I've Been Reading Plato, and I Have Questions

Or, what I have to show for the last few months

A 2nd century papyrus scroll of Plato’s Republic, from the Oxyrhynchus Papyri Oxyrhynchus Online

A quick primer on Greek philosophy: Parmenides thought everything was one and unchanging. Heraclitus thought everything was many and in constant flux. Plato thought both of them were right. That’s most of the story.

To flesh it out, we need to discuss Plato’s theory of forms. Parmenides and Heraclitus both seemed to be noticing things about reality that are persistently true-everything changes, but there seems to be some underlying, unifying logic to the change, something it’s happening in reference to. Nature has patterns. Where do the patterns come from?

For Plato, there needed to be some source of the structure by which matter is organized. For it to be true, in our world, that one and one make two, the principle of addition itself must have some sort of precedent-there must be some rulebook of sorts that the universe plays by. To Plato, the easiest explanation of this rulebook was the concept of forms of things-universal properties which things seem to partake in, but which have a separate ontological status as invisible, unchanging, self-contained “forms”. Plato applied this logic to a great many things, including material structures like trees, animals, and rocks, as well as intelligible concepts such as justice, love, truth, and the good.

It should be unsurprising to the reader that this theory has encountered some compelling problems. The first and most persistent objection is that the forms seem given to infinite regress, a problem Plato himself raises in the self-scrutinizing dialogue Parmenides.

Parmenides: How do you feel about this? I imagine your ground for believing in a single form in each case is this. When it seems to you that a number of things are large, there seems, I suppose, to be a certain single character which is the same when you look at them all; hence you think that largeness is a single thing.

Socrates: True.

Parmenides: But now take largeness itself and the other things which are large. Suppose you look at all these in the same way in your mind’s eye, will not another unity make its appearance-a largeness by virtue of which they all appear large?

Socrates: So it would seem.

Parmenides: If so, a second form of largeness will present itself, over and above largeness itself and the things that share in it. and again, covering all these, yet another, which will make all of them large. So each of your forms will no longer be one, but an indefinite number.

As formulated by Aristotle in Metaphysics, this is known as the Third-Man Problem. If the nature of the problem isn’t clear yet, suppose in all the known universe there are three things that are large-A, B, and C. All partake of largeness, so by Plato’s theory of one over many, there must be a form of largeness.

But if the form of largeness itself exists, it must itself be large, itself being the quality which A, B, and C share. Therefore, largeness itself is large, and thus largeness can be said to be set over A, B, C, and largeness itself.

Largeness

A, B, C, Largeness

Obviously, this is a problem-largeness can’t be set over or partake in itself. By the logic of the theory wherein all large things partake of largeness, there must be at least two forms of largeness-the form of largeness, and the form of largeness of which the form of largeness partakes. There are, then, two forms of largeness, so we get something more like this:

Largeness 1

A, B, C, Largeness

You see where this is going-largeness 1 is prone to the exact same problems as largeness, as would largeness 2, and 3, ad infinitum. Our world of non-physical entities is getting exponentially larger, and we are getting no closer to a root cause for the property of largeness in nature.¹

There are other problems, of course-just how reducible is the property of forms, and where do we reduce it to? There’s a form of cups, fine. Is there a form of solo cups? Is there a form of red solo cups vs. transparent solo cups? Is there a form of the red solo cup at the bottom of the stack, and another for all the others up to the top, and one for the stack itself? At the very least, the forms must partake of one another to some extent-the form of a solo cup necessarily partakes of the form of a cup, the form of a cup partakes of the form of a vessel, and the form of a vessel of the form of manmade artifacts, and so on (the same can be done in the opposite direction-the cup partakes of plastic, the plastic of carbon atom chains, the carbon atom chains of carbon atoms, etc.) It thus seems that forms cannot be, as Plato asserts in the Timaeus, one-they must be composites of each other somehow, and this would seem to creep towards giving them the convoluted complexity they are supposed to correct about the material world. This seems obvious, when you think about it-greatness and largeness are not the same, but certainly for something to be said to be great means there is some largeness to it-a large amount of excellence, or character, or heroism. Likewise, it would be strange to say that someone is heroic, but not great; heroism and greatness are distinct and would presumably have their own forms, but heroism necessitates some greatness, and greatness necessitates some largeness, and thus heroism too would seem to partake of largeness.

We can play this game all day. Obviously it’s pretty easy to find critiques to a theory philosophers have had 2400 years to pick away at-though these are all problems Plato himself was aware of and at any rate didn’t dismiss out of hand. And my point isn’t to repudiate Plato-at least not entirely-so much as to question whether we’ve actually gotten any closer to answering the question he hoped his theory would answer: why are things this way, and not some other?

These are questions we tend to relegate to science and mathematics. But neither can ever seem to actually answer the question of “if not forms, what else?” We put faith in the explanatory power of the study of nature, but Plato will always have a place in the argument, because he is rightly skeptical of it. His intuition is the same one which motivates the Cosmological Argument for God’s existence: if there is to be an uncaused cause at the bottom of everything, it must be something of a sort which is not subject to the normal apparent course of causality. Like the Cosmological Argument, it fails not because of any flaw in its reasoning, but because it seeks to answer questions which we don’t actually know that we have the wherewithal to answer, either in terms of data or sheer lack of reasoning power.

There are several areas, however, where Plato’s philosophy retains much of its power. His discussion of the soul, ethics, and political economy, primarily in Republic but also in many other dialogues, still carries enormous weight and relevance-but I won’t address them here. One area where the theory of forms has retained much of its power is in the philosophy of mathematics. Mathematical platonism, in its treatment by the Stanford Encyclopedia of Philosophy, drops the capital ‘P’ to denote its differences from Plato’s own doctrine, but the core idea is rigorously Platonic-mathematical objects exist, are wholly abstract, non-causal, invisible, and not located in space or time. They have ontological status despite having no material basis.

This doctrine obviously clashes quite considerably with my own naturalist position, and remains a point of uncertainty for me. It does seem to be the case that mathematical principles of some kind exist, but can’t readily be explained away as emergent properties. Whereas redness can be derived from the interaction between objects which are constituted such as to produce an impression we recognize as red with our eyes, there is no relation between observer and observed which gives rise to a property like the presence of pi in the circumference of circles-rather, there is something about mathematical objects which seems to have an ontological status totally independent of their corporeal manifestations.

That being said, there is one problem with pure platonism: while we seem to find certain mathematical principles well-verified by nature, countless potential mathematical systems can be constructed, based on more or less any set of axioms we could imagine. It seems apparent that the mathematics we use is constructed on the basis of some basic empirical observations of nature, and the math has been adjusted to account for this. This makes math appear more like a language than an underlying architecture of the universe-one we tweak and alter to better fit our changing picture of that universe. It just happens that math is so dazzlingly accurate, given the right set of axioms, in the predictions it’s capable of making, that we start wondering if we’re not reading it rather than writing it. Does this mean two sorts of math exist, that which is invented and that which is genuinely discovered, the latter alone potentially having status as a set of genuinely platonic universals?

Basically, I don’t see why not, but I’m also inclined against the platonist position. Here is how dialectical naturalism would theoretically account for mathematics:

In nature, we encounter matter which is given to some form-all matter has form, and all form, whether sensible or intelligible, depends upon matter to exist. Mathematics is a precise and predictive method of describing form, and those apparently abstract mathematical objects which we seem to find in nature are actually irrevocably bound up with the matter which constitutes them.

It seems plausible that, if we are able to conceive of fictional mathematics, our representations of “real” mathematics is not of a fundamentally different character. Just as we can tell a story of events which actually happened to us and ones that didn’t and still be telling a story in both instances, it could be that we create an abstract representation of nature and call it mathematics, but that the axioms which are present in nature as essential components of form inseparable from matter give rise to an abstract mathematics in the mind.

This doesn’t seem impossible, and isn’t incompatible with the contention that abstract objects exist but are contingent on material ones. But there is no compelling evidence for it, and intuition does seem to lead most people to suspect that, if every person dropped dead tomorrow, 9 would still be larger than 2 in some objectively true way.

To know what the real nature of mathematics is, then, would seem to require knowledge of the origin of nature itself-and I’m already way out of my depth even talking about math to begin with!

1. There have been relatively recent claims in academia that the third man argument has been resolved, originating from Constance Meinwald’s paper Good-bye to the Third Man. As I understand it, the basic form of the argument is that Plato seems to have two forms of predication-one where something partakes of a form (predication pros ta alla), and another where a form partakes of another form (predication pros heauto). In this formulation, largeness may not partake of largeness, but of size or some similar thing, and thus infinite regress can be avoided. The debate about whether this interpretation is accurate is mostly resolved (Plato does indeed seem to use two different forms of participation), but whether it actually resolves the third man problem remains heavily contested. I’m agnostic at present, but I wouldn’t say it brings me any closer to calling myself a Platonist.