The man who mistook his wife for a hat

  Hermann von Helmholtz, speaking of musical perception, says that though compound tones can be analysed, and broken down into their components, they are normally heard as qualities, unique qualities of tone, indivisible wholes. He speaks here of a 'synthetic perception' which transcends analysis, and is the unanalysable essence of all musical sense. He compares such tones to faces, and speculates that we may recognise them in somewhat the same, personal way. In brief, he half suggests that musical tones, and certainly tunes, are, in fact, 'faces' for the ear, and are recognised, felt, immediately as 'persons' (or 'personeities'), a recognition involving warmth, emotion, personal relation.

  So it seems to be with those who love numbers. These too become recognisable as such-in a single, intuitive, personal 'I know you!'* The mathematician Wim Klein has put this well:

  *Particularly fascinating and fundamental problems are raised by the perception and recognition of faces-for there is much evidence that we recognise faces (at least

  (continued)

  'Numbers are friends for me, more or less. It doesn't mean the same for you, does it-3,844? For you it's just a three and an eight and a four and a four. But I say, "Hi! 62 squared."

  I believe the twins, seemingly so isolated, live in a world full of friends, that they have millions, billions, of numbers to which they say 'Hi!' and which, I am sure, say 'Hi!' back. But none of the numbers is arbitrary-like 62 squared-nor (and this is the mystery) is it arrived at by any of the usual methods, or any method so far as I can make out. The twins seem to employ a direct cognition-like angels. They see, directly, a universe and heaven of numbers. And this, however singular, however bizarre-but what right have we to call it 'pathological'?-provides a singular self-sufficiency and serenity to their lives, and one which it might be tragic to interfere with, or break.

  This serenity was, in fact, interrupted and broken up ten years later, when it was felt that the twins should be separated-'for their own good', to prevent their 'unhealthy communication together', and in order that they could 'come out and face the world … in an appropriate, socially acceptable way' (as the medical and sociological jargon had it). They were separated, then, in 1977, with results that might be considered as either gratifying or dire. Both have been moved now into 'halfway houses', and do menial jobs, for pocket money, under close supervision. They are able to take buses, if carefully directed and given a token, and to keep themselves moderately presentable and clean, though their moronic and psychotic character is still recognisable at a glance.

  This is the positive side-but there is a negative side too (not mentioned in their charts, because it was never recognised in the first place). Deprived of their numerical 'communion' with each other, and of time and opportunity for any 'contemplation' or 'communion' at all-they are always being hurried and jostled

  (continued)familiar faces) directly-and not by any process of piecemeal analysis or aggregation. This, as we have seen, is most dramatically shown in 'prosopagnosia', in which, as a consequence of a lesion in the right occipital cortex, patients become unable to recognise faces as such, and have to employ an elaborate, absurd, and indirect route, involving a bit-by-bit analysis of meaningless and separate features (Chapter One).

  from one job to another-they seem to have lost their strange numerical power, and with this the chief joy and sense of their lives. But this is considered a small price to pay, no doubt, for their having become quasi-independent and 'socially acceptable'.

  One is reminded somewhat of the treatment meted out to Nadia-an autistic child with a phenomenal gift for drawing (see below, p. 219). Nadia too was subjected to a therapeutic regime 'to find ways in which her potentialities in other directions could be maximised'. The net effect was that she started talking-and stopped drawing. Nigel Dennis comments: 'We are left with a genius who has had her genius removed, leaving nothing behind but a general defectiveness. What are we supposed to think about such a curious cure?'

  It should be added-this is a point dwelt on by F.W.H. Myers, whose consideration of number prodigies opens his chapter on 'Genius'-that the faculty is 'strange', and may disappear spontaneously, though it is, as often, lifelong. In the case of the twins, of course, it was not just a 'faculty', but the personal and emotional centre of their lives. And now they are separated, now it is gone, there is no longer any sense or centre to their lives. *

  Postscript

  When he was shown the manuscript of this paper, Israel Rosen-field pointed out that there are other arithmetics, higher and simpler than the 'conventional' arithmetic of operations, and wondered whether the twins' singular powers (and limitations) might not reflect their use of such a 'modular' arithmetic. In a note to me, he has speculated that modular algorithms, of the sort described by Ian Stewart in Concepts of Modern Mathematics (1975) may explain the twins' calendrical abilities:

  Their ability to determine the days of the week within an eighty-

  *On the other hand, should this discussion be thought too singular or perverse, it is important to note that in the case of the twins studied by Luria, their separation was essential for their own development, 'unlocked' them from a meaningless and sterile babble and bind, and permitted them to develop as healthy and creative people.

  thousand-year period suggests a rather simple algorithm. One divides the total number of days between 'now' and 'then' by seven. If there is no remainder, then that date falls on the same day as 'now'; if the remainder is one, then that date is one day later; and so on. Notice that modular arithmetic is cyclic: it consists of repetitive patterns. Perhaps the twins were visualising these patterns, either in the form of easily constructed charts, or some kind of 'landscape' like the spiral of integers shown on page 30 of Stewart's book.

  This leaves unanswered why the twins communicate in primes. But calendar arithmetic requires the prime of seven. And if one is thinking of modular arithmetic in general, modular division will produce neat cyclic patterns only if one uses prime numbers. Since the prime number seven helps the twins to retrieve dates, and consequently the events of particular days in their lives, other primes, they may have found, produce similar patterns to those that are so important for their acts of recollection. (When they say about the matchsticks '111-37 three times', note they are taking the prime 37, and multiplying by three.) In fact, only the prime patterns could be 'visualised'. The different patterns produced by the different prime numbers (for example, multiplication tables) may be the pieces of visual information that they are communicating to each other when they repeat a given prime number. In short, modular arithmetic may help them to retrieve their past, and consequently the patterns created in using these calculations (which only occur with primes) may take on a particular significance for the twins.

  By the use of such a modular arithmetic, Ian Stewart points out, one may rapidly arrive at a unique solution in situations that defeat any 'ordinary' arithmetic-in particular homing in (by the so-called 'pigeon-hole principle') on extremely large and (by conventional methods) incomputable primes.

  If such methods, such visualisations, are regarded as algorithms, they are algorithms of a very peculiar sort-organised, not algebraically, but spatially, as trees, spirals, architectures, 'thought-scapes'-configurations in a formal yet quasi-sensory mental space.

  I have been excited by Israel Rosenfield's comments, and Ian Stewart's expositions of 'higher' (and especially modular) arithmetics, for these seem to promise, if not a 'solution', at least a powerful illumination of otherwise inexplicable powers, like those of the twins.

  Such higher or deeper arithmetics were conceived, in principle, by Gauss in his Disquisitiones Arithmeticae, in 1801, but they have only been turned to practical realities in recent years. One has to wonder whether there may not be a 'conventional' arithmetic (that is, an arithmetic of operations)-often irritating to teacher and student, 'unnatural', and hard to learn-and also a deep arithmetic of the kind described by Gauss, which may be truly innate to
the brain, as innate as Chomsky's 'deep' syntax and generative grammars. Such an arithmetic, in minds like the twins', could be dynamic and almost alive-globular clusters and nebulae of numbers whorling and evolving in an ever-expanding mental sky.

  As already mentioned, after publication of 'The Twins' I received a great deal of communication both personal and scientific. Some dealt with the specific themes of 'seeing' or apprehending numbers, some with the sense or significance which might attach to this phenomenon, some with the general character of autistic dispositions and sensibilities and how they might be fostered or inhibited, and some with the question of identical twins. Especially interesting were the letters from parents of such children, the rarest and most remarkable from parents who had themselves been forced into reflection and research and who had succeeded in combining the deepest feeling and involvement with a profound objectivity. In this category were the Parks, highly gifted parents of a highly gifted, but autistic, child (see C.C. Park, 1967, and D. Park, 1974, pp. 313-23). The Parks' child 'Ella' was a talented drawer and was also highly gifted with numbers, especially in her earlier years. She was fascinated by the 'order' of numbers, especially primes. This peculiar feel for primes is evidently not uncommon. C.C. Park wrote to me of another autistic child she knew, who covered sheets of paper with numbers written down 'compulsively'. 'All were primes,' she noted, and added: 'They are windows into an-

  other world.' Later she mentioned a recent experience with a young autistic man who was also fascinated by factors and primes, and how he instantly perceived these as 'special'. Indeed the word 'special' must be used to elicit a reaction:

  'Anything special, Joe, about that number (4875)?' Joe: 'It's just divisible by 13 and 25.' Of another (7241): 'It's divisible by 13 and 557.' And of 8741: 'It's a prime number.'

  Park comments: 'No one in his family reinforces his primes; they are a solitary pleasure.'

  It is not clear, in these cases, how the answers are arrived at almost instantaneously: whether they are 'worked out', 'known' (remembered), or-somehow-just 'seen'. What is clear is the peculiar sense of pleasure and significance attaching to primes. Some of this seems to go with a sense of formal beauty and symmetry, but some with a peculiar associational 'meaning' or 'potency'. This was often called 'magical' in Ella's case: numbers, especially primes, called up special thoughts, images, feelings, relationships-some almost too 'special' or 'magical' to be mentioned. This is well described in David Park's paper (op. cit).

  Kurt Godel, in a wholly general way, has discussed how numbers, especially primes, can serve as 'markers'-for ideas, people, places, whatever; and such a Godelian marking would pave the way for an 'arithmetisation' or 'numeralisation' of the world (see E. Nagel and J.R. Newman, 1958). If this does occur, it is possible that the twins, and others like them, do not merely live in a world of numbers, but in a world, in the world, as numbers, their number-meditation or play being a sort of existential meditation-and, if one can understand it, or find the key (as David Park sometimes does), a strange and precise communication too.

  24

  The Autist Artist

  'Draw this,' I said, and gave Jose my pocket watch.

  He was about 21, said to be hopelessly retarded, and had earlier had one of the violent seizures from which he suffers. He was thin, fragile-looking.

  His distraction, his restlessness, suddenly ceased. He took the watch carefully, as if it were a talisman or jewel, laid it before him, and stared at it in motionless concentration.

  'He's an idiot,' the attendant broke in. 'Don't even ask him. He

  don't know what it is-he can't tell time. He can't even talk. They says he's 'autistic', but he's just an idiot.' Jose turned pale, perhaps more at the attendant's tone than at his words-the attendant had said earlier that Jose didn't use words.

  'Go on,' I said. 'I know you can do it.'

  Jose drew with an absolute stillness, concentrating completely on the little clock before him, everything else shut out. Now, for the first time, he was bold, without hesitation, composed, not distracted. He drew swiftly but minutely, with a clear line, without erasures.

  I nearly always ask patients, if it is possible for them, to write and draw, partly as a rough-and-ready index of various competences, but also as an expression of 'character' or 'style'.

  Jose had drawn the watch with remarkable fidelity, putting in every feature (at least every essential feature-he did not put in 'Westclox, shock resistant, made in USA), not just 'the time' (though this was faithfully registered as 11:31), but every second as well, and the inset seconds dial, and, not least, the knurled winder and trapezoid clip of the watch, used to attach it to a chain. The clip was strikingly amplified, though everything else remained in due proportion. And the figures, now that I came to look at them, were of different sizes, different shapes, different styles-some thick, some thin; some aligned, some inset; some plain and some elaborated, even a bit 'gothic'. And the inset second hand, rather inconspicuous in the original, had been given a striking prominence, like the small inner dials of star clocks, or astrolabes.

  The general grasp of the thing, its 'feel', had been strikingly brought out-all the more strikingly if, as the attendant said, Jose had no idea of time. And otherwise there was an odd mixture of close, even obsessive, accuracy, with curious (and, I felt, droll) elaborations and variations.

  I was puzzled by this, haunted by it as I drove home. An 'idiot'? Autism? No. Something else was going on here.

  I was not called to see Jose again. The first call, on a Sunday evening, had been for an emergency. He had been having seizures the entire weekend, and I had prescribed changes in his anticonvulsants, over the phone, in the afternoon. Now that his seizures

  were 'controlled', further neurological advice was not requested. But I was still troubled by the problems presented by the clock, and felt an unresolved sense of mystery about it. I needed to see him again. So I arranged a further visit, and to see his entire chart-I had been given only a consultation slip, not very informative, when I saw him before.

  Jose came casually into the clinic-he had no idea (and perhaps did not care) why he'd been called-but his face lit up with a smile when he saw me. The dull, indifferent look, the mask I remembered, was lifted. There was a sudden, shy smile, like a glimpse through a door.

  'I have been thinking about you, Jose,' I said. He might not understand my words, but he understood my tone. 'I want to see more drawing'-and I gave him my pen.

  What should I ask him to draw this time? I had, as always, a copy of Arizona Highways with me, a richly illustrated magazine which I especially delight in, and which I carry around for neurological purposes, for testing my patients. The cover depicted an idyllic scene of people canoeing on a lake, against a backdrop of

  mountains and sunset. Jose started with the foreground, a mass of near-black silhouetted against the water, outlined this with extreme accuracy, and started to block it in. But this was clearly a job for a paintbrush, not a fine pen. 'Skip it,' I said, then pointing, 'Go on to the canoe.' Rapidly, unhesitatingly, Jose outlined the silhouetted figures and the canoe. He looked at them, then looked away, their forms fixed in his mind-then swiftly blocked them in with the side of the pen.

  Here again, and more impressively, because an entire scene was involved, I was amazed at the swiftness and the minute accuracy of reproduction, the more so since Jose had gazed at the canoe and then away, having taken it in. This argued strongly against any mere copying-the attendant had said earlier, 'He's just a Xerox'-and suggested that he had apprehended it as an image, exhibiting a striking power not just of copying but of perception. For the image had a dramatic quality not present in the original. The tiny figures, enlarged, were more intense, more alive, had a feeling of involvement and purpose not at all clear in the original. All the hallmarks of what Richard Wollheim calls 'icon-icity'-subjectivity, intentionality, dramatisation-were present. Thus, over and above the powers of m
ere facsimile, striking as

  these were, he seemed to have clear powers of imagination and creativity. It was not a canoe but his canoe that emerged in the drawing.

  I turned to another page in the magazine, to an article on trout fishing, with a pastel watercolour of a trout stream, a background of rocks and trees, and in the foreground a rainbow trout about to take a fly. 'Draw this,' I said, pointing to the fish. He gazed at it intently, seemed to smile to himself, and then turned away-and now, with obvious enjoyment, his smile growing broader and broader, he drew a fish of his own.

  I smiled myself, involuntarily, as he drew it, because now, feeling comfortable with me, he was letting himself go, and what was emerging, slyly, was not just a fish, but a fish with a 'character' of sorts.

  The original had lacked character, had looked lifeless, two-dimensional, even stuffed. Jose's fish, by contrast, tilted and poised, was richly three-dimensional, far more like a real fish than the original. It was not only verisimilitude and animation that had been added but something else, something richly expressive, though not wholly fishlike: a great, cavernous, whalelike mouth; a slightly crocodilian snout; an eye, one had to say, which was distinctly human, and with altogether a positively roguish look. It was a very

  funny fish-no wonder he had smiled-a sort of fish-person, a nursery character, like the frog-footman in Alice.

  Now I had something to go on. The picture of the clock had startled me, stimulated my interest, but did not, in itself, allow any thoughts or conclusions. The canoe had shown that Jose had an impressive visual memory, and more. The fish showed a lively and distinctive imagination, a sense of humour, and something akin to fairy-tale art. Certainly not great art, it was 'primitive', perhaps it was child-art; but, without doubt, it was art of a sort. And imagination, playfulness, art are precisely what one does not expect in idiots, or idiots savants, or in the autistic either. Such at least is the prevailing opinion.