systematically discussed always going from the more general to the particular, and never generalizing a particular result.

And :

In 1950 Dieudonné published, signing with the name of Bourbaki, an article that came to be identified as the group’s manifesto, “The Architecture of Mathematics”. Faced with the unprecedented growth and diversification of knowledge in the discipline over the preceding decades, Dieudonné raised once again the well-known question of the unity of mathematics.

Mathematics is a strongly unified branch of knowledge in spite of appearances, he claimed, and now it is clear that the basis of this unity is the use of the axiomatic method as the work of David Hilbert had clearly revealed starting from the beginning of the century.

Mathematics should be seen, Dieudonné added, as a hierarchy of structures at the heart of which lie the so called “mother structures”:

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 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 additon to Leo Corry's papers, cab be useful also his book :

Leo Corry, Modern Algebra and the Rise of Mathematical Structures (2nd ed - 2003).

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  • $\begingroup$ Nice quotes. While you are at it you could add Corry's comments on Bourbaki's blocking of Groethendieck's attempt to publish a volume on algebraic infinitesimals. $\endgroup$ – Mikhail Katz May 13 '14 at 7:56
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    $\begingroup$ @user72694 - The "massive effort" of Bourbaki's systhematization achieved a "landmark" in the shaping of modern math; of course, it had also an aspect of "dogmatism" : what was "outside" Bourbaki's framework was descouraged... $\endgroup$ – Mauro ALLEGRANZA May 13 '14 at 8:00
  • $\begingroup$ Precisely, with one of the larger things that were "outside" being category theory, as Corry discusses in detail. $\endgroup$ – Mikhail Katz May 13 '14 at 8:14
  • $\begingroup$ Is it fair to say "Bourbaki's style" is heavily deductive whereas "Russian style" is inductive? $\endgroup$ – Geremia Jun 29 '17 at 20:25
  • $\begingroup$ @Geremia - I do not think so... maybe Russian math is more interested to application (to e.g. physiscs) while Bourbaki's one is more "abstract". $\endgroup$ – Mauro ALLEGRANZA Jun 30 '17 at 6:00

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