10. A chaotic end?
1. Introduction
I am not thinking of the ‘practical’ consequences of mathematics...at present I will say only that if a chess problem
is, in the crude sense, ‘useless’, then that is equally true of the best mathematics; that very little of mathematics is
useful practically, and that that little is profoundly dull. (Hardy 1940, p. 29)
Mathematical formalism, however, whose medium is number, the most abstract form of the immediate,...holds
thinking firmly to mere immediacy. Factuality wins the day; cognition is restricted to its repetition; and thought
becomes mere tautology. (Adorno and Horkheimer 1979, p. 27)
In 1936, the year when the Spanish Civil War broke out, Lancelot Hogben wrote a book called
Mathematics for the Million—an unexpected best-seller, which still sells today. The aim was to
educate the masses in mathematics, since:
The mathematician and the plain man each need one another. Maybe the Western world is about to be plunged
irrevocably into barbarism. If it escapes this fate, the men and women of the leisure state which is now within
our grasp will regard the democratization of mathematics as a decisive step in the advance of civilization. (Hogben
1936, p. 20)
Hogben’s judgement stands in opposition to that of Hardy (what is good in mathematics is not
useful); and equally to that of Adorno and Horkheimer, also writing in the 1940s (mathematization
is everywhere, and to mathematize ideas is to deprive them of their creativity). The mathematics
which he aimed to democratize was not easy—it included algebra, the calculus, and statistics—but
both in content and in presentation it would have been found trivial by Hardy, and not surprisingly,
it included no recent results. How has the democratization of mathematics fared during the last
70 years—and is anyone still convinced that it is either possible or desirable?
It seems difficult, in mathematics, to approach the history of the present—or even of the recent
past. School history programmes and TV history channels thrive on the wars and oppressions of
the last 100 years; but they have a clear narrative line to help them, and ample resources in the
form of film and picture archives. With regard to the history of any science, and of mathematics
in particular, there seems to be too much of it; it is too difficult, too diffuse, and it is hard to put
the various narratives together to point a moral or adorn a tale. The historian begins by being
grateful to Andrew Wiles, who, having proved the outstanding theorem of mathematics (‘Fermat’s
Last Theorem’) in the closing years of the twentieth century, has at least provided the story with
a neat, if provisional ending. However, it is a rather special kind of ending, centred on the rarefied
world of universities and extremely hard pure mathematics. It could serve as a model for what, in
mathematics, is undemocratic. A contemporary Hogben could, at a pinch, include a chapter on
the exposition of Gödel’s Theorem (proved in 1931), but might find it pointless except as a lead-in
to Turing’s work, and so, as we shall see, to computers. He would have no time at all for Wiles’s