240 A History ofMathematics
Idealism—the conviction that the war was not mere national defence but the defence of
civilization—powered the enlistment of many mathematicians, and other scientists, in the war
effort. Conscientious objectors were rare, and although André Weil unusually decided not to enlist,
he was prepared (as Oppenheimer was)^3 to cite the warlike Hindu scriptures in support of the war
effort:
The law is not ‘Thou shalt not kill’, a precept which Judaism and Christianity have inscribed—to what avail?—in their
commandments. TheGitabegins with Arjuna, ‘filled with the deepest compassion’, stopping his chariot between two
armies, and ends with his acceptance of Krishna’s injunction to go to combat unflinchingly...Arjuna belongs to
a caste of warriors, so hisdharmais to go to combat. (Weil 1992, p. 124)Indeed, from the Second World War onwards, mathematicians were to find that theirdharmadid
not always involve simply sitting in libraries and proving theorems.4. Abstraction and ‘Bourbaki’
On these foundations, I state that I can build up the whole of the mathematics of the present day; and if there is
anything original in my procedure, it lies solely in the fact that, instead of being content with such a statement,
I proceed to prove in the same way as Diogenes proved the existence of motion^4 ; and my proof will become more and
more complete as my treatise grows. (Bourbaki 1948)
For Bourbaki the fields to encourage were few, and the fields to discourage were many. (Mandelbrot 1989)The drive to a more abstract view of mathematics, which has been both admired and deplored
as peculiar to the twentieth century, had its roots early on in the foundational enterprise. The
schools of axiom-builders often claimed that it was not important what their axioms referred to:
Hilbert was quoted as saying that one should be able to replace the words ‘points, lines, planes’ in
the axioms of geometry with ‘tables, chairs, beer-mugs’; and Russell characterized mathematics as
the science ‘in which we do not know what we are talking about, nor whether what we are saying
is true’. However, if we think of the abstract viewpoint as one in which one aims systematically
to lose sight of any actual real-worldobjectsto which the discourse refers, and to concentrate on
therelationsandstructureswhich connect those objects, then the high point of abstraction came
in the 1940s and 1950s, and the leading spirits in carrying through a programme for making
all mathematics more abstract were a strange revolutionary band of young French university
teachers who formed a semi-secret society under the collective name ‘N. Bourbaki’. Typically, the
name derived from a juvenile prank—the invention of a mathematician whose name was borrowed
from a Greek general under Napoleon III.
A high-spirited male clique from typically French élite-school backgrounds, the Bourbakists
(Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné, André Weil, and a later ‘second
generation’ after the war) did not set out to call the foundations of mathematics in question. Their
aim was more straightforward, and more understandable: they felt that they had received atrocious
and antique teaching. They had, as an alternative, learned of new ideas from Germany, particularly
those of Hilbert and Emmy Noether as embodied in van der Waerden’s ground-breaking new
algebra textbook; and they decided to produce a series of textbooks which would form a complete
new course of mathematics from the ground up. Did they aim for their books to be set texts?
- Oppenheimer’s famous description of the atomic bomb as ‘brighter than a thousand suns’ is from theBhagavad-Gita.
- That is, by walking.