Oliver Sacks and the Amazing Twins

By George Johnson | February 25, 2013 9:54 pm

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

CATEGORIZED UNDER: Mind & Brain, select, top-posts
  • eyeofnewt

    I have long been fascinated by accounts of the twins and their uncanny capacity to “experience” primality at what seems like a perceptual, rather than conceptual, level. I remember an early account of the savant twins in which 1 twin held out his arms at a 60 degree angle, trisecting the space in front of him, in an effort to explain how he “counted” a set of spilled matchsticks by partitioning them into 3 sets. I suspect that the underlying mechanism may have something in common with synesthesia. Allan Snyder has suggested that the capacity for equipartitioning in time and space may be the substrate for savant skills
    http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1689812/pdf/10212449.pdf, and has come up with some ingenious experiments using transcranial magnetic stimulation, where disinhibiting left temporal brain function seems to facilitate some forms of mathematical insight
    http://www.sciencedirect.com/science/article/pii/S0304394012003618. Alex Bäcker has hypothesized a possible underlying neurobiological mechanism for equipartitioning

  • Buddy199

    This is very similar to terminal lucidity, a phenomenon where terminally patients whose mental facilities have been severely compromised by stroke, Alzheimer’s or coma suddenly recover normal mental function for a brief period before expiring. See “The Irreducible Mind”, by Kelly and Kelly.

    • stolzy

      Actually, I find it to be quite dissimilar to that.

  • stolzy

    I’ve always found Sachs to be a thoughtful, respectful, and simpatico thinker and writer. Well grounded in science, but open and sober about it’s natural boundaries and responsibly curious as to the terrain beyond it’s reach. It’s not insignificant that in Sach’s account the twins ‘make space’ for him and invite him into their circle after verifying his initial 8-digit offering. This human approach of Sachs’ is exactly what is needed and also what is missing in investigations into consciousness phenomena, which by-and-large is automatically accorded only a neurologically measurable reality. Thank goodness for people like Sachs! He clearly points to what he perceives as the self-imposed limitations of the usual stance towards neurologic investigations in his linked piece.

    Yes, it would have been interesting to hear of further, deeper scrutiny, along the lines that George suggests. I, for example, would be quite interested to know to what extent the fact that they were twins played into the phenomena in question. But who is to say that these were in line with Sachs’ motivations at the moment, or appropriate to the feel of the situation? Who is to say that the rapport which Sachs established is irrelevant and that he could be profitably replaced with any old double-blind mentality clinicians or researchers?

    I find George’s characterizations such as ‘romantic’ and ‘cling’ when describing Sachs’ observations to be dismissive and actually saying something more about the way he thinks, namely materialistically. Which is not to say that I quite enjoy reading and listening to his stuff.

  • http://www.facebook.com/profile.php?id=100003752878746 Mark Smith


    “We know 2 can’t be a factor since 7244985277, like all primes, is odd”

    should be

    “We know 2 can’t be a factor since 7244985277, like all primes other than 2, is odd”

  • http://www.facebook.com/people/Dena-Torcivia/1227887910 Dena Torcivia

    I am more amazed that the twins were separated and after that lost their abilities. How sad.

  • http://www.facebook.com/elizabeth.alexandermacisaac Elizabeth Alexander-macisaac

    Am so glad that fake numbers were not introduced….I think tha the integrity of the developing and healing relationship between the 3 would have been compromised/ fractured…. Kudos to Oliver.


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Whether a subtle new pattern shows up in an experiment on the Higgs boson, an epidemiological report about a suspected cancer cluster, or a double-blind trial purporting to demonstrate ESP, it can be maddeningly difficult to distinguish between what we see and what we think we see. "Fire in the Mind" takes a look at the big questions behind today’s science news.

About George Johnson

George Johnson writes about science for the New York Times, National Geographic Magazine, Slate, and other publications. His nine books include The Cancer Chronicles: Unlocking Medicine's Deepest Mystery (August 2013), The Ten Most Beautiful Experiments, A Shortcut Through Time, and Fire in the Mind. He is a winner of the AAAS Science Journalism Award and has twice been a finalist for the Royal Society science book prize. Co-founder and director of the Santa Fe Science Writing Workshop, he can be found on the Web at talaya.net. Twitter @byGeorgeJohnson.


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