A History of Mathematics From Mesopotamia to Modernity

220 A History ofMathematics

C

D

f(A)

A

g(A)

Fig. 2Brouwer fixed point theorem.

We have therefore a contradiction from supposing that there are no fixed points. Are we allowed

to deduce that there is one? Brouwer the prover of the theorem would have said yes, but Brouwer

the philosopher was emphatically against the idea. The theorem (see Weyl above) gives you no idea

of where the fixed point is.

Unlike the axiom of choice, the law of the excluded middle is venerable and would be hard to

give up. Yet it was simple for the intuitionists to find examples where it seemed not to work, often

relating to conjectures which were unsolved such as Fermat’s Last Theorem; an example from

Weyl is given in Appendix B(i). It is interesting since the conjecture which it draws on (whether

any numbers of form 2^2

n

+1 are prime forn>4) is still unsettled, and is the object of searches

taking years of computer time and superspeedy search programs. Hence one might say there is still

a question of belief for mathematicians, at least for those who are interested: is it the case thateither

the statement ‘no such numbers are prime’or‘one of the numbers is prime’ is true? An intuitionist

would say that you cannot assert such an either/or, since implicitly it means that you can search

through all the integers and decide the question.

It is important to realize how radical the intuitionists were in their aims. Where the work of

making the calculus rigorous left all theresultsin place, only changing the proofs so that they

made sense without the infinitely small, Brouwer and Weyl were forced into a situation where they

declared large parts of contemporary mathematics to be unacceptable and meaningless. Brouwer’s

concise formulation of his programme, to which he gave the name ‘intuitionist’ some time after

1907, is in Appendix B(ii). Note that he bans two things: the free use of sets and the law of the

excluded middle.

The problem for the intuitionists, as time went on, was that their destructive programme was

much easier to understand than their various attempts to be constructive. Weyl agreed that the

‘honest’ real numbers were the ones which one could calculate by a well-defined procedure;

examples include

√

2,π, cos(0.5),oreven (^) i∞= 0 ( 1 /is)forsrational; roughly what Turing not
long after was to call ‘computable numbers’. Yet he also wanted to keep, in his words, ‘the con-
tinuous ‘spatial soup’ that is poured between these [Euclidian] points’ (In Mancosu 1998, p. 132).
Bogged down in an attempt to preserve something of the ‘continuum’, even the infinite, intuitionist
mathematics, which was seen as the way of the future by many in the 1920s, became a difficult
and rather arcane specialist study by the 1930s—and survives as such.
There is a tempting analogy with the famous study of 1920s German physics by Paul Forman
(1971). Forman argued that physics, in the Weimar republic, had to adapt to a milieu in which