The mystery of whether there is a natural resonance between music and our brains, as I mentioned in a post last week, brings up an even deeper question: whether mathematics itself is neurologically innate, giving the mind (or some minds) direct access to the structure of the universe. Thinking about that recently led me back to one of Oliver Sack’s most astonishing essays. It appeared in his collection The Man Who Mistook His Wife for a Hat, and is about two twins, idiot savants who appeared to have an almost supernatural ability to quickly tell if a number is prime.
Prime numbers are those that cannot be broken down into factors — smaller numbers that can be multiplied together to produce the larger one. They have been described as the atoms of the number system. 11 and 13 are obviously prime while 12 and 14 are not. But with larger numbers our brains are quickly flummoxed. Is 7244985277 prime? I just typed the digits by twitching my fingers along the top row of my keyboard. To test the number by hand I would have to start at the beginning of the number system and begin trying out the possible divisors.
There are shortcuts to avoid testing every single one. We know 2 can’t be a factor since 7244985277, like all primes, is odd. For the same reason we can rule out all even factors. And you only have to test factors up to the square root of a number. (The factors of 100 are 2 x 50, 4 x 25, 5 x 20, and 10 x 10. Testing beyond 10 would be redundant.)
There are ways to pare down the calculations even further. Numbers ending in 5 can’t be prime, and there are tricks for seeing if a number is divisible by 3, 7, 0r other small factors. Mathematicians have come up with other more sophisticated algorithms. But that still leaves long nights of mental drudgery. It took until the late 1800s for mathematicians to dig out a prime as large as 39 digits — and another half a century to get up to 44 digits.
Now I can check my number with the Primomatic (it can be broken into 2659 and 2724703). Testing by hand a number that long could take anywhere from hours to months of arithmetic.
In Sacks’s account, the twins — who were variously diagnosed as autistic, psychotic, or severely retarded — are said to have been able to perceive within minutes whether a 20-digit number, twice as long as the one I came up with, was prime.
It makes for a wonderful story with allusions to Borges and the great neuropsychologist Alexander Luria. Sacks tells how he met the twins in 1966 at a state mental hospital. With IQs of 60 they could barely do simple arithmetic, he reports, but they were already known as calendrical calculators. Given a date far in the future they could quickly tell you what day of the week it would fall on.
Their eyes move and fix in a peculiar way as they do this — as if they were unrolling, or scrutinizing, an inner landscape, a mental calendar.
Eerie as it seems, there are calculational tricks for doing that, though Sacks insists they were beyond the ability of the twins. But what he goes on to describe — and he was apparently the only one ever to witness this — is far more amazing, defying what is currently understood about the nature of computation and the brain.
One day he came upon the brothers sitting together in a corner “with a mysterious, secret smile on their faces.” One twin would say a long number and the other would nod and smile in appreciation. Then he would offer an equally long number of his own. “They looked, at first, like two connoisseurs wine-tasting, sharing rare tastes, rare appreciations.”
As this point they were trading six-digit numbers. Sacks took notes, and when he got home he looked the numbers up in a book of mathematical tables and found that they were primes. Though the twins had the ability to remember and repeat long streams of numbers, there was no reason to believe that they had somehow gained access to a table of primes. They appeared, Sacks suggests, to be somehow grokking the numbers from some Platonic realm where numerical truths reside.
Sacks brought the book with him the next day to the hospital, and when he found the brothers playing their game he sidled up and circumspectly offered his own contribution, an eight-digit number from the table of primes.
They both turned toward 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 or 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, second, 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.
They drew apart slightly, making room for me, a new number playmate, a third in their world. . . .
After several minutes of quiet one of the twins came up with a nine-digit number, and the other thought for a while and matched it with his own. Sacks looked in his book and offered a 10-digit prime.
There was again, and for still longer, a wondering, still silence; and then John, after a prodigious internal contemplation, brought out a twelve-figure number.
At this point Sacks could no longer check their work. His book, he writes, only went as high as 10-digits. But the twins kept on going and after an hour they were exchanging 20-digit numbers, also untestable.
And — this is the part of the story that drives me crazy — Sacks apparently didn’t write the numbers down.
A Dutch mathematician and skeptic, Pepijn van Erp, has suggested how, with a bit of luck, the twins might have chanced upon the six-digit primes that first drew Sacks’s attention. But why, he wonders, didn’t Sacks think to test the twins’ powers by slipping in “fake primes” — numbers like the one I clacked out above that look like they might be prime but are not? Would they have smiled in wonder at those too? And if only Sacks had called on some mathematicians and psychologists to help him discreetly perform more tests. Instead he seems to have put the matter aside until he wrote about it almost 20 years later.
By that time the twins had long been separated and moved to halfway houses. They lost their numerical powers, and there was no way to check the story. That left Sacks free to cling to his romantic interpretation: “The twins seem to employ a direct cognition—like angels,” he wrote. “They see, directly, a universe and heaven of numbers.”
Part 2: Idiot Savants and Prime Numbers