Alan Turing: The Enigma The Centenary Edition

by Andrew Hodges

  From this sphere of cells, Brewster explained, the animal would take shape:

  If it is an animal like ourselves, this body stuff, before it becomes a body, is a round ball. A furrow doubles in along the place where the back is to be, and becomes the spinal cord. A rod strings itself along underneath this, and becomes the back-bone. The front end of the spinal cord grows faster than the rest, becomes larger, and is the brain. The brain buds out into the eyes. The outer surface of the body, not yet turned into skin, buds inward and makes the ear. Four outgrowths come down from the forehead to make the face. The limbs begin as shapeless knobs, and grow out slowly into arms and legs. …

  Alan had thought about embryology all the time, fascinated by the fact that how such growth was determined was something ‘nobody has yet made the smallest beginning at finding out.’ There had been little advance since Growth and Form, the 1917 classic that he had read before the war. The 1920s had made it possible to invoke the Uncertainty Principle to suggest that life was intrinsically unknowable, like the simultaneous measurement of position and velocity in quantum mechanics.42 As with minds, there was an aura of religion and magic around the subject that attracted the attention of his scepticism. It was a fresh field. C.H. Waddington’s 1940 standard work43 on embryology did no more than list experiments on growing tissue, to explain in what circumstances it seemed to know what to do next.

  The greatest puzzle was that of how biological matter could assemble itself into patterns which were so enormous compared with the size of the cells. How could an assemblage of cells ‘know’ that it must settle into a fivefold symmetry, to make a starfish? How was this symmetry communicated across millions of cells? How could the Fibonacci pattern of a fir cone be imposed in its harmonious, regular way upon the growing plant? How could matter take shape or, as biological Greek had it, what was the secret of morphogenesis? Suggestive words like ‘morphogenetic field’, vague as the Life Force, were employed by biologists to describe the way that embryonic tissue seemed to be endowed with an invisible pattern which subsequently dictated its harmonious development. It had been conjectured that these ‘fields’ could be described in chemical terms – but there was no theory of how this could be.* Polanyi believed that there was no explanation except by a guiding esprit de corps; the inexplicability of embryonic form was one of his many arguments against determinism.45 Conversely, Alan told Robin that his new ideas were intended to ‘defeat the Argument from Design’.

  Alan was familiar with Schrödinger’s 1943 lecture, What is Life, which deduced the crucial idea that genetic information must be stored at molecular level, and that the quantum theory of molecular bonding could explain how such information could be preserved for thousands of millions of years. At Cambridge, Watson and Crick were busy in the race against their rivals to establish whether this was really so, and how. But the Turing problem was not that of following up Schrödinger’s suggestion, but that of finding a parallel explanation of how, granted the production of molecules by the genes, a chemical soup could possibly give rise to a biological pattern. He was asking how the information in the genes could be translated into action. Like Schrödinger’s contribution, what he did was based on mathematical and physical principle, not on experiment; it was a work of scientific imagination.

  There were other suggestions in the literature for the nature of the ‘morphogenetic field’, but at some point Alan decided to accept the idea that it was defined by some variation of chemical concentrations, and to see how far he could get on the basis of that one idea. It took him back to the days of the iodates and sulphites, to the mathematics of chemical reactions. But the new problem was of another order. It was not merely to examine substance A changing into substance B, but to discover circumstances in which a mixture of chemical solutions, diffusing and reacting with each other, could settle into a pattern, a pulsating pattern of chemical waves; waves of concentration into which the developing tissue would harden; waves which would encompass millions of cells, organising them into a symmetrical order far greater in scale. This was the fundamental idea, parallel to Schrödinger’s – that a chemical soup could contain the information required to define a large-scale chemical pattern in space.

  There was one central, fundamental problem. It was exemplified in the phenomenon of gastrulation. This was the process described and illustrated in Natural Wonders, in which a perfect sphere of cells would suddenly develop a groove, determining the head and tail ends of the emergent animal. The problem was this: if the sphere were symmetrical, and the chemical equations were symmetrical, without knowledge of left or right, up or down, where did this decision come from? It was just such a phenomenon that inspired Polanyi to claim that some immaterial force must be at work.

  In some way information was being created at this point, and this went against what was normally expected. When the lump of sugar has been dissolved in the tea, no information remains, at the chemical level, as to where it was. But in certain phenomena, those of crystallisation for instance, the reverse process could occur. Pattern could be created, rather than destroyed. The explanation lay in the interplay of more than one level of scientific description. In the chemical description, in which only average concentrations and pressures would be considered, no spatial direction would be preferred to any other. But at a more detailed, Laplacian level, the individual motions of the molecules would not be perfectly symmetrical, and under certain conditions, like that of the crystallising liquid, could serve to pick out a direction in space. The example Alan chose as illustration was drawn from his electrical experience:46

  The situation is very similar to that which arises in connexion with electrical oscillators. It is usually easy to understand how an oscillator keeps going when once it has started, but on a first acquaintance it is not obvious how the oscillation begins. The explanation is that there are random disturbances always present in the circuit. Any disturbance whose frequency is the natural frequency of the oscillator will tend to set it going. The ultimate fate of the system will be a state of oscillation at its appropriate frequency, and with an amplitude (and a wave form) which are also determined by the circuit. The phase of the oscillation alone is determined by the disturbance.

  He set up a system of oscillating circuits in his office, and used to show people how they would gradually all come to resonate with each other.

  Such a process of toppling, or crystallising, or falling into some pattern of oscillation, could be described as the resolution of an unstable equilibrium. In the case of the developing sphere of cells, it would have to be shown that in some way, through a change in temperature or the presence of a catalyst, the stable chemical balance could suddenly become unstable. It would be the chemical equivalent of piling the last straw on the camel’s back. Alan’s own analogy was that of a mouse climbing up a pendulum.

  Here was an idea which might explain something about how the information in the genes could be translated into physiology. The problem of growth as a whole would be far, far more complicated than this. But analysis of this moment of creation might yield a clue as to how the harmony and symmetry of biological structures could suddenly appear, as if by magic, out of nothing.

  To examine this moment of crisis mathematically, one had to approximate over and over again. He had to ignore the internal structure of the cells, and forget that the cells would themselves be moving and splitting as the patterning process took place. There was also an obvious limitation to the chemical model. How was it that the human heart was always on the left hand side? If this symmetry-breaking of the primordial sphere were determined at random, then hearts would be equally distributed between left and right. He had to leave this problem on one side, with a conjecture that at some point the asymmetry of the molecules themselves would play a part.

  The process of gastrulation, essentially as revealed in Natural Wonders.

  But with these reservations, his approach was to take the model and try it out. As he wrote,47 in a classic statement of
the scientific method:

  … a mathematical model of the growing embryo will be described. This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge.

  The result was applied mathematics par excellence. Just as the simple idea of the Turing machine had sent him into fields beyond the boundaries of Cambridge mathematics, so now this simple idea in physical chemistry took him into a region of new mathematical problems. But this time at least it was all his own work. No one else could make a mess of it for him.

  Even with immense simplifications, the mathematical equations corresponding to a soup of only four interacting chemical solutions, were still too intractable. The problem was that chemical reactions were ‘non-linear’. The equations for electricity and magnetism were ‘linear’, meaning that if two electromagnetic systems were superimposed (as for example when two radio transmitters radiated simultaneously), then the effects would simply be additive. The two transmitters would not interfere with each other. But chemistry was quite different. Double the concentration of reactants, and the reaction might go four times as fast. Superimpose two solutions, and anything might happen! Such ‘non-linear’ problems had to be solved as a whole, and not by the mathematical methods familiar in electromagnetic theory, those of describing the system as the sum of many little bits. However, that critical moment of budding, at the instant when the unstable system crystallised into a pattern, could be treated as if it were a ‘linear’ process – a fact familiar to applied mathematicians, and one which gave him a first handle upon the problem of growth.

  So he had his hands on another central problem of life, this time not of the mind, but of the body, although both questions were related to the brain. He quite literally had his hands upon it, for he had always enjoyed examining plants when on his walks and runs, and now he began a more serious collection of wild flowers from the Cheshire countryside, looking them up in his battered British Flora,48 pressing them into scrapbooks, marking their locations in large scale maps, and making measurements. The natural world was overflowing with examples of pattern; it was like codebreaking, with millions of messages waiting to be decrypted. Like codebreaking, the field was open-ended; with his chemical model he had one sharp tool to apply to it, but that was only the beginning.

  Mrs Webb was just the latest person to be given a talk on the Fibonacci spiral pattern of the fir cone, the pattern which showed itself also in the seeds in sunflower heads, and the leaf arrangements of the common plants. It was the problem of explaining its occurrence in nature that he set himself as a serious challenge. But this required the analysis of a two-dimensional surface, and he chose to leave it while he first considered in detail some rather simpler cases.

  In a chapter called ‘Nature’s Repair Shop’, Brewster had dwelt upon the regeneration of the Hydra, the small fresh water worm, which could grow a new head or new tail from any chopped-off section. Alan took the hydra, with its simple tubular form, and simplified it yet again, by neglecting its length, and concentrating upon the idea of a ring of cells. Then, he found, taking a model of just two interacting chemicals reacting and diffusing around this ring, that he was able to give a theoretical analysis of all the different possibilities for the moment of budding. And the idea, although admittedly in a grossly oversimplified and hypothetical way, actually worked. It appeared that under certain conditions the chemicals would gather into stationary waves of concentration, defining a number of lobes on the ring. These, it could be imagined, would form the basis for the pattern of tentacles. The analysis also showed the possibility of waves gathering into asymmetrical lumps of concentration, which reminded him of the irregular blotches and stripes on animal hides. With this last idea he did some experimental numerical work. By the end of 1950 the prototype computer had been closed down, and the scientists at the university were waiting for the new one to arrive from Ferranti, so this work was done on a desk calculator. It produced a dappling pattern rather resembling that of a Jersey cow. He was beginning to do something once more.

  At Christmas 1950 Alan met J.Z. Young again, to follow up the discussion about brain-cells that they had had in October 1949. Young had just given the Reith Lectures for 1950,49 presenting a rather aggressive statement of the claims of neuro-physiology to explain behaviour. Young later recalled50 Alan’s

  … kindly teddy-bear quality as he tried to make understandable to others, ideas that were still only forming in his own mind. To me, as a non-mathematician, his exposition was often difficult to follow, accompanied as it was by funny little diagrams on the blackboard and frequently by generalizations, which seemed as if they were his attempt to press his ideas on me. Also, of course, there was his rather frightening attention to everything one said. He would puzzle out its implications often for many hours or days afterwards. It made me wonder whether one was right to tell him anything at all because he took it all so seriously.

  They talked about the physiological basis of memory and of pattern recognition. Young wrote:51

  Dear Turing,

  I have been thinking more about your abstractions and hope that I grasp what you want of them. Although I know so little about it I should not despair of the matching process doing the trick. You have certainly missed a point if you suppose that to name a bus it must first be matched with everything from tea-pots to clouds. The brain surely has ways of shortening this process by the process – I take it – you call abstracting. Our weakness is that we have so little idea of the clues and code it uses. My whole thesis is that the variety of objects etc. are recognised by use of comparison with a relatively limited number of models. No doubt the process is a serial one, perhaps a filtering out of recognised features at each stage and then feeding back the rest through the system.

  This probably does not make much sense in exact terms and the only evidence for it is that people do group their reactions around relatively simple models – circle, god, father, machine, state, etc.

  Can we get anywhere by determining the storage capacity given by 1010 neurons if arranged in various ways and assuming facilitation of pathways by use? Is there any finite number of sorts of arrangement that they could have? For example, each with 100 possible outputs to others arranged a) at random through the whole or b) with decreasing frequency with distance. Given any particular plan of feedback can one compare the storage capacity of these plans, assuming say a given increase of probability of re-use of a pathway with each time of use?

  This is all very vague. If you have any ideas about the next important sorts of question to ask do let me know. Would it be a great help if we could give some sort of specification of the destinations of the output (within the cortex) of each cell? I feel we ought to be able to disentangle the pattern somehow.

  Yours, John Young.

  Alan’s reply made clear the connection between his interests in the logical and the physical structure of the brain:

  8th February 1951

  Dear Young,

  I think very likely our disagreements are mainly about the use of words. I was of course fully aware that the brain would not have to do comparisons of an object with everything from teapots to clouds, and that the identification would be broken up into stages, but if the method is carried very far I should not be inclined to describe the resulting process as one of ‘matching’.

  Your problem about storage capacity achievable by means of N (1010 say) neurons with M (100 say) outlets and facilitation is capable of solution which is quite as accurate as the problem requires. If I understand it right, the idea is that by different trainings certain of the paths could be made effective and the others ineffective. How much information could be stored in the brain in this way? The answer is simply MN binary digits, for there are MN paths each capable of two states. If you allowed each path to have eight states (whatever that might mean) you would get 3MN. …

  I am afra
id I am very far from the stage where I feel inclined to start asking any anatomical questions. According to my notions of how to set about it that will not occur until quite a late stage when I have a fairly definite theory about how things are done.

  At present I am not working on the problem at all, but on my mathematical theory of embryology, which I think I described to you at one time. This is yielding to treatment, and it will so far as I can see, give satisfactory explanations of –

  (i) Gastrulation

  (ii) Polygonally symmetrical structures, e.g. starfish, flowers.

  (iii) Leaf arrangement, in particular the way the Fibonacci series (0,1,1,2,3,5,8,13. …) comes to be involved.

  (iv) Colour patterns on animals, e.g. stripes, spots and dappling.

  (v) Pattern on nearly spherical structures such as some Radiolaria, but this is more difficult and doubtful.

  I am really doing this now because it is yielding more easily to treatment. I think it is not altogether unconnected with the other problem. The brain structure has to be one which can be achieved by the genetical embryological mechanism, and I hope that this theory that I am now working on may make clearer what restrictions this really implies. What you tell me about growth of neurons under stimulation is very interesting in this connection. It suggests means by which the neurons might be made to grow so as to form a particular circuit, rather than to reach a particular place.