The Music of Pythagoras (24 page)

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 Pythagorean 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.”

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.
*

Disappointingly, Aristotle did not really answer the question whether Plato’s view of the relationship between the ideal and the material world was derivative of the Pythagoreans, or original, or somewhere in between. What Aristotle concluded has hung in the air for centuries, with the answer depending on what he meant by one ambiguous Greek sentence. Burkert cut to the heart of the matter:

Again and again it becomes clear that the Pythagorean doctrine cannot be expressed in Aristotle’s terminology. Their numbers are “mathematical” and yet, in view of their spatial, concrete
nature, they are not. They “seem” to be conceived as matter and yet they are something like Form. They are, in themselves, Being, and yet are not
quite
so.
⁵

Guthrie put it more simply: “By the use of his own terminology, Aristotle imports an unnecessary confusion into the thought of the early Pythagoreans. It is no use his putting the question whether they employ numbers as the ‘material’ or the ‘formal’ causes of things, since they were innocent of the distinction.”
⁶

Aristotle gave what is probably the most reliable description of the pre-Platonic Pythagorean concept of “music of the spheres,” grumbling that “it does not contain the truth,” though he admitted it was “ingeniously and brilliantly formulated.” He explained that the Pythagoreans realized that all harmonious-sounding musical intervals were the result of certain numerical ratios in the tuning of an instrument, so “number”
was
“harmony.” The same numerical ratios determined the arrangement of the cosmic bodies, resulting in a “harmony of the spheres.” Here, said Aristotle, was “what puzzled the Pythagoreans and made them postulate a musical harmony for the moving bodies”:

It seems that bodies so great must inevitably produce a sound by their movement. Even bodies on Earth do that, although they are not so great in bulk or moving at so high a speed, or so many in number and enormous in size, all moving at a tremendous speed. It is unthinkable that they should fail to produce a noise of surpassing loudness. Taking this as their hypothesis, and also that the speeds of the stars, judging from their distances, are in the ratios of the musical consonances, they affirm that the sound of the stars as they revolve is concordant.

Some heavenly bodies appear to move faster than others. Aristotle wrote that the Pythagoreans had arrived at the idea that the faster the motion, the higher the pitch it produced, and they had taken this into consideration when allowing the ratios of the relative distances between the bodies to correspond to musical intervals. With the full complement of heavenly bodies, the result was a complete octave of the diatonic scale.
*

What surprises is that Aristotle or anyone could think the eight notes of the scale heard simultaneously would be harmonious. The sound would not be beautiful. There would be cacophony in the heavens. Humans should be glad they cannot hear it. Pity Pythagoras, who, legend says, could! The explanation cannot be that
harmonia
did not imply audible sound, for Aristotle thought the Pythagoreans believed planetary movement produced actual tones. He never explained how it could be beautiful, but he did give what he thought was the Pythagorean explanation—different from Archytas’—for why ordinary humans do not hear it:

To solve the difficulty that no one is aware of this sound, they account for it by saying that the sound is with us right from birth and has thus no contrasting silence to show it up; for voice and silence are perceived by contrast with each other, and so all mankind is undergoing an experience like that of a coppersmith, who becomes by long habit indifferent to the din around him.

A
T THE TIME
of Aristotle and in later antiquity, it was generally assumed that if one mentioned “Pythagorean mathematics,” an educated person would know what that meant, but in fact the meaning was vague, apparently referring to a tradition that thought it inspiring to discover hidden, true relationships of the sort that, once found, seemed inevitable. Since the evidence about what sixth- and fifth-century Pythagorean mathematics were like is so sparse, we are at a loss to know how authentically Pythagorean this so-called Pythagorean mathematics was. To modern eyes, its vestiges seem feeble by comparison with Euclid’s
Elements
, which appeared around 300
B.C
. Did it really reflect a naive mathematics of Pythagoras himself, and his associates? Or was it “a dilute, popularized selection from what had been originally a rigorous mathematical system”?
⁷
Perhaps it was a hodgepodge of what survived from early, primitive mathematical thought from several sources, mistakenly lumped under the heading “Pythagorean”? Maybe a much more authentically Pythagorean, lively
mathematici
heritage had moved through Archytas to influence Euclid, while this older, calcified, fading mathematics limped alongside, still bearing the name “Pythagorean.”

There are also differences of opinion about whether there is valid reason to call the five regular solids that Plato featured “Pythagorean” solids.
⁸
*
The issue is not a simple one, for “knowing about” the solids, or “discovering” them, or “trying to figure them out,” are not the same as “giving them a full mathematical description” or being able to prove that they are the only possible perfect solids. It is uncertain which achievement deserves to be rewarded with having one’s name attached to it.