236 A History ofMathematics
proof, which is perfectly in the Hardy mode: 100 pages long, understood only by a small circle,
and, while possibly applicable in its results (see later), certainly not in the methods it takes to
get there.
And yet mathematics, at all levels of subtlety and difficulty, is everywhere present in the world
we live in. One may ridicule the failures of ‘smart bombs’ to hit their targets, or of satellites to
find the weapons they are supposedly searching for; but the technology which they rely on, and
that which guides robots on Mars, and makes it possible to find optimal routes for motorists, is all
underpinned by mathematics of various kinds. The computer, which we shall have to consider, is an
essential part of much of this technology, and is in part a spin-off of the early twentieth century’s
preoccupation with logic and the constructible. But there are many more diverse inputs. When an
architectural design programme translates building specifications (breeze-blocks, windows, doors,
joists) into three-dimensional views of the projected construction, it isusingthe ability of the
computer to translate keystrokes or mouse movements into the eighteenth-century language of
Monge’s descriptive geometry (see Chapter 8). At the more advanced level ofTomb Raideror in the
animations ofThe Matrix, the same trick is being worked for jumps and turns in three dimensions,
using the classical dynamics developed by d’Alembert and Euler. When we buy, pay bills, or consult
our bank balance on a ‘secure website’ our data are encrypted using (perhaps) the properties of
large prime numbers, or even elliptic curves, which also play a part in the design of CDs; this most
classical part of mathematics, (Wiles’s preoccupation, one could say) is now intensively modernized
and even—for obvious reasons—subject to intellectual property law. If, rather than historians, we
were simply surveyors of the field of what mathematics is doing in 2004, it would be hard to
avoid intoxication, an endless list, and, somewhere along the line, a word like ‘awesome’. We
could hardly any longer agree with Hardy’s assertion that practical mathematics is by its nature
trivial. On the other hand, the pessimists among us might, like the 1940s Marxists Horkheimer
and Adorno, conclude that what is creative about human thought has been lost in its universal
mathematization.
From the historian’s viewpoint, then, what stands out about the development of mathematics in
the later twentieth century? First, we have an extremely rapidgrowthin the uses of mathematics,
the number of people employed, and the amount of work done; this really begins after the Second
World War. Second, as far as pure mathematics is concerned, we have seen a tendency towards
increasingabstraction, which has now perhaps peaked, but which was very influential in the mid-
century. Third, we have the rise ofnew forms of ‘applied’ mathematics; most obviously computer
science and statistics, but including a number of others (operational research, control theory,...).
There have been other major and important developments, but these will be enough to be going on
with. Even to connect such a brief list into some kind of coherent account will obviously involve
leaving out a great deal, and the variety of what is left may still seem confusing. It’s the mathematics
which we have now, for better or worse, and for that reason alone it deserves our attention.
2. Literature
The historians of mathematics, who are so dedicated and scholarly on the Greeks and the Chinese,
have not been as productive on the present. A vast amount of historical work has been pro-
duced on the Second World War; even the collapse of the Soviet Union a mere 16 years ago