COPElAND | 65
however, intuition is no longer needed: from there on, everything is just a matter of follow-
ing the rules.
In his 1938 doctoral dissertation Turing said, referring to Hilbert and his followers:^45
In pre-Gödel times it was thought by some that . . . all the intuitive judgments of mathematics
could be replaced by a finite number of these rules [and axioms].
Turing argued that, on the contrary, mathematics is just too unruly for intuition’s required role
to be limited in the way Hilbert wished. Summing up his argument, he explained: ‘We have
been trying to see how far it is possible to eliminate intuition’; and his conclusion was that
in fact the need for intuition turns out to be boundless. Hilbert might have responded that
Turing’s notion of an untamed and vaultingly powerful faculty of intuition belonged to what
he called ‘mysterious arts’.^46 Turing could have replied that Hilbert’s thinking was simply too
narrow, in consequence of his old-fashioned quest for ‘complete certitude’ in mathematics.
Conclusion
Hilbert’s views on the nature of mathematics, although innovative at the time, were the product
of a more uncomplicated mathematical era, now long gone: Gödel and Turing changed math-
ematics forever. Nevertheless, Hilbert’s ‘proof theory’—probably his proudest invention—is
still very much alive and well today, and is a core topic in every university curriculum on math-
ematical logic.
Hilbert certainly felt the force of Turing’s attack on his thinking about the computational
nature of mathematics. The following bold statement, made in 1928 in the first edition of his
famous book Grundzüge der Theoretischen Logik (Principles of Mathematical Logic, written with
Wilhelm Ackermann) was omitted from the much-revised second edition, published in 1938:^47
it is to be expected that a systematic, so to speak computational treatment of the logical formu-
lae is possible . . .