A History of Mathematics From Mesopotamia to Modernity

Modernity and itsAnxieties 215

Mathematicians can only feel flattered by William Everdell’s breathless placing of them as the
forerunners of Rimbaud, Freud, Joyce, Picasso et al., even if Dedekind’s statement of what he did
seems something of a let-down. He was aware of this himself:

[T]he majority of my readers will probably be disappointed in learning that by this commonplace remark the secret of
continuity is to be revealed. (Dedekind 1872, in Fauvel and Gray 18.C.1, p. 575)

Nonetheless, the work of Richard Dedekind and his more adventurous friend Georg Cantor on
numbers, the continuous and the infinite, did lead to a reshaping of mathematics if not the whole
world-view. Indeed, after a relatively short time it brought about the ‘crisis of foundations’, which
began some time around 1903, became acute in the 1920s, and was in some sense killed off,
if in no way settled, by the work of Kurt Gödel in 1931. The problems which arose were about
sets; and a first reasonable question is, how did mathematics, which as long as we have known it
has been about numbers and geometry, come to concern itself with sets? It has to be understood
that now even more than before, the world of mathematics was becoming fragmented, and these
concerns were not those of the average university teacher, let alone the engineer or statistician.
We are concerned for the moment with a relatively small research élite working mainly in France
and Germany, and the crisis as it developed came out of their attempts to make some sense of the
calculus which, as we have seen (Chapter 7) made very little sense as theory although it worked
well in practice.
Dedekind’s statement on what he could prove stands as an important pointer. To discuss what
problems his definition was meant to solve would take us too far back but the fundamental idea
was, in the words of one commentator:

to find definitions from which the basic theorems on limits could be proved. (R. Bunn, in Grattan-Guinness (ed.) 1980,
p. 222)

Briefly, you needed limits to define both derivatives and integrals properly; and hence to deal
with the problems of the calculus, and with numerous other problems, notably the behaviour of
Fourier series, which had arisen since. Dedekind’s definition of real numbers, as the necessary
foundation for the calculus of limits, is reproduced in Appendix A. It is more popular and easier
to understand than Cantor’s—though still not much taught in calculus courses—and as such it
is indeed a founding document of modernism in mathematics, if nowhere else. Given the setR
of all rational numbers (i.e. fractions—^13 ,^75 , and so on), which for the time being we consider

unproblematic, Dedekind considers the problem of characterizing, say,

√

2. This is not rational—
   there is a ‘hole’ in the rational numbers where the square root of 2 should be. The idea is todefine
   the real number to be the hole. Less mystically, we consider the ‘cut’ defined by the two sets:

L={x∈R:x<0orx^2 < 2 }; U={x∈R:x>0 andx^2 > 2 }

(See Fig. 1.) Everything inLis less than

√

2, everything inUis greater;

√

2 is the missing point
between.

a

a

bc de f

Fig. 1Dedekind cut. The numberαdivides the left-hand class L (which contains rational numbersa,b,c) from the right-hand class
(which containsd,e,f).