A History of Mathematics From Mesopotamia to Modernity

Modernity and itsAnxieties 219

to Euclid’s postulate 5: no one liked it, but it became clear (and Zermelo pointed out) that a great

many people^4 had, without acknowledgement, been using it. The usual statement is:

Given a set of sets{Xα}indexed byα∈A, there exists a functionfon the indexing setAsuch that for eachα,f(α)∈Xα.

In other words, givenanycollection of sets, you can pick out—think of it as ‘electing’—one

representative from each of them. This is easy to agree if the setAis finite, although it might be

far from practical if (say) it contained 10^25 elements. It is when it is infinite that it begins to look

dubious.

At this point, the problems probably disturbed some mathematicians intensely, but they did

not seriously divide them. Again, Gray finds a point in O. Perron’s 1911 inaugural lecture to

demonstrate the existence of argument, indeed doubt about the adequacy of procedures:

Indeed, there is one branch of mathematics today over which opinion is divided, and some consider right what

others reject. This is the so-called set theory, in which the certainty of mathematical deduction seems to be becoming

completely lost. (Perron, cited in Gray 2004, p. 41)

It was the attack initiated by L. E. J. Brouwer on a much more fundamental principle, the ‘Law of

the Excluded Middle’, which created a situation in which mathematicians became intemperate and,

for a short period, made the world of mathematics more exciting than it had been since the time

of Newton and Leibniz. This is not surprising, because the Law of the Excluded Middle underpins

the kind of mathematics which derives from the Greeks. From the simple, and apparently harmless

statement:

EitherPis true, orPis false.

applied to a propositionP, the Greeks derived their peculiar method of ‘proof by contradiction’,

which is still such a favourite. To prove thatPis true, you suppose that it is not. By a chain of

deduction, you derive a contradiction (‘Which is absurd’). Therefore the assumption thatPis not

true must have been wrong, and hence it must be true.

This principle was used constantly by Euclid: to take a random example, as early as book I

proposition 7 on isosceles triangles (‘If in a triangle two angles equal one another, then the sides

opposite the equal angles also equal one another’). You suppose one of the two sides greater, and

derive an absurd conclusion.

Many students, to be sure, feel uncomfortable about this kind of proof, but they learn to consider

it acceptable. What was considered doubtful, even wrong, what Weyl (under the shadow of the

approaching German hyper-inflation) described as ‘paper money’ was the use of the law in existence

proofs applied to infinite sets. Ironically, one of the neatest examples of such a proof is due to

Brouwer himself, and is still an essential element in beginning topology courses. This is theBrouwer

Fixed Point Theorem, which asserts:

LetDbe the disk{(x,y):x^2 +y^2 ≤ 1 }, and letfbe a continuous mapping fromDtoD(i.e. for(x,y)∈D,f(x,y)is also

inD, and depends continuously on(x,y)). Then there exists afixed point: for some(x 0 ,y 0 ),f(x 0 ,y 0 )=(x 0 ,y 0 ).

The proof of this proceeds by supposing that there is no fixed point (Fig. 2); we join eachf(A)toA,
and continue to the boundary circle C, which it hits atg(A). The mappingg, which fixes C, is shown
to be ‘impossible’ by methods which had been developed a little earlier (see ‘topology’ below).

4. Particularly in France, where opposition was strongest.