As with the Gregorian reform and the birth of George Washington, understanding such calendrical complexities really does help us to grasp some puzzling aspects of everyday life that otherwise persist as annoying confusions among the hundreds of little daily bothers that provoke the conventional response of a harried life: “Why the hell does it work this way? Someday I just have to look it up and find out”—and then we never do.
For example, didn’t you always wonder why Chanukah creeps backward through the December calendar by about ten days every year? Then, just when you thought that Chanukah would sneak into November, it suddenly shoots forward the next year into late December, falling even after Christmas. Blame the Metonic Cycle. With respect to a Gregorian date, any Jewish date must move backward in any short lunar year of 354 days (twelve of nineteen in the Metonic Cycle), but shoot forward in long years of 384 days that add a leap month.
On the other hand, you may have noticed that Ramadan just keeps moving backward on our Gregorian calendar, and therefore can occur at any time during the solar year. The Islamic calendar is also lunar, but does not use the correction of the Metonic Cycle. The shortfalls therefore accumulate continually, and all fixed Islamic dates move constantly back on the Gregorian calendar.
In Christian history, the need to reconcile solar and lunar cycles has centered on one of the most complex and persistently vexatious problems in the history of calendrics: the calculation of Easter. Books, indeed libraries, have been written on the subject, and great scholars have devoted their lives to devising rules and procedures for getting this cardinal day right. I shall not even begin to probe the details here but only wish to state, as a good summary for this part (for Easter, in its nutshell, epitomizes all the issues here discussed), that Easter became more problematic than any other calendrical day, or any other movable feast, because its definition includes both lunar and solar elements, and its date cannot be determined until we know how to reconcile all the great, and distressingly fractional, cosmic cycles. For Easter falls on the Sunday following the first full moon (the lunar component) after the vernal equinox (the solar contribution).
Guernica (1937), Pablo Picasso. (illustration credit 3.4)
An Epilogue
I have always and dearly loved calendrical questions because they display all our foibles in revealing miniature. Where else can we note, so vividly revealed, such an intimate combination of all the tricks that recalcitrant nature plays upon us, linked with all the fallacies of reason, and all the impediments of habit and emotion, that make the fulfillment of our urge to understand even more difficult—in other words, of both the external and the internal pitfalls to knowledge. Yet we press on regardless—and we do manage to get somewhere.
I think that I love humanity all the more—the scholar’s hangup, I suppose—when our urge to know transcends mere practical advantage. Societies that both fish and farm need to reconcile the incommensurate cycles of years and lunations. Since nature permits no clean and crisp correlation, people had to devise the cumbersome, baroque Metonic Cycle. And this achievement by several independent societies can only be called heroic.
I recognize this functional need to know, and I surely honor it as a driving force in human history. But when Paleolithic Og looked out of his cave and up at the heavens—and asked why the moon had phases, not because he could use the information to boost his success in gathering shellfish at the nearby shore, but because he just wanted to resolve a mystery, and because he sensed, however dimly, that something we might call recurrent order, and regard as beautiful for this reason alone, must lie behind the overt pattern—well, then calendrical questions became sublime, and so did humanity as well.
If we regard millennial passion in particular, and calendrical fascination in general, as driven by the pleasure of ordering and the joy of understanding, then this strange little subject—so often regarded as the province of drones or eccentrics, but surely not of grand or expansive thinkers—becomes a wonderful microcosm for everything that makes human beings so distinctive, so potentially noble, and often so actually funny. Socrates and Charlie Chaplin reached equal heights of sublimity.
I hate to end with such waffling generality, with such a risibly inadequate stab at lyricism—so let me finish instead with a little story about an ordinary person who has done something heroic in the domain of calendrics, and who loves the millennium with all his heart. His tale belongs to a classic genre in the annals of calendrics—day-date calculation, a subject that cannot be equaled (hence its classic status) as an illustration of interaction between human foibles and divine failure: that is, as something made difficult both because we chose peculiar definitions rather than eminently available and sensible alternatives, and also because nature made matters even worse by arranging the cosmic cycles of days and years in such poor and irrational correspondence.
The Pirate King, in Gilbert and Sullivan’s leap year tale, begins his explanation of Frederick’s calendrical dilemma with the following intonation: “For some ridiculous reason, to which, however, I have no desire to be disloyal …” I feel the same way about the day-date problem. For some ridiculous and arbitrary reason, our culture decided to parse days into groups of seven called weeks—a unit with no correspondence whatever to anything cyclical in nature. Because the year runs for 365 days, we end up with fifty-two weeks in a year—and one bothersome extra day left over.
If the year contained a nonfractional number of weeks (I am setting aside the leap year problem for a moment), we would have no day-date problem, for each date of the year would always fall on the same day of the week. But we made matters unduly difficult by running the year through a series of weeks, and then leaving one day over each year—for the day of the week must now shift for each date in each new year. A date that falls on a Tuesday in 1997 must switch to a Wednesday for 1998, Thursday in 1999, and so on. (We will get to those leap years in a moment.)
All this wouldn’t matter a damn—except that we care. Don’t ask me why, but we take an uncommon interest in the day of the week that our birthday, or some other date of importance to us, occupied in years other than the one now running (which we can easily look up on a calendar). We are especially concerned, in our culture, about the day on the actual date of our birth. Ask almost anyone if they know the weekday of their birth (most people don’t), and they will immediately dredge some doggerel out of their infantile consciousness of nursery rhymes: “Monday’s child is fair of face, Tuesday’s child is full of grace …” (Believe me, I am not conjecturing, or making this up. As will soon become clear, I speak from empirical experience and have witnessed the repetition of this scene dozens of times. I was also born on a Wednesday, and “Wednesday’s child is full of woe.”)
The day-date problem could be solved with relative ease, if the only difficulty lay in this inconvenience of human definition—and the consequent need to add one each year. Anyone could then look up this year’s weekday for any date on a calendar, figure out the interval between this year and any other year of interest, divide by seven, and then subtract the remainder from this year’s weekday.
But nature now intervenes to impose an additional difficulty based on the fractionality of days in the year, and the consequent need to designate leap years. The year contains fifty-two weeks and one extra day—except for leap years, when the year runs for fifty-two weeks and two extra days. Therefore, to tell the day of the week for any date in a past (or future) year, you must first find the date on this year’s calendar. You then need to make two calculations: first, to take human foibles into account and correct for the extra day added each year because the year contains fifty-two weeks plus one day; and second, to acknowledge natural complexities and make another correction for the extra day added in any leap year (not forgetting, if you are calculating across centuries, the Gregorian rule for omitting leap years at century boundaries not divisible by 400).
The entire procedure therefore becomes pretty complex—and the exercise of figuring ou
t the day of the week for dates in distant years (and doing so quickly enough to keep oneself and others interested) goes by the name of “day-date calculation.” The subject has also generated a surprisingly long and learned literature. Some people are prodigious day-date calculators and can instantly (and without error) tell you the day of the week for any date in any year, often ranging widely over centuries and millennia with apparent ease.
As one staple of this literature, some of the most famous, and most proficient, day-date calculators have been mentally retarded or autistic people with general mental skills and accomplishments so limited that no one can figure out how they could possibly develop such an odd, complex, and arcane skill. What could be more marvelous, magical, or even miraculous? Day-date calculation seems hard enough to contemplate for ordinary mortals; how can people with such great limitations possibly manage such a thing? What does their achievement tell us about the nature of human intelligence—not to mention human courage? My final section tells the story of such a person.
PART TWO:
FIVE WEEKS
Poets, extolling the connectedness of all things, have said that the fall of a flower’s petal must disturb a distant star. Let us all be thankful that universal integration is not so tight, for we would not even exist in a cosmos of such intricate binding.
Georges Cuvier, the greatest French naturalist of the early nineteenth century, argued that evolution could not occur because all parts of the body are too highly integrated. If one part changed, absolutely every other part would have to alter in a corresponding manner to produce a new but equally elegant configuration for some different mode of life. Since we cannot imagine such comprehensive change of every single part, each to the perfection of a new optimality, organisms cannot evolve.
Half of Cuvier’s argument is undeniably sound. If evolution required such comprehensive alteration, such a process might well be impossible. But parts of bodies are largely modular and dissociable to a great extent. Alpha Centauri (not to mention more distant stars) didn’t blink the slightest notice when little Susie pulled those petals off the daisy—“He loves me, he loves me not …” And even though the foot bone’s connected to the ankle bone, evolution can change the number of stripes on a snail’s shell without altering the number of teeth on its radula (jaw).
The functions of the brain, and human intelligence in general, also tend to be quite modular and dissociable. No g-factor, or unitary measure of “general intelligence,” lurks within the brain, capable of ranking people according to their inherited quantity of a coherent thing, measured by a single number called IQ. (See my critique of this position in my earlier book The Mismeasure of Man.) Rather, intelligence is a vernacular word that we apply to the large set of relatively independent mental attributes that build, in their entirety, something we call “mind.”
The best, and classical, illustration of the relative independence of mental attributes lies in the stunning phenomenon illustrated by people who were once labeled with the stunningly insensitive name idiot savant—that is, globally retarded people with a highly precise, separable, and definable skill developed to a degree that would surprise us enough in a person of normal intelligence but that strikes us as simply miraculous in a person otherwise so limited. Some savants can do lightning calculation, multiplying and dividing long strings of numbers instantaneously and with unfailing accuracy—but cannot make change from a dollar or even understand the concept. (Dustin Hoffman played such a character with great sensitivity in Rain Man.) Others can draw pictures, accurate to the finest detail, of complex scenes that they have viewed but once and for a fleeting moment—yet cannot read, write, or speak.
These people fascinate us for two very different reasons. We gasp because they are so unusual, and extremes always fascinate us (the biggest, the fiercest, the ugliest, the most brilliant). We need not be ashamed of this quintessentially human propensity. But savants also compel our attention because we feel that they may be able to teach us something important about the nature of normal intelligence—for we often learn most about an average by understanding the reason for an extreme deviation.
We have favored two broad interpretations of these savants (each too simple, and probably both wrong, but still representing a reasonable first pass at formulating the problem). Do these people acquire their extraordinary skill because they discover one thing they can do—and then work so very hard, and so assiduously, at developing it? In this case, any of us could probably master the savant’s skill, but we would never choose to devote so much time to one mental operation. (In this alternative, the savant’s brain does not differ from ours in the module devoted to his hypertrophied skill—and the phenomenon teaches us something about the nature of dedication.)
Or do these people develop their skill because deficiencies in one part of the brain’s structure may be balanced by unusual development in another part? In this case, most of us could not learn the savant’s skill even if we chose a path of single-minded devotion to such an activity. (In this alternative, the savant’s brain may differ from ours in the module regulating his special skill—and the study of this phenomenon may teach us something important about the physical nature of mentality.)
In any case, day-date calculation represents one of the most famous, and most frequent, of so-called “splinter skills” manifested by many savants. The subject has generated a great deal of study, well summarized in two recent books (Steven B. Smith, The Great Mental Calculators, Columbia University Press, 1983; and Darold A. Treffert, Extraordinary People: Understanding “Idiot Savants,” Harper and Row, 1989). One obvious question has dominated the literature about mentally retarded and autistic day-date calculators: How do they do it?
The most obvious approach—simply asking a savant how he performs his day-date calculations—does not work. Few of us can give any decent explanation of how we accomplish the things we do best, for our truly unusual accomplishments seem automatic to us. (Sports heroes are famously unable to describe their extraordinary skills—“Well, um, er, I just keep my eye on the ball and …” Intellectuals do no better in elucidating their literary or mathematical accomplishments—“Well, um, I had a dream, and I saw these six snakes, and …”) Savants, if they speak at all, will tend to say “I just do it”—and most of us could describe our special skills no better.
The literature has considered two basic modes—and results are typically inconclusive in illustrating the usual variety of reasons for human achievements. That is, some savants do it one way, others the other way, yet others in combination, and still others in a manner as yet undetermined. First, a savant might have extraordinary, even truly eidetic, skills in memorization. A day-date calculator might then simply memorize the calendars for a certain number of years and read the right day of the week for any date in any year directly out of memory. Second, a savant might develop an algorithm or rule of calculation, and then apply the rule so often, and with such concentration and dedication, that his calculation becomes extremely rapid and “second nature.” At some point, the procedure may start to feel automatic.
Some savant day-date calculators do use memory alone—and this method can be spotted because practitioners tend to memorize only a limited number of years. A savant who can do day-date calculation from, say, 1980 to 2020—but has no clue about dates in earlier or later years—has probably memorized forty years’ worth of calendars (as researchers might be able to ascertain by checking a subject’s bookshelf or asking if he owns a perpetual calendar for a limited number of years).
But many savant day-date calculators, including the young man described herein, use algorithms of their own invention. Some of these people, including my subject, can calculate effortlessly, and apparently instantaneously, sometimes across thousands of years, past or future, and with no apparent difference in the time needed to calculate a date two years or two hundred years from the present. The statement that some savants use algorithms still leaves two mysteries and complexities unaddres
sed—and these also figure prominently in literature on the subject. First, day-date calculation, as I showed in the last section, is a two-step process. You need, first, to know the day of the week for some reference year—usually the current year as given on a calendar. Then you can apply your algorithm to calculate the difference between your reference year and the year in question. Thus, no matter how good your algorithm, you still need to put some basic reference into memory. (Of course, you could begin any application by looking up the day of the week for this year on a portable calendar—but no self-respecting day-date calculator would use such a crutch.)
Second, and of most potential interest for insight into human mentality in general, the best algorithmic calculators, including my subject, do their reckoning far too quickly to be using their algorithm in an explicit manner. As a striking example, a graduate student studying George and Charles, the famous mathematical twins (and prodigious day-date calculators) so brilliantly and poignantly described by Oliver Sacks (in a chapter in The Man Who Mistook His Wife for a Hat), decided to try to equal their skills in day-date calculation by applying their method with the same singlemindedness manifested by many savants. He found that he could do the calculation, but he could not come close to their speed for a long time. Finally, and in a manner that he could never describe accurately, the technique just “clicked” and started to feel automatic. The student could then match the twins. Darold Treffert’s book quotes a report by Dr. Bernard Rimland on this experiment:
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