It all comes down to the triangles

One of the most puzzling passages in Plato’s Timaeus describes the formation of the elements out of triangles. Without illustrations, it is nearly impossible to follow, but he is claiming that each of the four classical elements is made up of particles shaped like one of the Platonic solids — i.e., solid shapes with all equilateral faces. They will be familiar to fans of D&D, and I often provide my students with paper templates that they can cut out and fold into the requisite shapes. Every time I teach the Timaeus, at least one student actually does take the time and brings sample Platonic solids to class to show everyone. Extra credit is duly awarded.

One’s first temptation, of course, is simply to skip that section as a bizarre indulgence. Over the years, though, I’ve come to see it as absolutely essential for understanding Plato’s project in the Timaeus. And it is quite literally central to the text, coming roughly halfway through the page count — perhaps just a coincidence, though we are dealing with an author who leaves nothing to chance. More structurally speaking, it takes place after Timaeus announces that he needs to make a second beginning to account for the role of matter in the universe, or as he calls them, “the creations of necessity” (48a, following Robin Waterfield’s translation for the Oxford World’s Classics edition). His speech had begun with a series of binary oppositions that I always find very productive to write on the board: being/becoming, knowledge/opinion, eternal/temporal, etc. (28a). Now he introduces a new governing opposition: reason vs. necessity.

Already, we can see the distance between Plato’s approach and that of modern science, where reason and necessity always go together in the form of uniform, comprehensible, and ideally mathematical, scientific laws. For Plato, though, the material world is the real of unaccountable brute facts. Matter is simply there, seemingly co-eternal with the creator god, and even the demiurge is constrained to work within its limitations. Understanding the chaotic, changing nature of this matter requires the introduction of the famous concept of khora, translated as “the receptacle,” which can be grasped not through knowledge or opinion, but only through “a kind of bastard reasoning” (52b). This is the passage that of course launched a thousand essays by Derrida and his epigones, and it is also one that reduces my students to stony silence. Gen ed-seeking underclassmen are apparently unmoved by the play of différance.

In reality, though, the whole thing with the receptacle is just a setup for the account of how the elements came into being, something that Timaeus boasts (rightly, as far as I can tell) “no one before has ever explained” (48b). What happens is that the matter moving chaotically in the receptacle isn’t purely chaotic. Instead, as in a centrifuge, it “shakes out” into rough regions, which seem to correspond to the zones of the various elements that are more familiar from Aristotelian cosmology — “layers” for earth, water, air, and fire (53a). The various clumps of matter have some incohate commonality that draws them together. But those vague proto-properties are not enough to account for how the elements as we know them interact in such systematic and predictable ways. In order to produce that disciplined regularity, the creator god moulds the raw matter into geometrical shapes based on the most solid of polygons, the triangle. This common underlying mathematical basis is what allows them to transform into each other in a familiar cycle, and also accounts, for instance, for why earth is more solid (it’s a cube) whereas fire can cut through everything (it’s the pointiest shape).

It’s a really clever and charming theory, and the students find it fun to think about. In fact, it’s one of those moments when you can really see the light going on in everyone’s mind: “Ooooh! I see!” And it really gets to the heart of Plato’s project of demonstrating how the raw material of the world has been modelled on eternal perfection — even the lowliest elemental particles reflect geometric truths! Not in themselves, however. Regularity and order and mathematical precision for Plato can only be the product of a rational intelligence. But in order to create a material world, that creator god must work within the limits of an existing raw material — in this case, making positive use of matter’s brute, unaccountable tendency to “clump,” which proves important not only for suggesting the construction of distinct elements but also for designing the human body and its interface with the outside world (the fire implanted in the eyes attracts the fire of sunlight, etc.).

Is it “right”? Obviously not. Does it somehow anticipate modern science? You could make the case, but I’m not interested in that question for the purposes of my class — or at least I try to train my students to be less interested in that question, so they can grasp the plausibility and even beauty of an account of the natural world that is, in the first instance, completely foreign to our contemporary worldview.

(You can see the syllabus for my class here if you’re curious.)

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