16 Apr 2009

The Computation of Becoming and Eternal Recurrence in Nietzsche's Pythagoreans

by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]
[Computation Entry Directory]
[Nietzsche Entry Directory]
[Pythagoras, Entry Directory]

[In 1872, 1873, 1876, and perhaps also 1869-70, Nietzche delivered his lecture series on the pre-Platonic philosophers at the University of Basel. The following is a summary of his lecture on the Pythagoreans.]

Friedrich Nietzsche

The Pre-Platonic Philosophers

Chapter 16: The Pythagoreans

To understand the fundamental principles of the Pythagoreans, we will depart from an Eleatic notion.

Say you have a paper and pen. The paper is not the pen. And the pen is not the paper. If they were the same thing, then you would have just one thing. But we have two things. Both the pen and paper are real. They have being. But because one is not the other, a certain not-ness is involved. The being of the one is not the being of the other.

Now consider if there was only one thing in the cosmos. There would be nothing to compare it to. But it would still have being. However, because there is nothing else for it to relate-to, we cannot say that it is not-this-thing or not-that-thing. The one single thing has being, but not not-being.

So we see that if we have multiplicities, we become involved with not-being. We might take not-being to have deeper cosmological value. This Not-Being would be "identical to Anaximander's Unlimited, the absolutely Indefinite, that which has no qualities at all, which is contrasted to the absolutely definite (πέρας, peras)." (131d) The One originates from both the definite and indefinite. This means that "it is equal and unequal, limited and unlimited, without qualities and having qualities." The One comes from two contrary principles. But already we are saying that the One implies a multiplicity.
Out of oneness is produced the series of arithmetic (monadic) numbers and then geometric numbers or magnitudes (spatial things). Thus oneness is something that has come to be, and hence there is also multiplicity. If we have first of all points, lines, surfaces, and bodies, then we also have material objects: number is the genuine essence of things. (131-132)
In this way, the Pythagoreans defended mathematical science against the Eleatism which argued that there is neither multiplicity nor change. Pythagoreans believed that both the one and the many exist: "the existence of the one was maintained in order also to deduce multiplicity from it."

On the basis of this deduction, they conclude that reality is fundamentally quantitative and not qualitative.
And indeed they believed to have recognized the true essence of each thing in its numerical relations. Hence fundamentally qualities do not exist; only quantities [do,] yet not quantities of elements (water, fire, and so forth) but rather limitations to the Unlimited, to the άπειρον (apeiron): as such it resembles Aristotle's merely potential being of matter (ύλη, hyle). (132c)
Aristotle provides a table of Pythagorean oppositions:

Limit[ed] / Unlimited
Odd / Even
One / Plurality
Right / Left
Male / Female
Resting / Moving
Straight / Curved
Light / Dark
Good / Bad
Square / Oblong

The Pythagoreans claim that everything qualitative is really quantitative. They conclude this on the basis of acoustics. We take a stringed musical instrument called a monochord. It has one string. Under the string is a 'bridge' mechanism that shortens the string to whatever division we wish. What we will do is keep one string at its full length, while we adjust the string of another monochord to shorter sizes. This will increase the pitch of the second string. But they will play together simultaneously. We continuously slide the second string shorter and shorter. Throughout most of its variation, there is a somewhat noisy combination of sounds between the two monochords. But as our bridge moves to certain places, there emerges a cleaner sound that is qualitatively different than either string taken apart from each other. When we play one string at half and another in its full length, we obtain a noticeably clean and qualitatively different sound. This is the octave.

Each successive halving of the string produces higher octaves. The ratio of the larger string to the shorter is 1:2. Then we will move backwards from the middle of the string to find the other lesser obvious harmonies that come before it. We find the next one when we have only shortened the string by a third. This leaves two thirds left. So this ratio is 2:3. The next one is 3:4. These are the most noticeable harmonic ratios that the monochord can calculate. We find that when we divide 2:3 by 3:4, we obtain 8:9. This is the "whole tone."

The monochord instrument was called the κανών (canon).
Its movable bridge is the μαγάδιον (magadion).
The 2:3 ratio is the λογοσ ήμιόλιος (logos emiolios).
This ratio produces a harmonic interval called the fifth (διά πέντε, dia pente)
And the 3:4 ratio (έπίτριτοσ, epitritos) produces the fourth (διά τεσσάρων, dia tessaron).
The 8:9 ratio (επόγδοος λογοσ, epogdoos logos) is the whole tone (τόνος).
The consonant intervals (σύμφωνα, symphona) are 1:2, 2:3, and 3:4, give us the sacred numbers, 1, 2, 3, 4.

The larger circles on the image below is associated with Nicomachus the Pythagorean's Manual of Harmony. The three largest internal circles show those consonant ratios. Notice the hand of god tuning the string. [Click on the image for an enlargement. Credits posted at the end.]

When we consider music, we can better see how the universe is constituted by number. What we call music is "actual only in our auditory nerves and brains." Outside our nervous experience of music, it is made-up solely of numerical relations. The sounds last a quantity of time. They have quantities of volume or amplitude. The qualitative differences between tones are really quantitative ratios of wave sizes (and speeds). And the rhythm of music is as well a numerical computation. (133d)

We can furthermore see the whole world as being made-up of numbers. Chemistry and physics "rigorously strive to find the mathematical formula for absolutely impenetrable forces. In this sense, our science is Pythagorean!" (133d)

Here lies the Pythagorean contribution. Other theories explained qualitative differences in the world in terms of associations and dissociations of elements. The Pythagoreans demonstrate that "qualitative differentiation resides solely in differences of proportion." (134d)

The Pythagoreans explained all of reality on the basis of number. Because the number 10 is perfect, they concluded there must be 10 heavenly bodies. When only nine bodies were found, they posited the "counterearth" as the tenth. They argued that the odd is limited and the even is unlimited. We can add 1 to an odd number to get an even. And we can add 1 to an even to get an odd. So oneness is both even and odd, and thus it is both limited and unlimited. This is why we can have all other numbers: they are repetitions of the One. "Oneness originates number, and the universe consists of numbers." (135bc)

We may divide all numbers into the even and the odd. And numbers contain the properties of evenness and oddness. 5 for example is made up of 2 and 3. Three is made up of 1 and 2. Two (which already has the property of evenness) is made-up of 1 and 1. In fact, even and odd were the "general conditions of existence for things." Now, when something is an even number, it can be divided evenly into two other whole numbers. But this is not so for odd numbers. We begin with 20 for example. We divide it evenly into 10 and 10. Each of these we divide into 5 and 5. But now we can no longer divide into whole numbers. On account of this limitation, the Pythagoreans equated the odd with the Limited, and the even with the Unlimited. Because of their special mathematical properties, the odds were considered perfect. So, whenever the Pythagoreans perceived opposite qualities, they there considered the superior to be limited and odd and the inferior to be unlimited and even. (136a)

Because all numbers are mixtures of even and odd, some bond is needed to keep them together. This bonding force is harmony.
Everything is number, everything is harmony, because every definite number is a harmony of the even and the odd (136b)
We noted that the strongest harmonic ratio is 1:2. Here we see the even-odd opposition, and the one-many opposition, that are kept together by the forces of harmony. (136b)

Consider a square. It has right angles. To make a square number we multiply a number by itself, or "like times like." This square-ness property was what the Pythagoreans called justice [perhaps in accordance with 'eye-for-an-eye'].

They thought that numbers repeat after ten, so "it seemed that all powers of number were contained within the decas; it signifies greatness, omnipotence, the completion of all things, beginning and feminine guide to divine and earthly life. It is perfection." (136-137)

Geometrically, they considered one to be a point, two to be a line, three to be a surface, and four to be a solid. These compose the extensional shapes. In this way, physical bodies are again just number. What makes fire be fire is that it is made-up of many small tetrads. Air is just octrads, and so on.

The Pythagoreans held that the universe is a sphere. At the middle is "the central fire." Around the fire are ten celestial bodies that are "coiled from west to east, their round dance [occurring] in the widest distance in the heaven of fixed stars." (138a) They said that the earth is spherical. And, they did not place the earth at the center of the celestial rotations.

The stars are in motion, which creates harmonic sounds. For, "every rapidly moved solid emits a sound." The stars produce an octave. We cannot hear it, because we have heard it from birth, so we do not know what it is like to not hear it.

They considered the milky way to be a circle of fire that wraps around the heavenly spheres, keeping them together. Outside the circumferential fire lies the Unlimited.

There are three rings of stars: Olympus, Cosmos, and Uranos. Olympus contains the elements in all their purity, namely, the unlimited and the limited. In Cosmos is found ordered motion. And Uranos is the realm of Becoming and Passing Away. (139c)

The stars in these rings revolve in circles, and their movement is eternal return.
Whenever the stars once more attain the same position, not only the same people but also the same behavior will again occur. (139c)
The human soul is the harmony of its body. We find reason in the brain, life and sensation in the heart, rooting and germination in the navel, and productivity in the reproductive parts. (140a)

Without number, we can know nothing. Number can never express untruth; "it alone makes the relation of things knowable. Everything must be either limited, unlimited, or both; without boundaries, however, nothing would be knowable." (140a)

The Pythagoreans considered the limiting force to be Heraclitean fire
whose task is to now resolve the Indefinite into nothing but definite numerical relations; a calculating force [eine rechnende Kraft] is essential. Had they taken the expression Logos from Heraclitus, they would have meant by it precisely proportio (that is, producing proportions, as the Limited-πέρας sets boundaries.) (140d, some emphasis mine)
So the Indefinite of Anaximander becomes definite things with their own qualities when the Logos or Proportio of Heraclitus' fire organizes the indefinite stuff into numerical relations, like the elemental shapes, the harmonious speeds of the heavenly bodies, and so forth. (140d) Order emerges from chaos on account of computation.
the basic idea is the matter considered to be entirely without quality becomes this and that various quality by way of numerical relations alone. So Anaximander's problem is answered. Becoming appeared as a calculating! We are reminded of Leibniz's saying that music is "an unconscious exercise in arithmetic in which the mind does not know it is counting." The Pythagoreans could not, of course, also have said of the world what actually calculates! (140-141)

In a following entry, we will show that the eternal return is what actually performs the calcuation of becoming.

Nietzsche, Friedrich. The Pre-Platonic Philosophers. Transl. Greg Whitlock. Chicago: University of Illinois Press, 2001.
Preview of this chapter in its entirety may still be available here:

Image from:


  1. neato

    did Nietzsche write a book called The Pre-Platonic Philosophers?

  2. Sorry, I forgot to mention, this is from a lecture series Nietzsche gave as a professor at Basel. Thanks for asking, I added this information to the entry.

    The translator Greg Whitlock quotes Walter Kaufmann as writing "In the summer of 1872, in 1873, and in 1876, Nietzsche, then a professor at the University of Basel, lectured on 'The Pre-Platonic Philosophers.' (qt xiv) Whitlock later writes "I believe Nietzsche held the pre-Platonic lecture series for the first time in the winter semester of 1869-70." (xxii)

    Hopefully this link works. It is a google book preview of the entire chapter on the Pythagoreans.