PYTHAGORAS OF SAMOS
The life of Pythagoras can be told rather briefly. Born in Samos, an island in the Aegean off the Ionian coast, where he is said to have lived until his fortieth year, perhaps 532/1 B.C., he then fled the tyranny of Polycrates. He then went to Croton in southern Italy where he was well received and, according to tradition, exercised no little political influence. His pupils there were said to have numbered some three hundred. The citizens of Croton finally revolted and set fire to a house in which the elder Pythagoreans were meeting, but Pythagoras himself escaped. He went to Metapontium where he died many years later.
The Pythagorean community was structured more like life in a religious order. The community embraced men and women. There was a common doctrine but it was not to be divulged to outsiders; indeed, we have no mention of a Pythagorean writing anything until Philolaus at the end of the fifth century before our era. Pythagoras himself wrote nothing, but the practise of the society was to attribute every doctrine to its founder. Renowned for their secrecy, the Pythagoreans and Pythagoras himself early became the object of mystery and speculation. Plato and Aristotle, consequently, are not in the habit of saying that Pythagoras said such-and-such, but that he is said to have said such-and-such, or, more usually, that the Pythagoreans or some Pythagoreans say such-and-such. There seems to be no doubt, however, that Pythagoras did live.
One very early testimony is that of Xenophanes who says of Pythagoras....
"Once they say that he was passing by when a puppy was being whipped, and he took pity and said: 'Stop, do not beat it; for it is the soul of a friend that I recognized when I heard it giving tongue."
Thus, Pythagoras is said to have held the doctrine of the transmigration of souls; indeed, he is said to have remembered four previous incarnations of his own! Much later, Porphyry summarized his doctrine thus: (1) he believed in the immortality of the soul; (2) that it changes into other kinds of living things; (3) that events occur in definite cycles such that the time will come when I will once more be writing these words and you will be reading them. That is, nihil novi sub sole: nothing is new and unique; (4) that all living things should be regarded as akin. There is a persistent tradition, beginning with Herodotus, that such doctrines were imported into Greece from abroad. Herodotus claimed that the doctrine of the transmigration of souls was borrowed from the Egyptians; but the Egyptians seem never to have held the belief themselves.
The belief that human souls could show up in other living things is connected with certain taboos or prohibitions observed by the Pythagoreans, such as abstention from meat and an injunction against associating with butchers. The testimony on these points in conflicting, however, since Pythagoras is said to have sacrificed an ox when he discovered the Pythagorean theorem. Later writers listed rules of conduct which were said to guide the Pythagorean community -- rules such as abstention from beans, and of smoothing out the impression left on one's bed, not wearing rings, not letting swallows nest under one's roof, etc. One that still has a peculiar force is this.... "Speak not of Pythagorean matters without light."
This is but half of the Pythagorean story, however; for to this mystic fervor was coupled an interest in science, particularly mathematics. Indeed, the society itself divided into two groups after the death of Pythagoras, the "Acousmatics" (hearers) and the "Mathematicians" (knowers). The former probably concentrated on the religious aspect of the society, while the latter devoted themselves to the more scientific aspect.
The figure of Pythagoras is a shadowy one, not, as with the philosophers considered earlier, because of scanty information, but almost by design. He is the founder, the master, to whom all doctrines are attributed. (The Pythagoreans were famous for introducing statements with, "He himself said so.") And who is regarded as more than human, the son of Hermes. In one legend, for example, he is described as revealing his golden thigh. Soon the historical figure is lost behind the stories and our knowledge of what he taught is reduced to a view of the kinship of all things and an interest in mathematics which, apart from some mystical interpretations on the power of numbers, seems genuinely scientific. There seems to be as well the identification of things with numbers, leading perhaps to a belief in the harmony of all things -- a belief called into question by the discovery of the incommensurability of the diagonal and side of a square.
A remark attributed to Pythagoras by Diogenes Laertius describes perhaps for the first time an important aspect of what had been begun by Thales and was carried on by subsequent thinkers....
"Life, he said, is like a festival; just as some come to the festival to compete, some to ply their trade, but the best people come as spectators, so in life the slavish men go hunting for fame or gain, the philosophers for the truth."
"Life, he said, is like a festival; just as some come to the festival to compete, some to ply their trade, but the best people come as spectators, so in life the slavish men go hunting for fame or gain, the philosophers for the truth."
We have here a distinction between the practical pursuits of men, the mark of which is activity and striving, and the pursuit of truth, described in terms of seeing or understanding for its own sake.
Because the philosopher wants to see, he must purge himself. The Pythagoreans held that as medicine purges the body, so does music purge the soul; and music -- proportioned sound -- is number. Number is the nature of all things. We must in this connection consider a lengthy passage from Aristotle....
"Contemporaneously with these philosophers, and before them, the Pythagoreans, as they are called, devoted themselves to mathematics; they were the first to advance this study, and having been brought up in it they thought its principles were the principles of all things. Since of these principles numbers are by nature the first, and in numbers they seemed to see many resemblances to the things that exist and come into being -- more than in fire and earth and water (such and such a modification of numbers being justice, another being soul and reason, another being opportunity -- and similarly almost all other things being numerically expressible); since, again, they saw that the attributes and the ratios of the musical scales were expressible in numbers; since, then, all other things seemed in their whole nature to be modeled after numbers, and numbers seemed so be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number."
Here Aristotle expresses what he learned of the Pythagoreans. They were interested in mathematics, Aristotle says, and this indicates an interest in the abstract, the formal, a science which does not have for its object the sensible things around us. Now the Pythagoreans, Aristotle says, thought of numbers as the stuff out of which things are made, as the Ionians had spoken of air and water as the primal matter out of which all things are fashioned. This is a difficult transition, and Aristotle gives us a few preliminary clues as to how it should be understood. Justice is a number, as is soul, and all other things; they are different arrangements of units and are thus made up of numbers. When Aristotle says that the Pythagoreans noticed that the attributes of the musical scale were expressible in numbers, he is speaking in terms of the recognition of a distinction which was most likely not known at the outset of the Pythagorean school. There is an unavoidable tension in the Aristotelian passage between the view that number is material cause and that it is somehow formal, applied to natural things, but itself different from natural things. Because they had not adequately distinguished between material and formal causes, the Pythagoreans seem to be making the same thing do service as both kinds of cause; number is that out of which things are made, and the particular arrangement of the elements is their nature. Consequently, we are faced with a doctrine according to which there is no distinction between natural science and mathematics, according to which the study of number tells us about the natural world as natural world. For the elements of number are the elements of all things.
What is meant by the elements of number?
"Evidently, then, these thinkers also consider that number is the principle both as matter for things and as forming their modifications and their permanent states, and hold that the elements of number are the even and the odd, and of these the former is unlimited, and the latter limited; and the 1 proceeds from both of these (for it is both even and odd), and number from the 1; and the whole heaven, as has been said, is numbers."
In this continuation of the previously quoted passage, Aristotle recognizes that number is matter and form for the Pythagoreans. What we must understand, if we are to grasp the identification of physics and mathematics, is the notion of oddness and evenness as the "elements of number," the relation between these and the number one and the numbers proper which follow from it....
In this continuation of the previously quoted passage, Aristotle recognizes that number is matter and form for the Pythagoreans. What we must understand, if we are to grasp the identification of physics and mathematics, is the notion of oddness and evenness as the "elements of number," the relation between these and the number one and the numbers proper which follow from it....
"Further, the Pythagoreans identify the infinite with the even. For this, they say, when it is taken in and limited by the odd, provides things with the element of infinity. An indication of this is what happens with numbers. If the gnomons are placed round the one and without the one, in the one construction the figure that results is always different, in the other it is always the same."
When we grasp that numbers are conceived in terms of different configurations of units in space, we can see how the Pythagoreans could have come to believe that the elements of number are the elements of all things. The crude way of making this identification was to take pebbles and form with them a picture of an object and, by counting the pebbles used, assign the number-nature of the object.
From Aristotle's account, we know that according to the Pythagoreans numbers have magnitude....
"Now the Pythagoreans also believe in one kind of number -- the mathematical; only they say it is not separate but sensible substances are formed out of it. For they construct the whole universe out of numbers -- only not numbers consisting of abstract units; they suppose the units to have spatial magnitude . . ."
The illustrations we have seen indicate the identification of the arithmetical unit and the geometrical point; the point, however, while considered to be indivisible, is not without extension. Because of this, physical things could be looked on as in someway composed of such units as of their matter.
Before looking into the cosmology which followed from this view of mathematics, however, we must consider an important point made by Aristotle in his account of Pythagoreanism at the outset of his Metaphysics....
"Other members of this same school say there are ten principles, which they arrange in two columns of cognates -- limit and unlimited, odd and even, one and many, right and left, male and female, resting and moving, straight and curved, light and darkness, good and bad, square and oblong. In this way Alcmaeon of Croton seems also to have conceived the matter, and either he got his view from them or they got it from him . . . For he says most human affairs go in pairs, meaning not definite contraries such as the Pythagoreans speak of, but any chance contrarieties, e.g., white and black, sweet and bitter, good and bad, great and small. He threw out indefinite suggestions about the other contrarieties, but the Pythagoreans declared both how many and which their contraries are."
Alcmaeon of Croton, who is thought to have flourished at the beginning of the fifth century, was primarily concerned with medical matters, and one indication of his interest in contraries is to be found in his view that health is a balance of moist and dry, cold and hot, sweet and bitter, and so forth, and that sickness is the result of one contrary getting the upper hand. Certain Pythagoreans, as Aristotle says, were more systematic in pursuing the recognition of the role of opposites in the world and tried to summarize in ten oppositions the major types. We have already seen the reason for linking limit, odd and square, on the one hand, and unlimited, even and oblong, on the other. Even numbers are unlimited or infinite not, as Simplicius held, because the even number is infinitely divisible -- this is manifestly absurd -- but only in the sense that nothing prevents their being divisible into halves. The odd number is limited since, by adding one to an even number it prevents such equal division and, like three, the first odd number, has "a beginning, a middle and an end." Explanations of other contraries in the right or left column of opposites emerge when we look at Pythagorean cosmology.
Now this has in common with Ionian thought the fact that it is a way of explaining how the world began; "they are describing the making of a cosmos and mean what they say in a physical sense," as Aristotle adds. Aristotle is contrasting the Pythagorean view with the Platonic one according to which there are subsistent numbers, existing apart from physical bodies. Aristotle is under no illusions about the Pythagorean view of the extent of reality. Already in his account of their doctrine at the outset of the Metaphysicshe wrote....
"They employ less ordinary principles or elements than the physical philosophers, the reason being that they took them from non-sensible things (for the objects of mathematics, except those of astronomy, are without motion); yet all their discussions and investigations are concerned with Nature. They describe the generation of the Heaven, observing what takes place in its parts, their attributes and behavior, and they use up their causes and principles upon this task, which implies that they agree with the physicists that the real is just all that is perceptible and contained in what they call 'the Heaven'."
We must keep in mind the identification made by Pythagoras himself of sounds and numerical ratios -- identification, not application -- lest we delude ourselves into thinking that the Pythagoreans have turned from the objects which concerned Ionian philosophy to other, more real entities. It is an appraisal of physical reality in both cases, not a conscious change of objects. The first stage of the cosmogonical process which Aristotle attributes to the Pythagoreans, consists of the formation of the first unit, though elsewhere he objects that the Pythagoreans are at a loss to describe the nature of his formation. Subsequently, the unit draws in the unlimited and by imposing limits on it produces other units. As to the formation of the first unit, Aristotle mentions several possibilities: it could have been formed of planes or of surface, of seed or of some other elements. If composed of planes or surfaces, the first unit would be a solid. The supposition that the constituents of this first unit are seed would fit in with the location of "male" in the same column of opposites as limit. Its complement, the female, is the unlimited which it "draws in." The picture delineated becomes very much like the cosmogony of Anaximines when we learn that the unlimited is air and that the first unit breathes it in. Air or void is drawn in and keeps things apart, for it seems likely that the first unit grows and splits and is kept apart by the void or air. The continuation of this growth results in the universe we know.
The Pythagorean view of the universe represents a significant shift from the geocentric view of Ionian philosophy. Fire, not earth, is the center of things and the earth is one of the stars for which night and day is caused by its circular motion around the central fire. They are said to have invented a planet, the so-called counter-earth, to bring the number of planets to ten, the perfect number. The counter-earth follows the earth in its path around the sun, always remaining invisible to us because of the bulk of the earth. It is thought that the notion of the counter-earth dates from the time of Philolaus; another astronomical theory, that of the "harmony of the spheres" is considered to be of earlier origin in the school....
"From all this it is clear that the theory that the movement of the stars produces a harmony, i.e., that the sounds they make are concordant, in spite of the grace and originality with which it has been stated, is nevertheless untrue. Some thinkers suppose that the motion of bodies of that size must produce a noise, since on our earth the motion of bodies far inferior in size and in speed of movement has that effect. Also, when the sun and the moon, they say, and all the stars, so great in number and in size, are moving with so rapid a motion, how should they not produce a sound immensely great? Starting from this argument and from the observation that their speed, as measured by their distances, are in the same ratios as musical concordances, they assert that the sound given forth by the circular movement of the stars is a harmony."
We do not hear this sound only because we have always heard it and are unable to contrast it with any opposed silence.
The Pythagoreans have lumped together unit and a point with magnitude, from which point the line is generated and so on to solids. Just as no differentiation is made between number and extension, so no distinction is recognized between mathematical and physical bodies. The coming into being of our world is likened to the generation of the number series and the series of solids. The universe has grown from a primal unit which breathes in air or void and then splits up, imposing limits on the previously unlimited. The unit is considered to be male, the unlimited female. Earth is not at the center of the universe, but swings in a circular motion around a central fire, which motion produces day and night. In their movements, the heavenly bodies produce a wonderful music which has been singing in our ears since birth.... and so is imperceptible by us.
Ralph McInerny (selected).