In 1966, Oliver Sacks met “the twins,” John and Michael, in a state hospital, at which time they were both twenty-six years old. At this time, the twins were already well known—they’d appeared on TV many times. They had been diagnosed as intellectually disabled, but also as autistic or psychotic, and they were at the same time known for their incredible mental gifts. However, in the twenty years since Sacks’s first meeting with the twins, there have been few, if any, studies conducted upon them. Sacks argues that the original neurological examinations of the twins were too reductive in their format. By focusing too exclusive on the superficial aspects of the twins’ existences, they reduced the twins’ “psychology, methods, and lives … almost to nothing.” To understand the twins more fully, Sacks says, one must study them as subjects, and observe them from day to day.
In this chapter, Sacks not only studies the lives of John and Michael, whose awesome mathematical talents briefly made them into celebrities; he also critiques the neurological community, which, according to Sacks, is far too simplistic in its attempts to understand intellectual impairment. Previous neurologists analyzed John and Michael’s mental abilities, but only in the narrowest, most quantitative senses (in much the same way that neurologists in the 1960s failed to identify cases of Tourette’s because they didn’t try to get a sense for their patients’ personalities or day-to-day behaviors).
The twins, Sacks has found in the course of his 1966 tests, might seem grotesque at first—they’re unusually small, with high voices, disproportionately large heads, and poor vision. The twins have jaw-dropping intellectual abilities, however—they can determine the day of the week for any date in history, past or future, and can remember long numbers perfectly. The twins can also remember the weather on any day in their lives. Strangely, however, the twins can’t perform even the most basic arithmetic.
Much like Martin A. and Rebecca, the twins are prodigies in spite of their mental deficits in certain areas. Sacks tries to understand the twins’ talent without reducing it to a mere party trick.
There have been many cases throughout history of people with phenomenal mathematical abilities. However, what makes the twins so interesting for Sacks is that they can perform complex mathematical calculations without understanding what they’re doing. Once, Sacks spilled matches on the ground, only to have the twins inform him that he’d spilled exactly 111 matches. The twins claimed that they hadn’t counted the matches at all—they’d simply “seen” the correct number. The twins then repeated “thirty-seven” three times—which, Sacks realized, suggested that they’d broken up the 111 matches into three groups of thirty-seven, without understanding the concepts of multiplication or factoring.
Sacks’s description would seem to suggest that, in a sense, the twins are capable of doing arithmetic; they just don’t understand what they’re doing—for this reason, they can count to 111 by dividing up the number into three equal groups of thirty-seven. But this makes it unclear whether the twins really can “see” 111 in a fraction of a second (the way most human beings can “see” the number three without actually counting it) or whether they need to use arithmetic to count it. It’s also been noted by critics that the matches Sacks spilled came from the twins themselves, who may have counted them beforehand (or found a special beauty in the number 111).
On another occasion, Sacks watched the twins exchanging long, six-figure numbers. After each number, the twins would smile as if savoring a fine wine. Sacks wondered if the numbers had any meaning. Only later did he realize that all the numbers the twins had named were primes (i.e., numbers that can only divided by themselves and one). The next day, Sacks joined in the twins’ game, naming seven-, eight-, and eventually ten-digit prime numbers. After each one, the twins would pause for a moment, and then smile, as if verifying that Sacks’s numbers were, indeed, prime. Even though there is no simple method for calculating prime numbers, the twins clearly had some method of calculating them.
Some doctors have criticized Sacks’s description of the twins in this passage for containing less insight into their mental functions than it seems to. However, Sacks’s point seems to be that the twins’ understanding of mathematics, contrary to what many would assume, is highly sophisticated—their passion for mathematics is no less than that of a university professor. Sacks doesn’t try to explain or de-code the twins’ mathematical methods; rather, he regards their methods with respect and even awe.
Sacks draws an analogy between the twins’ mathematical talents and the talents of a great musician. Like a musician, the twins had an intuitive understanding of the “sense” of different numbers, even if they had little to no understanding of arithmetic. It was as if they could see the “harmony” in different numbers. The twins seemed to live in a world of numbers—and, as a result, they seemed to have developed an almost aesthetic sense for the beauty and proportion of different numbers, not unlike a musician’s sense for the beauty and proportion of different melodies. For the twins, numbers were both awesome, godlike entities and also intimate friends.
Sacks is hardly the first neurologist to draw a comparison between mathematical ability and musical ability—some of the greatest musicians were mathematical prodigies, and many eminent mathematicians have compared their work to music. In all, Sacks argues that the twins’ understanding of mathematics is mature and highly sophisticated—contrary to what earlier neurologists had concluded.
Ten years after Sacks met with the twins, it was decided that they should be separated to prevent their “unhealthy communication.” Now they both work in menial jobs for little money. Tragically, since being separated and working menial jobs, the twins seem to have lost their numerical powers, “the chief joy … of their lives.” But, Sacks comments, “This is considered a small price to pay, no doubt, for their having become quasi-independent and ‘socially acceptable.’”
We can sense Sacks’s bitterness here: he’s quietly angry and disappointed that the twins have been separated and that, as a result, they’ve lost their one great source of happiness, their mathematical ability. Sacks’s point seems to be that societal definitions of what is and isn’t “acceptable” are arbitrary and, in the case of people like the twins, actively cruel. In forcing the twins to behave “normally” (get a job, etc.), society has ruined their lives.
In the Postscript, Sacks notes one hypothesis regarding the twins, proposed by the researcher Israel Rosenfield: although the twins couldn’t do basic arithmetical operations like multiplication, perhaps they were capable of modular arithmetic—for example, following the cyclical algorithm necessary to calculate days of the week. And perhaps the fact that the twins could calculate the number of matches—111—by dividing them up into three groups of thirty-seven suggests that the twins could do arithmetical operations, if only with primes. Sacks accepts that Rosenfield’s hypothesis may be true.
Here Sacks explores the twins’ mathematical ability in more detail than he attempted in the chapter itself. Sacks seems to acknowledge that more research is needed into the mathematical ability of prodigies like the twins, whether they’re capable of arithmetical thinking or not (and whether their arithmetical abilities only apply to certain special numbers, such as the prime numbers).
After publishing the original version of this chapter as an article, Sacks received many comments and letters from other doctors. Some parents wrote to him about their children’s abilities to visualize numbers. Sacks realized that prime numbers have played a special role in the lives of several mathematically gifted, autistic children. It’s still unclear how, precisely, gifted children calculate primes and prime factors—even the gifted patients themselves can’t explain how they do it. The great mathematician Kurt Gödel argued that, in the future, primes could serve as a way of marking different ideas, places, and people, paving the way for the total numeralization of the world. If Gödel’s hypothesis ever comes true, then people like the twins may truly be able to live in a world of numbers.
Prime numbers have been a subject of great fascination for thousands of years, across a wide variety of cultures, and the twins aren’t the first to attach some divine significance to them (in both ancient Chinese and ancient Greek culture, for instance, primes were thought to be mystical entities). Sacks’s closing remarks about Gödel convey the arbitrariness of society’s definitions of what is and isn’t “normal.” In another lifetime, one could imagine the twins’ abilities to remember prime numbers being very useful and highly sought-after.