222 A History ofMathematics
one which with hindsight is more ambitious than anyone before or since has thought necessary
or possible. The proposal was as follows. One would define mathematics to be (a) a set of formulae
constructed on simple lines and (b) a finite set of axioms which characterized some of those
formulae as true; plus standard rules for deduction. If you take the axioms to be those which define
basic properties of natural numbers (and Hilbert did, as did Russell and Whitehead inPrincipia
Mathematica(1910–1913)), then you can of course deduce arithmetic of a simple kind; and if you
add some infinite processes, you can define real numbers à la Dedekind and deduce calculus and
geometry. To a certain extent, youdo not carewhat the axioms, or formulae refer to. The question is
not about what numbers are, but about how they behave; and this is why Hilbert was (by others)
called a ‘Formalist’. ‘In the beginning was the sign’, was a typical quote, which has been interpreted
in any number of ways. (Part of ) his statement of aims, which at the time he seems to have believed
completed, is reproduced in Appendix C.
You now, in Hilbert’s programme, shift to a different register called ‘metamathematics’. (One
could make a parallel with other twentieth-century studies where the activity, once pursued for its
own sake, becomes the object of study.) Looking at the process of forming provable formulae, you
ask two questions:1.Completeness. Is it possible, given any formulaP, to determine that it is true or false? An
obvious example is the formula:There exist natural numbersx,y,zandp>2 such thatxp+yp=zpnecessarily capable of being proved true or false? (True, we know the answer now, but Hilbert did
not.) If so, the system is called complete.
2.Consistency. Is it impossible, in the system, to deduce bothPand ‘notP’? If so, the system is
called ‘consistent’.We have stressed the immense ambition of this programme, but the field was completely new,
and Hilbert had an often justified confidence in the rapid progress of mathematics. Around 1929,
his student von Neumann seemed to have a proof, which only needed a little patching to make
it work.
The next part of the story is well known. In 1931 a young Austrian, Kurt Gödel, announced
a proof that neither completeness nor consistency were provable; more strictly, that they could
not be established by the finite methods which (under pressure from the intuitionists) were seen
as necessary. It was a perfect piece of Hilbert-type mathematics—and Gödel was much more
a formalist than an intuitionist—but it effectively destroyed the programme.When Hilbert first learned about Gödel’s work from Bernays, he was ‘somewhat angry’...The boundless confidence
in the power of human thought which had led him inexorably to this last great work of his career now made it almost
impossible for him to accept Gödel’s result emotionally. (Reid 1970, p. 98)Retired, and with his last attempt to make mathematics secure defeated at least provisionally,
Hilbert had to watch in disbelief as his best colleagues and students—Courant, Landau, Noether,
Weyl, and von Neumann, were either forced out of Göttingen by the Nazis or left because they were
unable to endure life under the Third Reich. He survived a few more years in Göttingen among