FOM: review of "The Ignorance of Bourbaki" (from Math. Reviews)

[Stephen G Simpson]

94a:03004a 03A05 00A30 01A60 
Mathias, A. R. D.(4-CAMB)
The ignorance of Bourbaki. (English. English summary) 
Physis Riv. Internaz. Storia Sci. (N.S.) 28 (1991), no. 3, 887--904 (1992). 

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Mathias, A. R. D.(PL-WASW)
The ignorance of Bourbaki. (English) 
Math. Intelligencer 14 (1992), no. 3, 4--13. 

This is an iconoclastic paper. The title itself points out the
ignorance of the great and poly-headed Bourbaki. Ignorance of what? Of
Godel's incompleteness results and logical progress after the
1920s. It is true that the Bourbachiste papers on the foundations of
mathematics, even those published late in the forties, do not mention
the name of Godel and present a pre-Godelian understanding of logical
work. Many quotations from papers written by Henri Cartan and Jean
Dieudonne support the author's criticism. As well as this, one must
add that Bourbaki ignored Tarski's efforts to define, after 1926, a
new conception of the link between mathematics and logic, and his
attempts to show how they can compound to give new results or new
understanding of some classical ones. Before 1950, on the whole, model
theory was unknown among mathematicians. The disregard for logic in
the mathematical world was not limited to France, as the author
suggests. But in France it was surely very strong, and it still exists
today. Even in the past few years, many leading French mathematicians
have continued to have a negative attitude towards logic and to think
that this "formal" subject cannot help to solve "concrete"
mathematical problems. This paper may rightly draw the attention, at
least of the historians of mathematics, to this regrettable attitude
and to some reasons for it: on one hand we had Poincare's fierce
opposition and mocking attitude to set theory, and, one must add, his
criticism of "Hilbert's logic" and his defense of intuition against
logic; on the other hand we have Bourbaki's desire to confine logic
once and for all to the first chapter of the Elements of Mathematics
series and then forget about it, without seeing that this is made
impossible by Godel's and Tarski's work. Logical issues actually
permeate the whole body of mathematics, as is now evident from
model-theoretic proofs of many mathematical results.

Some people say Bourbaki is dead. But many of the same people who
criticize Bourbaki's "formalism" persist in supporting his fossilized
view of logic. This fact is worth noting and the author's paper
deserves attention.

The two versions of the paper are identical. 

                  Reviewed by Hourya Sinaceur 

© Copyright American Mathematical Society 1994, 1997