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!’[23] The mathematician Wim Klein has put this welclass="underline"
‘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 from one job to another—they seem to have lost their strange numerical power, and with this the chief joy and sense of their lives. But this is considered a small price to pay, no doubt, for their having become quasi-independent and ‘socially acceptable’.
One is reminded somewhat of the treatment meted out to Nadia—an autistic child with a phenomenal gift for drawing (see below, p. 219). Nadia too was subjected to a therapeutic regime ‘to find ways in which her potentialities in other directions could be maximised’. The net effect was that she started talking—and stopped drawing. Nigel Dennis comments: ‘We are left with a genius who has had her genius removed, leaving nothing behind but a general defectiveness. What are we supposed to think about such a curious cure?’
It should be added—this is a point dwelt on by F.W.H. Myers, whose consideration of number prodigies opens his chapter on ‘Genius’—that the faculty is ‘strange’, and may disappear spontaneously, though it is, as often, lifelong. In the case of the twins, of course, it was not just a ‘faculty’, but the personal and emotional centre of their lives. And now they are separated, now it is gone, there is no longer any sense or centre to their lives.[24]
When he was shown the manuscript of this paper, Israel Rosen-field pointed out that there are other arithmetics, higher and simpler than the ‘conventional’ arithmetic of operations, and wondered whether the twins’ singular powers (and limitations) might not reflect their use of such a ‘modular’ arithmetic. In a note to me, he has speculated that modular algorithms, of the sort described by Ian Stewart in Concepts of Modern Mathematics (1975) may explain the twins’ calendrical abilities:
Their ability to determine the days of the week within an eighty-thousand-year period suggests a rather simple algorithm. One divides the total number of days between ‘now’ and ‘then’ by seven. If there is no remainder, then that date falls on the same day as ‘now’; if the remainder is one, then that date is one day later; and so on. Notice that modular arithmetic is cyclic: it consists of repetitive patterns. Perhaps the twins were visualising these patterns, either in the form of easily constructed charts, or some kind of ‘landscape’ like the spiral of integers shown on page 30 of Stewart’s book.
This leaves unanswered why the twins communicate in primes. But calendar arithmetic requires the prime of seven. And if one is thinking of modular arithmetic in general, modular division will produce neat cyclic patterns only if one uses prime numbers. Since the prime number seven helps the twins to retrieve dates, and consequently the events of particular days in their lives, other primes, they may have found, produce similar patterns to those that are so important for their acts of recollection. (When they say about the matchsticks ‘111—37 three times’, note they are taking the prime 37, and multiplying by three.) In fact, only the prime patterns could be ‘visualised’. The different patterns produced by the different prime numbers (for example, multiplication tables) may be the pieces of visual information that they are communicating to each other when they repeat a given prime number. In short, modular arithmetic may help them to retrieve their past, and consequently the patterns created in using these calculations (which only occur with primes) may take on a particular significance for the twins.
By the use of such a modular arithmetic, Ian Stewart points out, one may rapidly arrive at a unique solution in situations that defeat any ‘ordinary’ arithmetic—in particular homing in (by the so-called ‘pigeon-hole principle’) on extremely large and (by conventional methods) incomputable primes.
If such methods, such visualisations, are regarded as algorithms, they are algorithms of a very peculiar sort—organised, not algebraically, but spatially, as trees, spirals, architectures, ‘thought-scapes’—configurations in a formal yet quasi-sensory mental space.
23
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 familiar faces) directly—and not by any process of piecemeal analysis or aggregation. This, as we have seen, is most dramatically shown in ‘prosopagnosia’, in which, as a consequence of a lesion in the right occipital cortex, patients become unable to recognise faces as such, and have to employ an elaborate, absurd, and indirect route, involving a bit-by-bit analysis of meaningless and separate features (Chapter One).
24
On the other hand, should this discussion be thought too singular or perverse, it is important to note that in the case of the twins studied by Luria, their separation was essential for their own development, ‘unlocked’ them from a meaningless and sterile babble and bind, and permitted them to develop as healthy and creative people.