The Mind’s Best Work 
by D.N. Perkins.
Harvard, 314 pp., £12.95, November 1981, 0 06 745762 2
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The Mathematical Experience 
by Philip Davis and Reuben Hersh.
Harvester, 440 pp., £12.95, November 1981, 0 7108 0364 8
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Why are some people creative to the point of genius, even though they may not appear especially intelligent, or in any other way remarkable? Creativity is a long-standing puzzle which has received many trite and no very convincing accounts. Explanations range from the Divine Spark to slogging hard work; from unconscious problem-solving to super-conscious awareness; from the darkness of profound dreams to the brightness of extreme wakefulness; from slow gestations to instant insights. A merit of this book is that it looks at these various accounts – in the light of historically testified and autobiographical examples – with a good pinch (I almost said a Lot) of salt. The author, D. N. Perkins, a Co-Director of Project Zero at Harvard, includes puns and situation jokes as examples. Here he is following Arthur Koestler, but he does not altogether accept Koestler’s notion of Bisociation, expounded in The Act of Creation. This is the notion that we generally act within reference frames, and that creating involves relating normally independent matrices. Perkins’s objection – which is a central theme – is that creativity is not a special kind of thinking. For him, there is no essence, no special mental functions, for creativity: but there is strongly held motivation to achieve ends that may be new. For him, it is a matter of searching for and discovering surprising relations with an eye on fairly specific end-results, though with nothing very special required to produce even the unique results of genius.

An analogy for Perkins’s view of creativity might be an athlete with well-developed muscles and exceptional neural co-ordination, who does not have muscles we do not have, or anything essentially different from the rest of us. He may be better endowed to start with, and he will certainly have to train to develop his potential: but his unique achievements are based on a happy combination of common abilities together with exceptional motivation. This is how Perkins sees the creativity even of genius. He does not entirely reject Koestler’s Bisociation notion, but he points out, I think fairly, that it is not an essential of invention. And relating across normally separate reference frames is a common mental act in normal life, and so cannot by itself explain special creativity. To give an explanation of remarkable creative achievement we would need to know what is special about the particular associations, or Bisociations – which takes us back, almost, to the original question of what is special about invention and genius.

As with any explanation, to suppose that there is some special process, or thing, or essence, is ad hoc and generally unhelpful. The history of explanations very strongly suggests that postulates are only useful when they can be quite generally applied. When allowed to apply only to a few special cases, they give no predictive power and no understanding. On the other hand, the converse view that special cases can be explained in terms of what applies to many other, and seemingly very different, cases leads to reductionism. This is strongly resisted by many people who want favoured things or people to be unique. There are echoes here, surely, of theology. For the Greeks, the Muses were intelligences which communicated to poets and artists. This extraterrestrial theory of creativity is an extreme form of anti-reductionism which is by no means dead. For the reductionist camp, sufficiently detailed knowledge of normal brain processes, and strategies of normal thinking and perception, should allow exceptional creations to be at least in principle explained in terms of what applies to normal abilities. If we take this view, or aim, to the extreme, the highest creativity may be explained by – and accomplished by – computer programs not essentially different from programs capable of mundane thinking and perception. At present there are no adequate programs even for what comes easily to us all, but it is thought that research along these lines should lead to understanding, and even to the production of superhuman genius machines. There is no concern here with mechanical invention or technology, and classical treatments such as J. Jewkes, D. Sawers and R. Stillerman’s The Sources of Invention (1958) are not cited. Perkins does not examine the question of Artificial Intelligence in any detail, which gives his book rather an old-fashioned look. But his discussion of cases of creation, discovery and invention – such as Coleridge and the man from Porlock, and Poe’s Raven – is interesting. One of his propositions is: ‘No general plan guides invention because innovation is contrary to plan and is of so many varied sorts.’ Another: ‘The master of a creative activity is fluent.’ The examples are mainly literary, with a few well-known examples from the other arts and a few from science. What is lacking is an appreciation of the potential creativity of Artificial Intelligence; and, perhaps more surprisingly, any detailed accounts of psychological experiments on thinking and perception, in animals or humans. This is odd, as there are references to the experimental psychology literature on cognition, and, as I have said, Perkins, very sensibly, regards geniuses as much like the rest of us – except for their happy combination of abilities, their unusual motivation and their life-long drive to achieve things that do not yet exist.

Philip Davis and Reuben Hersh have written an attractive book which reveals the fascination of mathematics as an activity, and the seductions of its claims to indubitability. These have inspired, challenged and confounded views of the world and of the mind throughout philosophical history. Here we find the myth of Euclid, Platonism v. Aristotelianism, relations between pure and applied mathematics, the creation of new mathematics, the significance of paradoxes, the status of infinity. Here also are discussions of the psychology of mathematics: of how and what mathematicians think, what they do and do not believe, and what helps and hinders their special ability to grasp and make use of concepts lying outside sensory experience. There are splendid stories, such as the errors and remarkable insights leading to Fourier analysis, which is the basis of understanding phenomena as diverse as heat flow, vibrations and, recently, the spatial resolution of the eye. In spite of gaps and errors in his reasoning, Joseph Fourier’s generalisations were appreciated in his time, and received the highest honours, though it was said: ‘It was no doubt partially because of his very disregard for rigour that he was able to take conceptual steps which were inherently impossible to men of more critical genius.’ And Fourier was right – even though he never stated, let alone proved, a correct theorem about his Fourier series of sines and cosines. This allows any graph to be described analytically – or generated, as in the computer synthesis of musical sounds which may never have been heard before.

The issues of the status of mathematics are central for considering the nature of thinking and the status of knowledge, and go back to Classical philosophy. Thus Platonists regard mathematical objects (and it is a deep question just what is an object) as things already existing: so that numbers exist apart from our minds, and we discover, rather than invent, mathematics and logic. Constructivists regard the Natural numbers as the basis of all mathematics, and not to be analysed, reduced to anything simpler, or explained. The Formalists reject both these views, holding that mathematics is a game, with rules which are the rules of inference by which we transform one formula to another: any meaning that formulas may have is not within mathematics and is no concern of mathematics. Platonists require that intuition can unite our perceptual awareness with a mathematical timeless reality. But what is this intuition? How can we develop it, and know when to trust it? Do our minds really have such an intuitive sense? This position is frankly a theology. For the Formalists, the power to understand and create is in symbolic models, and models may, in principle, be represented and handled by machines. So, just as computers can play chess, inventing new moves as they go along, so may machines be mathematicians. Does it follow, then, that we (or at least mathematicians!) are machines? This depends, of course, on what we mean, or should mean, by ‘machine’, and here again the philosophy of mathematics is significant for the philosophy of mind and the psychology of creativity.

These two American authors have a strong historical sense and considerable wit: so this is not for the non-mathematician an arid book. Most of it can be understood by the layman, though technical points sometimes pop up without warning and go away unexplained. The book ends with Four-Dimensional Intuition. Here ‘intuition’ does not mean a path to truth, so much as appreciation or understanding. The intriguing question is: can computer graphics help to give intuitive appreciation of dimensions of space beyond the three dimensions of the world, as these exist for us in the perception of objects with the senses? Can it help us to ‘see’ a four-dimensional hypercube? The number of edges (32), vertices (16) and faces (this is complicated) can be calculated: but can we learn to ‘see’ it in four-dimensional space? Here there is a striking account of the experience of one of the authors. When shown a film of hypercube geometry, although impressed by the achievement, he did not get a feel of the hypercube. But when it was presented on a computer screen, and he was allowed to change its orientation interactively with three knobs (the system which made it possible to make the film), then:

I tried turning the hypercube around, moving it away, bringing it up close, turning it around another way. Suddenly I could feel it! The hypercube had leapt into palpable reality, as I learned how to manipulate it, feeling in my finger tips the power to change what I saw and change it back again. The active control at the computer console created a union of kinaesthetics and visual thinking which brought the hypercube up to the level of intuitive understanding.

This reminds one (though it is not referred to) of the rare reported experiences of people who have been born blind, or have been blind since infancy, and who then recover their sight through an eye operation when adult. Take the case of a man (‘S. B.’) who had corneal transplants when he was 52, having been effectively blind since a baby. He was shown, in a science museum, a lathe, which he knew about when blind but had never seen:

We led him to the glass case, which was closed, and asked him to tell us what was in it. He was quite unable to say anything about it, except that he thought the nearest part was a handle. (He pointed to the handle of the transverse feed.) He complained that he could not see the cutting edge, or the metal being worked, or anything else about it, and appeared rather agitated. We then asked a museum attendant (as previously arranged) for the case to be opened, and S. B. was allowed to touch the lathe, with his eyes tight shut. Then he stood back a little and opened his eyes, and said: ‘Now I’ve felt it I can see.’ He then named several of the parts correctly and explained how they would work ...

It seems that mathematics allows us to create and to see relations and generalisations which can apply to the world of sense, and be of great practical importance. It seems also that how mathematical discoveries are made and how they are proved are extremely different: and the difference may be said to run, roughly, from an all too human chaos to an inhuman order. It is clear that computers can carry out mathematical operations incomparably faster and more accurately than humans: yet they may lack ‘intuition’. Whatever this is, it seems to give a kind of global view for seizing upon distant things and creating new relations and models. And now it seems that by interacting with computers we can not only solve new problems but develop intuitions to see still further beyond the limitations of our senses into new, not always welcome and even impossible worlds. This is something like recovering from blindness.

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