238 A History ofMathematics
3. The Second World War
Pure mathematics (in the narrowest sense) has a meaning only as a means of education to a formal character-building
that is consciously employed for service to the entire people.
Mathematical character-building: that is, cultivation of the masculine principle in spiritual life. (E. A. Weiss, cited
in Segal 2003, p. 193)
To assemble sufficient aircraft to implement the strategy required the diversion of several squadrons from Bomber
Command to Coastal Command, a proposal that was fiercely resisted by Air Chief Marshal ‘Bomber’ Harris, who
demanded of Churchill, ‘Are we fighting this war with weapons or the slide rule?’ Churchill puffed on his cigar and
replied, ‘That’s a good idea. Let’s try the slide rule.’ The results of the strategy turned out almost exactly as Blackett
and his colleagues had predicted. (Anecdote reported on http://www.orsoc.org.uk/conf/black.htm))
The involvement of mathematicians, and scientists in general, in the Second World War is in itself
a vast field of study. In mathematics one naturally begins by thinking of the allied effort, the origins
of operational research, the computer, the Enigma Code. All these are important, and were to lead
to an increasing collaboration of mathematicians with the militaryafterthe war, which chapters
in Nasar (1998) describe in some detail. However, the story begins some time earlier, with the rise
of Nazism in Germany. Our two quotes set up a somewhat facile contrast, with the Nazis involved
in cloudy rhetoric about the national spirit while the down-to-earth Allies see the point of using
the slide-rule. Easy as this is, it is not a gross distortion.
Like many others, German mathematicians who were Jewish (Landau and Courant) or com-
munist (Zorn) or in the case of Emmy Noether both, were surprised when they found that the Nazi
régime actually proposed to dispense with their services, however distinguished they might be. For
Noether it was easy—she did not have a titular post. For Landau, considerable manoeuvring was
necessary, including a student ‘boycott’ of his classes; but he was eventually forcibly retired.
The combined expulsion of Jews from their university positions and exodus of many non-Jews
(like Weyl and von Neumann) who found Nazi Germany uncongenial was catastrophic for math-
ematics in Germany. At the same time, in keeping with the Nazi programme, there was an attempt
to define what German mathematics should be. This posed a challenge, given the nature of the
subject. It was too easy to seeallmathematics as part of the abstract intellectualism which the
Nazis wished to overthrow, which of course would be unfortunate for any mathematicians how-
ever patriotic who wished to hold on to jobs in universities. Drawing on some of Brouwer’s ideas
(although Dutch, he was a strong German nationalist in the period), a group of mathematicians,
some convinced Nazis or at least right-wing nationalists, some simply opportunists, presented an
image of two opposing kinds of mathematics: crudely, Nordic/concrete/intuitive versus Jewish
(or French)/abstract/cerebral. As the main spokesman Bieberbach put it:
[T]he whole dispute over the foundations of mathematics is a dispute of contrary psychological types, therefore in the
first place, a dispute between races. (Quoted in Segal 2003, p. 365)The turn towards intuitionism as a model for ‘German mathematics’ was not by any means uni-
versal, but for a short time it was influential, particularly under the ascendancy of Bieberbach,
a time-server who succeeded in dominating the depleted German scene in the early 1930s. The
(Jewish analyst) Landau’s definition ofπin his classic textbook as ‘twice the smallest positivexfor
which cos(x)=0’ (where cos(x)is defined as a series, 1−(x^2 / 2 !)+(x^4 / 4 !)−···) was easy to
characterize as such a retreat from the ‘natural-intuitive’.