The man who mistook his wife for a hat (29 page)

   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 al-

   gorithm, 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.

   As I observed them, quietly-this was easy to do, because I had an office on the ward where the twins were housed-I observed them in countless other sorts of number games or number communion, the nature of which I could not ascertain or even guess at.

   But it seems likely, or certain, that they are dealing with 'real' properties or qualities-for the arbitrary, such as random numbers, gives them no pleasure, or scarcely any, at all. It is clear that they must have 'sense' in their numbers-in the same way, perhaps, as a musician must have harmony. Indeed I find myself comparing them to musicians-or to Martin (Chapter Twenty-two), also retarded, who found in the serene and magnificent architectonics of Bach a sensible manifestation of the ultimate harmony and order of the world, wholly inaccessible to him conceptually because of his intellectual limitations.

   'Whoever is harmonically composed,' writes Sir Thomas Browne, 'delights in harmony . . . and a profound contemplation of the First Composer. There is something in it of Divinity more than

   the ear discovers; it is an Hieroglyphical and shadowed Lesson of the whole World … a sensible fit of that harmony which intellectually sounds in the ears of God . . . The soul … is harmon-ical, and hath its nearest sympathy unto Musick.'

   Richard Wollheim in
The Thread of Life
(1984) makes an absolute distinction between calculations and what he calls 'iconic' mental states, and he anticipates a possible objection to this distinction.

   Someone might dispute the fact that all calculations are non-iconic on the grounds that, when he calculates, sometimes, he does so by visualising the calculation on a page. But this is not a counter-example. For what is represented in such cases is not the calculation itself, but a representation of it; it is
numbers
that are calculated, but what is visualised are
numerals,
which represent numbers.

   Leibniz, on the other hand, makes a tantalising analogy between numbers and music: 'The pleasure we obtain from music comes from
counting,
but counting unconsciously. Music is nothing but unconscious arithmetic'

   What, so far as we can ascertain, is the situation with the twins, and perhaps others? Ernst Toch, the composer-his grandson Lawrence Weschler tells me-could readily hold in his mind after a single hearing a very long string of numbers; but he did this by 'converting' the string of numbers to a tune (a melody he himself shaped 'corresponding' to the numbers). Jedediah Buxton, one of the most ponderous but tenacious calculators of all time, and a man who had a veritable, even pathological, passion for calculation and counting (he would become, in his own words, 'drunk with reckoning'), would 'convert' music and drama to numbers. 'During the dance,' a contemporary account of him recorded in 1754, 'he fixed his attention upon the number of steps; he declared after a fine piece of musick, that the innumerable sounds produced by the music had perplexed him beyond measure, and he attended even to Mr Garrick only to count the words that he uttered, in which he said he perfectly succeeded.'

   Here is a pretty, if extreme, pair of examples-the musician

   who turns numbers into music, and the counter who turns music into numbers. One could scarcely have, one feels, more opposite sorts of mind, or, at least, more opposite modes of mind.*

   I believe the twins, who have an extraordinary 'feeling' for numbers, without being able to calculate at all, are allied not to Buxton but to Toch in this matter. Except-and this we ordinary people find so difficult to imagine-except that they do not 'convert' numbers into music, but actually feel them, in themselves, as 'forms', as 'tones', like the multitudinous forms that compose nature itself. They are not calculators, and their numeracy is 'iconic'. They summon up, they dwell among, strange scenes of numbers; they wander freely in great landscapes of numbers; they create, dra-maturgically, a whole world made of numbers. They have, I believe, a most singular imagination-and not the least of its singularities is that it can imagine only numbers. They do not seem to 'operate' with numbers, non-iconically, like a calculator; they 'see' them, directly, as a vast natural scene.

   And if one asks, are there analogies, at least, to such an 'icon-icity', one would find this, I think, in certain scientific minds. Dmitri Mendeleev, for example, carried around with him, written on cards, the numerical properties of elements, until they became utterly 'familiar' to him-so familiar that he no longer thought of them as aggregates of properties, but (so he tells us) 'as familiar faces'. He now saw the elements, iconically, physiognomically, as 'faces'-faces that related, like members of a family, and that made up,
in toto,
periodically arranged, the whole formal face of the universe. Such a scientific mind is essentially 'iconic', and 'sees' all nature as faces and scenes, perhaps as music as well. This 'vision', this inner vision, suffused with the phenomenal, none the less has an integral relation with the physical, and returning it, from the psychical to the physical, constitutes the secondary, or external, work of such science. (The philosopher seeks to hear within himself the echoes of the world symphony,' writes Nietzsche, 'and to re-project them in the form of concepts.') The twins, though

   *Something comparable to Buxton's mode, which perhaps appears the more 'unnatural' of the two, was shown by my patient Miriam H. in
Awakenings
when she had 'arithmomanic' attacks.

   morons, hear the world symphony, I conjecture, but hear it entirely in the form of numbers.

   The soul is 'harmonical' whatever one's IQ and for some, like physical scientists and mathematicians, the sense of harmony, perhaps, is chiefly intellectual. And yet I cannot think of anything intellectual that is not, in some way, also sensible-indeed the very word 'sense' always has this double connotation. Sensible, and in some sense 'personal' as well, for one cannot feel anything, find anything 'sensible', unless it is, in some way, related or re-latable to oneself. Thus the mighty architectonics of Bach provide, as they did for Martin A., 'an Hieroglyphical and shadowed Lesson of the whole World', but they are also, recognisably, uniquely, dearly, Bach; and this too was felt, poignantly, by Martin A., and related by him to the love he bore his father.

   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. One's soul is 'harmonical' whatever one's IQ, and 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 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. This is their way-as music is Martin's way-of apprehending the First Composer.

   But numbers are not just awesome for them, they are friends too-perhaps the only friends they have known in their isolated, autistic lives. This is a rather common sentiment among people who have a talent for numbers-and Steven Smith, while seeing 'method' as all-important, gives many delightful examples of it: George Parker Bidder, who wrote of his early number-childhood, 'I became perfectly familiar with numbers up to 100; they became as it were my friends, and I knew all their relations and acquaintances'; or the contemporary Shyam Marathe, from India-'When I say that numbers are my friends, I mean that I have some time in the past dealt with that particular number in a variety of ways, and on many occasions have found new and fascinating qualities hidden in it . . . So, if in a calculation I come across a known number, I immediately look to him as a friend.'

   Hermann von Helmholtz, speaking of musical perception, says that though compound tones
can
be analysed, and broken down into their components, they are normally heard as qualities, unique qualities of tone, indivisible wholes. He speaks here of a 'synthetic perception' which transcends analysis, and is the unanalysable essence of all musical sense. He compares such tones to faces, and speculates that we may recognise them in somewhat the same, personal way. In brief, he half suggests that musical tones, and certainly tunes,
are,
in fact, 'faces' for the ear, and are recognised, felt, immediately as 'persons' (or 'personeities'), a recognition involving warmth, emotion, personal relation.

   So it seems to be with those who love numbers. These too become recognisable as such-in a single, intuitive, personal 'I know you!'* The mathematician Wim Klein has put this well:

   *Particularly fascinating and fundamental problems are raised by the perception and recognition of faces-for there is much evidence that we recognise faces (at least

   (continued)

   'Numbers are friends for me, more or less. It doesn't mean the same for you, does it-3,844? For you it's just a three and an eight and a four and a four. But I say, "Hi! 62 squared."

   I believe the twins, seemingly so isolated, live in a world full of friends, that they have millions, billions, of numbers to which they say 'Hi!' and which, I am sure, say 'Hi!' back. But none of the numbers is arbitrary-like 62 squared-nor (and this is the mystery) is it arrived at by any of the usual methods, or any method so far as I can make out. The twins seem to employ a direct cognition-like angels. They see, directly, a universe and heaven of numbers. And this, however singular, however bizarre-but what right have we to call it 'pathological'?-provides a singular self-sufficiency and serenity to their lives, and one which it might be tragic to interfere with, or break.

   This serenity was, in fact, interrupted and broken up ten years later, when it was felt that the twins should be separated-'for their own good', to prevent their 'unhealthy communication together', and in order that they could 'come out and face the world … in an appropriate, socially acceptable way' (as the medical and sociological jargon had it). They were separated, then, in 1977, with results that might be considered as either gratifying or dire. Both have been moved now into 'halfway houses', and do menial jobs, for pocket money, under close supervision. They are able to take buses, if carefully directed and given a token, and to keep themselves moderately presentable and clean, though their moronic and psychotic character is still recognisable at a glance.

   This is the positive side-but there is a negative side too (not mentioned in their charts, because it was never recognised in the first place). Deprived of their numerical 'communion' with each other, and of time and opportunity for any 'contemplation' or 'communion' at all-they are always being hurried and jostled