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 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 'documentary' 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 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 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 '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.