by Paul Stepahin • August 25, 2015
In The Man Who Mistook His Wife for a Hat , Oliver Sacks writes about a pair of autistic savant twins. They had been diagnosed with a host of mental challenges- psychosis, autism, severe retardation. They could not understand arithmetic. And yet they were prodigious ‘calculators.’ For example, if given any calendar date in the span of 40,000 years they could instantly tell you what day of the week it landed on. It might be wondered how, without a serviceable notion of calculation, the twins were able to compute these weekdays so quickly and accurately. When asked how they managed this the twins would make the enigmatic claim that they could simply “see” the answer.
I find the idea of ‘seeing’ in this sense to be profoundly evocative. The weekday that corresponds to a random date is an obtainable fact, to be sure, but for most of us this fact resides at the far end of some systematic process of calculation. So what can it mean to “see” the answer? Well, one way to see the answer would be to simply get better at mental calculation through practice. As your arithmetic nimbleness improved the thicket of reckoning that obscures the answer would become more transparent. At some point, your arrival at the answer could become so rapid that you might have the feeling you’re seeing it instantly.
But this is not what is going on with the twins. Their seeing is not facilitated by a proficiency; they are incapable of even basic arithmetic. Instead they have some faculty of mind, some natural intuition, which is as effective as calculation but which feels to them as natural as looking. This is what I find most intriguing: they are mundane mystics, they may see farther than we can but they only report back that the Declaration of Independence was signed on a Thursday. Their powers could hardly be less interesting. What is interesting is that they are right, that whatever they’re seeing we can verify that they see it with perfect accuracy. It’s hard not to wonder what this seeing feels like. Sacks writes, “ One may observe, though this is not usually mentioned in the reports, that their eyes move and fix in a peculiar way as they (calculate)—as if they were unrolling, or scrutinizing, an inner landscape,”
He then relates the following story. “… (T)hey 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 savor 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.”
Sacks later determined that the numbers being exchanged between the twins were all primes. The prime numbers, recall, are those numbers that are only divisible by themselves and one, a definition that conceals all significance behind a bland technicality. All numbers are divisible by themselves and one. If we grant that inevitability the definition becomes much cleaner: The prime numbers are those numbers that are indivisible. We cannot know exactly how this indivisibility manifested itself to the minds of the twins but by some inscrutable consideration they were able to detect and find pleasure in it. Sacks later attempts to join in their game with a book of prime numbers, offering for their consideration an eight digit 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 delightful 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.”
The twins had eidetic memory. They could remember with perfect clarity every event of every day of their lives. They also had prolific visual number discrimination; a box of matchsticks is dropped, spilling its contents, and the twins instantly and accurately declare that there are 111 of them. So it could be that they are able to see 111 objects in their mind at once, to move and arrange those objects. The next thing they said was, “37 37 37.” Note that 111=37+37+37, as if the 111 objects in their heads spontaneously resolved into three groups of 37.
We can compose our own ways of arranging numbers of objects to test for primality. 3 is prime. Imagine that I arrange three dots into a column. Next, I try to arrange the dots evenly into two columns. It doesn’t work obviously, one column has two dots and the other has only one. With three columns I can distribute them evenly, one dot per column. But notice that I’ve gotten all the way from a vertical column of dots to a horizontal row of dots. The first shows that 3=1×3 and the second shows that 3=3×1. Altogether, this shows us that 3 is prime by the formal definition: it is only divisible by itself and one.
Let’s try the same thing with 4 dots. When we try to make two columns we find that we can successfully make a rectangle because 4=2×2. This rectangle tells us that 4 is not prime.
I call this the “spill test,” we systematically spill dots over into an increasing number of columns, as level in height as we can make them. Any divisible number will eventually make a rectangle whose dimensions are equal to its divisors. Only prime numbers will spill all the way from a vertical column to a horizontal row without making a single rectangle along the way.
Mathematics may be universal, but calculation is a base human activity. The notation we use to do calculations is arbitrary and our process in performing a calculation is incidental and mechanistic. It may not be clear at first whether primes are simply byproducts of our arbitrary notation or whether they belong in some more fundamental way to the bedrock of mathematics. It is the latter. Primes remain one of the deepest and most productively mysterious objects in number theory and mathematics generally. Their properties and beguiling irregularities have provided us with many practical applications as well, including many of our digital security protocols.
What the twins provide is a more romantic basis for perceiving the significance of the primes. Take away the capacity for arithmetic. To a mind, bent sufficiently from the mean, lost to calculation but eidetic when it comes to numeracy, the primes will show. And as every other sense is subject to its fascinations, like music to the ear or food to the tongue, the sense of numeracy will find in these strange objects its own kind of singular gratification.