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.¹ They had even, I suspected, found their way into science fiction, a little “fictionalized,” but essentially as portrayed in the accounts that had been published.²

The twins, who were then twenty-six years old, had been in institutions since the age of seven, variously diagnosed as autistic, psychotic, or severely retarded. Most of the accounts concluded that, as idiots savants go, there was “nothing much to them”—except for their remarkable “documentary” 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 recently published, comprehensive, and imaginative book, The Great Mental Calculators.³ 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 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 Sixty 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, get to know the twins, and 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 Tweedledee and Tweedledum, 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 glasses 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 embarrassing, appearances on TV.

The “facts,” under these circumstances, 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 scrutinizing, an inner landscape, a mental calendar. They have the look of “seeing,” of intense visualization, 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 Friedrich 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 large 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’s novel Thorns 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 “documentary” kind, in which there is no personal reference, no personal relation, no living center 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”—“visualizing”—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 analogies 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 organization 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 the two-year-old Mozart 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 Mozart’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 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 I 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 of numbers, not as formal properties, 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 the young Mozart 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, either isolated or in relation. It was isolation, rather than a sense of relation, that was chiefly exhibited when they unfurled their implacable, haphazard “documentary.” But might not such prodigious powers of visualization—a power essentially concrete, and quite distinct from conceptualization—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, recognize, relate, and compare, in an entirely sensual and nonintellectual 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 glasses 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 passion 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 savor 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, mesmerized, 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,” savored, 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—brain-damaged, speech-damaged identical twins, and how they would play and prattle with each other, in a primitive, babble-like half-language of their own.⁴ 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 passion 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 toward 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 or 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, second, 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 surreptitious look in my book, added my own rather dishonest contribution, a ten-figure prime I found in my book.

There was again, and for still 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 assume 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 algorithm, 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 nonetheless. (But see the box on this page.)

Again I thought of Dase, whom I had read of years before, in F.W.H. Myers’s enchanting book, Human Personality.

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 Ass’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 (it had been purely a matter of luck that I had eavesdropped, and detected them in primes).

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.⁵ Indeed I find myself comparing them to musicians—or to a patient of mine, Joseph, also retarded, 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 God…. The soul…is harmonical, and hath its nearest sympathy unto Musick.”

Richard Wollheim in The Thread of Life makes an absolute distinction between calculations and what he calls “iconic” mental states, and he anticipates 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 visualizing 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 visualized are numerals, which represent numbers.

Leibniz, on the other hand, makes a tantalizing analogy 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, passion 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.⁶

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 multitudinous tones that compose paintings or music, the multitudinous 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, dramaturgically, 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, noniconically, like a calculator; they “see” them, directly, as a vast natural scene.

And if one asks, are there analogies, at least, to such an “iconicity,” 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, nonetheless has an integral relation with the physical, and returning it, from the psychical to the physical, constitutes 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 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 relatable to oneself. Thus the mighty architectonics of Bach provide, as they did for Joseph, “an Hieroglyphical and shadowed Lesson of the whole World,” but they are also, recognizably, uniquely, dearly, Bach; and this too was felt, poignantly, by Joseph, 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 organized, mysteriously, by the power of number. This is beautifully expressed in the prologue to Bertrand Russell’s Autobiography:

With equal passion 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 Joseph’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 sometime 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 analyzed, 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 unanalyzable essence of all musical sense. He compares such tones to faces, and speculates that we may recognize 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 recognized, 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 recognizable as such—in a single, intuitive, personal “I know you!”⁷ The mathematician Wim Klein has put this well: “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, I am sure, which say “Hi!” back. But none of the numbers is arbitrary—like 62 squared—or (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 recognizable at a glance.

This is the positive side—but there is a negative side too (not mentioned in their charts, because it was never recognized 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 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.⁸ Nadia too was subjected to a therapeutic regime “to find ways in which her potentialities in other directions could be maximized.” 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 center of their lives. And now they are separated, now it is gone, there is no longer any sense or center to their lives.⁹