The Music of Pythagoras (14 page)

There are very few school mathematics tablets dating from 1600–1350
B.C
., and another evidential gap 1100–800
B.C
. The historian and Assyriologist Eleanor Robson, who has given these issues more thought than perhaps any other modern scholar, listed several possible explanations, but concluded that “the collapse of the Old Babylonian state in 1600
B.C.E
. entailed a massive rupture of all sorts of scribal culture. Much of Sumerian literature was lost from the stream of tradition, it seems, and most of Old Babylonian mathematics too.”
^12
†

Although Robson believes that the later Babylonians were probably ignorant of the achievements of Old Babylonian mathematics, it is likely that useful fallout from that lost knowledge, such as a triple that was handy for finding right angles, would have remained in use in Mesopotamia and elsewhere for centuries, without those who utilized it remembering the hidden relationship among the numbers.
¹³
And even if Pythagoras never visited Babylon, Greece was no wasteland when it came to building and surveying: Eupalos’ astounding water tunnel on Samos was built in Pythagoras’ century, as were many magnificent Greek temples. Though Pythagoras and his followers were not the first to know the theorem, their discovery might have been an independent discovery, or linked only by some surviving vestige of the more ancient, lost knowledge.

Pythagoreans in possession of the triple 3–4–5, wherever they learned it, and recognizing its usefulness, would not have let matters rest there. And if they set their minds to looking for a meaningful connection among the three numbers, it would arguably not have taken long to find the theorem. They could not have done it with pebbles, which they used as counters rather than as units to measure distances, but the same visualization that made pebbles interesting would soon have arrived at something like the diagram above, where squaring the numbers in the triple reveals the hidden relationship.

If the Pythagoreans found this relationship, having already discovered the harmonic ratios, they must have felt as though lightning had struck twice, for here was another stunning example of the hidden numerical rationality of the universe. Believing so strongly in a unity of all things, they would have been quick to jump to the correct conclusion that this same pattern of hidden connections had to apply to all right triangles—perhaps even to the incorrect conclusion that it had to be true of all triangles.

The second part of the tradition that has Pythagoras and his early followers discovering the theorem was that afterward a sword of Damocles hung over their heads. The universe had a cruel surprise in store for them, “incommensurability.” Most right triangles have no whole-number triplet like 3–4–5. For example, according to the theorem, a right triangle with sides measuring 3 inches and 3 inches, each of which squared is 9, must have a third side—a hypotenuse—the length of which squared is 18.

However, it is no simple matter to find the square root of 18 and the length of that hypotenuse, for the square root of 18 does not exist among whole numbers or fractions. An isosceles triangle like this one was a nightmare for a community of scholars who believed in a rational universe based beautifully and neatly on numbers. They could see that it did indeed exist and was a right triangle. It was not something hypothetical hanging fuzzily out in conceptual space. It was the triangle they got when they drew a diagonal from corner to corner of a square. But no subdivision of the length of the sides (neither inch, nor centimeter, nor mile, nor any fraction thereof) divided evenly into the length of the diagonal. More generally, though it might seem that for any two lengths you might try to measure there would be some unit that would divide into both of them and come out even with no remainder—maybe not the inch or centimeter or any length that has a name, but some unit, however small—the fact is that this is not the way reality works in this universe. Nor does the problem of incommensurability exist only with isosceles triangles. It was also true of the “gate” measured on the Babylonian tablet. The Babylonians knew about the triples and also apparently accepted that the units measuring the diagonal most of the time did not come out even.

The early Pythagoreans may well have discovered the problem, but it is far less likely that they found the solution—irrational numbers—or that they would have liked it if they had.
*
Irrational numbers are not neat or beautiful like whole numbers. An irrational number has an
infinitely long string of digits to the right of the decimal point with no regularly repeating digit or group of digits.

The suggestion that the only information Pythagoras learned elsewhere was a vestige of the theorem—the triple 3–4–5—has in its favor that it solves a problem with the sequence of the Pythagoreans’ discoveries. In order to have had a devastating crisis of faith resulting from the discovery of incommensurability, they had to have had the faith first, not the crisis, and indeed the tradition is strong that this was the order in which the discoveries were made. However, it is difficult to imagine anyone discovering the theorem from scratch (without the triple) while not simultaneously discovering the problem. Another possible sequence would have the Pythagoreans discovering incommensurability first, as they struggled with right triangles, and only later realizing that a few right triangles were, in fact, not incommensurable; but that is not the traditional sequence. A similar issue undermines the possibility that Pythagoras learned the whole theorem elsewhere: He would also have learned about incommensurability, so that it could have come as no surprise later.

Eleanor Robson is convinced, from evidence in the mathematics itself, that Old Babylonian mathematics was not the arithmetical precursor to early Greek mathematics; but a triple used in construction and surveying—the origin of which no one remembered—hardly represented the bulk of Old Babylonian mathematics.
¹⁴
It could have waited on the shelf while the Pythagoreans investigated harp string lengths and discovered musical ratios. The study of those was a new kind of thinking about numbers, what Aristoxenus meant when he wrote: “The numbers were withdrawn from the use of merchants and honored for themselves.” A love of this sort of thinking could have led the Pythagoreans, after their musical discovery, to consider the 3–4–5 triple more carefully. Granted, this particular triangle was, by Pythagorean standards, not very interesting. Five does not show up in the basic ratios of music; 3, 4, and 5 do not add up to 10 or make the
tetractus
. No one is going to swear by this triangle! But the hidden connection . . .
that
was another reason to fall to one’s knees, and perhaps to have a huge crisis of faith when you began looking at other right triangles.

Some Pythagoreans reputedly found other beauty in the triplet: They designated 5 as “marriage.” The 5-unit side of this right triangle connected the 3-unit side and the 4-unit side. Thus, 5, or “marriage,” connected 3 (which is odd) and 4 (even). “Odd” was male and “limiting”;
“even” was female and “limitless.” So this triangle was a manifestation of the harmony reconciling limiting and limitless. In the modern world, we associate such weak links with a different sort of mind from that which would come up with the Pythagorean theorem. In the ancient world, whose people were taking the first tentative steps toward understanding nature and the cosmos and the human condition, that distinction is invalid.

If Pythagoras discovered or knew the rule, did he prove it? Most historians of mathematics believe that the concept of “proof” as later understood was unknown before the Alexandrian Euclid introduced it in his
Elements
, around 300
B.C
. The decision that something would “be true for every right triangle,” for someone living as early as Pythagoras, would probably have been made on grounds other than a Euclidian proof. It would either have been an unsupported assumption or a guess, or a decision made in a scientific rather than a mathematical way—by testing it, as many times and with as many different examples as possible. The idea that mathematical statements should apply generally, though taken for granted today and implied in the Babylonian work, was not usually part of the ancient mind-set before Pythagoras. It is considered to be one of the great contributions of early Greek mathematics and probably a contribution of Pythagoras and his followers. With them, it might have been only an assumption based on their belief in the unity of all being, not something they could demonstrate at all or even thought it necessary to demonstrate.

There are, nevertheless, simple proofs of the theorem that some would like to attribute to the Pythagoreans, and one argument that they used such proofs is that these same thought sequences are good ways to discover the theorem, even if you had no concept of “proofs” after the fact. There is a geometry lesson in Plato’s
Meno
that some think is traceable to Pythagoras. The clue is that Plato used it to demonstrate the “recollection” of what one learned before birth, an idea related to the Pythagorean doctrine of memory of past lives. The triangle in the
Meno
proof is the troublesome isosceles triangle, but the proof sidesteps the problem of incommensurability by using no numbers. It is admittedly difficult to imagine Pythagoreans being satisfied with any “truth” that used no numbers. It would have seemed the universe was thumbing its nose at them, with this triangle that provided such a clear and unequivocal demonstration of their rule, and that contained incommensurability.
The discussion of Plato’s proof fits better in the context of a later chapter. Bronowski, in the book from his television series, showed another numberless proof that he believed Pythagoras may have used. Bronowski’s clever proof is in the Appendix.

The right triangle was not the only pitfall in Pythagorean thinking where incommensurability lurked, but it was the most obvious. The scholarly argument about whether the Pythagoreans discovered it, and whether that caused a crisis of faith in the rationality of the universe, rambles on until it resembles that string of digits after the decimal place in an irrational number. However, in truth, an intelligent person thinking along Pythagorean lines and dealing with right triangles could hardly have missed discovering incommensurability. But only someone who reverenced numbers and the rationality of the universe would have been deeply troubled. Some have thought that Pythagoras and his followers reacted by retreating to a geometry without numbers—that what had early been an “arithmetized geometry” was reformulated in a nonarithmetical way, and this carried over into Euclid. In spite of the passage in Plato’s
Meno
, and the suggestion that it reflected Pythagoras’ proof of his theorem, nothing could seem more blatantly un-Pythagorean than a retreat from numbers!
¹⁵

Porphyry would have been pleased to learn that the earlier Mesopotamians knew about right triangles, the triples, and the theorem. His choice of possibilities would almost certainly have been that the theorem was known earlier but that Pythagoras’ was an independent discovery, for he believed that several ancient peoples—he named the Indians, Egyptians, and Hebrews—possessed primeval, universal wisdom (
prisca sapientia
was the later term), and Pythagoras was the first to possess it in the Greek world.
¹⁶
The theorem is so intrinsic to nature, so beautifully simple, that it would be odd if no earlier triangle user in prehistory or antiquity got curious and figured it out.

What about a more startling suggestion: that Pythagoras had nothing whatsoever to do with the discovery? Could it be that it was later credited to him only because such legends tend to become associated with famous people? Over two thousand five hundred years, numerous achievements that were not remotely Pythagorean have been carelessly credited to Pythagoras. “Pythagorean” or “of Pythagoras” have become descriptive words connoting something clever that shows mathematical insight, with an overlay of wisdom, fairness, or morality.
A “Pythagorean cup,” sold on Samos, punishes the immoderate drinker who fills it above a marked line, by allowing the entire cup of wine to drain out the bottom. Modern citizens of Samos are surprised—or at least pretend to be—that anyone would doubt this was an invention of Pythagoras. A “Pythagorean” formula predicts which baseball teams in America are likely to win. No one is insulted by doubts about that one.