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.
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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'.
