A History of Mathematics From Mesopotamia to Modernity

254 A History ofMathematics

for many mathematicians there is a strong temptation to see Fritiof Capra’s intoxicated description

of the Tao of modern physics as the literal truth.

9. Fermat’s Last Theorem

I think I’ll stop here. (Andrew Wiles, Cambridge, 1993)

Before beginning I would have to put in three years of intensive study, and I haven’t that much time to squander on

a probable failure. (Hilbert 1920, on being asked why he did not attempt to prove Fermat’s Last Theorem)

And so to Andrew Wiles. One would like to say, after all the international excitement, that his proof
was in some way peripheral to this story, an isolated result. This is obviously not so, although the
reasons are quite complex. The fact that Wiles was stimulated in childhood by E. T. Bell’s romantic
personalized anecdotal bookMen of Mathematicsto nurse an ambition to solve the problem is in
itself an index of the power which a certain view of the history of mathematics can exercise.
The Last Theorem (FLT for short in what follows) states, if you have not seen it before, that the
equation

xn+yn=zn

has no solutions(x,y,z)in non-zero integers, forn> 213 ; it was claimed by Fermat in the 1630s,

but never proved, and has remained a challenge ever since. Andrew Wiles (of Princeton, however

much the English like to claim him as their own) announced its proof in 1993; flaws were found

in the proof, but a corrected version with help from Richard Taylor appeared in 1994 and is now

accepted. His famous ending to his Cambridge lecture quoted above (translation: I have solved

the most difficult problem there is, but I am too modest to say so) has become a favourite tag

for mathematicians.

Without going into detail on the history of FLT, we should note its role in the development of

ideals and factorization theory by Kummer and Dedekind in the mid-nineteenth century. There is

an illuminating study on this by Catherine Goldstein (1995) which shows up the historicity of the

problem; the difference between the seventeenth-century context of Fermat and the nineteenth-

century one of Kummer:

At the end of the eighteenth century, number theory was still no more than a flower-filled country lane, disdainfully

ignored by the great mathematical roads. Jean-Étienne Montucla, the first historian of mathematics, was still able to

write: ‘Geometry is still the general and only key to mathematics.’ A woman, Sophie Germain, prevented by her sex

from following a course of higher education, was still able successfully to solve certain cases of Fermat’s problem by

elementary methods and to maintain a real exchange with Gauss...Whatever the always keen interest Gauss always

had in numbers, one is still very far from Kummer, who began his researches on this field as soon as he was appointed

to a university post. (Goldstein 1995, p. 367)

The context of the late twentieth century is different again, of course. FLT, far from beingthe
problem (if it had ever been so) had become marginal, and it plays little part in the work of such
key number theorists as Hardy, Ramanujan, and André Weil. Interest in what one might have

13. There are of course plenty of solutions forn=2; these are the ‘Pythagorean triples’.