But in describing the Turing machines ten years before, he had also justified his formalisation of the idea of ‘mechanical’ with a complementary argument, that of the ‘instruction note’. This put the emphasis not on the internal workings of the brain, but upon the explicit instructions that a human worker could follow blindly. In 1936 such ‘instruction notes’ had entered his experience through the rules of Sherborne School, other social conventions, and of course in the mathematical formulae that one could apply ‘without thinking’. But in 1945 a great deal of water had flowed under the bridge, and the ‘instruction notes’ that had been somewhat fanciful in 1936, just as were the theoretical logical machines, had become exceedingly concrete and practical. The cornucopian abundance was one of messages ‘based on a machine and broken on a machine’, and these machines were Turing machines, in which the logical transformation of symbols was what mattered, not physical power. And in designing such machines, and in working out processes that could be given to people acting like machines – the ‘slaves’ – they had effectively been writing elaborate ‘instruction notes’.
This was a different, but not incompatible, approach to the idea of ‘brain’. It was the interplay between the two approaches that perhaps fascinated Alan most – just as at Bletchley there had been a constant play between human intelligence and the use of machines or ‘slave’ methods. His ‘weight of evidence’ theory had shown how to transfer certain kinds of human recognition, judgment and decision into an ‘instruction note’ form. His chess-playing methods did the same thing – as did the games on the Colossi – and posed the question as to where a line could be drawn between the ‘intelligent’ and the ‘mechanical’. His view, expressed in terms of the imitation principle, was that there was no such line, and neither did he ever draw a sharp distinction between the ‘states of mind’ approach and the ‘instruction note’ approach to the problem of reconciling the appearances of freedom and of determinism.
All these questions remained to be explored, for the exigencies of the German cipher machines had barely scratched the surface of what could be done. It was yet to be seen how much could be achieved by writing ‘instruction notes’, and yet to be seen whether a machine could behave like a brain in developing ‘thinking spots’ for itself. As he had stressed in his discussions with Donald Michie, it had to be shown that a machine could learn. To explore these questions it would be necessary to have machines on which to experiment. But the almost incredible fact was that it would require only one machine, for the performance of any and all such experiments. For a universal Turing machine could imitate the behaviour of any Turing machine whatever.
In 1936 the Universal Turing Machine had played a purely theoretical part in his attack upon the Hilbert Entscheidungsproblem. But in 1945 it had a very much more practical potential. For the Bombes and Colossi and all the other machines and mechanical processes were parasitic beasts, dependent upon the whims and blindness of the German cryptographers. A change of mind on the other side of the Channel would mean that all the engineering that had been required to construct them would suddenly become useless. It had happened right from the start, with the Polish ‘fingerprint’ file, their perforated sheets and their simple Bombe, and it had nearly led to catastrophe in the blackout of 1942. The construction of special machines had led the cryptanalysts into one problem after another with the acquisition and application of new technology. But a universal machine, if only it could be realised in practice, would require no fresh engineering, only fresh tables, encoded as ‘description numbers’ and placed upon its ‘tape’. Such a machine could replace not only Bombes, Colossi, decision trees and all the other mechanical Bletchley tasks, but the whole laborious work of computation into which mathematicians had been conscripted by the war. The zeta-function machine, the calculation of roots of seventh-order equations, the large sets of equations arising in electrical-circuit theory – they could all alike be performed by a single machine. It was a vision beyond the comprehension of most people in 1945, but not beyond Alan Turing. As he would write later in 1945:18
There will positively be no internal alterations to be made even if we wish suddenly to switch from calculating the energy levels of the neon atom to the enumeration of groups of order 720.
Or as he would put it in 1948,19
We do not need to have an infinity of different machines doing different jobs. A single one will suffice. The engineering problem of producing various machines for various jobs is replaced by the office work of ‘programming’ the universal machine to do these jobs.
From this point of view, a ‘brain’ would not be just some bigger or better machine, some superior kind of Colossus. It did not develop out of an experience of things, but out of a consciousness of underlying ideas. A universal machine would not just be a machine; it would be all machines. It would replace not only the physical Bletchley machinery, but all that was routine – almost all that those ten thousand people had been doing. And not even the ‘intelligent’ work of the high-level analysts would be sacrosanct. For a universal machine could also play out the workings of human brains. Whatever a brain did, any brain, could in principle be placed as a ‘description number’ on the tape of a Universal Machine. This was his vision.
But there was nothing in the paper design of the Universal Turing Machine that suggested it could be made a practical proposition. In particular, there was nothing about its speed of operation. The tables of Computable Numbers could be realised by people sending postcards to each other, without affecting the theoretical argument. But if a universal machine were to be of any practical use, it would have to be able to run through millions of steps in a reasonable time. This demand for speed could only be met by electronic components. And this was where the revolution of 1943 had made all the difference in the world.
More precisely, the point was that electronic components could be regarded as operating upon discrete, on-or-off, quantities, and so could realise a Turing machine. This he had learnt in 1942, and thereafter he had known all about the Robinsons, the X-system, and the Rockex; he had also picked up a fund of radar knowledge from his new friends at Hanslope. But above all there were the two developments that had begun in 1943. Whatever its usefulness to the war effort, the technical success of the Colossi told him that thousands of electronic valves could be used in conjunction – something that few could have believed in until it had been done. And then he had worked with his own bare hands on the Delilah. There had been a method in his madness all along. By working in these second-rate conditions, on a device that officialdom had not called for, he had proved that he could carry off an electronic project of his own. Coordinated with his theoretical ideas and his experience of mechanical methods, this direct knowledge of electronic technology formed the last link in his plans. He had learned how to build a brain – not an electric brain, as he might possibly have imagined before the war – but an electronic brain. It was thus that ‘round about 1944’, Alan’s mother heard him talking20 about ‘his plans for the construction of a universal [machine] and of the service such a machine might render to psychology in the study of the human brain.’
There was a further fundamental consideration besides that of discreteness, reliability and speed: that of sheer size. There would have to be room on the ‘tape’ of a universal machine both for the ‘description numbers’ of the machines it had to imitate, and its workings. The abstract universal machine of 1936 was equipped with a ‘tape’ of infinite length, meaning that although at any stage the amount of tape used would be finite, it was assumed that as more space was required it could always be made available. In a practical machine, space would always be limited in extent – and for that reason no physical machine could actually realise a truly universal machine. Still, Alan had suggested in Computable Numbers that human memory was finite in extent. If this were so then the human brain itself could hold only a limited number of ‘tables of behaviour’, and a sufficiently large tape could contain them all. The
finiteness of any practical machine would not, on this argument, debar it from having a brain-like quality. The question was, however, how much ‘tape’ would be required for a machine that could actually be built: enough to make it interesting, but not more than would be technically feasible. And how could such storage be arranged without inconceivable expense in terms of electronic valves?
This practical question was one more up Don Bayley’s street. As the European war ground to its end, and the problems of the Delilah were essentially solved, it became clear that Alan’s interest had turned to ‘the brain’. He described to his assistant the universal machine of Computable Numbers, and its ‘tape’ on which instructions would be stored. They began thinking together about ways in which to realise a ‘tape’ that could store such information. And thus it was that in this remote station of the new Sigint empire, working with one assistant in a small hut, and thinking in his spare time, an English homosexual atheist mathematician had conceived of the computer.*
That was not how the world was to see it, and the world was not being entirely unfair. Alan Turing’s invention had to take its place in a historical context, in which he was neither the first to think about constructing universal machines, nor the only one to arrive in 1945 at an electronic version of the universal machine of Computable Numbers.
There were, of course, all manner of thought-saving machines in existence, going back to the invention of the abacus. Broadly these could be classed into two categories, ‘analogue’ and ‘digital’. The two machines on which Alan worked just before the war were examples of each kind. The zeta-function machine depended on measuring the moment of a collection of rotating wheels. This physical quantity was to be the ‘analogue’ of the mathematical quantity being calculated. On the other hand, the binary multiplier had depended upon nothing but observations of ‘on’ and ‘off’. It was a machine not for measuring quantities, but for organising symbols. In practice, there might be both analogue and digital aspects to a machine. There was not a hard-and-fast distinction. The Bombe, for instance, certainly operated on symbols, and so was essentially ‘digital’, but its mode of operation depended upon the accurate physical motion of the rotors, and their analogy with the enciphering Enigma. Even counting on one’s fingers, by definition ‘digital’, would have an aspect of physical analogy with the objects being counted. However, there was a practical consideration which provided the effective distinction between an analogue and a digital approach. It was the question of what happened when increased accuracy was sought.
His projected zeta-function machine would have well illustrated the point. It was designed to calculate the zeta-function to within a certain accuracy of measurement. If he had then found that this accuracy was insufficient for his purpose of investigating the Riemann Hypothesis, and needed another decimal place, then it would have meant a complete re-engineering of the physical equipment – with much larger gear-wheels, or a much more delicate balance. Every successive increase in accuracy would demand new equipment. In contrast, if the values of the zeta-function were found by ‘digital’ methods – by pencil and paper and desk calculators – then an increase in accuracy might well entail a hundred times more work, but would not need any more physical apparatus. This limitation in physical accuracy was the problem with the pre-war ‘differential analysers’, which existed to set up analogies (in terms of electrical amplitudes) for certain systems of differential equations. It was this question which set up the great divide between ‘analogue’ and ‘digital’.
Alan was naturally drawn towards the ‘digital’ machine, because the Turing machines of Computable Numbers were precisely the abstract versions of such machines. His predisposition would have been reinforced by long experience with ‘digital’ problems in cryptanalysis – problems of which those working on numerical questions would be entirely ignorant, by virtue of the secrecy surrounding them. He was certainly not ignorant of analogue approaches to problem-solving. Apart from the zeta-function machine, the Delilah had an important ‘analogue’ aspect. It depended crucially on accurate measurement and transmission of the amplitudes, in contrast to the X-system, which made them ‘digital’. He might well have admitted that for certain problems, the analogue solution could not remotely be rivalled by a digital method. Putting a model aircraft in a wind tunnel would immediately produce a picture of stresses and vortices that centuries of calculation would never obtain. In 1945 there was plenty of scope for debating the relative practical usefulness of analogue and digital devices, and the priorities for construction. But so far as Alan Turing was concerned, this was a debate for other people. He was committed to the digital approach, flowing out of the Turing machine concept, and with its potential universality at the centre. No analogue machine could lay claim to universality, such devices being constructed to be physical analogies of particular problems. It followed that his ideas had to find their place among, and compete with, the prevailing developments of digital calculators.
There had been machines to add and multiply numbers, the digital equivalents of the slide rule, since the seventeenth century. Alan had a desk calculator at Hanslope, and used it for the calculation of circuit properties. It was a very long step indeed from such devices to the idea of a practical universal machine. But as Alan knew by this time, that step had been made a hundred years before, by the British mathematician Charles Babbage (1791–1871). He used to speak to Don Bayley of Babbage, and knew something of what Babbage had planned.
After working on a ‘Difference Engine’, to mechanise the particular numerical method used in the construction of mathematical tables, Babbage had conceived (by 1837) of an Analytical Engine, whose essential property was that of mechanising any mathematical operation. It embodied the crucial idea of replacing the engineering of different machines for different tasks, by the office work of producing new instructions for the same machine. Babbage did not have a theory like that of Computable Numbers, to argue for universality, and his attention was focussed upon operations using numbers in decimal notation. Yet he did perceive that its mechanism could serve to effect operations upon symbols of any kind whatever,* and in this and other ways the Analytical Engine came close in its conception to the Universal Turing Machine. Babbage wanted a ‘scanner’, in effect, working on a stream of instructions, and putting them into operation. He hit on the idea of coding the instructions on punched cards, such as then were used for the weaving of complicated patterns in brocade. His plans also called for storing numbers in the form of positions of gear wheels. Each instruction card would cause an arithmetical operation such as ‘subtract the number in location 5 from that in location 8, and put the result in location 16.’ This required machinery he called the ‘mill’ to do the arithmetical operations, but the crucial innovation of Babbage’s plans did not lie in the efficient mechanisation of adding and multiplying. It lay in his perception that it was the mechanisation of the organising or logical control of the arithmetic that mattered.
In particular, Babbage had the vital idea that it must be possible to move forwards or backwards among the stream of instruction cards, skipping or repeating, according to criteria which were to be tested by the machine itself in the course of its calculation. This idea, that of ‘conditional branching’, was his most advanced. It was equivalent to the freedom allowed to Turing machines, that of changing ‘configuration’ according to what was read on the tape, and it was this that made Babbage’s planned machine a universal one, as he himself was well aware.
Without ‘conditional branching’, the ability to mechanise the word IF, the grandest calculator would be no more than a glorified adding machine. It might be thought of as an assembly line, everything being laid down from start to finish, and there being no possibility of interference in the process once started. The facility of ‘conditional branching’, in this model, would be analogous to specifying not only the routine tasks of the workers, but the testing, deciding and controlling operations of the management. Babbage was well-plac
ed to perceive this idea, his book On the Economy of Machinery and Manufactures being the foundation of modern management.
These ideas were a hundred years ahead of their time, and were never to be embodied in working machinery during Babbage’s lifetime. Government funding did not solve the problems created by his grossly over-ambitious specifications; the project was not advanced by Babbage’s contempt for committees, administrators, and other scientists; nor did his own efforts to bring mechanical engineering to an entirely new standard, and his absorption in every theoretical and practical aspect of the work, overcome these difficulties.
Indeed, it was exactly a hundred years from the conception of the Analytical Engine, until there were substantially new developments either in the theory or in the construction of such a universal machine. On the side of theory, 1937 saw the publication of Computable Numbers, which made all these ideas precise, explicit, and conscious. On the practical side, there had been the inevitable Looking Glass war as the revived and expanding electrical industry of the 1930s provided rival powers with new opportunities.
The first development had, in fact, occurred in 1937 Germany, at the Berlin home of K. Zuse, an engineer who had rediscovered many of Babbage’s ideas, though not that of conditional branching. Like the Babbage machine his first design, which was actually built in 1938, was mechanical and not electrical. But he had avoided the thousands of meshing ten-spoke gear wheels that Babbage had demanded, by the simple expedient of having his machine work in binary arithmetic. This was not a deep theoretical advance, but from any practical point of view it was an immense simplification. It was also a liberation from the usual engineer’s assumption that numbers had to be represented in the decimal way. Alan had used the same idea at the same time in his 1937 electric multiplier. Zuse had quickly moved on to construct further versions of his machine which made use of electromagnetic relays rather than mechanical elements, and with collaborators experimented with electronics before the end of the war. Zuse calculators were used in aircraft engineering but not in code-breaking; it was argued that the war would be over too soon. Short-term Nazism left Zuse in 1945 desperately trying to save his work from destruction.
Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game Page 47