The Pythagoreans

Most people know of Pythagoras by the theorem that bears his name; we will speak of that below. Of Pythagoras himself we know relatively little. He came from the Greek island of Samos, but left there around 530 BCE to escape political tyranny. From Samos he went to Croton, in southern Italy and for a time was a respected and influential figure. His political influence was not altogether well-received, however, and he fled Croton for the city of Metapontium, where he died.

Pythagoras was the leader of a religious fraternity -- perhaps some would have said cult -- while he was in Croton. The core of the religious doctrine was that the soul -- the pure and noble part of us -- is mired in the body, from which it longs to escape. Consequently, Pythagoreans engaged in various practices that were intended to purify the soul so that it could escape from the body. Some of these practices were dietary; Pythagoreans were vegetarians and also, oddly enough, abstained from beans. (Why? I have heard various accounts. The most intriguing has to do with the effects of beans on the lower digestive tract. The Pythagoreans, like many of their contemporaries, though of the soul as essentially airy in nature -- gaseous if you will. Consumption of beans, therefore, could lead to a loss of soul. I do not know if this is true, but being rather fond of chile and black bean soup, not to mention Boston Baked Beans, I sincerely hope not.) According to Pythagoras, the soul is immortal, and he famously subscribed to a doctrine of transmigration or reincarnation. As in some species of Eastern thought, he held that the soul could be reborn not just in human garb, but in the body of an animal. This was the reason for Pythagorean vegetarianism; animals are kin to us; they have souls too.

Pythagoras's contemporary, Xenophanes, wrote:

    They say that once when a puppy was being whipped, Pythagoras, who was passing by, took pity on it, saying "Stop! Do not beat it! It is the soul of a friend; I recognize his voice."

The Pythagoreans followed a number of rules that seem to have grown from a mixture of motives. Here is a sample from Kirk and Raven

Let the rules to be pondered be these:

    1. When you are going out to a temple, worship first, and on your way neither say nor do anything else connected with your daily life.
    2. On a journey, neither enter a temple nor worship at all, not even if you are passing the very doors.
    3. Sacrifice and worship without shoes on.
    4. Turn aside from highways and walk by footpaths.
    8. Stir not the fire with iron.
    10. Help a man who is loading freight, but not one who is unloading.
    11. Putting on your shoes, start with the right foot; washing your feet, with the left.
    12. Speak not of Pythagorean matters without light.
    21. Let not a swallow nest under your roof.
    22. Do not wear a ring.
    26. Be not possessed by irrepressible mirth.
    29. When your rise form bed roll the bed-clothes together and smooth out the places where you lay.
    34. Leave not the mark of the pot in the ashes.
    37. Abstain from beans.
    39. Abstain from living things.

Some of these rules express the Pythagorean view of the kinship of all life; others cannot. Kirk and Raven write of these others:

    A few, such as 'be not possessed of irrepressible mirth...' would seem to be nothing more than commonplace ethical and religious reflections. A larger group... are probably descended from primitive folk-taboo. Others, such as 'sacrifice and worship without your shoes on'... clearly concern ritual purity. And finally, some, such as 'when you rise from bed, roll the bed clothes together and smooth out the place where you lay,' seem to owe their origin to sympathetic magic.

As Kirk and Raven note, if this were all there were to Pythagoreanism, it would be indistinguishable from a mystery religion of a sort that flourished in that time. But the Pythagorean concern with mathematics and science distinguishes it from the mystery cults and constitutes its peculiar contribution to the history of ideas. According to one ancient source (Diogenes Laertius) Pythagoras was the first to all himself a philosopher -- a lover of wisdom -- and rational inquiry played a key part in the Pythagorean life: it was part of the process of purifying the soul. Mathematics was at the heart of this rational inquiry.

This may seem strange at first, but Robinson points out that it is really not so strange after all. In mathematical contemplation, we come into direct contact with the eternal; mathematical truth does not depend on the vagaries of matter. And anyone who has ever appreciated the beauty of mathematics must surely be able to grasp at least a glimpse of what the Pythagoreans had in mind.

The most famous of the Pythagorean mathematical discoveries, attributed to Pythagoras himself, is the Pythagorean theorem, which tells us that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. Some ancient authorities clam that when Pythagoras made this discovery, he sacrificed an ox. However, Plutarch claims that the sacrifice was given after the discovery of a much more difficult theorem: the proof that it is possible to construct (i.e., with straight-edge and compass) a triangle equal in area to one given triangle and similar to another given triangle. Be that as it may, the likely form of the original proof of Pythagoras's theorem is characteristic of the way in which Pythagorean mathematics relied on rations. Here is the proof that Proclus attributes to Pythagoras many centuries later. I have borrowed here from Robinson.

In the triangle ABC above, the angle at A is a right angle. The line from A to D is perpendicular to BC. This means that the triangles ABC, ABD and ACD are all right triangles. Further, each triangle shares some vertex with each other. For example, the vertex B is common to triangles ABC and ABD. Since all triangles sum to 180 degrees -- a Pythagorean discovery -- this means that all three triangles are similar and hence have their sides in the same ratios.

Now consider:


AB    BC
--  = --
BD    AB
Why? Because AB is the hypotenuse of triangle ABD and BD is its short side. But BC is the hypotenuse of triangle ABD and AB its short side. So from the above equality, we get

ABxAB = BCxBD

Similar reasoning shows that ACxAC = CDxBC

ABxAB, of course, is AB2. So we have

AB2 + AC2 = BCxBD + BCxCD

= BC(BD + CD)

But BD + CD is, as the diagram shows, just BC. So our equation becomes

AB2 + AC2 = BC2

That is, the square of the hypotenuse of the large triangle is the sum of the squares of the two remaining sides.

Ratios were also an essential part of Pythagorean musical theory. The Pythagoreans discovered that the musical intervals of the octave, fifth and fourth (think, respectively, of the first two notes of "Somewhere Over the Rainbow,", the first and second "twinkle" in "Twinkle, Twinkle, Little Star ," and the opening "Day is done..." in "Taps") we characterized by the intervals 2:1, 3:2 and 4:3. What this means in physical terms is that if you stop a string at the mid-point (ratio of string to plucked portion 2:1) you raise the pitch an octave. If you stop it at the 2/3 point and pluck the longer portio n(ratio 3:2) the pitch is raised by a fifth. And if you stop it at the 3/4 mark (ratio 4:3) and pluck the longer portion, the pitch is raised by a fourth. What was exciting abut this discovery was that it showed that a salient feature of the physical world was the expression of a mathematical relationship. And while it may be obvious to us, it was by no means obvious 2500 years ago that mathematics could describe the workings of the world in this way. Robinson writes:

    These [intervals] correspond to the octave, the major fifth and the major fourth [NOTE: we would say perfect fifth and fourth] -- the chief "consonances" of Greek music. But they do not merely correspond to them; they make them what they are. And it was in recognizing this that the genius of Pythagoras lay, for it opened his eyes to the possibility that all order is at bottom capable of being understood and expressed in terms of number.

The fact that the numbers 1, 2, 3 an 4 suffice to characterize these relationships may have been important for another Pythagorean concept: the role of the number 10. For the Pythagoreans, this was a sacred number. One ancient commentator (Aetius) writes:

    Ten is of the very nature of number. All Greeks and all barbarians alike count up to ten, and having reached ten revert again to the unit. And again, Pythagoras maintains, the power of the number ten lies in the number four, the tetrad.

The reference to the tetrad has to do with the way the Pythagoreans represented the number 10. As is obvious, it is the sum of the first four numbers:

1 + 2 + 3 + 4 = 10

and the Pythagoreans represented this fact by this figure:

 
                               *
                             *   *
                           *   *   *
                         *   *   *   *
known as the Tetractys of the Decad. So 4 in some sense contains the essence of 10.

Other facts about the way in which the Pythagoreans represented numbers are important for their cosmological views. The Pythagoreans made a fundamental distinction between odd and even numbers -- a distinction that they took to be of cosmic significance. Numbers were displayed in the form of gnomons -- a gnomon is a carpenter's square. The odd numbers formed a series thus:

                                         
                                    * * * *             
                         * * *            *                
                * *          *            *
         *        *          *            *   etc.
and the even numbers thus:

                                       
                                        * * * * *  
                          * * * *               *             
                * * *           *               *  
        * *         *           *               *  etc.
The odd series, as we can see plainly, has a regularity to it that the even series does not. Each successive gnomon, after the "one," is similar to the ones that came before: it is square. Not so for the even number series.

We can add gnomons to one another to produce a series of figures that also interested the Pythagoreans. The two diagrams below illustrate.

                                                                  
          * * * *                             
          * * *|*                            
          * * *|*                          
          * * *|*                           
                                       
          * * * * *                             
          * * * *|*                            
          * * * *|*                            
          * * * *|*
The top figure illustrates an important property of the odd numbers: the sequence of sums: is the sequence of squares of the successive integers:

1, 4, 9, 16...

The difference between the odd and the even numbers was literally of cosmic significance to the Pythagoreans. For them, numbers were literally the stuff out of which the universe was constructed. This may be hard for us to imagine, but at this point in intellectual history, the concept of something that exists outside of space and time had not yet evolved. The Pythagoreans seem to have thought of numbers as physical, rather like points. The Universe contained two principles, the Unlimited and the Limited. The unlimited seems to have been associated with the Void, and Aristotle wrote:

    The Pythagoreans, too, said that void exists, and that it enters the universe from the infinite breath as if it were being inhaled. It is the void which keeps things distinct, being a kind of separation and division of thngs that are next to each other. This is true firt and foremost of numbers, for the void keeps them distinct.

It seems that the Pythaogoreans thought of numbers as being like the stones that they laid out on flat surfaces to make the squares and rectangles and gnomons in our preceding diagrams. The void, then, is indeed, what makes distinct numbers possible; without it there would be no distinction among the elements that compose the number. In fact, Robinson insists that the void is as much part of the number as is the set of points.

But what does evenness have to do with unlimitedness and oddness with limit? Kirk and Raven understand it in terms of the fact that as we build up larger and larger square numers by adding the successive gnomons associated with the odd numbers, the resulitng figure or shape is still square. However, as we add the gnoms associated with the even numbers, we get a rectangle whose ratio of length to breadth is ever-changing. No two of these figures are similar; rather, we have an infinite series of distinct rectangles. Thus, evenness bespeaks unlimitedness.

The Limited and the Unlimited are one pair from a table of ten pairs of opposites

limit unlimited
odd even
one many
right left
male female
rest motion
straight curved
light darkness
good bad
square oblong

It seems that the first member of each pair was somehow supposed to be superior to the second, though it is not clear how strictly this should be taken. As we have already noted, the unlimited in the form of the void is necessary for the very exisence of number. And while odd numbers have a special role as introducing liit, the numbers 4 and 10, both of which are even, were understod as having a special perfection.

In any case, the development of the universe itself is a combination of the development of number and the work of the void in keeping things distinct. The one -- the Unit -- came into being by a process that is nowehre detailed in any of our information about the Pythagoreans. This Unit is depicted as drawing in and limiting the unlimited. And whether as a result of this drawing in or not, the Unit grows and divides, giving rise to Two and successively, to the rest of the numbers. The process is one that reminds me, at least, of the process of emanation that we will encounter later on when we discuss the Neo-Platonists. Three-dimensionality also comes into being as the numbers develop. A line is determined by two points; a surface is determined by three points; and a volume is determined by four points. Whether the process was one of generating the figures point, line, triangle, tetrahedron or of generating point, line, plane, solid is less clear from the sources.

But what do we make of all this? The picture is quite different from that presented to us by the natural philosophers. The universe is not based on any ordinary physical substance. And while the Void may be reminiscent of Anaximander's apeiron, there is no attempt to extract the usual elements from the Void. Rather, we have a mathematical cosmos produced by a process of mathematical progression. Number is all, and all the important properties of things are determined numerically. In fact the cosmos itself reverberates with the harmony of the spheres, a result of the mathematical relationships among the orbits of the planets. (This is the "music of the spheres.")

The results of this approach can be both prescient and absurd. It seems that because the Pythaogoreans saw 10 as the perfect number, they believed that the number of heavenly ojects should be 10. But the only ones known are Earth, Moon, Sun, Mercury, Venus, Mars, Jupiter and the sphere of the fixed stars -- a total of nine. So they apparently posited another body -- counter-earth -- to bring the number to 10. The reason we don't see this body is that the sun lies directly between earth and counter-earth. Justice was also understood as having a numerical character. There seems to have been some association betwen justice and the number 4, which is the first square number, but more importantly -- and more characteristically -- justice was associated with proportion, or with the gemometric mean. (b is the geometric mean between a and b if a/b = b/c.) In fact, proportion was the underlying concept in Pythagorean views of friendship, of governance and of the soul. The soul, indeed, was conceived of as a sort of atunement.

The philsopher Ludwig Wittgenstein once referred to certain sorts of philosophical speculation as language gone in holiday. You may feel that Pythagoreanism is number gone on holiday. However, even if the details were not accepted, the spirit of Pythagoreanism -- the attempt to understand the world in terms of number -- exerted a deep and lasting influence on Western thought. And the more mystical aspects of Pythagorean attitudes toward number played an important role in the development of various strains of magical theory, as we shall see.

© Copyright Allen Stairs, 1998
stairs@glue.umd.edu

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