'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 'documentary'.
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 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 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 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 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 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 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 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 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 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.
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 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 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 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 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 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.*
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 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 analogies, 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, 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
*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 what
ever 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 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 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.'