Alan Turing: The Enigma (26 page)

(B) A possible change (b) of observed squares, together with a possible change of state of mind.

The operation actually performed is determined, as has been suggested [above] by the state of mind of the computer and the observed symbols. In particular, they determine the state of mind of the computer after the operation is carried out.

‘We may now construct a machine,’ Alan wrote, ‘to do the work of this computer.’ The drift of his argument, indeed, was obvious, with each ‘state of mind’ of the human computer being represented by a configuration of the corresponding machine.

These ‘states of mind’ being a weak point of the argument, he added an alternative justification of the idea that his machines could perform any ‘definite method’ which did not need them:

We suppose [still] that the computation is carried out on a tape; but we avoid introducing the ‘state of mind’ by considering a more physical and definite counterpart of it. It is always possible for the computer to break off from his work, to go away and forget all about it, and later to come back and go on with it. If he does this he must leave a note of instructions (written in some standard form) explaining how the work is to be continued. This note is the counterpart of the ‘state of mind’. We will suppose that the computer works in such a desultory manner that he never does more than one step at a sitting. The note of
instructions must enable him to carry out one step and write the next note. Thus the state of progress of the computation at any stage is completely determined by the note of instructions and the symbols on the tape…

But these arguments were quite different. Indeed, they were complementary. The first put the spotlight upon the range of thought within the individual – the number of ‘states of mind’. The second conceived of the individual as a mindless executor of given directives. Both approached the contradiction of free will and determinism, but one from the side of internal will, the other from that of external constraints. These approaches were not explored in the paper, but left as seeds for future growth.
^*

Alan had been stimulated by Hilbert’s decision problem, or the
Entscheidungs problem
as it was in German. He had not only answered it, but had done much more. Indeed, he would entitle his paper ‘On Computable Numbers,
with an application
to the Entscheidungs problem.’ It was as though Newman’s lectures had tapped some stream of enquiry which had been flowing all the time and which found in this question an opportunity to emerge. He had
done something
, for he had resolved a central question about mathematics, and had done so by crashing in as an unknown and unsophisticated outsider. But it was not only a matter of abstract mathematics, not only a play of symbols, for it involved thinking about what people did in the physical world. It was not exactly science, in the sense of making observations and predictions. All he had done was to set up a new model, a new framework. It was a play of imagination like that of Einstein or von Neumann, doubting the axioms rather than measuring effects. Even his model was not really new, for there were plenty of ideas, even in
Natural Wonders
, about the brain as a machine, a telephone exchange or an office system. What he had done was to combine such a naive mechanistic picture of the mind with the precise logic of pure mathematics. His machines – soon to be called
Turing machines
– offered a bridge, a connection between abstract symbols, and the physical world. Indeed, his imagery was, for Cambridge, almost shockingly industrial.

Obviously there was a connection between the Turing machine and his
earlier concern with the problem of Laplacian determinism. The relationship was indirect. For one thing, it might be argued that the ‘spirit’ he had thought about was not the ‘mind’ that performed intellectual tasks. For another, the description of the Turing machines had nothing to do with physics. Nevertheless, he had gone out of his way to set down a thesis of ‘finitely many mental states’, a thesis implying a material basis to the mind, rather than stick to the safer ‘instruction note’ argument. And it would appear that by 1936 he had indeed ceased to believe in the ideas that he had described to Mrs Morcom as ‘helpful’ as late as 1933 – ideas of spiritual survival and spiritual communication. He would soon emerge as a forceful exponent of the materialist view and identify himself as an atheist. Christopher Morcom had died a second death, and
Computable Numbers
marked his passing.

Beneath the change there lay a deeper consistency and constancy. He had worried about how to reconcile ideas of will and spirit with the scientific description of matter, precisely because he felt so keenly the power of the materialist view, and yet also the miracle of individual mind. The puzzle remained the same, but now he was approaching it from the other side. Instead of trying to defeat determinism, he would try to account for the appearances of freedom. There had to be a reason for it. Christopher had diverted him from the outlook of
Natural Wonders
, but now he had returned.

There was another point of constancy, in that he was still looking for some definite, down-to-earth resolution of the paradox of determinism and free will, not a wordy philosophical one. Earlier, in this search, he had favoured Eddington’s idea about the atoms in the brain. He would remain very interested in quantum mechanics and its interpretation, a problem that von Neumann had by no means resolved, but the Jabberwocky would not be
his
problem. For now he had found his own
métier
, by formulating a new way of thinking about the world. In principle, quantum physics might include everything, but in practice to say anything about the world would require many different levels of description. The Darwinian ‘determinism’ of natural selection depended upon the ‘random’ mutation of individual genes; the determinism of chemistry was expressed in a framework where the motion of individual molecules was ‘random’. The Central Limit Theorem was an example of how order could arise out of the most general kind of disorder. A cipher system would be an example of how disorder could arise by means of a determinate system. Science, as Eddington took care to observe, recognised many different determinisms, many different freedoms. The point was that in the Turing machine, Alan had created his own determinism of the automatic machine, operating within the
logical
framework he held to be appropriate to the discussion of the mind.

He had worked entirely on his own, not once discussing the construction of his ‘machines’ with Newman. He had a few words with Richard
Braithwaite at the High Table one day on the subject of Gödel’s theorem. Another time he put a question about the Cantor method to Alister Watson, a young King’s Fellow (a communist, as it happened) who had turned from mathematics to philosophy. He described his ideas to David Champernowne, who got the gist of the universal machine, and said rather mockingly that it would require the Albert Hall to house its construction. This was fair comment on Alan’s design in
Computable Numbers
, for if he had any thoughts of making it a practical proposition they did not show in the paper.
³⁸
Just south of the Albert Hall, in the Science Museum, were lurking the remains of Babbage’s ‘Analytical Engine’, a projected universal machine of a hundred years before. Quite probably Alan had seen them, and yet if so, they had no detectable influence upon his ideas or language. His ‘machine’ had no obvious model in anything that existed in 1936, except in general terms of the new electrical industries, with their teleprinters, television ‘scanning’, and automatic telephone exchange connections. It was his own invention.

A long paper, full of ideas, with a great deal of technical work and evidence of more left unpublished,
Computable Numbers
must have dominated Alan’s life from spring 1935 through the following year. In the middle of April 1936, returning from Easter at Guildford, he called on Newman and gave him the draft typescript.

There were many questions to be asked about the discoveries that Gödel and he had made, and what they meant for the description of mind. There was a profound ambiguity to this final settlement of Hilbert’s programme, though it certainly ended the hope of a too naive rationalism, that of solving every problem by a given calculus. To some, including Gödel himself, the failure to prove consistency and completeness would indicate a new demonstration of the superiority of mind to mechanism. But on the other hand, the Turing machine opened the door to a new branch of deterministic science. It was a model in which the most complex procedures could be built out of the elementary bricks of states and positions, reading and writing. It suggested a wonderful mathematical game, that of expressing any ‘definite method’ whatever in a standard form.

Alan had proved that there was no ‘miraculous machine’ that could solve all mathematical problems, but in the process he had discovered something almost equally miraculous, the idea of a universal machine that could take over the work of
any
machine. And he had argued that anything performed by a human computer could be done by a machine. So there could be a single machine which, by reading the descriptions of other machines placed upon its ‘tape’, could perform the equivalent of human mental activity. A single machine, to replace the human computer! An electric brain!

The death of George V, meanwhile, marked a transition from protest at the old order to fear of what the new might hold in store. Germany had
already defeated the new Enlightenment; had already injected iron into the idealist soul. March 1936 saw the re-occupation of the Rhineland: it meant that the future lay with militarism. Who then could have seen the connection with the fate of an obscure Cambridge mathematician? Yet connection there was. For one day Hitler was to lose the Rhineland, and it would be then, and only then, that the universal machine could emerge into the world of practical action. The idea had come out of Alan Turing’s private loss. But between the idea and its embodiment had to come the sacrifice of millions. Nor would the sacrifices end with Hitler; there was no solution to the world’s
Entscheidungs problem
.

^*
John Bennett was a boy in the house, who himself died later in 1930 on a lone winter trek across the Rockies.

^*
For comparison: a skilled worker earned about £160 per annum; unemployment benefit ran at £40 per annum for a single man.

^*
W. Sierpinski, a prominent twentieth century Polish pure mathematician.

^*
‘Mays’ were the semi-official second year examinations.

^*
Joynson Hicks, the reactionary Home Secretary.

^*
This gave him a tenuous link with his mother, who had shares in a Bethnal Green housing association. Alan’s reaction was approval that they planned the flats for the families who needed them rather than
vice versa
.

^†
‘Regarding Aunt J’s funeral’, Alan wrote in January 1934 to his mother, ‘I am not v. keen on going, and I think it would be consummate hypocrisy if I did. But if you think anyone will be the better for my attending I will see whether it can be managed.’

^*
Alan also considered Ibsen’s plays ‘remarkably good’.

^*
The analogy is not intended to be exact; Hilbert space and quantum mechanical ‘states’ differ in an essential way from anything in ordinary experience.

^*
The word ‘group’, as used in mathematics, has a technical meaning quite distinct from its use in ordinary language. It refers to the idea of a set of operations, but only when that set of operations meets certain precise conditions. These may be illustrated by considering the rotations of a sphere. If A, B and C are three different rotations, then one can see that:

(i) there exists a rotation which exactly reverses the effect of A.

(ii) there exists a rotation which has exactly the same effect as performing A, and then B.

Let this rotation be called ‘AB’. Then

(iii) AB, followed by C, has the same effect as A, followed by BC.

These are essentially the conditions required for the rotations to form a ‘group’. Abstract group theory then arose by taking these conditions, representing them appropriately with symbols, and then abandoning the original concrete embodiment. The resulting theory might profitably be applied to rotations, as indeed it was, in quantum mechanics. It could also apply to the apparently unrelated field of ciphering. (Ciphers enjoy the ‘group’ properties: a cipher must have a well-defined decipherment operation which reverses it, and if two ciphering operations are performed in succession, the result is another cipher.) But by the 1930s it was accepted that ‘groups’ could be explored in the abstract, without any concrete representation or application in mind.

^*
There is nothing ‘real’ about ‘real numbers’. The term is a historical accident, arising from the equally misleading terms ‘complex numbers’ and ‘imaginary numbers’. The reader not familiar with these expressions could think of ‘real numbers’ as ‘lengths defined with a hypothetical infinite precision.’

^*
Alan acquired a copy, soon heavily annotated, of Hilbert and Courant’s
Methoden der Mathematischen Physik
in July 1933.

^*
The author of one of the books which described the Central Limit Theorem.