Modernity and itsAnxieties 221
precision and abstraction were seen as impoverished forms of thought, associated with Germany’s
defeat; and that the stress on uncertainty and subjectivity in the new physics of Heisenberg and
Schrödinger deflected such criticisms. Intuitionism—which had in fact started some time before—
perhaps owed some of its popularity to a similar reaction.
Exercise 2.
(a) A shop contains an infinite number of pairs of socks S 1 ,S 2 ,.... I want to choose one sock from
each pair.
(1) Why will I need the Axiom of Choice to do it?
(2) Why will I not need the axiom if I am dealing with pairs of shoes?
(b) Show using the Law of the Excluded Middle that there exist numbers x,y such that x,y are irrational
but xyis rational. [Hint: Start with x=
√
2 .]What would be an intuitionist view of this argument?
(c) (Bolzano-Weierstrass theorem—hard!). Let x 1 ,x 2 ,...be an infinite sequence of numbers in the unit
interval[0, 1]. (1) Show that there is a subsequence xi 1 ,xi 2 ,...which tends to a limit x. (2) What,
from the intuitionist point of view, has gone wrong here?^5
5. Hilbert
I remember how enthralled I was by the first mathematics class I ever attended [at the University]...It was Hilbert’s
famous course on the transcendence ofeandπ. (Weyl, quoted in Reid 1970, p. 201)
In mathematics...we find two tendencies present. On the one hand, the tendency towards abstraction seeks to
crystallise the logical relations inherent in the maze of materials...being studied, and to correlate the material in
a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more
immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning
of their relations. (Hilbert 1999)
We have delayed mentioning David Hilbert perhaps longer than we should, because although
a central figure in the crisis, he was much more. His broad achievements and immense influence
have made him something of a folk-hero, at least among mathematicians, although we are unlikely
to see a film of his relatively uneventful life. For 30 years he dominated mathematics at Göttingen,
and made Göttingen the centre of the world; and one would have to go a long way today to find
a teacher who could transfix students on the subject of the transcendence ofeandπ,^6 if that
particularly late nineteenth-century subject is still taught. He has been well served by Constance
Reid’s biography (1970), with an excellent mathematical section by his favourite student Weyl.
Genial, productive, liberal, he remoulded the style of mathematics in algebra (particularly), and
in the foundational disputes which were to be such a central preoccupation, where he stood in
direct opposition to the intuitionists—and here Weyl’s defection was to be a cause of distress, if not
permanently so.
Rather than a film, Hilbert could make a good subject for a Greek tragedy, of downfall resulting
from an excess of ambition. We have already seen his announcement in 1900 of his belief that any
problem could be solved, to which Brouwer took such exception. The attacks of the intuitionists
and the notable weaknesses in set theory forced him into constructing an ingenious position; but
- This exercise and the preceding one have been borrowed, with their solutions, from Assad J. Kfoury at BU.
- That is, that neither of them satisfies an algebraic equationanxn+an− 1 xn−^1 +···+a 0 wherean,...,a 0 are integers.