AChaoticEnd? 243
of teaching was the only right one. It never extended even over all of France, but where it was
entrenched, it could be fairly intolerant. If installed as head of an unreformed mathematics
department, a Bourbakist was capable of purging the library of most of its books and writ-
ing out orders for replacements—Bourbaki, naturally, but also the seminars of Henri Cartan,
the works of van der Waerden, Eilenberg, MacLane, Steenrod, and subscriptions to theAnnals
of Mathematics, and the publications of the American Mathematical Society. More disastrous
was the brief incursion of Bourbakism into the French high-school curriculum in the 1960s
(paralleled by what, in the United States was called ‘new math’); the idea of replacing times
tables by theorems aboutZcaused confusion among teachers and students and was eventually
withdrawn.
We have expressed doubts about the responsibility of mathematicians for ‘modernism’. In the
more minor case of the philosophical movement called structuralism, the case is clearer. In a text
for the ‘Que-sais-je’ series explaining structuralism to the general public (1970), Jean Piaget cited
Bourbaki as his first example before proceeding to the social sciences, adding that ‘the structural
models of Lévi-Strauss, the acknowledged master of present-day social and cultural anthropology,
are a direct adaptation of general algebra’ (p. 17). The influence appears directly in an anecdote of
André Weil:
In New York, I had met the sociologist [sic] Lévi-Strauss, and we had hit it off quite well. I had solved for him a problem
of combinatorics concerning marriage-rules in a tribe of Australian aborigines. (Weil 1992, p. 185)
Mathematics never consciously progressed to post-structuralism; nor did Lévi-Strauss repay the
debt by teaching the Bourbakists some useful elements of anthropology.
Exercise 1.(From Bourbaki, Algebra, chapter I, §1, §7.)
(i) Show that the only triplets(m,n,p)of natural numbers = 0 , such that(mn)p=m(np)
are:(1,n,p),
n and p being arbitrary;(m,n,1)where m,n are arbitrary; and(m,2,2)where m is arbitrary.
(ii) If G is a finite group of order n, prove that the number of automorphisms of G is≤nlogn/log 2
(show that there exists a system of generators{a 1 ,...,am}of G such that aidoes not belong to the
subgroup generated by a 1 ,a 2 ,...,ai− 1 for 2 ≤i≤m; deduce that 2 m≤n, and that the number of
automorphisms of G is≤nm).
5. The computer
The distinctive characteristic of the Analytical Engine, (from the earlier Difference Engine)...is the introduction into
it of the principle which Jacquard devised for regulating, by means of punched cards, the most complicated patterns in
the fabrication of brocaded stuffs...We may say most aptly that the Analytical Engine weaves algebraical patterns just
as the Jacquard-loom weaves flowers and leaves. (Lovelace, Note A, on Menabrea’s description of Babbage’s engine
(1843), in Fauvel and Gray 19.B.4, p. 392)
Anautomatic computing systemis a (usually highly composite) device, which can carry out instructions to perform
calculations of a considerable order of complexity—for example to solve a non-linear partial differential equation
in 2 or 3 independent variables numerically. (von Neumann 1945, p. 7)
It may appear somewhat surprising that this can be done. How can one expect a machine to do all this multitudinous
variety of things? The answer is that we should consider the machine as doing something quite simple, namely carrying
out orders given to it in a standard form which it is able to understand. (Alan Turing, ‘Intelligent Machinery’, cited in
Hodges 1985, p. 318)