by Ellen Kaplan
We’ll therefore begin not with a square of side but , and area therefore 7200. We now proceed exactly as above, but fortunately with easier calculations.
We fit into this square the largest whole number square we can. Since 84ay = 7056, we put this in and see that a gnomon of 144 is left.
We now have to remind ourselves that this is really , or, in the way we transcribe Babylonian, 1;24. The Babylonians just wrote 1 24. Look familiar? This is how the approximation in that palm-sized Yale tablet began—and how it continued follows from just what we did above. If you’d like the details, see the appendix to this chapter.
We have wandered among a variety of Old Babylonian tools for extracting square roots—some of them blunt, some showing the signs of having lain so long in the earth, some still with a bright edge on them. Our faith in imagining, if not reasoning, backward has been affirmed by what we’ve found in three Middle Eastern mathematicians of the first century A.D.
What has this told us about ancient Babylonian culture, and hence about what forms the precursors of the Pythagorean Insight took among them? It is time to read the messages left for us by those two ungainly methods. Robson remarks of the second (which yielded such absurd results) that it may have remained in use because of the ease with which it could be calculated.26 This sounds like looking for your lost key under the lamppost—a damning judgment on the vast, ramshackle society, taking their lack of concern with utility and application not as idiosyncratic but idiotic. Yet might it not show just what their working on examples, rather than from exemplars, did: a suiting of each How to the lovely impertinence of What. In this context general truths, like the Pythagorean Insight, were of as little value as they are to cooks, who shop by smell and choose by touch and use the pot with the broken handle because they know from long experience exactly how it will behave.
We said at the beginning that theirs seemed a culture of algorithm—but in fact it was founded on recipes, which are always modified by locality and detail. Bakers need to know which flour they’re using, the humidity in the room, and its height above sea level, not to mention the temperature of their own hands. The Old Babylonian scribes had to know which particular numbers were in question and exactly what shape they lay in; they neither thought about nor cared for the generality that most tellingly characterizes mathematical structure (resting on the vital differences between each and every, next and all).
The Pythagorean Theorem is a signpost on our way toward the ever more general, and the Babylonian treatment of it—so sensitive to examples—shows vividly how they drifted away from this direction. Yes, they generalized means (such as their brilliant box), but not ends. That sensitivity to initial conditions, you’d have guessed, could only move thought forward. Yet as today’s physicists and the invention of fractals have stunningly shown us, its product is—chaos.
APPENDIX
We left our small slab with its approximation to at 1;24. To go on, we want to fill up this gnomon as best we can with a smaller gnomon, wide, whose parts will be two rectangles of area · 84 each, plus a small square in the corner, with area . Its total area will therefore be
As before, we drop the small square , which we can’t deal with and don’t need anyway. Since this gnomon must have an area less than 144, we want the largest such that
and we are compelled to choose = 51. Our approximation now is , or (in the way we transcribe Babylonian) 1; 24, 51.
If you feel you have aged a lifetime in the last few pages, it is in the good cause of time-machining yourself back to Babylon. Here’s the last stage of the journey. Our remaining area is now
which simplifies to .4775. We now want an integral width y for our next gnomon so that its area will be less than or equal to this remainder—i.e., so that , and so y must be less than 10.129 . . .
Hence y is 10, and our third approximation is
exactly what you find incised on YBC 7289.27
CHAPTER THREE
Through the Veil
Something happened—but not what you’d expect. The theorem we’ve been vainly chasing through the Middle East emerged, armed with a proof, from the Ionian Sea, more than midway through the sixth century B.C. It wasn’t just about right triangles whose sides were in the proportions of 3-4-5 or 5-12-13: it was about infinitely many right triangles, all at once.
Were the course of human affairs rational, this proof would have been the one you saw Huxley put in the hands of his Guido, since it follows so readily from the Old Babylonian play with gnomons in a box (simply divide the gnomon’s rectangles into triangles, move them around as they moved around other pieces—and there you are). You could then have argued that someone, in the last days or aftermath of their empire, seized on the pleasure they took in generalizing techniques and carried it to its natural end of generalized insights—and it spread abroad. But in fact the proof that surfaced wasn’t anything like Guido’s.
Let’s take literally that “rational” in our feigned reconstruction and insert Guido’s proof in the thousand-year void of evidence between the Old Babylonians, with whose math we now feel ourselves on familiar terms, and this new culture rising in Magna Graecia.1 Heath gives cogent grounds for doubting this was the proof that the Pythagoreans had.2 If you label those two termini, a millennium apart, as a and b, the shadowy existence of this proof is an x, making the equality of ratios . Guido’s proof is, as it were, a “mean proportional” between two knowns, a way of letting the distant ends approach one another.
You might think that doing this was a kind of smoothing out, something between story and history.az But our supposition has an aim different from making a real series of events approximate to the ideal. By keeping the Guido touchstone at hand as a wholly different proof takes shape, we will see more vividly how radical was the shift from the Old Babylonian viewpoint. It gives a context for asking not only why Guido’s proof wouldn’t have harmonized with the new cast of thought, but why there had been any proof at all, rather than simply asserting or exemplifying a2 + b2 = c2.
Who brought this concern with mathematics and proof in general, this striking shard of it in particular, to a distant outpost of civilization?
Pythagoras, born on the island of Samos, fetched up as a young man in Croton (modern Crotona in Calabria) around 530 B.C.3 He lived on the cusp between myth and legend, from which the slide into either is sudden and steep. Had he previously been wandering in the wild lands north of Thrace, where the Hyperborean Apollo was worshipped? Or had he been the Hyperborean Apollo himself? Had he been also, or instead, in India, or in Egypt? Perhaps in all, and at the same time, since he was known to be capable of appearing in several places at once.4 The common theme in the many tales about him, however, is his theory of the soul’s migration through the veil of mortality. For the body to be simultaneously in different parts of space has magic to it, but for the soul to travel through time, taking on successively this persona and that, rings of divinity, since it appeals to our deeper Finnegan desires.
Yet what has this to do with the theorem that bears his name, or mathematics generally? The story, like Robert Frost’s woods, is lovely, dark, and deep. It takes some telling, because while there is more evidence than there was for the Babylonians, it is obscured by the secrecy so dear to the Pythagoreans and the tangle of rival claims made by their followers and detractors. The solvents and tracer dyes of modern textual analysis have done much to let us look through the verbiage, and with the tweezers and microscopes of scholarship, the rough pieces of evidence begin to fit into a picture. A striking feature is this: Pythagoras felt compelled to prove his theory of metempsychosis.
Why wouldn’t a shaman’s authority have sufficed? Perhaps because what the eye can’t see the heart may not grieve for, but the mind will continue to worry about. And perhaps the brotherhood that began to form around Pythagoras—first at Croton, then 150 miles away, along the coast, in Tarentum (modern Taranto)—already had those skeptical traits for which the Greeks were famous. Not too skept
ical, though, since the proofs that convinced them consisted in his recalling publicly some of his past incarnations. While in Argos, for example, Pythagoras saw hanging in a sanctuary the shield of the Trojan hero Euphorbus (which Menelaus had taken when he killed him, half a millennium before), and recognized it as his own.5 How he knew that he had also once been a peacock, and a son of Hermes, and the commissary Pyrandus (who had been stoned to death), and a Delian fisherman, and the beautiful prostitute Alco, has not come down to us; but since he had also once been Aethalides, the Argonauts’ herald whose soul could forget nothing, his power of recollection seems not unreasonable.6
At this point human rather than superhuman traits advance the story. Had Pythagoras been nothing more than a shaman with a very good memory, what would there have been to do but worship him—perhaps as a reincarnation of Apollo? But if the transmigrations of his soul meant that ours too survived death, then a cult of hope could form around him; and if the soul was embodied in lives as various as his had been, perhaps one of these hopes could be that the right practices would make the next incarnation better than the present one. The right practices: worship turns into ritual, satisfying our need to do something active for our salvation.
Apollo was the god of catharsis—purification—so these rituals took the form of ever more strict observances to purge your life of the impure. Always put on the right shoe first; do not dip your hand into holy water, nor travel by the main road; food that falls from the table belongs to the heroes: don’t eat it. Never sacrifice a white cock; don’t stir the fire with a knife; don’t break bread; don’t look in a mirror by firelight; always pour libations over the cup’s handle; don’t turn around when crossing a border; don’t eat beans, nor sit on a bushel measure.7 Hundreds of these “acusmata”—things heard—filled the days of the growing Pythagorean brotherhood with anxiety. You just couldn’t be careful enough: demons were lurking everywhere to trip you up. On the other hand, ascetic demands (initiates had to keep silent for perhaps as long as five years)8 and repetitive practices might induce not only formal feelings but a trancelike state, in which your previous lives were recalled. They would certainly bind the community together in proud exclusivity, as these observances became fixed rules of life.
Although the holiness of the Pythagoreans made them the political power of Magna Graecia, defeating the opulent rival city of Sybaris,9 at the same time a split developed within the community. The pure never feel themselves sufficiently pure. A clue to something more profound seemed to lie in numerical acusmata. Each number had its own significance: 1 was being, 2 was female, 3 male, 4 justice, 5 marriage, 10 the perfect number, since it was both the sum of 1, 2, 3, and 4, and the harmony in which the Sirens sang—and so on. Some of them therefore found that thinking about number and shape extracted the mind from the material world: imperfection was bypassed, exceptions gave way to rules, this particular number and that to even and odd, examples to exemplars. Our architectural instinct surfaced in them and became central, revealing that the soul which took on various material shapes was ultimately form. Studying form was thus the true purification.
Since intense study in general, and in particular of structures rather than things, lessens the sense of self and so seems to free soul from body or let it wander from one to another, this group of mathematikoi, as they were called—we’ll call them Knowers—looked down on the acusmatikoi—the Hearers—as people who obeyed without understanding. The Hearers may have practiced at night recalling the day’s events, in hopes of strengthening their powers enough to remember previous lives, but the Knowers studied eternal geometry, number theory, music.ba
This set the paradigm for what has become our ingrained distinction between opinion and knowledge (later elevated to knowledge versus wisdom), whose two strands were eventually synthesized in Plato’s theory of recollection.
Even more profoundly, contemplating pure form encouraged among the Knowers a sense of distinct souls belonging to a One, and developed the metaphor of a reality behind appearances. When a philosophy emphasizing Being emerged, they would be ready for it. The evolution of their thought from extracted selves to abstract unity—from the soul’s reincarnation to its immortality—took the Pythagoreans past the shaman Pythagoras. This accounts for the startling paradox delivered by modern scholarship: Pythagoras had nothing to do with the theorem that bears his name! The only theorem he can rightly claim kinship with is Stigler’s: no scientific discovery is named after its original discoverer.bb
But haven’t we ancient authority for Pythagoras sacrificing a hundred oxen to celebrate his proof? Lewis Carroll (writing as his alter ego, the Reverend Dodgson) first questioned this: “Sacrificing a hecatomb of oxen—a method of doing honor to Science that has always seemed to me slightly exaggerated and uncalled-for. One can imagine oneself, even in these degenerate days, marking the epoch of some brilliant scientific discovery by inviting a convivial friend or two, to join one in a beefsteak and a bottle of wine. But a hecatomb of oxen! It would produce a quite inconvenient supply of beef.”10
The problem with Ancient Authority is that—unlike mathematics—it aims to establish truth by deducing important results from worthy names rather than from worthy premises. Those numerical acusmata may well have been sayings of Pythagoras, and the view of numbers as archetypal symbols, read for their cosmic significance, might indeed be his. But what had he to do with the games that numbers play in and for themselves?
Pythagoras emerges from current studies as a charismatic cult leader, who could, for example, miraculously tell his followers what had happened among them during his three-year absence (because he had been hiding in a basement, where his mother passed notes down to him about each day’s doings).11 The river Cosas did not rise as he crossed it and hail him by name.12 He didn’t bite a deadly serpent to death in Tuscany, nor stroke a white eagle in Croton. He never foretold the advent in Caulonia of a white she-bear.13 These studies have even taken from him the golden thigh he displayed to the audience at the Olympian Games, and he hasn’t a leg left to stand on.bc
Demolishing the reputation of Pythagoras is the new orthodoxy, and in its zeal even seeks to devalue the work of the Knowers. What has been attributed to them likely came instead from the Greek world at large, this argument runs. That “likely”, however, reflects more fashion than fact, since we usually can’t tell at this remove who conjectured and who proved what. Something we might call Travelers’ Internet surely played a larger role than we tend to give it credit for: that spread of chatter via seafarers and adventurers that brought Babylonian doings through the veil to the Western world, and revealed what Thales in Miletus or Hippocrates from Chios was thinking.
When we can follow the course of rumor, we see the Knowers (where, in one another’s presence, invention was on the boil) at the focus of this more diaphanous community, as in their struggling over the problem of how, given a cubical altar, to construct another with twice its volume.
This was asked by Apollo through his oracle at Delos; partially solved by Hippocrates years later and a hundred sea miles away; then completed by Archytas in Tarentum another seven hundred miles and some fifty years distant from him.bd What will soon very much bear on our story is the nature of Hippocrates’s approach. A prevalent mathematical concern was how to make a square with twice the area of a given square (as we would say, how, given a2, to find the side of a square with area 2a2). They saw that this was equivalent (again, in our terms) to finding a mean proportional between a and 2a—an x, that is, such that
for then x2 = 2a 2, and x= a. Working by analogy, Hippocrates suggested that for doubling a cube, you would need to insert two mean proportionals, x and y: If
then ay = x2 and xy = 2a2, so ,giving us x3 = 2a3. We would therefore have . We would have it, that is, if we knew how to construct such a length—and it was this that waited fifty years for Archytas and his successors. Keep this story in mind. Keep in mind too that “keeping in mind”, rather than before his eyes, was just wh
at Hippocrates did with his proto-algebra. For these manipulations with mean proportionals are invisible; the mind sees symbols, the eye only what they stand for.
Walter Burkert, the leading light in the field of Pythagorean scholarship, makes a sharper attack on the Pythagorean brotherhood. Distinguishing between people engrossed in numerology (shorthand for the metaphorical acusmata) and people engrossed in mathematics, he argues that none of the former would have been able or even have wanted to practice the latter.
How right Burkert would be, were the society of mathematicians made in the image of our last century’s logical positivists, those steely scientists so pure of purpose and honed by logic as to be all but computers draped in flesh. You need only look at the variety of weekend practices that actual mathematicians engage in, however, to be assured that their nature, like ours, is a twisted wood that binds reason together with passion. Their weekday lives, as well, are shot through with longings, insights, superstitions, far-fetched conclusions drawn from wild premises, irrational hope and rational despair.
Well, but you think: as mathematicians they are surely immune to mystical excesses. We offer you the Name Worshippers. Those who founded the early twentieth-century Moscow School of Mathematics, Dmitrii Fedorovich Egorov and Nikolai Nikolaevich Luzin, belonged—even during the materialistic Soviet Union—to a secret group that believed in the profound power of chanting the Jesus Prayer (made up only of the names Jesus, God). Franny, in J. D. Salinger’s novel Franny and Zooey, explains it well: “If you keep saying that prayer over and over again—you only have to just do it with your lips at first—then eventually what happens, the prayer becomes self-active. Something happens after a while. I don’t know what, but something happens, and the words get synchronized with the person’s heartbeats, and then you’re actually praying without ceasing. Which has a really tremendous, mystical effect on your whole outlook. I mean that’s the whole point of it, more or less. I mean you do it to purify your whole outlook and get an absolutely new conception of what everything’s about.”