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 clusters 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 another 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 associational ‘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 Godel, in a wholly general way, has discussed how numbers, especially primes, can serve as ‘markers’—for ideas, people, places, whatever; and such a Godelian 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 retarded, 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.