Borges and Mathematics

8. The Pythagorean Twins

8 The Pythagorean Twins In May 2003 I had the opportunity to review Oliver Sacks’The Man Who Mistook His Wife for a Hat for the Argentine newspaper La Nación. Among this extraordinary collection of clinical tales, one of the most astonishing for any mathematician is “The Twins,” which reveals an unexpected source of “biological,” or more precisely, “neurophysiological” evidence for the formulation of a critical, stillunanswered question in the history of mathematics about prime numbers. Sacks relates that “The Twins . . . had been variously diagnosed as autistic, psychotic, or profoundly retarded” (195). In 1966, when Sacks began observing them, most of the reports concluded, as is often the case with “idiot savants,” that there was nothing special about them “except for their remarkable ‘documentary’memories of the tiniest visual details of their own existence, and their use of an unconscious, calendrical algorithm that enabled them to say at once on what day of the week a date far in the past or future would fall” (195). This ability, incidentally, earned them some television appearances .| 91 | | 92 | BORGES AND MATHEMATICS “The reality,” says Sacks, “is far stranger, far more complex . . . than any of these studies suggests” (196). Following is part of his description, written from a naturalist’s perspective: 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. . . . 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. Sacks again emphasizes the extent of the twins’ memory: [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’. . . . 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 PYTHAGOREAN TWINS | 93 | 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. . . ? (198-99) At this point Sacks describes his first contact with the twins’ “natural” powers. 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. . . .’ ‘And why did you murmur “37”, and repeat it three times?’I asked the twins. They said in unison, ‘37, 37, 37, 111’. . . . That they should see 111—‘111-ness’—in a flash was extraordinary , but perhaps no more extraordinary than Oakley’s ‘G sharp...