64 | 7 HIlBERT AND HIS fAmOUS PROBlEm
going to fit into the usual student–supervisor pattern. Unfortunately, no spark seems to have
developed between the two, and ultimately Church was probably a disappointment to Turing.
For his thesis topic Turing chose another of Hilbert’s pontifical claims about the nature of
mathematics—and once again he argued that mathematical reality is not as Hilbert thought.^34
This time the disagreement concerned mathematical intuition.
Intuiting the truth
Most people can see by intuition that elementary geometrical propositions are true—for
instance, that a straight line and a circle can intersect no more than twice. There is no need to
go through a train of logical steps to convince yourself that this proposition is true. Hilbert gave
2 + 3 = 3 + 2 and 3 > 2 as examples of arithmetical propositions that are ‘immediately intuit-
a b l e ’.^35 The hallmark of intuition is this simple ‘seeing’, in contrast to following a conscious train
of logical deductions (if this then that, and if that then such-and-such). ‘The activity of the
intuition’, Turing said, ‘consists in making spontaneous judgments which are not the result of
conscious trains of reasoning’.^36 The more skilled the mathematician is, the greater his or her
ability to apprehend truths by intuition.
Turing and Hilbert both emphasized the importance of intuition. Hilbert declared:^37
we are convinced that certain intuitive concepts and insights are necessary for scientific knowl-
edge, and logic alone is not sufficient.
Nevertheless Hilbert wanted both to tame and to deflate intuition. He said that mathemati-
cal intuition, far from supplying knowledge of an abstract mathematical realm as others had
thought—a realm containing abstract objects such as numbers, many-dimensional figures and
(Turing might have added) endless paper tapes—has instead a much more down-to-earth role,
supplying knowledge concerning ‘concrete signs’ and sequences of these.^38 Hilbert’s examples
of concrete signs included ‘1 + 1’, ‘1 + 1 + 1’ and ‘a = b’; and he used the sign ‘2’ to abbreviate
the pattern ‘1 + 1’, ‘3’ to abbreviate ‘1 + 1 + 1’, and so on.^39 His concrete signs are nothing other
than the pencil or ink marks that a human computer could write when calculating. Hilbert
insisted, provocatively and with great originality, that ‘the objects of number theory’ are ‘the
signs themselves’; and he urged that this standpoint suffices for ‘the construction of the whole
of mathematics’.^40
With these simple but profound statements Hilbert deflated mathematical intuition: numbers
are simulated by pencil marks, and intuition consists in perceiving truths about the concrete
signs and the patterns in which the signs stand. The structure of these patterns ‘is immedi-
ately clear and recognizable’ to us, Hilbert said.^41 For example, we can immediately grasp that
appending ‘+ 1’ to ‘1 + 1’ produces ‘1 + 1 + 1’: or, in other words, that adding ‘1’ to ‘2’ gives ‘3’.
Hilbert also wanted to limit the role of intuition as far as is possible. Intuition often serves
to give us certainty that axioms are true (although even so, he recognized that sometimes
mathematicians disagree about axioms, and said that there might on occasion be ‘a cer-
tain amount of arbitrariness in the choice of axioms’).^42 Other mathematical propositions
are proved from the finite list of axioms by means of ‘formulable rules that are completely
definite’.^43 These are rules that any human computer can apply, such as that B follows from
A and A → B (A implies B) taken together. The rules are, like the axioms, finite in number,
and they too are seen to be correct by intuition.^44 Once the axioms and rules are in place,