excerpts from Chapter 23 of Oliver Sacks' The Man Who Mistook His Wife For a Hat

[Thanks to Andreas Knauf for bringing this to my attention. Note that this is an actual psychiatric case study, not a work of fiction.]

"It is hard for me to hear this story without feeling awe and astonishment at the workings of the brain. But I wonder: Do my nonmathematical friends have the same response? Do they have any inkling how bizarre, how prodigious and even otherworldly was the singular talent th twins so naturally enjoyed? Are they aware that mathematicians have been struggling for centuries to come up with away to do what John and Michael did spontaneously: to generated and recognize prime numbers? Or can most people do little more than shrug and perhaps secretly imagine that a real mathematician would find what the twins did no more taxing or worthy of attention than performing long division in one's head?"

E. Bombieri from "Prime Territory: Exploring the Infinite Landscape at the Base of the Number System" (The Sciences, Sept/Oct 1992)

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],[2]). They had even, I suspected, found their way into science fiction, a little 'fictionalised', but essentially as portrayed in the accounts that had been published [3].

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 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 comprehensive and imaginative book, The Great Mental Calculators (1983). 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.

* * *
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 no '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, 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 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.

The 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 - brain-damaged, speech-damaged identical twins, and how they would play and prattle with each other, in a primitive, babble-like language of their own [4]. John and Michael were not even using words or half-words - simply throwing numbers at each other.

* * *
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 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.

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 in twelve years mad 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.
* * *
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.

* * *
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. 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 the 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.

* * *

(postscript)

* * *
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 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 [5] and [6]). The Park's 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 [6].

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 [7]). 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.


References

[1] D.J. Hamblin, "They are 'idiots savants' - wizards of the calendar", Life 60 (18 March 1966) 106-108.

[2] W.A. Horwitz, "Identical twin 'idiots savants' - calendar calculators", American Journal of Psychiatry 121 (1965) 1075-1079.

[3] R. Silverberg, Thorns, notably pp.11-17.

[4] A.R. Luria and F.I. Yudowich, Speech and the Development of Mental Processes in the Child (English translation, 1959).

[5] C.C. Park, The Siege: the First Eight Years of an Autistic Child, 1967.

[6] D. Park and P. Youderian, "Light and number: ordering principles in the world of an autistic child", Journal of Autism and Childhood Schizophrenia 4 (4) (1974) 313-323.

[7] E. Nagel and J.R. Newmann, Godel's Proof, 1958.


Also, see Japanese psychologist M. Yamaguchi's note in the Journal of Autism and Developmental Disorders regarding the authenticity of Sacks' claims.


An explanation for the twins' abilities regarding large prime numbers has been proposed by Finnish physicist Matti Pitkanen as part of his TGD-inpsired theory of consciousness. See Chapter 2 of his online book.


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