23.
The Twins When I first met the twins, John and Michael, in 1966 in a state hospital, they were already well known. They had been on radio and television, and made the subject of detailed scientific and popular reports. * They had even, I suspected, found their way into science fiction, a little "fictionalised", but essentially as portrayed in the accounts that had been published. + The twins, who were then twenty-six years old, had been in inst.i.tutions since the age of seven, variously diagnosed as autistic, psychotic or severely r.e.t.a.r.ded. Most of the accounts concluded that, as idiots savants go, there was "nothing much to them"- except for their remarkable "doc.u.mentary" memories of the tiniest visual details of their own experience, and their use of an unconscious, calendrical algorithm that enabled them to say at once on what day of the week a date far in the past or future would fall. This is the view taken by Steven Smith, in his comprehensive and imaginative book, The Great Mental Calculators (1983). There have been, to my knowledge, no further studies of the twins since the mid-Sixties, the brief interest they aroused being quenched by the apparent "solution" of the problems they presented.
But this, I believe, is a misapprehension, perhaps a natural enough one in view of the stereotyped approach, the fixed format of questions, the concentration on one "task" or another, with which the original investigators approached the twins, and by which they *W.A. Horwitz, etal. (1965), Hamblin (1966).
+See Robert Silverberg"s novel Thorns (1967), notably pp. 11-17.
reduced them-their psychology, their methods, their lives-almost to nothing.
The reality is far stranger, far more complex, far less explicable, than any of these studies suggest, but it is not even to be glimpsed by aggressive formal "testing", or the usual 60 Minutes-like interviewing of the twins.
Not that any of these studies, or TV performances, is "wrong". They are quite reasonable, often informative, as far as they go, but they confine themselves to the obvious and testable "surface," and do not go to the depths-do not even hint, or perhaps guess, that there are depths below.
One indeed gets no hint of any depths unless one ceases to test the twins, to regard them as "subjects". One must lay aside the urge to limit and test, and get to know the twins-observe them, openly, quietly, without presuppositions, but with a full and sympathetic phenomenological openness, as they live and think and interact quietly, pursuing their own lives, spontaneously, in their singular way. Then one finds there is something exceedingly mysterious at work, powers and depths of a perhaps fundamental sort, which I have not been able to "solve" in the eighteen years that I have known them.
They are, indeed, unprepossessing at first encounter-a sort of grotesque Tweedledum and Tweedledee, indistinguishable, mirror images, identical in face, in body movements, in personality, in mind, identical too in their stigmata of brain and tissue damage. They are undersized, with disturbing disproportions in head and hands, high-arched palates, high-arched feet, monotonous squeaky voices, a variety of peculiar tics and mannerisms, and a very high, degenerative myopia, requiring gla.s.ses so thick that their eyes seem distorted, giving them the appearance of absurd little professors, peering and pointing, with a misplaced, obsessed, and absurd concentration. And this impression is fortified as soon as one quizzes them-or allows them, as they are apt to do, like pantomime puppets, to start spontaneously on one of their "routines".
This is the picture that has been presented in published articles, and on stage-they tend to be "featured" in the annual show in the hospital I work in-and in their not infrequent, and rather embarra.s.sing, appearances on TV.
The "facts", under these circ.u.mstances, are established to monotony. The twins say, "Give us a date-any time in the last or next forty thousand years." You give them a date, and, almost instantly, they tell you what day of the week it would be. "Another date!" they cry, and the performance is repeated. They will also tell you the date of Easter during the same period of 80,000 years. One may observe, though this is not usually mentioned in the reports, that their eyes move and fix in a peculiar way as they do this-as if they were unrolling, or scrutinising, an inner landscape, a mental calendar. They have the look of "seeing", of intense visualisation, although it has been concluded that what is involved is pure calculation.
Their memory for digits is remarkable-and possibly unlimited. They will repeat a number of three digits, of thirty digits, of three hundred digits, with equal ease. This too has been attributed to a "method".
But when one comes to test their ability to calculate-the typical forte of arithmetical prodigies and "mental calculators"-they do astonishingly badly, as badly as their IQs of sixty might lead one to think. They cannot do simple addition or subtraction with any accuracy, and cannot even comprehend what multiplication or division means. What is this: "calculators" who cannot calculate, and lack even the most rudimentary powers of arithmetic?
And yet they are called "calendar calculators"-and it has been inferred and accepted, on next to no grounds, that what is involved is not memory at all, but the use of an unconscious algorithm for calendar calculations. When one recollects how even Carl Fried-rich Gauss, at once one of the greatest of mathematicians, and of calculators too, had the utmost difficulty in working out an algorithm for the date of Easter, it is scarcely credible that these twins, incapable of even the simplest arithmetical methods, could have inferred, worked out, and be using such an algorithm. A great many calculators, it is true, do have a larger repertoire of methods and algorithms they have worked out for themselves, and perhaps this predisposed W.A. Horwitz et al. to conclude this was true of the twins too. Steven Smith, taking these early studies at face value, comments: Something mysterious, though commonplace, is operating here- the mysterious human ability to form unconscious algorithms on the basis of examples.
If this were the beginning and end of it, they might indeed be seen as commonplace, and not mysterious at all-for the computing of algorithms, which can be done well by machine, is essentially mechanical, and comes into the spheres of "problems", but not "mysteries".
And yet, even in some of their performances, their "tricks", there is a quality that takes one aback. They can tell one the weather, and the events, of any day in their lives-any day from about their fourth year on. Their way of talking-well conveyed by Robert Silverberg in his portrayal of the character Melangio-is at once childlike, detailed, without emotion. Give them a date, and their eyes roll for a moment, and then fixate, and in a flat, monotonous voice they tell you of the weather, the bare political events they would have heard of, and the events of their own lives-this last often including the painful or poignant anguish of childhood, the contempt, the jeers, the mortifications they endured, but all delivered in an even and unvarying tone, without the least hint of any personal inflection or emotion. Here, clearly, one is dealing with memories that seem of a "doc.u.mentary" kind, in which there is no personal reference, no personal relation, no living centre whatever.
It might be said that personal involvement, emotion, has been edited out of these memories, in the sort of defensive way one may observe in obsessive or schizoid types (and the twins must certainly be considered obsessive and schizoid). But it could be said, equally, and indeed more plausibly, that memories of this kind never had any personal character, for this indeed is a cardinal characteristic of eidetic memory such as this.
But what needs to be stressed-and this is insufficiently remarked on by their studiers, though perfectly obvious to a naive listener prepared to be amazed-is the magnitude of the twins"
memory, its apparently limitless (if childish and commonplace) extent, and with this the way in which memories are retrieved. And if you ask them how they can hold so much in their minds- a three-hundred-figure digit, or the trillion events of four decades-they say, very simply, "We see it." And "seeing"-"visualising"-of extraordinary intensity, limitless range, and perfect fidelity, seems to be the key to this. It seems a native physiological capacity of their minds, in a way which has some a.n.a.logies to that by which A.R. Luria"s famous patient, described in The Mind of a Mnemonist, "saw", though perhaps the twins lack the rich synesthesia and conscious organisation of the Mnemonist"s memories. But there is no doubt, in my mind at least, that there is available to the twins a prodigious panorama, a sort of landscape or physiognomy, of all they have ever heard, or seen, or thought, or done, and that in the blink of an eye, externally obvious as a brief rolling and fixation of the eyes, they are able (with the "mind"s eye") to retrieve and "see" nearly anything that lies in this vast landscape.
Such powers of memory are most uncommon, but they are hardly unique. We know little or nothing about why the twins or anyone else have them. Is there then anything in the twins that is of deeper interest, as I have been hinting? I believe there is.
It is recorded of Sir Herbert Oakley, the nineteenth-century Edinburgh professor of music, that once, taken to a farm, he heard a pig squeak and instantly cried "G sharp!" Someone ran to the piano, and G sharp it was. My own first sight of the "natural" powers, and "natural" mode, of the twins came in a similar, spontaneous, and (I could not help feeling) rather comic, manner.
A box of matches on their table fell, and discharged its contents on the floor: "111," they both cried simultaneously; and then, in a murmur, John said "37". Michael repeated this, John said it a third time and stopped. I counted the matches-it took me some time-and there were 111.
"How could you count the matches so quickly?" I asked. "We didn"t count," they said. "We saw the 111."
Similar tales are told of Zacharias Dase, the number prodigy, who would instantly call out "183" or "79" if a pile of peas was poured out, and indicate as best he could-he was also a dullard- that he did not count the peas, but just "saw" their number, as a whole, in a flash.
"And why did you murmur "37," and repeat it three times?" I asked the twins. They said in unison, "37, 37, 37, 111."
And this, if possible, I found even more puzzling. That they should see 111-"111-ness"-in a flash was extraordinary, but perhaps no more extraordinary than Oakley"s "G sharp"-a sort of "absolute pitch", so to speak, for numbers. But they had then gone on to "factor" the number 111-without having any method, without even "knowing" (in the ordinary way) what factors meant. Had I not already observed that they were incapable of the simplest calculations, and didn"t "understand" (or seem to understand) what multiplication or division was? Yet now, spontaneously, they had divided a compound number into three equal parts.
"How did you work that out?" I said, rather hotly. They indicated, as best they could, in poor, insufficient terms-but perhaps there are no words to correspond to such things-that they did not "work it out", but just "saw" it, in a flash. John made a gesture with two outstretched fingers and his thumb, which seemed to suggest that they had spontaneously trisected the number, or that it "came apart" of its own accord, into these three equal parts, by a sort of spontaneous, numerical "fission". They seemed surprised at my surprise-as if/ were somehow blind; and John"s gesture conveyed an extraordinary sense of immediate, felt reality. Is it possible, I said to myself, that they can somehow "see" the properties, not in a conceptual, abstract way, but as qualities, felt, sensuous, in some immediate, concrete way? And not simply isolated qualities-like "111-ness"-but qualities of relationship? Perhaps in somewhat the same way as Sir Herbert Oakley might have said "a third," or "a fifth".
I had already come to feel, through their "seeing" events and dates, that they could hold in their minds, did hold, an immense mnemonic tapestry, a vast (or possibly infinite) landscape in which everything could be seen, cither isolated or in relation. It was isolation, rather than a sense of relation, that was chiefly exhibited when they unfurled their implacable, haphazard "doc.u.mentary".
But might not such prodigious powers of visualisation-powers essentially concrete, and quite distinct from conceptualisation- might not such powers give them the potential of seeing relations, formal relations, relations of form, arbitrary or significant? If they could see "111-ness" at a glance (if they could see an entire "constellation" of numbers), might they not also "see", at a glance-see, recognise, relate and compare, in an entirely sensual and non-intellectual way-enormously complex formations and constellations of numbers? A ridiculous, even disabling power. I thought of Borges"s "Funes": We, at one glance, can perceive three gla.s.ses on a table; Funes, all the leaves and tendrils and fruit that make up a grape vine ... A circle drawn on a blackboard, a right angle, a lozenge- all these are forms we can fully and intuitively grasp; Ireneo could do the same with the stormy mane of a pony, with a herd of cattle on a hill ... I don"t know how many stars he could see in the sky.
Could the twins, who seemed to have a peculiar pa.s.sion and grasp of numbers-could these twins, who had seen "111-ness" at a glance, perhaps see in their minds a numerical "vine", with all the number-leaves, number-tendrils, number-fruit, that made it up? A strange, perhaps absurd, almost impossible thought-but what they had already shown me was so strange as to be almost beyond comprehension. And it was, for all I knew, the merest hint of what they might do.
I thought about the matter, but it hardly bore thinking about. And then I forgot it. Forgot it until a second, spontaneous scene, a magical scene, which I blundered into, completely by chance.
This second time they were seated in a corner together, with a mysterious, secret smile on their faces, a smile I had never seen before, enjoying the strange pleasure and peace they now seemed to have. I crept up quietly, so as not to disturb them. They seemed to be locked in a singular, purely numerical, converse. John would say a number-a six-figure number. Michael would catch the number, nod, smile and seem to savour it. Then he, in turn, would say another six-figure number, and now it was John who received, and appreciated it richly. They looked, at first, like two connoisseurs wine-tasting, sharing rare tastes, rare appreciations. I sat still, unseen by them, mesmerised, bewildered.
What were they doing? What on earth was going on? I could make nothing of it. It was perhaps a sort of game, but it had a gravity and an intensity, a sort of serene and meditative and almost holy intensity, which I had never seen in any ordinary game before, and which I certainly had never seen before in the usually agitated and distracted twins. I contented myself with noting down the numbers they uttered-the numbers that manifestly gave them such delight, and which they "contemplated", savoured, shared, in communion.
Had the numbers any meaning, I wondered on the way home, had they any "real" or universal sense, or (if any at all) a merely whimsical or private sense, like the secret and silly "languages" brothers and sisters sometimes work out for themselves? And, as I drove home, I thought of Luria"s twins-Liosha and Yura-braindamaged, speech-damaged identical twins, and how they would play and prattle with each other, in a primitive, babble-like language of their own (Luria and Yudovich, 1959). John and Michael were not even using words or half-words-simply throwing numbers at each other. Were these "Borgesian" or "Funesian" numbers, mere numeric vines, or pony manes, or constellations, private number-forms-a sort of number argot-known to the twins alone?
As soon as I got home I pulled out tables of powers, factors, logarithms and primes-mementos and relics of an odd, isolated period in my own childhood, when I too was something of a number brooder, a number "see-er", and had a peculiar pa.s.sion for numbers. I already had a hunch-and now I confirmed it. All the numbers, the six-figure numbers, which the twins had exchanged were primes-i.e., numbers that could be evenly divided by no other whole number than itself or one. Had they somehow seen or possessed such a book as mine-or were they, in some unimaginable way, themselves "seeing" primes, in somewhat the same way as they had "seen" 111-ness, or triple 37-ness? Certainly they could not be calculating them-they could calculate nothing.
I returned to the ward the next day, carrying the precious book of primes with me. I again found them closeted in their numerical communion, but this time, without saying anything, I quietly joined them. They were taken aback at first, but when I made no interruption, they resumed their "game" of six-figure primes. After a few minutes I decided to join in, and ventured a number, an eight-figure prime. They both turned towards me, then suddenly became still, with a look of intense concentration and perhaps wonder on their faces. There was a long pause-the longest I had ever known them to make, it must have lasted a half-minute or more-and then suddenly, simultaneously, they both broke into smiles.
They had, after some unimaginable internal process of testing, suddenly seen my own eight-digit number as a prime-and this was manifestly a great joy, a double joy, to them; first because I had introduced a delightful new plaything, a prime of an order they had never previously encountered; and, secondly, because it was evident that I had seen what they were doing, that I liked it, that I admired it, and that I could join in myself.
They drew apart slightly, making room for me, a new number playmate, a third in their world. Then John, who always took the lead, thought for a very long time-it must have been at least five minutes, though I dared not move, and scarcely breathed-and brought out a nine-figure number; and after a similar time his twin, Michael, responded with a similar one. And then I, in my turn, after a surrept.i.tious look in my book, added my own rather dishonest contribution, a ten-figure prime I found in my book.
There was again, and for even longer, a wondering, still silence; and then John, after a prodigious internal contemplation, brought out a twelve-figure number. I had no way of checking this, and could not respond, because my own book-which, as far as I knew, was unique of its kind-did not go beyond ten-figure primes. But Michael was up to it, though it took him five minutes-and an hour later the twins were swapping twenty-figure primes, at least I a.s.sume this was so, for I had no way of checking it. Nor was there any easy way, in 1966, unless one had the use of a sophisticated computer. And even then, it would have been difficult, for whether one uses Eratosthenes" sieve, or any other al- gorithm, there is no simple method of calculating primes. There is no simple method, for primes of this order-and yet the twins were doing it. (But see the Postscript.) Again I thought of Dase, whom I had read of years before, in F.W.H. Myers"s enchanting book Human Personality (1903).
We know that Dase (perhaps the most successful of such prodigies) was singularly devoid of mathematical grasp . . . Yet he in twelve years made tables of factors and prime numbers for the seventh and nearly the whole of the eighth million-a task which few men could have accomplished, without mechanical aid, in an ordinary lifetime.
He may thus be ranked, Myers concludes, as the only man who has ever done valuable service to Mathematics without being able to cross the a.s.s"s Bridge.
What is not made clear, by Myers, and perhaps was not clear, is whether Dase had any method for the tables he made up, or whether, as hinted in his simple "number-seeing" experiments, he somehow "saw" these great primes, as apparently the twins did.
As I observed them, quietly-this was easy to do, because I had an office on the ward where the twins were housed-I observed them in countless other sorts of number games or number communion, the nature of which I could not ascertain or even guess at.
But it seems likely, or certain, that they are dealing with "real" properties or qualities-for the arbitrary, such as random numbers, gives them no pleasure, or scarcely any, at all. It is clear that they must have "sense" in their numbers-in the same way, perhaps, as a musician must have harmony. Indeed I find myself comparing them to musicians-or to Martin (Chapter Twenty-two), also r.e.t.a.r.ded, who found in the serene and magnificent architectonics of Bach a sensible manifestation of the ultimate harmony and order of the world, wholly inaccessible to him conceptually because of his intellectual limitations.
"Whoever is harmonically composed," writes Sir Thomas Browne, "delights in harmony . . . and a profound contemplation of the First Composer. There is something in it of Divinity more than the ear discovers; it is an Hieroglyphical and shadowed Lesson of the whole World ... a sensible fit of that harmony which intellectually sounds in the ears of G.o.d . . . The soul ... is harmon-ical, and hath its nearest sympathy unto Musick."
Richard Wollheim in The Thread of Life (1984) makes an absolute distinction between calculations and what he calls "iconic" mental states, and he antic.i.p.ates a possible objection to this distinction.
Someone might dispute the fact that all calculations are non-iconic on the grounds that, when he calculates, sometimes, he does so by visualising the calculation on a page. But this is not a counter-example. For what is represented in such cases is not the calculation itself, but a representation of it; it is numbers that are calculated, but what is visualised are numerals, which represent numbers.
Leibniz, on the other hand, makes a tantalising a.n.a.logy between numbers and music: "The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic"
What, so far as we can ascertain, is the situation with the twins, and perhaps others? Ernst Toch, the composer-his grandson Lawrence Weschler tells me-could readily hold in his mind after a single hearing a very long string of numbers; but he did this by "converting" the string of numbers to a tune (a melody he himself shaped "corresponding" to the numbers). Jedediah Buxton, one of the most ponderous but tenacious calculators of all time, and a man who had a veritable, even pathological, pa.s.sion for calculation and counting (he would become, in his own words, "drunk with reckoning"), would "convert" music and drama to numbers. "During the dance," a contemporary account of him recorded in 1754, "he fixed his attention upon the number of steps; he declared after a fine piece of musick, that the innumerable sounds produced by the music had perplexed him beyond measure, and he attended even to Mr Garrick only to count the words that he uttered, in which he said he perfectly succeeded."
Here is a pretty, if extreme, pair of examples-the musician who turns numbers into music, and the counter who turns music into numbers. One could scarcely have, one feels, more opposite sorts of mind, or, at least, more opposite modes of mind.*
I believe the twins, who have an extraordinary "feeling" for numbers, without being able to calculate at all, are allied not to Buxton but to Toch in this matter. Except-and this we ordinary people find so difficult to imagine-except that they do not "convert" numbers into music, but actually feel them, in themselves, as "forms", as "tones", like the mult.i.tudinous forms that compose nature itself. They are not calculators, and their numeracy is "iconic". They summon up, they dwell among, strange scenes of numbers; they wander freely in great landscapes of numbers; they create, dra-maturgically, a whole world made of numbers. They have, I believe, a most singular imagination-and not the least of its singularities is that it can imagine only numbers. They do not seem to "operate" with numbers, non-iconically, like a calculator; they "see" them, directly, as a vast natural scene.
And if one asks, are there a.n.a.logies, at least, to such an "icon-icity", one would find this, I think, in certain scientific minds. Dmitri Mendeleev, for example, carried around with him, written on cards, the numerical properties of elements, until they became utterly "familiar" to him-so familiar that he no longer thought of them as aggregates of properties, but (so he tells us) "as familiar faces". He now saw the elements, iconically, physiognomically, as "faces"-faces that related, like members of a family, and that made up, in toto, periodically arranged, the whole formal face of the universe. Such a scientific mind is essentially "iconic", and "sees" all nature as faces and scenes, perhaps as music as well. This "vision", this inner vision, suffused with the phenomenal, none the less has an integral relation with the physical, and returning it, from the psychical to the physical, const.i.tutes the secondary, or external, work of such science. (The philosopher seeks to hear within himself the echoes of the world symphony," writes Nietzsche, "and to re-project them in the form of concepts.") The twins, though *Something comparable to Buxton"s mode, which perhaps appears the more "unnatural" of the two, was shown by my patient Miriam H. in Awakenings when she had "arithmomanic" attacks.
morons, hear the world symphony, I conjecture, but hear it entirely in the form of numbers.
The soul is "harmonical" whatever one"s IQ and for some, like physical scientists and mathematicians, the sense of harmony, perhaps, is chiefly intellectual. And yet I cannot think of anything intellectual that is not, in some way, also sensible-indeed the very word "sense" always has this double connotation. Sensible, and in some sense "personal" as well, for one cannot feel anything, find anything "sensible", unless it is, in some way, related or re-latable to oneself. Thus the mighty architectonics of Bach provide, as they did for Martin A., "an Hieroglyphical and shadowed Lesson of the whole World", but they are also, recognisably, uniquely, dearly, Bach; and this too was felt, poignantly, by Martin A., and related by him to the love he bore his father.
The twins, I believe, have not just a strange "faculty"-but a sensibility, a harmonic sensibility, perhaps allied to that of music. One might speak of it, very naturally, as a "Pythagorean" sensibility-and what is odd is not its existence, but that it is apparently so rare. One"s soul is "harmonical" whatever one"s IQ, and perhaps the need to find or feel some ultimate harmony or order is a universal of the mind, whatever its powers, and whatever form it takes. Mathematics has always been called the "queen of sciences", and mathematicians have always felt number as the great mystery, and the world as organised, mysteriously, by the power of number. This is beautifully expressed in the prologue to Bertrand Russell"s Autobiography: With equal pa.s.sion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway above the flux.
It is strange to compare these moron twins to an intellect, a spirit, like that of Bertrand Russell. And yet it is not, I think, so far-fetched. The twins live exclusively in a thought-world of numbers. They have no interest in the stars shining, or the hearts of men. And yet numbers for them, I believe, are not "just" numbers, but significances, signifiers whose "significand" is the world.
They do not approach numbers lightly, as most calculators do. They are not interested in, have no capacity for, cannot comprehend, calculations. They are, rather, serene contemplators of number-and approach numbers with a sense of reverence and awe. Numbers for them are holy, fraught with significance. This is their way-as music is Martin"s way-of apprehending the First Composer.
But numbers are not just awesome for them, they are friends too-perhaps the only friends they have known in their isolated, autistic lives. This is a rather common sentiment among people who have a talent for numbers-and Steven Smith, while seeing "method" as all-important, gives many delightful examples of it: George Parker Bidder, who wrote of his early number-childhood, "I became perfectly familiar with numbers up to 100; they became as it were my friends, and I knew all their relations and acquaintances"; or the contemporary Shyam Marathe, from India-"When I say that numbers are my friends, I mean that I have some time in the past dealt with that particular number in a variety of ways, and on many occasions have found new and fascinating qualities hidden in it . . . So, if in a calculation I come across a known number, I immediately look to him as a friend."
Hermann von Helmholtz, speaking of musical perception, says that though compound tones can be a.n.a.lysed, 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 a.n.a.lysis, and is the una.n.a.lysable 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 a.n.a.lysis 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 a.n.a.lysis 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 ma.n.u.script 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 repet.i.tive 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 s.p.a.ce.
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 cl.u.s.ters 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 a.s.sociational "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 G.o.del, in a wholly general way, has discussed how numbers, especially primes, can serve as "markers"-for ideas, people, places, whatever; and such a G.o.delian 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 r.e.t.a.r.ded, 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 ill.u.s.trated 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 ma.s.s 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 mere 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.