The Music of Pythagoras (23 page)

Tell me, boy, is not this our square of four feet? You understand?

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.

Fourth Century
B.C
.

W
HILE 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 time, 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.