The Ignorance of Bourbaki

Abstract

If one looks at the history of mathematics, one sees periods of bursting creativity, when new ideas are being developed in a competitive and therefore very hasty spirit; and periods when people find that the ideas so recently in vogue are inexact, incoherent, possibly inconsistent; in such periods there is an urge to consolidate past achievements. I said “the history of mathematics”: but mathematics is a complex sociological organism, and its growth takes place in different branches and in different countries, even different universities, in different ways and at different speeds. Sometimes national groups feel that mathematics in their country is in a bad way: you find an expression of that in the Introduction to later editions of Hardy’s Pure Mathematics, where he remarks that it was written with an enthusiasm intended to combat the insularity of British mathematics of the turn of the century, which had taken no account of the development of mathematics in France in the nineteenth century. Indeed in 1910 France could be proud of her succession of mathematicians such as Legendre, Laplace, Lagrange, Fourier, Cauchy, Galois, Dirichlet, Hadamard, Poincaré — a most impressive list of scholars of the highest distinction. But after the first World War, the feeling in France changed, and the young French mathematicians of the day began to consider that the torch of mathematical research had passed to Germany — where there were many great mathematicians building on the past work of Riemann, Frobenius, Dedekind, Kummer, Kronecker, Minkowski and Cantor, such as Klein, Hilbert, Weyl, Artin, Noether, Landau, and Hausdorff, — and that French mathematics had gone into a decline. So in 1935, a group of young French mathematicians resolved to restore discipline to their subject by writing a series of textbooks, under the joint pseudonym of Nicolas Bourbaki, that aimed to give definitive expositions with full French rigour to what they deemed to be the most important areas of pure mathematics. Now the question of mathematical rigour was very topical, a greater disaster than usual having occurred at the beginning of the twentieth century with the discovery by Russell of a major flaw in Frege’s proposed theory of classes. Frege wanted to form for any property Φ(y) the class {y | Φ(y)} of all objects y with the property Φ, and at the same time to count all such classes as objects to which such membership tests might be applied. If we write “a ∈ b” for “a is a member of b” and “a / ∈ b” for “a is not a member of b”, we may express Frege’s broad principle as follows. Denote {y | Φ(y)} by C: then for any object a, a ∈ C if and only if Φ(a). Russell, developing an idea of Cantor, noticed that if Φ(y) is taken to be the property y / ∈ y, of not being a member of oneself, then a contradiction results. For let B be the class of those objects that are not members of themselves; in symbols, B = {y | y / ∈ y}: then for any y, y ∈ B iff y / ∈ y; and so for the particular case when y is B, B ∈ B iff B / ∈ B. In response to this, there were some who wished to ditch all the more speculative areas of mathematics, which made use of the infinite and particularly of Cantor’s theories of cardinals and ordinals. Kronecker, Poincaré, Brouwer and Hermann Weyl should be mentioned here. But there were others — notably Hilbert — who wished to resist this wholesale amputation, and a programme was proposed aimed at formalising mathematics — the language, the axioms , the modes of reasoning etc — and at proving, by means the soundness of which could not possibly be doubted, that the resulting system was free of contradiction, that is, was consistent.

Cite this paper

@inproceedings{Mathias1990TheIO, title={The Ignorance of Bourbaki}, author={A R D Mathias and Paolo Mancosu and G{\'e}rard Bricogne}, year={1990} }