Pure Indetermination

From the Milesians to the Pythagoreans

Continuing on from our discussion of the Milesians, our next Presocratic school of thought is the Pythagoreans (which includes Pythagoras of Samos himself, Philolaus of Croton, Petron of Himera, and Eurytus of Croton). In my first reading of the Pythagoreans five years ago, I remember being intrigued by some of their ideas, but finding little purchase in their system as a whole. This time, however, was different. Specifically, in rereading Waterfield’s introduction, I found my attention arrested by a footnote citing K. S. Guthrie’s The Pythagorean Sourcebook, which asserts the following of the Pythagorean philosophy:

Things are numbers, or, if you like, the basis of nature is numerical, because solid bodies are built up of surfaces, surfaces of planes, planes of lines and lines of points, and in their geometric view of number the Pythagoreans saw no difference between points and units.1

Having watched Alain Badiou’s lecture “How to Begin with the Void” last year,2 this conflation of “points” with “units” caught my attention. Badiou constructs being from sets, equating what is with number (though his sets are sets of nothing, an impossibility for the Greeks3). The Pythagorean equation of points with units, for Badiou at least, is, then, not so far fetched. Where Badiou begins with Ø, the “pure name of nothingness,” the Pythagoreans begin with the One, which is in Badiou’s schema {Ø}, the “multiple with only one element” (this conception, too, would have been impossible for the Pythagoreans). The Ø is the name of “pure indetermination,” and as such this name is “purely abstract,” a “trace for the presence of the absence of nothingness.” But insofar as the Ø is absolutely indeterminate, it cannot be distinguished from any other nothing, dividing this void from that void. For Badiou, the Ø is, therefore, absolutely “unique,” and the name of this void is, as such, “one thing.” The void is singular, and is only fully cognizable when conceived in this way—otherwise, the Ø remains a “pure symbol,” “without any reason,” a name “that has no meaning at all.” But then, to conceive of the void is to conceive of “something which is” (while the void itself is not), something “which we can think, which we can realize.” In other words, to conceive of the void is to conceive of something that contains only the name of pure indetermination: which is the {Ø}. And so, the One is born. It is a one “with a very small meaning, a small one, but it’s a one,” and it is from this one, through a process of recursive “mediation” by the “name of nothingness” (the Ø) that the material of the finite (being, existence) is elaborated.4

We see, then, utilizing the resources of modern mathematics, that the construction of everyday matter on the basis of a profound mathematical abstraction is not a metaphysics to be sneered at. And we will return to Badiou next time, in discussing Heraclitus and Parmenides, to raise the matter of why we might want to begin with the void. But, for the time being, it would appear that there is some knowledge to be gleaned in returning to Pythagoreanism. Let us set Badiou aside in order to make the connection with the Milesians, so situating Pythagoras and his followers in a continuity.

First: Thales. Water is the “first principle,” and as such, it is the “source” of “all existing things.”5 Soul, however, not water, is “the principle of movement,” and the “universe is shot through with soul,” for which reason Thales “thought that all things were full of gods” (so also identifying soul and the divine).6 Because all things must be generated from and destroyed into water, soul, as the principle of movement, must be equiprimordial with water, the principle whereby being becomes beings, a becoming that is itself not reducible to being in its being.

Second: Anaximander. The boundless, the infinite, apeiron, is the first principle, and it is ontologically “different” from the “so-called elements” that constitute beings.7 By “necessity,” all “existing things” are born from and “die back into” the boundless (a reversibility that I wrestle with in note 25 of the previous essay).8 However, because these things are not equivalent with the boundless, their “creation” does not occur “as a result of any of the elements undergoing qualitative change, but as a result of the opposites being separated off by means of motion, which is eternal.”9 The boundless is the “one” that “contains oppositions,” and it is eternal motion that separates them.10 Furthermore, the boundless itself is not “subject to generation or destruction” like the opposites, and indeed does not “have an origin,” but rather is “the origin of everything else … contain[ing] everything and steer[ing] everything.”11 Recalling Thales, apeiron is also called “the divine,” because “it is immortal and imperishable,” but the location of divinity has here shifted from the soul to being, insofar as it is not motion that Anaximander calls divine but the boundless (indeed, soul is not even named in Anaximander’s invocation of motion).12

Third: Anaximenes. His first principle is “not boundless, but specific”—it is “air.”13 This air “manifests” as “different things” through a process of “rarefaction and condensation,” a process of “change” that is caused by “motion,” which is again “eternal.”14 Furthermore, for Anaximenes, air is a “god, which had been created, was infinitely huge, and was always in motion.”15 But for Anaximenes, “soul, which is air, holds us together, so the whole universe is surrounded by wind and air.”16 The first principle, air, and the principle of movement, soul, are thus identified with each other (and together, as one, called divine).

My conclusion in the prior essay was that the Milesians posit an originary dual in their metaphysics, an ontological intuition of the pair being-becoming. Being, for each, is one (water, boundless, air), and yet being is manifest in beings, the many (all existing things). As such, there must be a process of change or motion whereby being becomes beings, a process that is either qualitative (Thales and Anaximenes) or substantive (Anaximander). Because this process must always be, insofar as the becoming itself could not become without first being, the Milesians posit this becoming as another principle equiprimordial with being (soul, motion, god). However, in the case of Anaximenes, the last of the Milesians, we see the principle of being (air) almost entirely conflated with the principle of becoming (air as the god always in motion), and the equiprimordiality of being and becoming put into question (and indeed, the whole Milesian project).17

But perhaps the problem encountered here is with the dogged commitment to the substantiality of the first principle, not with the Milesian intuition itself. As I have already argued, Anaximander is the strongest of the three Milesians in his positing of an abstraction for his first principle, apeiron. To be accepted as first principles, water and air require baroque descriptions of generation and destruction to account for the existence of change and the many. The boundless, however, undergoes a substantive change, with the “opposites being separated off” from it, but is not changed in itself. There is no qualitative transformation of the boundless into this or that existent; rather, the boundless is the abstract principle that makes possible substantive difference (rather than the substantive principle that makes possible only abstract, i.e., qualitative, difference). It is for this reason that I previously invoked Simondon’s conception of the “preindividual,” abstract being without determination, which is then “dephased” or “resolved” into individuated being.18 This is not a qualitative change, insofar as the individual cannot revert to the preindividual state (a unilateral determination as Laruelle would say), nor is it total, insofar as the preindividual is not “exhaust[ed] with one stroke.”19 The boundless is named as such precisely because it is so: infinite, i.e., indeterminate.

Here, then, we find the passage in our continuity from the Milesians to the Pythagoreans. If we maintain the possibility of an abstract principle for material reality, Aristotle’s confusion over the Pythagorean claim that “the elements of numbers are the elements of all things” dissipates.20 The fact that the Pythagoreans “collected together all the properties of numbers and harmonies which were arguably conformable to the attributes and parts of the universe, and to its organization as a whole, and fitted them into place” is not so peculiar.21

The Pythagoreans begin with number, the first “elements” of which are “the even and the odd,” the “even … unlimited and the odd limited.”22 The one follows, “formed from both even and odd, since [the Pythagoreans believed] it is both even and odd.”23 All the numbers, then, are “formed from one,” and these “numbers constitute the whole universe.”24 This is the basic ontological framework of Pythagoreanism.

Aristotle notes that the Pythagoreans maintained “two causes” at the beginning of being, but were “idiosyncratic” in this belief insofar as the “limited and the unlimited and the one were not separate natures, on a par with fire or earth or something,” i.e., some material thing, “but the unlimited itself and the one itself were taken to be the substance of the things of which they are predicated.”25 For the Pythagoreans, then, there is the unlimited (the boundless) and there is limit (the one). This is a remarkable intuition. The Pythagoreans begin with indetermination, then through the determining limit that is the one the one-as-such comes to be, from which all else is generated (in Badiou’s terms, this is the progression Ø → {Ø} → 1, where the ‘determining limit that is the one’ is first the “small one” of the {Ø} before growing in meaning to become the numeral 1). “This is,” truly, “why they said that number was the substance of everything.”26 Aristotle puzzles over how the “first spatially extended unit was put together,” how stuff, “existing things,” could come from number,27 but as with Simondon’s application of modern physics to ancient dilemmas, Badiou is able to overcome this difficulty through modern mathematics, demonstrating how set theory is able to bootstrap nothing into something by calling itself in the recursive function presented above. This is perhaps not intuitive in the domain of everyday life, but in the domains of Pythagorean metaphysics and contemporary pure mathematics, there is no problem.

But whither the soul? The Pythagoreans, like the Milesians, posit an originary dual, but they replace the second term, motion or soul, with limit. Unlike motion, which implies variation and change, limit implies finality and closure. Is our continuity broken? I would contend no, and rather that it is here where the Badiousian connection becomes explicit. Aristotle writes in his Physics:

The Pythagoreans also claim that there is such a thing as void. According to them, it enters the universe from the infinite breath because the universe breathes in void as well as breath. What void does, they say, is differentiate things; they think of void as being a kind of separation and distinction when one thing comes after another. This happens first among the numbers, because on their view it is the void that distinguishes one number from another.28

But Aristotle has it backwards. As Waterfield remarks in an endnote, the “Pythagoreans conceived of numbers as arrays of dots (see n. 6 on p. 93); the dots are the limiting principle, the space between them the unlimited void.”29 The void is not that which separates and distinguishes, because it is the one, the unit-point, that is the first limit, the first determinate thing. It is the one, the {Ø}, that separates and distinguishes the void. Generally, there are not first things, which are then differentiated by void, but rather, things are determinations of the void, limits of the unlimited. There is the void—the boundless, the indeterminate, the preindividual—and there is limit—opposition, determination, individuation. Limit as a sort of closure of the unlimited is, at the same time, the movement that is the becoming of beings.

After all this, the soul at last returns, but now in a different role. The Pythagoreans recognize that, in maintaining their two causes, unlimited and limit, the universe consists of opposites, and the “dissimilar and incompatible and incommensurate ha[ve] to be connected by [a] kind of harmony, if they are to persist in an ordered universe” (because, we recall, the one, of which all things are constituted, is both even and odd, unlimited and limit).30 So, when the universe breathes in void, it is also breathing in breath, which we have already seen equated with air and the soul by the Milesians, and this breath or “soul” is “a kind of attunement (harmonia), on the grounds that attunement is a mixture and compound of opposites, and the body is made up of opposites.”31 Soul, rather than being that which separates or causes change, is for the Pythagoreans the harmony of the opposites.

And the Pythagoreans do not stop here. They do not relegate soul to the domain of the purely human, but rather place it in the domain of the real itself. Soul is also known to them as “reason,” which along with “substance” is “identified with 1.”32 The result of this double identification is not the reification of a correlation between cognition and reality, the entrenchment of a representational regime, but a profound abstraction of the natural and naturalization of the abstract: “everything which is known has number, because otherwise it is impossible for anything to be the object of thought or knowledge.”33 Number is what is knowable and number is what is. In other words, cognition is the real itself.34

There are some further cosmological extrapolations that are of some interest to us: that the “universe is single,”35 that it is “spherical” in shape and surrounded by the “unlimited”,36 and that the “first thing to be harmonized, the one, in the centre of the sphere, is called the hearth,”37 which is the “altar, bond, and measure of nature.”38 But such speculation is secondary to the remarkable abstraction we have already covered in detail here. With the Pythagoreans, we see a significant development beyond Anaximander’s intuition of the boundless which mathematizes, and so abstracts further, his material principle in such a way that shelters Pythagorean thought from the later distortions of Aristotle and his prime matter. If we are to grapple with the likes of Simondon and Badiou today, it is essential to draw this continuum from Anaximander to Pythagoras, preserving the abstraction of pure indetermination at the genesis of being.


  1. K. S. Guthrie, The Pythagorean Sourcebook and Library (Grand Rapids, MI: Phanes Press, 1987 [1920]), cited in footnote 4, The First Philosophers: The Presocratics and the Sophists, trans. Robin Waterfield (Oxford, UK: Oxford World Classics, 2000), 90. 

  2. Alain Badiou, “Infinity and Set Theory: How to Begin with the Void,” European Graduate School Video Lectures, YouTube, January 12, 2012 [2011], https://youtu.be/I1G_SI1-W-4

  3. Nils-Bertil Wallin, “The History of Zero,” YaleGlobal Online, November 19, 2002, https://web.archive.org/web/20160825124525/http://yaleglobal.yale.edu/about/zero.jsp

  4. As Badiou demonstrates, number-as-such can be constructed from Ø. If we begin with Ø, 1 = {Ø}, 2 = {Ø, {Ø}}, 3 = {Ø, {Ø}, {Ø, {Ø}}}, and so on. Thus, “in some sense, all numbers are variations concerning zero. All numbers are made of the void, are composed of the void, and so the real of numbers is in some sense composition with only the mark of nothingness. Numbers are made of nothingness.” 

  5. Aristotle, Metaphysics, 983b6-32, in Waterfield, The First Philosophers, 12. 

  6. Aristotle, On the Soul, 405a19-21, 411a7-9, in Waterfield, The First Philosophers, 13. 

  7. Simplicius, Commentary on Aristotle’s Physics, in Waterfield, The First Philosophers, 14. 

  8. Simplicius, Commentary on Aristotle’s Physics, in Waterfield, The First Philosophers, 14. 

  9. Simplicius, Commentary on Aristotle’s Physics, in Waterfield, The First Philosophers, 14. 

  10. Aristotle, Physics, 187a12-23, in Waterfield, The First Philosophers, 15. 

  11. Aristotle, Physics, 203b7-15, in Waterfield, The First Philosophers, 15. 

  12. Aristotle, Physics, 203b7-15, in Waterfield, The First Philosophers, 16. 

  13. Simplicius, Commentary on Aristotle’s Physics, in Waterfield, The First Philosophers, 17. 

  14. Simplicius, Commentary on Aristotle’s Physics, in Waterfield, The First Philosophers, 17-18. 

  15. Cicero, On the Nature of the Gods, in Waterfield, The First Philosophers, 18. 

  16. Simplicius, Commentary on Aristotle’s Physics, in Waterfield, The First Philosophers, 18. 

  17. Cicero’s testimonial is fascinating, but in this is likely to blame. If Anaximenes’s first principle is air, and air is a god, but also this god was “created,” and yet also “always in motion,” how, then is “motion eternal” for Anaximenes, as Simplicius asserts? Motion would, as such, be more primordial than air. This feels like a potential corruption on Cicero’s part, subordinating the material first principle that so characterizes the Milesians to Roman divinity. Regardless, there is an incoherence in the testimonials and fragments of Anaximenes’s system that render if difficult to assess today. 

  18. Gilbert Simondon, “The Position of the Problem of Ontogenesis,” trans. Gregory Flanders, Parrhesia 7 (2009), 4-16. 

  19. Simondon, “Problem of Ontogenesis,” 5. 

  20. Aristotle, Metaphysics, 985b23-986a26, in Waterfield, The First Philosophers, 102. 

  21. Aristotle, Metaphysics, 985b23-986a26, in Waterfield, The First Philosophers, 102. 

  22. Aristotle, Metaphysics, 985b23-986a26, in Waterfield, The First Philosophers, 103. 

  23. Aristotle, Metaphysics, 985b23-986a26, in Waterfield, The First Philosophers, 103. 

  24. Aristotle, Metaphysics, 985b23-986a26, in Waterfield, The First Philosophers, 103. 

  25. Aristotle, Metaphysics, 987a13-19, in Waterfield, The First Philosophers, 103. 

  26. Aristotle, Metaphysics, 987a13-19, in Waterfield, The First Philosophers, 103. 

  27. Aristotle, Metaphysics, 1080b16-21, in Waterfield, The First Philosophers, 103, and Aristotle, Metaphysics, 1090a20-5, in Waterfield, The First Philosophers, 103. 

  28. Aristotle, Physics, 213b22-7, in Waterfield, The First Philosophers, 110. 

  29. Waterfield, The First Philosophers, 321. 

  30. Philolaus in John of Stobi, Anthology, in Waterfield, The First Philosophers, 111. 

  31. Aristotle, On the Soul, 407b27-32, in Waterfield, The First Philosophers, 112. 

  32. Aristotle in Alexander of Aphrodisias, Commentary on Aristotle’s Metaphysics, in Waterfield, The First Philosophers, 104. 

  33. Philolaus in John of Stobi, Anthology, in Waterfield, The First Philosophers, 111. 

  34. This is in no way George Berkeley’s “esse est percipi” but rather something like Sartre’s invocation of Husserl that “[a]ll consciousness is consciousness of something” or Riccardo Manzotti’s argument that consciousness and the world are identical. See Jean-Paul Sartre, Being and Nothingness: An Essay on Phenomenological Ontology, trans. Hazel E. Barnes (London, UK: Routledge, 2003 [1943]) and Riccardo Manzotti, The Spread Mind: Why Consciousness and the World Are One (New York, NY: OR Books, 2018). 

  35. Philolaus in John of Stobi, Anthology, in Waterfield, The First Philosophers, 111. 

  36. Waterfield, The First Philosophers, 92. 

  37. Philolaus in John of Stobi, Anthology, in Waterfield, The First Philosophers, 111. 

  38. Aëtius, Opinions, in Waterfield, The First Philosophers, 109. 

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