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 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. (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.