The Music of Pythagoras
Page 16
BOY: Yes.
SOCRATES: Now we can add another equal to it like this?
(Draws.)
BOY: Yes.
SOCRATES: And a third here, equal to each of the others?
(Draws.)
BOY: Yes.
SOCRATES: And then we can fill in this one in the corner?
(Draws.)
BOY: Yes.
SOCRATES: Then here we have four equal squares?
BOY: Yes.
SOCRATES: And how many times the size of the first square is the whole?
BOY: Four times.
SOCRATES: And we want one double the size. You remember?
BOY: Yes.
SOCRATES: Now, do these lines going from corner to corner cut each of these squares in half? (Draws.)
BOY: Yes.
SOCRATES: And these are four equal [diagonal] lines enclosing this [central] area?
BOY: They are.
SOCRATES: Now think, how big is this [central] area?
BOY: I don’t understand.
SOCRATES: Here are four squares. Has not each [diagonal] line cut off the inner half of each of them?
BOY: Yes.
SOCRATES: And how many such halves are there in this [central area]?
BOY: Four.
SOCRATES: And how many in [one of the original squares]?
BOY: Two.
SOCRATES: And what is the relation of four to two?
BOY: Double.
SOCRATES: How big is this figure then?
BOY: Eight feet.
SOCRATES: On what base?
BOY: This one. (Indicates one of the diagonal lines.)
SOCRATES: The line which goes from corner to corner of the square of four feet?
BOY: Yes.
SOCRATES: The technical name for it is “diagonal”; so if we use that name, it is your personal opinion that the square on the diagonal of the original square is double its area?
BOY: That is so, Socrates.
SOCRATES: What do you think, Meno? Has he answered with any opinions that were not his own?
MENO: No, they were all his.
SOCRATES: Yet he did not know, as we agreed a few minutes ago.
MENO: True.
SOCRATES: But these opinions were somewhere in him, were they not?
MENO: Yes.
SOCRATES: So a man who does not know has in himself true opinions on a subject without having knowledge.
MENO: It would appear so.
SOCRATES: At present these opinions, being newly aroused, have a dream-like quality. But if the same questions are put to him on many occasions and in different ways, you can see that in the end he will have knowledge on the subject as accurate as anybody’s.
MENO: Probably.
SOCRATES: This knowledge will not come from teaching but from questioning. He will recover it for himself.
MENO: Yes.
SOCRATES: And the spontaneous recovery of knowledge that is in him is recollection, isn’t it?
MENO: Yes.
SOCRATES: Either then he has at some time acquired the knowledge which he now has, or he has always possessed it. If he always possessed it, he must always have known; if on the other hand he acquired it at some previous time, it cannot have been in this life, unless somebody has taught him geometry. He will behave in the same way with all geometrical knowledge, and every other subject. Has anyone taught him all these? You ought to know. He has been brought up in your household.
MENO: Yes, I know that no one has ever taught him.
SOCRATES: And has he these opinions, or hasn’t he?
MENO: It seems we can’t deny it.
SOCRATES: Then if he did not acquire them in this life, isn’t it immediately clear that he possessed and had learned them during some other period?
MENO: It seems so.
SOCRATES: When he was not in human shape?
MENO: Yes.
A modern attorney would probably object that Socrates was “leading the witness.” But Plato was not talking about knowledge the boy had hidden somewhere in his mind because he had witnessed it or been taught it in a previous life: the date of an event or the length of a road—knowledge of the changeable world. Plato meant inborn knowledge of truths that do not change—universal and immutable truths of the Forms, in this case truths of geometry. The point of Plato’s lesson scene was that at each stage of questioning, the boy knew whether what Socrates was suggesting was correct. Such recollection of the “eternal Forms” came not from past lives at all but from experiences of the disembodied soul.
Many who first encounter proofs in a setting other than a smotheringly dry presentation are struck by this deep, mysterious sense of recognition of something they already knew. Indeed there are truths that have been “rediscovered” time and time again (the Pythagorean theorem may be one of them) by ancient people and by more recent individuals who were unaware they were repeating a former discovery. Socrates’ demonstration was an extremely Pythagorean lesson, for it united the two Pythagorean themes: the immortality of the soul and the mathematical structure of the world.
Other dialogues and his Republic show that Plato’s mind was much taken up with the doctrines of recollection, reincarnation, and immortality. His Phaedo ends shortly after Socrates’ death, with Phaedo pausing on his journey home from Athens in a Pythagorean community in Phlius to tell Echecrates and other Pythagoreans about the philosopher’s last words. In a discussion centering on immortality and reincarnation, Phaedo repeats Socrates’ quote from an Orphic poem that Socrates had thought spoke of philosophy’s power to raise one to the level of the gods. In his Phaedrus, Plato wrote that human “love” was recollection of the experience of Beauty as an eternal Form.
In his “Myth of Er,” at the end of The Republic, Plato most clearly revealed his belief in reincarnation, although, true to his doctrine that knowledge of ultimate truth is unattainable, he used the term “myth” to indicate that he could not vouch for the absolute truth of the lessons it taught. In the “myth” he imagined what happens when one life has ended and the next has not yet begun: Each soul chooses what it will be in the next life. Choices include “lives of all living creatures, as well as of all conditions of men.” Orpheus chooses to be a swan so as not to be born of a woman—for frenzied Bacchic women had torn him apart in a former life—while a soul who has lived previously as a swan chooses to be a man. The harmony of the spheres was also on Plato’s mind. The souls see a vision, a magnificent model of the cosmos. On each of the circles in which the planets and other bodies orbit stands “a Siren, who was carried round with its movement, uttering a single sound on one note, so that all the eight made up the concords of a single scale.” Though Earth, in Plato’s cosmos, sat dead center, and there was no central fire or counter-earth, the “Myth of Er” was suffused with Pythagorean ideas.
When the members of Plato’s Academy before and after his death in 348/347 B.C. thought about Pythagoras and called themselves Pythagorean, they had in mind mainly Pythagoras as seen through Plato’s eyes. However, to say that Pythagoras was reinvented as a “late Platonist,” as some scholars insist, is to be too glib and overconfident about where to draw the lines between original Pythagorean thought, Pythagorean thought shortly after Pythagoras’ death, Archytas, Plato, and Plato’s pupils, some of whom attributed their own ideas to more ancient Pythagoreans and even to Pythagoras. As time passed, the line between Platonism and what called itself Pythagorean became increasingly difficult to discern. Eventually the two were indistinguishable.
CHAPTER 10
From Aristotle to Euclid
Fourth Century B.C.
WHILE MOST SCHOLARS WERE content to view Pythagorean teachings through Plato’s eyes and not eager to differentiate between Plato’s philosophy and the thinking of pre-Platonic Pythagoreans, one person was still curious. That was Aristotle. Born in 384, he was two generations younger than Plato and at age seventeen had come to Athens to study at Plato’s Academy. Plato was away at the ti
me, on one of his jaunts to Sicily. Twenty years later, when Plato died at age eighty in 348, Aristotle was only thirty-seven and, perhaps because of his youth, was not chosen to succeed Plato as scholarch of the Academy. Instead, though by then hardly anyone failed to recognize that Aristotle was one of the most gifted men around, Plato’s nephew Speusippus got the job. Aristotle left Athens and eventually returned to found his own school, the Lyceum. His debt to Plato was clear throughout his work, but so was the fact that the two disagreed in significant ways. Aristotle was not happy with Plato’s concept of Forms. Plato thought the world as humans knew it was only an undependable reflection of a real world that humans could never know. Aristotle, by contrast, believed that the world humans perceive is the real world. He highly valued what could be learned about nature through use of the human senses, and what could be extrapolated from those perceptions. It would not have displeased Aristotle to find that Plato’s teachings were at least in part derivative of the Pythagoreans. In his Metaphysics, in a passage following his description of Pythagorean philosophies, Aristotle looked down his nose at Plato and invited his readers to do the same: “To the philosophies described, there succeeded the work of Plato, which in most respects followed these men, though it had some features of its own apart from the Italian philosophy.”1
To make such a statement, Aristotle had to be fairly confident he knew what the “Italian philosophy” was before it fell into Plato’s hands. His research was extensive and careful, including the work of Philolaus and Archytas and other sources we know little or nothing about, and he recorded the results in several books.* Unfortunately, those devoted entirely to the person of Pythagoras and Pythagorean teaching are lost, but because he spent so much time and effort on them, and referred elsewhere to his “more exact” discussions in them, there is no doubt Aristotle knew the subject well.† References and quotations from the lost books appear in the writings of authors who lived before the books disappeared, making it possible to peer, indirectly, at a few of the vanished pages.2 The result is a window into what Pythagoreans were thinking and teaching before Plato, helping, at least a little, to circumvent that frustrating impasse, the question of whether what later generations thought they knew about the Pythagoreans and their doctrine was only a Platonic interpretation.
Aristotle was one of the earliest, most dependable sources used by Iamblichus, Porphyry, and Diogenes Laertius. His information went back to shortly after Pythagoras’ death (within about fifty years), but in the books that have survived he never claimed that any specific teaching could or could not be attributed directly to Pythagoras. He also made no distinction between the ideas of Pythagoreans who lived close to the time of Pythagoras and those who lived nearer the time of Plato. He used a Greek form that Burkert says is the equivalent of putting words between quotation marks in modern literature—the “Pythagoreans”—though translating it as “the so-called Pythagoreans” would put too negative a spin on it.
Aristotle wrote that what set both Plato and the Pythagoreans apart from all other thinkers who had lived before Aristotle’s own time was their view of numbers as distinct from the everyday perceivable world. However, the Pythagoreans regarded numbers as far less independent of the everyday, perceivable world than Plato did. At the same time, for the Pythagoreans, numbers were also more “fundamental.” If these distinctions seem confusing, they were, even for Aristotle. His difficulty deciding and explaining what the Pythagoreans thought about numbers was not, at heart, a matter of being unable to find out. Rather, he could not think with their minds. The discussion he was insisting on having—about what was more fundamental, more abstract, or more or less distinct from sensible things—would not have taken place at all among the first Pythagoreans. Whether numbers were independent of physical reality, or how independent, were not questions they would have thought to ask.
In his attempt to squeeze the Pythagoreans into Plato’s and his own molds, Aristotle overinterpreted them and became particularly ill at ease with the idea that all things “are numbers.” The Pythagoreans, he reported with chagrin, believed that numbers were not merely the design of the universe. They were the building blocks, both the “material and formal causes” of things. Physical bodies were constructed of numbers. Aristotle threw up his hands: “They appear to be talking about some other universe and other bodies, not those that we perceive.”
As Aristotle understood the Pythagorean connection between numbers and creation, for numbers to exist, there first had to be the distinction between even and odd—the “elements” of number. The One had a share in both even and odd and “arose” out of this primal cosmic opposition.* The One was not an abstract concept. It was, physically, everything. Aristotle was puzzled by that idea, and unhappy with it.
Odd was “limited”; even was “unlimited.” As the unlimited “penetrated” the limited, the One became a 2 and then a 3 and then larger numbers.* This emergence of numerical organization resulted in the universe humans know. In Aristotle’s words (he was still rankled by the “substance” of the One):
They say clearly that when the One had been constructed—whether of planes or surface or seed or something they cannot express—then immediately the nearest part of the Unlimited began “to be drawn and limited by the Limited” . . . giving it [the Unlimited] numerical structure.
Aristotle had found that, at least in its broad outlines, the numerical creation of the universe was a pre-Platonic Pythagorean concept. However, he often regarded the Pythagoreans with a frown of frustration, like a professor faced with brilliant students who have disappointed him. Though he was, in fact, not consistent in the way he described Pythagorean ideas about numbers, and was never able to define what he thought “speak like a Pythagorean” and say “the One is substance” meant, it is clear that he feared theirs was a sadly earthbound, material view. “The Pythagoreans introduced principles,” said he, that could have led them beyond the perceptible world to the higher realms of Being, but then they only used them for what is perceptible, and “squandered” their principles on the world itself as though nothing else existed besides “what the sky encloses.”3
His was, in truth, an earthbound interpretation of the Pythagoreans. Their attempt to give numbers a physical role in creation may look as naive to us as it did to him, but they faced difficult questions: What were numbers, really? What was their role—their power—in creating, sustaining, and controlling the physical universe? Those questions have never been answered. Humans have all but given up on them. If numbers underlie, even constrain, physical reality, as the Pythagoreans thought was the case, then where, precisely, is the connection? How do mathematics and geometry exert their grip on the universe? The Pythagoreans tried to find ways to answer such questions, and at the root of their thinking, spanning the years that led to the time of Aristotle, lay that first realization that “what the sky encloses” was much more mysteriously and wondrously interconnected and infused with rationality than anyone had recognized before.4 A cosmos governed by numbers—no matter how everyday-perceptible it also was, or whether you could figure out how it got built—was a mind-haunted cosmos. Where to go from there, with this treasure that had fallen into their hands? That was new, unknown territory, and the Pythagorean exploration of it was always a work in progress.
In an age when abstract thinking was supposed to be more prevalent than in the sixth century B.C., Aristotle seems, in his interpretation of the Pythagoreans and his frustration with some of their ideas, to have been insisting, for them, that they thought of numbers only as something concrete and physical. He was apparently blind to any other way of interpreting their thoughts and would allow them little sophistication and subtlety. Complicating this issue, the Greeks used the same word for “same” and “similar,” making it difficult even to have a meaningful disagreement about whether the Pythagoreans meant a number was something or was “something like it” or was a symbol for it.
Aristotle summed up his interpretation of the Pytha
gorean view of numbers more sympathetically in two statements: “Having been brought up in it [mathematics], they came to believe that its principles are the principles of existing things.” And (transmitted through Iamblichus) “Whoever wishes to comprehend the true nature of actual things, should turn his attention to these things, the numbers and proportions, because it is by them that everything is made clear.” As Burkert paraphrased Aristotle’s Metaphysics: “Number is that about things which can, with a claim to truth, be expressed; nothing is known without number.”
One approach the Pythagoreans had taken, Aristotle found, was to express the creation process in a “table of opposites.”
Limited
Unlimited (recall that the One, when it arises, will have a share in both)
odd
even (recall that the One will be both odd and even)
One
plurality
right
left
male
female
resting
moving
straight
crooked
light
darkness
good
bad
square
oblong
Nothing in the table could be linked with Plato’s Indefinite Dyad in a clear way. In Plato’s creation scheme the One and the Indefinite Dyad were there first, with limit and unlimited “inherent in their nature.” On these points, if Aristotle’s interpretation was correct, Plato chose not to follow the Pythagoreans, misunderstood them, or transformed their ideas to suit himself.
The Pythagoreans apparently thought creation had to involve both drawing together (of the limiting and the unlimited) and separation (as numbers and pairs of opposites arose from the One), and the universe could only exist if things were different from one another—an idea found in many ancient creation accounts. In Genesis, God separated light from darkness, the water above the earth from the water below the earth, and sea from dry land; Adam and Eve ate from the tree of the “knowledge of good and evil.” In Aristotle’s interpretation of the Pythagoreans, the One was not undifferentiated unity, like the unlimited. It was harmony of many different things whose differences were necessary in order for anything to exist in the way humans experience the world.*