The Pythagorean Doctrine of Number

The Pythagorean Doctrine of Number

In today’s essay I would like to continue our investigation of examination of the core doctrines of Pythagoreanism. Specifically, this essay will explore the Pythagorean theory of number, harmony, and cosmos. That number would play an important role in Pythagoreanism should probably not surprise us, given the famous theorem bearing the Pythagorean name and the fact that Pythagoras’s inner circle of students was called the mathematikoi. Yet, if we take time to consider the details of the Pythagorean theory of number, we will come to be astonished by its power, beauty, and historical impact. For Pythagoras was one of the first western philosophers to posit that the visible physical world depends upon an invisible world of number, a world grasped by pure reason rather than physical sight.

Three Arguments for the Ontological Priority of Number

One of our earliest and most detailed surviving accounts of the Pythagorean doctrine of number comes from Aristotle. In Metaphysics book I, he claims that:

“The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this study, but also 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 modifications and the ratios of the musical scales were expressible in numbers;-since, then, all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to 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. And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangement of the heavens, they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent. E.g. as the number 10 is thought to be perfect and to comprise the whole nature of numbers, they say that the bodies which move through the heavens are ten, but as the visible bodies are only nine, to meet this they invent a tenth–the ‘counter-earth’. We have discussed these matters more exactly elsewhere.” (Aristotle, Metaphysics, I.5 trans. Ross).

Here Aristotle claims that Pythagoreans offered three arguments for their thesis that numbers are ontologically fundamental, “that its principles were the principles of all things”. Before examining these arguments, it would perhaps be beneficial to explain what it is for something to be ontologically fundamental. Briefly stated, one thing is more ontologically fundamental than another, if it is more real than that other thing. An illustration may be helpful here. Consider the example of an object and its shadow. A shadow, in some sense, exists. We can see it, chart its movements, measure its length, and determine how dark it is. But its existence is derivative. It depends upon the existence of the object casting the shadow, the light source illuminating the scene, and the background on which it falls. A physical object is, then, in this instance, more real than the shadow it casts and is thus more ontologically fundamental.

From a Pythagorean perspective, physical objects are themselves derivative entities existing in virtue of numbers. Just as shadows exist in virtue of physical objects, so physical objects exist in virtue of numbers.[1]

The First Principles Argument

Let’s now examine the three arguments for this claim that Aristotle attributes to the Pythagoreans.  The first argumentis that numerical principles are primary in nature. It is expressed in the claim that “of these principles numbers are by nature the first”. Let’s call this the first principles argument. The argument appears to turn on the fact that, at the most basic level of description, any natural object will prove to be a unity.  Rocks, flowers, and birds are what they are, and not something else. For example, a sapphire stone is one thing and not something else. It is distinct from the hand which might grasp it, the air which surrounds it, or a wall which might be seen in the background. The sone is a stone and not these other things. Yet, when we examine it more closely we see that the sapphire is also a multiplicity. For we can see that it is blue, and hard, and has a determinate shape. It is blue, and also hard, and also elliptical, etc. This stone then, as a natural object, is both a unity and a multiplicity. In traditional philosophical terminology, it is both a one and a many. The Pythagorean argument for the fundamentality of numbers then goes on to point out that our clearest understanding of oneness and manyness is provided by mathematics. In mathematics we grasp unity and multiplicity in their purest form as numbers, and since unity and multiplicity are the most basic features of natural objects, Pythagoreans conclude that the first principles of such objects are found in numbers.

The Similarity Argument

The second argument Aristotle attributes to the Pythagoreans turns on observed similarities between numbers and things. Let’s call this the similarity argument. The idea is that we can find similarities between numbers and other entities.  Porphyry, for example, records that Pythagoreans associated the number one with:

“Unity, Identity, Equality, the purpose of friendship, sympathy, and conservation of the Universe, which results from persistence in Sameness. For unity in the details harmonizes all the parts of a whole, as by the participation of the First Cause.” (Porphyry, Life of Pythagoras, § 49).

Since the goal of friendship is union with another in pursuit of shared values, it is similar to the unity signified by the number one. Likewise, since sympathy with another person is the ability to identify with his or her perspective — to be one in perspective — it is also akin to the number one. Pythagoreans bring forth many such correlations and support them with abstruse numerological speculations, associating two with change, diversity, and matter, three with actuality and temporality, and four with harmony, justice, and the union of body and soul. The argument then appears to be something like the following: There are similarities between features of entities and features of numbers. We have a better grasp of the features of numbers than of other entities. So we can use them to explain other entities, and, since numbers are explanatorily basic and similar to other entities in their core features, we should also think of them as ontologically basic.

The Harmony Argument

The third argument Aristotle attributes to the Pythagoreans is rooted in musical harmony. Let’s call this the harmony argument. In legend, Pythagoras is said to be the first person to have observed that musical harmony is grounded in mathematics. On one popular telling of the tale, Pythagoras was walking outside a blacksmith shop and heard various tones coming from hammers pounding away at objects. He then went and measured the hammers and noticed a correlation between their weight, and the tones they produced, with harmonies occurring at ratios of 1 to 2 (what we would today call the octave), 3 to 2 (today called the perfect fifth), and 4 to 3 (today called the perfect fourth). Other versions of the story say he made these discoveries by experimenting on the monochord, an instrument he is said to have invented. These stories highlight Pythagoras’s observation that the musical harmony heard with the ears flows from a deeper mathematical harmony perceived by the mind. Pythagoreans then extend this analysis to the harmony of nature. For the universe, according to Pythagoreans, is not a chaotic aggregate, but a harmonious, rationally ordered whole — a cosmos. And, since harmony is grounded in number, the cosmos itself is so grounded. Some legends claim that Pythagoras could literally hear this cosmic harmony. Porphyry, for example, records that:

“He [Pythagoras] himself could hear the Harmony of the Universe, and understood the universal music of the spheres, and of the stars which  move in concert with them, and which we cannot hear because of the limitations of our weak nature. This is testified to by these characteristic verses of Empedocles: ‘Amongst these was one in things sublimest skilled,/ his mind with all the wealth of learning filled./ whatever sages did invent, he sought; / and whilst his thoughts were on this work intent,/ all things existent, easily he viewed, through ten or twenty ages making search.” (Porphyry, Life of Pythagoras, § 30).[2]

These kinds of stories reinforce the argument by showing how natural harmony is akin to musical harmony, and since musical harmony is grounded in number, we can infer that the harmony of nature is similarly grounded.

Two Accounts of Numerical Grounding

Polarities: Even and Odd

Aristotle claims that in light of the first principles, similarity, and harmony arguments, Pythagoreans maintained:

“Since, then, all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to 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.” (Metaphysics, I.5, trans. Ross).

Pythagoreans thus depict reality as a vast harmonic whole dancing itself to completion according to the ratios set forth in the world of number. Numbers are, for Pythagoreans, the first principles and fundamental constituents of the universe. Yet the details of their account of how numbers ground reality are sketchy. In its most general formulation, the Pythagorean view, according to Aristotle, grounds reality in two contrasting principles, reminiscent of yin and yang in traditional Chinese philosophy.  Aristotle explains:

“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 plurality, right and left, male and female, resting and moving, straight and curved, light and darkness, good and bad, square and oblong” (Metaphysics, I.5 trans. Ross).

What we have then is a contrast between two sets, each of whose members are roughly equivalent to each other. The first set contains: {limit, odd, unity, right, male, resting, straight, light, good, and square}. And the second set contains: {unlimited, even, plurality, left, female, moving, curved, dark, bad, oblong}. These distinctions will likely sound bizarre to a contemporary audience. Why, for example, should we conceive of limit as good, and the unlimited as bad? Aren’t limitations and constrictions bad? Don’t they prevent us from progressing towards our goals and being all that we can be? To understand the Pythagorean view, it is helpful to work through the situation visually as they would have. Consider a rectangle built from an odd number. People in the ancient world would have done this using a device called a gnomon, or carpenter’s square. I start with one and produce a square. Then I add an odd number of units around it to produce another square, and then I add more to produce another square. Each time I have produced the same figure, a square. The ratio between the sides of the figure does not change no matter how many iterations I perform. The shape I produce is thus limited. Every time I perform the operation, a square will result. The shape is a unity, it is a square and not some other kind of rectangle. And it is at rest, not changing from one shape to another. And, to the classical mind, such constancy was associated with goodness. We can still see this today in the concept of integrity. The good man, for example, will act justly under a variety of situations. His character will not fluctuate with his mood or the expediency of the situation, but he will act with constancy and integrity.

Suppose we instead constructed a rectangle using an even number as a base. We would start with two, and the ratio between the sides would be 1 to 2. We could then construct another rectangle from that, this time with a ratio of 2 to 3. And if we constructed yet another, we would have a ratio of 3 to 4. Every time we construct a new rectangle, we get a different ratio between its sides. The odd is thus unlimited. We could construct an endless variety of rectangles from it. It generates a plurality and moves between various ratios in each iteration of the procedure. And, as a result, it is bad in the sense of being inconstant or lacking integrity. The resulting picture is one in which all of reality is built from this basic contrast between the odd and the even.

Limit, The Unlimited, and the One

Aristotle observes that other Pythagoreans attempted to provide a more detailed account of how the universe is grounded in number. According to this group:

“These thinkers also consider that number is the principle both as matter for things and as forming both their modifications and their permanent states, and hold that the elements of number are the evenand the odd, and that of these the latter is limited, and the former unlimited; and that the One proceeds from both of these (for it is both even and odd), and number from the One; and that the whole heaven, as has been said, is numbers.” (Metaphysics 1.5, trans. Ross).

From this terse description, we can extrapolate the following model. The position seems to be that there are two basic principles, the limitedand the unlimited. The limited constitutes the odd, and the unlimited the even. Together, these somehow generate the one, the basic numerical unit which combines within itself both odd and even, and hence both the limited and unlimited. The one then generates the rest of the number series, and number goes on to ground material reality.

Such a picture was elaborated upon by later Pythagoreans. For example, a text attributed to Iamblichus describes the one, or the monad, as follows:

“The monad is the non-spatial source of number. It is called ‘monad because of its stability, since it preserves the specific identity of any number with which it is conjoined. For instance, 3 x 1 = 3, 4 x 1 = 4: See how the approach of the monad to these numbers preserved the same identity and did not produced a different number.

            Everything has been organized by the monad, because it contains everything potentially: for even if they are not yet actual, nevertheless the monad holds seminally the principles which are within all numbers, including those which are within the dyad. For the monad is even and odd and even-odd” (Iamblichus, Theology of Arithmetic, 35).[3]

The grounding of reality in number was symbolized in the tetractys, a figure sacred to the Pythagoreans. This picture was of a triangle, composed of a series of points. Four at the bottom, then three, then two, and then one at the top. This figure was meant to signify the emanation of physical reality and the fundamental relations constitutive of cosmic harmony. Speusippus, a disciple of Plato who took over the academy after his death, describes the symbolism of the tetractys as follows:

“One is a point, two a line, three a triangle, and four a pyramid: these are the sources of the things which are of the same category as each of them” (Iamblichus, Theology of Arithmetic, 113).

So we have one indicating a single point, two a line connecting two points, three two dimensional shapes, and four three dimensional figures. And such three dimensional figures could ground physical objects by being associated with the four classical elements thought to constitute the sublunar world: earth, air, fire, and water. Aëtius, for example, notes:

“There being five solid figures, called the mathematical solids, Pythagoras says that earth is made from the cube, fire from the pyramid, air from the octahedron, and water from the icosahedron, and from the dodecahedron is made the sphere of the whole” (Aëtius cited in Gutherie, A History of Greek Philosophy Vol 1. The Earlier Presocratics and the Pythagoreans, 267.)[4]

Cosmology: Antichthon and the Central Fire

The number ten was so important that Pythagoreans, according to Aristotle, posited the existence of an unseen planet, antichthon or counter-earth, so that the heavenly bodies would be ten in number. Unlike other ancient cosmologies, Pythagoreans did not believe the earth to be the center of the universe. Rather, the earth and the rest of the solar system orbited a central fire. Pythagoreans believed that this cosmic center was the most important part of the universe. Aristotle records, for example, that:

“The Pythagoreans have a further reason. They hold that the most important part of the universe, which is the center, should be most strictly guarded, and name it, or rather the fire which occupies that place, the ‘Guard-house of Zeus’” (Aristotle, de Caelo, II.13.1, 293b1 trans. Stocks, modified).

Like the holy of holies in other religious traditions, it was specially guarded by the divine. For similar reasons, it was also called the tower and throne of Zeus. This central fire was for Pythagoreans the true light of the world, the sun’s beams being a mere reflection of its radiance. Philolaus, a notable Pythagorean, is, for example, said to have taught that:

“The sun is like glass. It receives the reflection of the fire in the cosmos and filters through to us both the light and the heat, so that in a sense there are two suns, the fiery substance in the cosmos and that which is reflected from the sun owing to its mirror-like character; unless one wishes to distinguish as a third the beam which is scattered in our direction by reflection from the mirror” (Aëtius II.20.12 cited in Gutherie, A History of Greek Philosophy Vol 1. The Earlier Presocratics and the Pythagoreans, 284-285.)

The sun then, with all its glory, is merely a mirror of this deeper light, one we cannot see directly, but which is responsible for all that we can see. Likewise, between us and this central fire, the Pythagoreans posited antichton, the counter-earth, thus bringing the number of heavenly bodies in the solar system to ten.

On such a Pythagorean view, our world runs in concert with an unseen one. This invisible world is near our own, yet never perceived directly as it orbits closer to the cosmic center and source of universal light. Though such a position is surely strange to our contemporary point of view, it is nonetheless a strangely familiar strangeness. For, in a variety of times and cultures, mankind has been aware of the uncanny, and at times, felt our world to be entwined with another, a fairy world which, though usually unnoticed, nonetheless crosses us pivotal times and places, and is somehow co-implicated in the destiny of our own. What we have then in the Pythagorean theory of number is a view in which the visible world is both haunted by and resting in the invisible. If we look to the sky above, we see the heavens turning about an unseen center and the sun reflecting a light whose source is inaccessible to our gaze. And if we look to the world below, we see nature and the entities it contains as depending on an invisible realm of number. Thus, for the Pythagorean, whichever way we turn, we find ourselves in the grip of the invisible. I’ll leave you with one of Rilke’s Sonnets to Orpheus:

“A god can do it. But will you tell me how/ a man can penetrate through the lyre’s strings?/ Our mind is split. And at the shadowed crossing/ of the heart-roads, there is no temple for Apollo./ Song, as you have taught it, is not desire,/ not wooing any grace that can be achieved;/ song is reality. Simple, for a god./ But when can we be real? When does he pour/ the earth, the stars, into us? Young man,/ it is not your loving, even if your mouth/ was forced wide open by your own voice—learn/ to forget that passionate music. It will end. True singing is of a different breath, about/ nothing. A gust inside the god. A wind.” (Rilke, Sonnets to Orpheus, I,3 trans. Stephen Mitchell).

[Ein Gott vermags. Wie aber, sag mir, soll
Mann ihm folgen durch die schmale Leier?
Sinn ist Zwiespalts. An der Kreuzung zweier
Herzwege steht kein Tempel für Apoll.

Gesang, wie du ihn lehrst, ist nicht Begehr,
nicht Werbung um ein endlich noch Erreichtes;
Gesang ist Dasein. Für den Gott ein Leichtes.
Wann aber sind wir? Und wann wendet er

an unser Sein die Erde und die Sterne?
Dies ists nicht, Jüngling, daß du liebst, wenn auch
die Stimme dann den Mund dir aufstößt,—lerne

vergessen, daß du aufsangst. Das verrinnt.
In Wahrheit singen, ist ein andrer Hauch.
Ein Hauch um nichts. Ein Wehn im Gott. Ein Wind.]

[1] This illustration was used later by Plato in his famous analogies of the cave and the divided line.

[2] See The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings that Relate to Pythagoras and Pythagorean Philosophy. Trans. Gutherie, Taylor, and Fairbanks. Ed. Fideler. (Grand Rapids: Phanes Press, 1987).  

[3] Iamblichus, The Theology of Arithmetic trans. Waterfield. (Grand Rapids: Phanes Press, 1988).

[4] W.K.C Gutherie, A History of Greek Philosophy Vol 1. The Earlier Presocratics and the Pythagoreans. (New York: Cambridge University Press, 1962).

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