246 A History ofMathematics
could be saved, how many expensive workers replaced^8 has been much faster than any of the
pioneers could have imagined. Mathematicians have, of course, continued to contribute directly
and indirectly; and the fact that your machine not only can calculate partial differential equations—
which you probably do rarely—but can respond when you type the letters ‘Dear’ by (a) storing an
encoded form of the letters in memory, (b) displaying them on your screen in Times New Roman,
and (c) asking you if you need help in writing a letter is, in itself, a form of debased practical
mathematics. We have come a long way from Mrs Turing’s typewriter.Exercise 2.(a) Devise a Turing machine which will change the natural number n (in binary digits) into
n+ 1 (e.g. 11 to 100, or 110 to 111). [Hint: Your machine will need to move to the left as it reads and
changes the number, for obvious reasons.] (b)What is the programme fragment above intended to do?6. Chaos: the less you know, the more you get
Then [Lorenz] walked down the hall to get a cup of coffee. When he returned an hour later, he saw something
unexpected, something that planted the seed of a new science. (Gleick 1987, p. 16)
It’s the paradigm shift of paradigm shifts. (Ralph Abraham, cited Gleick 1987, p. 52)The final nail in the coffin of a crude Marxist history of mathematics would seem to be provided
by chaos theory. Surely, the argument would run, if mathematicians hope to gain something for
their tedious profession their aim must be to persuade the Emperor that they can predict what is
coming and so make more crops grow, defeat famines, warn against attack. Yet, it would seem, they
can invent a theory whose main thrust is that, however precisely determined all these processes
may be, they are unknowable. A butterfly in Brazil can cause a typhoon on the Isle of Wight. There
is no point (or so some have said) in long-range weather forecasting—or, one might suppose, in
long-range anything. The Emperor might as well sack the mathematicians and watch the butterflies.
And yet, if you search for ‘chaos theory’+‘finance’+‘prediction’ on Google, you come up with
4300 hits, among them the following (which sounds definitely optimistic):
With chaos, financial understanding grows exponentially, creating new software and hardware to understand and
manage increasing risk. (Scholes 2002)There seems to be a great deal of interest out there. Is this simply an application of the old
deterministic mathematical principle that there’s one born every minute? It is easy to be cynical,
and chaos theory (as its name suggests) faces a continual risk of expanding beyond all reasonable
bounds; and so becoming too formless for the historian to describe it in as centred and confident
a way as (say) the Bourbakists. James Gleick’s book (1987) reflects this, ranging over a great
number of different bodies of work which call themselves chaos (or are so called by others), a great
panorama with no central landmark on which the reader can fix attention. Even so, in that the
main outlines of the subject were already fixed by the late 1980s, it is still a useful popular guide to
the variety of ways in which it can be viewed.
Moreover, through the 1970s and 1980s, chaos theory not only became fashionable among
mathematicians, and among the journalists like Gleick who try to find out what they are up to; it
- The analogy with the Jacquard loom and its impact on the handloom weavers (a typical image of hardfaced early capitalism) is
a striking one.