A History of Mathematics From Mesopotamia to Modernity

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was found to be an easily teachable and attractive subject on university courses. This could not have

been the case in the early days of Edward Lorenz’s surprise ‘discovery’ referred to in our first quote;

another case of the mathematician’s inspiration through coffee. The folklore account, which seems

reliable enough, makes clear that the use of computers was essential in the breakthrough. Lorenz

was solving complicated systems of differential equations by using a primitive (by our standards)

computer programme; and the discovery referred to was what is now called ‘sensitive dependence

on initial conditions’; or, popularly, ‘the butterfly effect’. A tiny variation in the starting pointf( 0 )

for the solution of an equation would lead to a large variation inf(t)astgrew, and Lorenz created

just such a tiny variation ‘by mistake’ (i.e. by leaving out the last three figures of the decimal for

f( 0 ), which the computer had stored in memory but not displayed) (Fig. 3).

Lorenz’s system was a particular one—a toy model for a weather system—and many classical

systems (think of a pendulum, or the motion of a planet) do not have this behaviour, otherwise the

edifice of mechanics becomes problematical. We only ever know the initial conditions approxim-

ately, and if a small error is going to increase beyond all bounds, then how is the model going to be

of any use?

All the same, once sensitive dependence was discovered, it made sense to try to understand the

phenomenon. Two things made the field much more easily accessible. The first was the idea, which

came from the mathematicians around Steve Smale at Berkeley, that one can replace the hard

study of differential equations by the much easier study ofmaps f:X→Xwhen iterated. [Usually

Xis the line, or the plane, or a suitable subset.] One considers what happens when one repeatsf

indefinitely. getting a sequence of maps:

f 1 =f; f 2 =f◦f; f 3 =f◦f◦f;...

The second was the arrival in the 1980s of the desktop computer, vastly simpler, quicker and
more graphically oriented than Lorenz’s model, with the help of which a second-year university
student (say) can study such equations and draw pictures. It is less fashionable now, but at the
time it was a favourite occupation to write and run programmes in one’s preferred language which
would compute the iterations of some map and draw gaudily coloured pictures of its behaviour on

Fig. 3The ‘butterfly effect’. Two solutions of Lorenz’s equation with very close starting points (at the left) eventually follow

completely different paths (at the right).