Each of us is one, but that very thing which each of us is, both of us are not; for we are not one but two. (Plato, Hippias Major 301d)
Don't you think it is the common good of everybody, more or less, for it to become evident how it is with each being? (Plato, Charmides 166d)
Near the end of a long career in which he had articulated his philosophical doctrines indirectly in the form of dialogues, and in private to a circle of students, Plato decided to make a direct, public statement in the form of a lecture that is known as On the Good. Unfortunately, it does not survive; we know its contents primarily from a memorable account deriving ultimately from Plato’s student Aristotle, who attended the lecture. Aristotle’s brief account dwells on the lecture’s reception, which was not positive. It was not that there was anything distasteful in Plato’s lecture; rather, it seems merely that it was very technical and confusing, and failed to speak directly to the issues his audience expected to hear about in a lecture on the Good. In particular, we are told that his extensive recourse to mathematics, and his final conclusion, that the Good is the One, or Unity, left his audience perplexed and, probably, bored.
There are certain aspects of this account which ought to cause us, before even attempting to reconstruct from Plato’s surviving works what he might have been saying, to question some assumptions so widespread in modern commentators as to go virtually unremarked upon and, certainly, unquestioned. These assumptions concern what in general Plato meant by ‘the One’. They are so widespread and unquestioned because they accord with the appropriation of Platonic thought by Christian monotheism, an appropriation which was by no means peaceful in antiquity. This struggle, however, is treated as a footnote in the history of philosophy, to the extent that the waning of Christian hegemony in the intellectual life of the West has led to no wholesale reconsideration of received notions about the sense and import of classical metaphysics. Modern thinkers, while distancing themselves from the monotheistic project per se, have nevertheless treated that project’s conception of the goals and sense of Hellenic philosophy as though it was more or less correct.
According to this view, when Plato spoke of the One, or of the Good, he was speaking of a singular item, either a singular item beyond everything, or a singular item encompassing everything. Scholars will plead that their interpretations are far more subtle and complex than this, but a radical alternative reading can, I believe, show just how profound is the distortion in our received understanding of the tradition of classical metaphysics. It is a simple matter to catalogue affirmations of polytheistic devotion in Plato’s works, and scholars have at last begun to offer unbiased accounts of Plato’s theology from the things he explicitly says about the Gods, rather than fabricating a theology for him conforming to modern prejudices. What remains, however, is to address the obfuscation crucial to the entire project of the monotheistic appropriation of classical metaphysics, namely the meaning and significance of ‘one’ (τὸ ἓν) in Greek philosophy.
The necessity for such clarification arises immediately from the language itself. Like any similar expression in Greek composed of the neuter definite article in front of an adjective or participle—such as to ison, ‘the equal’, to kalon, ‘the beautiful’—to hen is ambiguous. Such terms can refer to a particular thing exhibiting a certain property, as long as the gender is appropriate, or to everything that exhibits that property, or to the property itself, what we would usually designate by a term like ‘unity’, ‘equality’ or ‘beauty’. This abstract aspect can be underscored by adding the term auto, ‘itself’, though it does not require it. It is in the nature of Greek to be able to form such expressions at will, and Greek philosophy takes every advantage of this. This ambiguity is not limited to terms with the neuter article, either. To use one notorious example, ho theos, ‘the God’, refers ambiguously to a particular God indicated in the context, or as a means to talk about things true of Gods in general, in which case it is freely interchangeable with hoi theoi, ‘the Gods’. A philosophical inquiry into to hen or auto to hen, therefore, is most naturally understood as an inquiry into the nature of unity, what it is for something to be one thing or one something.
In philosophy, one hears of ‘the problem of the one and the many’, referring to an entire genre of inquiry already well known by Plato’s time. In fact, Plato is already keen to distinguish serious or interesting problems of this kind from vapid ones. Thus, in the Philebus, Socrates speaks of certain one-and-many problems as “childish”, “glib”, and “a hindrance to discussion” (Philebus 14d-e). In the Sophist, too, Plato speaks of “youngsters and elders whose learning has come to them late in life,” who “feast” upon trivial one-and-many problems (Sophist 251a-c). These specious treatments of the one and the many have as their common end, Plato says, that “the one is many and infinite and the many only one” (Philebus 14e) or that “each thing we posit as one we in turn treat as many and call by many names” (Sophist 251b). We see, therefore, that Plato is well aware of a tendency for one-and-many problems to be treated in an unserious way, a degree of critical insight that speaks to a development and refinement of the terms of this discussion over a period of time.
What, then, does Plato regard as a serious and worthy discussion to have about one and many? In the Philebus, Socrates explains that the interesting discussions come with respect to unities like ‘the Human’, meaning not this particular human, but what we call the species; or ‘the Bovine’; or ‘Beauty’; or, indeed, ‘the Good’ (15a). Plato makes clear in this way that he is interested in unities, ‘ones’, that lack a simple, unproblematic basis. The examples he offers proceed from the less problematic to those more so. Natural kinds like humans and oxen surely have something real to them; they reproduce themselves continuously and always show some common traits, amid much variation as well. A unit such as Beauty is more contested. It holds together sufficiently as a concept for us to talk about it, at least, and understand one another, though we take for granted that judgments of beauty are highly subjective. Finally, there is the Good—and we know to begin with that what is good for one thing is by no means good for another, so even if we might hope in the case of Beauty to arrive at canons of beauty, or, in a more sophisticated move, in rules for the judgment of beauty that allow for its subjectiveness, it’s going to be rather more difficult to figure out how to conceive a unity such as the Good, assuming that it’s really something more than just hot air.
Unities like these are problematic. At minimum, they pop up at different times and places, and with all sorts of variations that threaten their unity, while in cases like Beauty and Goodness their instances conflict directly with one another in non-trivial ways. But these problems clearly concern determinate unities and expect determinate answers, unlike the pseudo-problems of the ‘late learners’ and their ilk. How are we to get a handle on serious one-and-many problems? Plato’s answer in the Philebus is a method Socrates characterizes as both ancient and as “a gift from the Gods to humans” (16c, again at 16e), and which he says forms the basis of all arts and crafts. This method is based on the notion that “things ever said to exist are from One and Many, and have Limit and Unlimited inherent in them.” The method operationalizes this fundamental insight about beings. It works like this:
[W]e must always assume that there is in every case one idea concerning everything and must look for it—for we shall find that it is there—and if we get a grasp of this, we must look next for two, if there be two, and if not, for three or some other number; and again we must treat each of those units in the same way, until we can see not only that the original One is one and many and infinite, but just how many it is. And we must not apply the idea of the infinite to plurality until we have a view of the total number of it between infinity and one; then, and not before, we may let each One of all things pass on unhindered into infinity. (16de)
Let’s review this procedure. We’re curious about something; since we even have that much notion of it, we have some One to start with. Then we try to see if that one idea is somehow really two, or if not, three or some other number. It’s not an additive process. Rather, the initial One is either going to stay one, or its going to yield some distinct number: if it’s three, then it’s not two, apparently. It seems a bit like searching for the number that will divide another without remainder. Then the units that come from this process each get analyzed in the same way. Each One is one, and it’s also Many, that is, a discrete number, and it’s infinite, in other words, a continuum. The initial unity of the unit and its infinity or continuity mirror each other at the beginning and end of the procedure, while in between we have a discrete multiplicity. And Plato is quite clear that what’s wrong with the people drawn in by the spurious one-and-many problems is that they aren’t interested in discrete multiplicities, rather, “they put infinity immediately after unity; they disregard all that lies between them,” whereas it is a concern with finite multiplicities that distinguishes serious from vapid discussions (17a).
I’m not going to concern myself with the examples given in the Philebus of the method, things like finding the number of tones between two musical notes or devising an alphabet to represent the sounds of spoken language. Rather, I’d like to keep the focus on the most universal dimension of the method and think a little about what it means, all on its own, for the lecture on the Good. For one thing, we can see that any interpretation that would make Plato’s meaning in equating the One and the Good be that all things are one thing, and this is good would fall right into the trap Plato sees having caught his contemporary “wise men”, namely, becoming obsessed with the direct identity between the unity of the universe, on the one hand, and the infinity of things in the universe, on the other. Plato has thus, it would seem, already rejected a certain monism—and a corresponding monotheism—as at any rate not philosophically interesting, and so this interpretation of the meaning of the doctrine put forward in the lecture on the Good would seem to be ruled out at the start. In the method, ‘One’ is always some one, some one thing whose integrity or individuality we are testing, to see how it holds together, and what kind of discrete multiplicity or number (arithmos) can be elicited from it. This orientation is previewed early in the Philebus in an interesting fashion.
The Philebus begins from the question of what condition of the soul makes life happy. Protarchus, on behalf of Philebus, argues that it is pleasure; Socrates, that it is wisdom. Philebus, who leaves early, has made some sort of statement offstage, so to speak, concerning a Goddess. Socrates seizes upon this statement and says, “Let us begin with the very Goddess whom Philebus says is spoken of as Aphrodite but most truly named Pleasure [Hêdonê]” (12b). It is clear from what Socrates says that Philebus embellished his argument in favor of pleasure as the best disposition of the soul by stating that in upholding the primacy of pleasure, he was upholding the primacy of Aphrodite among the Olympians. He’s gone further, however, and apparently made some rhetorical claim amounting to the substitutability of the concept and the Goddess. Socrates takes issue with this, and proceeds to gently scold Philebus. Socrates says that for his own part, he possesses an awe in respect to the names of the Gods that is supreme and, indeed, superhuman; accordingly, Socrates says, “I call Aphrodite by that name which is pleasing [philon] to her” (12c). These words could even perhaps be read simply as “that name which is hers.”
Similarly, later in the Philebus (30d) we read that “in the nature of Zeus a royal soul and a royal intellect emerges through the power of the cause”—i.e., causality, agency—“and in other deities other noble qualities, according to which each is called what pleases them [philon … legesthai].” Here there is a chain formed by the Gods’ agency or action, their emergent qualities, and the names or epithets they receive, which are at once ‘their own’ and those which ‘please them’. In the Cratylus (400d-e), similarly, Socrates states that it is evident that the names the Gods call themselves are true; while our own knowledge of them falls short of this absolute standard, nevertheless “there is a second kind of correctness, as is customary in prayers, that they be named whatever and from whencesoever pleases them [chairousin], and these we call them, since we know no other.”
Philebus trifles with the names of the Gods in a way that is, according to Socrates, all too human, presuming to replace Gods with concepts. Socrates is going to analyze the concept of pleasure; he does not intend to subject Aphrodite to such an analysis. We can compare again a passage from the Cratylus, immediately following the one I quoted above, where Socrates states categorically that in the inquiry which is to follow, he will inquire into the human contribution to the names of the Gods, that is, into the results of the human cognition resulting from theophany, and not into the nature of the Gods themselves (Cratylus 401a). This is not solely a question of piety. Rather, we can see from the explanation of the divine method later in the Philebus that Gods and concepts are clearly different kinds of unit. Persons of all kinds, we know, can hold together in the kind of unity peculiar to a person as such contradictions that would disintegrate a mere concept. And indeed, in the course of Socrates’ and Protarchus’ discussion about the nature of pleasure, it appears that pleasures are sufficiently heterogeneous as to pull apart the apparent integrity of the notion of Pleasure.
And so, in the difference between Aphrodite and pleasure, we already have an illustration of the consequences of practicing the divine method, namely, we begin to discern different classes of unit, different kinds of ‘one’ with different kinds of integrity. When we consider the unity of a person, we know that certain sorts of manifest contradictions are consistent with their continued unity. Our friend changes in many ways, and may even have conflicting aspects enduring over time. Certain contradictions, however, would not be consistent with a corporeal person’s unity insofar as they have a discrete position in space and time. If we were to consider our friend, however, through the lens of the doctrine of reincarnation, then we would find that all of these restrictions have been lifted. My friend may have been a different kind of animal at some time, or a human with completely different traits. This, I would argue, is why Plato is so interested in reincarnation: because if we accept the thought experiment, it reveals a very important kind of unity: a unit the same while any of its particular attributes vary—an individuality, thus, beyond identity and difference.
This procedure of investigating unities, testing them, figuring out how they hold together, is at once the inquiry into the nature of unity itself or as such, the question What is unity? We are always also pursuing this when investigating the mode of unity possessed by this or that unit. So this investigation goes right on up to the unit that is Unity as such, or ‘the One Itself’, to hen auto. The nature of this unit is investigated in what may be Plato’s most important dialogue, the Parmenides. The investigation of the nature of unity in the Parmenides thus forms a crucial adjunct of the method outlined in the Philebus, and it also clearly gets us closest to what Plato could have meant by his statement that ‘the One is the Good’, because more than any other dialogue, it tells us what unity is in itself, and no dialogue is similarly forthcoming about the Good. What is primarily said about the Good Itself is in the Republic, which states that it is ‘beyond being’ or ‘beyond substance’ (epekeina tês ousias), a characterization traditionally explicated with reference to the account of the One in the Parmenides, for reasons that will become evident.
The Parmenides tells of an encounter between the young Socrates and the two great philosophers and life-partners, Parmenides and Zeno of Elea in Italy, who have come to Athens to celebrate the Panathenaia. Socrates engages Zeno with regard to Zeno’s famous paradoxes of plurality and motion, raising the notion of a theory of pure forms or ideas as a way of dealing with them. Such a theory is probably not to be attributed to Socrates alone, but seems to have been in the air, so to speak, especially in Pythagorean circles. The elderly Parmenides then proceeds to examine Socrates about the theory of ideas, showing a number of problems the theory faces, problems young Socrates is not quite ready to tackle. Parmenides acknowledges the necessity of something like the theory, though, simply so that we may orient ourselves in our discussions (what philosophers today call a heuristic device). But he says that it will be crucial to develop a rigorous dialectical procedure in connection with it. With regard to anything that one might take as an object of inquiry
if you suppose that it is or is not, or that it experiences any other affection, you must consider what happens to it and to any other particular things you may choose, and to a greater number and to all in the same way; and you must consider other things in relation to themselves and to anything else you may choose in any instance, whether you suppose that the subject of your hypothesis exists or does not exist, if you are to train yourself to see the truth perfectly. (Parm. 136bc, trans. Fowler)
In a sense, Parmenides advances here a kind of coherentism, or even holism of meaning, that is, it seems that understanding anything requires placing it in relation to many other things, or possibly everything else. We see right away from this that Parmenides does not seem to hold the hypothesis that there is only one thing in the universe, the thesis of ‘numerical’ or ‘existence monism’ often attributed to him. There are many things in the universe and their relations to one another are both complex, and worth getting to know. Parmenides agrees to provide a demonstration of the sort of procedure he is recommending, and suggests that “I begin with myself, taking my own hypothesis and discussing the consequences of the supposition that the one exists or that it does not exist” (137b). Parmenides thus characterizes his hypothesis, not as that all things are one thing, but that ‘the One exists’, that there is such a thing as unity itself.
I will not take up here the significance of this for our understanding of the extant fragments of Parmenides’ own thought. Rather, I will confine myself to the results of the discussion Parmenides proceeds to have with a boy even younger than Socrates, who happens to be named Aristotle—not, unfortunately, the philosopher of the same name. In this discussion, Parmenides first posits that the One Itself exists, and what the consequences are for it itself. This is naturally the part of the inquiry which concerns us most insofar as it speaks directly to the nature of ‘the One’. Its result is that every property one tries to assert of the One Itself must be denied, because if the One is also that, then it is no longer One, no longer itself. The final result, in fact, is that the One Itself cannot even be, or be one (141e). This result provokes a certain incredulity, and so Parmenides makes a fresh start with his young interlocutor, and posits instead this time a One which is explicitly a being, with all that comes with that; and in this Second Hypothesis, as it is known, everything indeed comes in, because the One turns out to embrace every attribute that was previously denied it, and its negation. Neither in the First nor in the Second Hypothesis, then, is there room for the One Itself to be a singular item. In the First, the One Itself is nothing; in the Second, the One’s own unity disintegrates in contradiction.
It becomes crucially important at this point to defend Plato from the charge of paradox mongering, because this is the implicit interpretation demanded by the monotheistic appropriation of Platonism. On a straightforward interpretation, nothing either in the First or the Second Hypothesis could answer to the monotheist notion of a singular God, either in its transcendent or in its immanent form, that is, it is neither a singular item beyond all things nor a singular item encompassing all things. But if the result of the entire procedure is nothing coherent, then the monotheist gets to proceed on the basis of what came to be known as ‘negative theology’, a notion which, in fact, was born largely out of the necessity of dealing with this very problem. The ‘negative theology’ interpretation of the Parmenides would have it that the One really must be, and be one thing, only, as Wittgenstein would say, in a very different context, “in a queer way”. The negations and contradictions, on this interpretation, merely express our own inability to conceive the One Itself in its eminence. On this interpretation, we might say, the One is, and is one, to the hilt.
But another interpretation is possible, in which the One ‘is’ in a way much queerer indeed, but also more rational. For everything said with regard to the One in the First Hypothesis would be correct with respect to any individual conceived as absolutely unique. ‘But there isn’t anything that’s absolutely unique’, someone will object. Indeed; and thus we proceed to the Second Hypothesis, where we see how un-individual and un-unique, any one that happens to be, must be, just insofar as it is. Neither side of this opposition can be eliminated. The hypothesis that there is such a thing as unity itself just yields these two poles of austere and generous unity, as Mary M. McCabe has termed it.
A unit, therefore, is in one sense austerely one, and is just itself, in the most inalienable fashion, and is also, in the other sense, wide open onto all other things. This is the nature of unity. Now, where have we heard something like this before? In the description of the divine method in the Philebus, only there it was formulated differently, stating that every unit was one, and also unlimited, but most importantly, was some discrete multiplicity. It is this latter part which the investigation in the Parmenides into the nature of unity itself leaves open. Let’s go back to the Parmenides, though, and consider a little further what we might make of the unity proposed in the First Hypothesis.
A truly ‘austere’ unit is utterly individual and unified, and hence utterly peculiar, that is, it is not comparable with anything else, for nothing about it can be considered separately from it, and potentially common with something else. Such a unit does not permit one to classify it. It is, of necessity, one of a kind, but one may not say that it has even this as a property, which would of course then render it not one. See, for instance, this passage from the First Hypothesis: after affirming that identity or sameness is “a nature separate from unity,” Parmenides states that “if the One was to be affected by anything separate from unity, it would be affected so as to be more than one, and that is impossible,” so “the One cannot be affected in the same way as another or as itself,” and cannot thus be “like another or like itself” (Parm. 140a). No unit works only like this, to be sure. But the kind that functions most like this austere unit is the kind of unit with a proper name, the kind of unit, namely, that isn’t a what, but a who.
Giving something a proper name is how we express its uniqueness, something we emphasize further by the categorical distinction we draw between ‘what’ and ‘who’. If I ask what something is, I expect to be answered with a term that expresses its real or potential commonality with some number of other entities, whereas if I ask who someone or somebody is, I expect to be answered with something designating this entity alone. Now, anything I can ask the who question about, I can also ask the what question about. On the other hand, we don’t generally ask the who question about just anything, but we recognize that one could give a proper name to a particular object of whatever sort.
In this who-and-what, proper-name-and-common-noun practice, we see two fundamental aspects of anything’s unity. In proper-name unity, we zero in immediately upon one unique entity, whereas in common-noun unity, we as it were fashion increasingly fine nets in which to catch a smaller and smaller number of entities, until we get down either to one, or to a set whose members are indiscernible according to the criteria of the ‘net’ we’re using. We can’t reduce the proper-name unity to the common-noun unity, or else we break it. A proper name by definition is not supposed to apply to more than one being. In practice, of course, things may have the same name, but that’s not how proper names work in principle. And even when we happen to arrive at a net that sorts a category down to a set with one member, we can’t ensure that there couldn’t be more than one being in the set unless we turn the sortal term (the ‘net’) in effect into a proper name, and then we’ve broken that. We can name a lion ‘Lion’ and mean just this peculiar one we’ve met, but the two uses of ‘lion’ no longer function the same way from then on, and we show this in English by using the capital letter.
So we have two aspects of unity, one that designates unique unity, and the other that designates commonalities of some sort. In later Platonists like Iamblichus, Proclus and Damascius, there is a set of prime units, called ‘henads’—a term that originates in the Philebus—who are unique, proper-named entities, namely the Gods themselves. These henads are ‘in’ the First Hypothesis of the Parmenides insofar as they are each a perfectly unique individual, while the classifications of them according to their properties yield the primary common terms for all of Being, which lies for its part in the Second Hypothesis.
We can see from the exchange near the beginning of the Philebus about Aphrodite that the distinction between proper names and common nouns was on Plato’s mind, even if, like other advanced issues in Plato’s thought, it was not discussed overtly or at length in the dialogues, but rather reserved for the private sessions of the Academy. And the issue of proper names is already in the Philebus linked to the consideration of the Gods, as well. If proper-named entities exhibit the primary modes of unity in primary fashion, what sort of entity best exhibits the formal properties of proper-naming? Certainly not ourselves, because so many kinds of whatness infuse our whoness as to overwhelm it, and to make our uniqueness, our proper-named unity, seem rather trivial by comparison. Indeed, it has been common enough, in the wake of the erasure of the very notion of a multiplicity of unique divine individuals, to attribute uniqueness as such purely to objects in space and time, or made out of some particular heap of stuff, or to things that can be uniquely designated conceptually, like the unique definition of a geometrical figure. To imagine a uniqueness beyond conceptual singularity, not inferior to it but superior, would in effect require us to imagine something like the Gods, even if we didn’t believe in such things. We would need to imagine individuals that could be both more peculiar, and more comprehensive, than mundane individuals can be. Such individuals would hold open the space of a positive or existential difference distinct from negative difference (heterotês).
But if we conceive the Gods as unique in this way, how do we understand everything about them which they have in common: powers or potencies, including that of being a God, and relations with one another, some of which place them before or after one another in a pseudo-temporal sequence? These things can be taken so as to reduce the multiplicity of the Gods to some single unit. But it’s not only on account of polytheistic piety that we don’t do that ourselves, or shouldn’t. As philosophers, we shouldn’t do so because while that would make the Gods go away, it won’t make the problem of the nature of unity go away. We would still have to recognize the metaphysical reality of the two different kinds of unity. Even if monotheism was all there had ever been, and all that any of us knew, this metaphysical problem would not go away, we just wouldn’t have an example in the world driving us to work through it as polytheism does, so we would have to investigate it through thought-experiments. Indeed, without polytheism to spur philosophers on, the sensitivity to this problem languished significantly, even divorced from theological considerations, because it was so easy to treat what I have termed proper-name unity as only applying to entities deficient in common-noun unity, like spatio-temporal particulars that come and go all the time and can barely hold themselves together—which is why particulars conceived in this way feature prominently in the skeptical, unserious one-and-many problems Plato already complained about. Monotheists can think of their God as a proper-name unity, but as long as they want to make arguments for why there can be only one God based upon the whatness, the conceptual content attributed to this God, the distinction between proper-name unity and common-noun unity will be consistently and deliberately blurred by them.
Instead of dissolving the Gods into their common powers and relations, their common-noun unity, the Platonists, by contrast, developed complex accounts of the declination of these powers and relations from the metaphysically postulated uniqueness of each of these ‘henads’. This satisfies the demands of piety, offering many solutions to practical problems that arise in polytheistic devotion, and also satisfies the demands of philosophy, by providing an account of how the second kind of unity (common-noun unity) emerges from the first kind of unity (proper-name unity).
So that’s the trajectory of this doctrine in later antiquity; right now, we need to return to Plato himself and think about how the foundations of these ideas could have furnished him with the doctrine he presented in his lecture on the Good, and what that doctrine might have looked like. It is clear that Plato did not, in his lecture, discuss the Gods, at least not in a central way, or else it would certainly have been mentioned in the reports. Rather, he must have discussed how unity and the good operate generally, in all things. This probably took the form of a discussion about how, for each thing, its unity was its primary good. We know that a good deal of Plato’s discussion concerned mathematics and astronomy, though, and we have not discussed these matters at all up to now. What would the role of mathematics and astronomy have been in Plato’s talk?
With respect to astronomy, we know that in his dialogue the Laws, the importance of astronomy is that celestial motion is akin to “the motion and revolution and calculations of reason” (Laws 897c). Thus the role of astronomy in Plato’s argument would have concerned the importance of a certain kind of motion in the cosmos, namely the kind that holds things together and fosters their orderly coexistence rather than their dispersion and disintegration, both individually and in harmonious conjunction. Plato would have sought to demonstrate thereby the way in which a single principle, the principle of unity, could govern all things in a just manner. Truly universal justice cannot, however, be such as to impose itself upon things as something separate from them. Hence we read in Plato’s Timaeus that “The best motion is that caused by itself in itself, for this is most akin to the motion of intelligence and of the All, while motion by another is worse” (89a).
Plato wishes to show that souls that are ordered in the right way individually will, just by virtue of that, also act collectively in the right way; and the best of souls, “which are good also with all virtue, we shall declare to be Gods, whether it be that they order the whole heaven by residing in bodies, as animals, or whatever the mode and method […] Is there anyone who agrees with this view who will stand hearing it denied that ‘all things are full of Gods’?” (Laws 899b). We can see that Plato is not concerned to privilege the Gods of one realm over the others. Rather, once we have shown the key role played in the heavens by a motion analogous to the motion of reason in the soul, then we recognize that this kind of soul, wherever it exists, in whatever form, embodied or otherwise, is divine to whatever extent, and is giving order not only to itself but to the whole cosmos in its peculiar way, from whatever its station. Plato is quite explicit about the plurality of the divine force: “[A]s soul thus controls and indwells in all things everywhere that are moved, must we not necessarily affirm that it controls Heaven also?—Yes—One soul, is it, or more than one [pleious]? I will answer for you—‘more than one’,” (896e).
The ‘motion’ of soul which interests Plato is circular motion, because rectilinear motion is more complex than circular motion and inherently finite. Circular motion is essentially of two kinds, one in which something rotates in place, centering itself, we might say, and the other, in which something revolves around something else, orienting itself to that; and souls will engage in both kinds of motion, integrating themselves and orienting themselves to the best things other than themselves.
Clearly things aren’t in general literally turning in circles around themselves and circling around other things, though some things, like the heavenly bodies, are, and hence the special interest in those things for the sake of the argument. This mechanical circular motion, however, is as it were a special case of a more general, metaphysical motion by which things are on the one hand centering the cosmos upon themselves, integrating it into themselves and giving internal order to themselves thereby, and at the same time recognizing the centrality of other things in other respects, and integrating themselves into the order of the totality ‘outside’. These two ‘motions’ are inextricably entwined. I cannot successfully compose myself as a human without a sense of humanity as such, and the place of a mortal being such as myself in the universe, and what is incumbent upon me as a result. In a polycentric universe everything is in one respect a center for all things, while in other respects it is at the periphery, and this to varying degrees and in diverse ways. With this recognition, the notion of an absolute center becomes unnecessary, as Giordano Bruno, himself a very astute Platonist, would see and apply to astronomy almost 2,000 years later.
That these circular motions are metaphysical does not make them any less real. On the contrary, Plato presumably felt that he had to introduce astronomy into his lecture on the Good in order to integrate the elements of motion and time into his account of the nature of unity and the kinds of unity in things. Things existing in time bind themselves together, individually and severally, through cycles, something which of course has been recognized since the dawn of civilization in every tradition. That the circular motions are metaphysical, in other words, does not mean they are merely metaphorical. Rather, they are, so to speak, motions of motions, in the way that many disparate motions are oriented to a common end, the goal of some process; and so too, in a futher move of formalization, such goal-oriented motions are in principle cyclical, even if in fact they are only carried out one time, or even remain incomplete, because just insofar as they are ideal, they could repeat. So here again, we have our two notions of unity. Our discernment of that which is formal in something, what I have termed ‘common-noun’ unity here, can be understood as that in it which is repeatable, and thus involves the notion of cycles in time, of something returning to presence or returning to itself, as souls do, whether they are rational souls, which reflect upon their actions and their nature, or irrational souls, who return to or revolve upon themselves by doing the things necessary to repeat themselves and therefore sustain themselves as individuals in and through the difference brought about by time or as species, replicating themselves through the individuals here at this or that time.
The final component of Plato’s lecture on the Good which I wish to analyze is the role of mathematics. What does mathematics mean for Plato, and why should the Good have so much to do with mathematics? To return to the Philebus, we find an important distinction between an arithmetic “of the people” and “of the philosophers” (Phil. 56d-e). In the former,
arithmeticians reckon unequal units [monadas], for instance, two armies and two oxen and two very small or incomparably large units; whereas others [i.e., the ‘philosophical’ mathematicians] refuse to agree with them unless each of countless units is declared to differ not at all from each and every other unit.
A similar doctrine is suggested in the Republic (526a), which speaks of ‘numbers’ in which the One is such that “each unity [is] equal to every other without the slightest difference and admitting no division into parts,” and Aristotle (Metaphysics 1080a) attributes to Platonists a doctrine of ‘incomparable’ [asymblêtos] units alongside the conventional units of the mathematicians. Aristotle explains the nature of these units in this way:
if Two is first after One, and Three follows Two, and so on with the other numbers, and the units [monades] within each number are comparable (for example the two units in the first Two are comparable with each other, the three units in the first Three are likewise comparable, and so on with the rest of the numbers), but the units of Two Itself are not comparable with those of Three Itself, and similarly for any two such numbers (and so mathematical number is counted thus: one, and then two, the latter resulting by the addition of another unit to one, and three results by the addition of another unit to two, and similarly with the other numbers; but these numbers are counted thus: One, and then Two, the latter being composed of units distinct from One, and then Three, without including Two as a part, and so on with the other numbers).
If we widen our perspective beyond the narrow sense that ‘number’ has for us, we realize that such an “incomparable unit” exhibits the mode of unity I characterized above as ‘proper-named’. For each of the attributes belonging to such a unit, which are here termed ‘monads’, are peculiar to that unit, and not comparable to attributes in another, even in the case where these attributes are as similar as the two monads in the number Two and two of the three monads in the number Three. We see the usefulness of the recourse to mathematics here, inasmuch as it presents the case in the starkest terms possible. How much more so, then, must the attributes of Aphrodite, for example, be incomparable, in the Platonic view, with those of any other Goddess or God, be they ever so similar in our eyes?
When Aristotle discusses these matters, he is usually speaking about numbers in the sense we generally understand this term today, though not always. Hence, the discussion in On the Soul concerns the doctrine, held by Xenocrates, one of Plato’s earliest successors at the Academy, that the soul is “a self-moving number” (De Anima 408b32). Here, clearly, something broader than mathematics as we know it is intended. We will better understand what is at stake in these discussions by broadening our sense of what is meant by the ‘units’ and ‘numbers’ in them.
For example, how are units that are not comparable, and thus do not differ from one another, in that sense at least, to be distinguished from one another? Such units, we read in Aristotle, can differ only in position (thesis) (De Anima 409a20) or in order (taxis) (Categories 5a30) or in succession (ephexês) (Physics 227a19-21, 29-31). This is clearly true of the natural number series, but it is also true of incomparable units in the sense of unique or peculiar existential individuals, namely that we must distinguish them by their relations to one another, since their attributes are peculiar to them. If we think of position, order, or succession in ways that might apply to any sort of entity, rather than mathematical or geometrical entities alone, we will see that this wider sense looks rather like the kinds of relations exhibited by elements in a story or a picture. Indeed, ‘number’, arithmos, always has in Greek a wider sense than the purely mathematical. To this we may compare the now-obsolete sense of numbering or calculation that was once included in English words for narration, such as ‘tale’ and ‘tell’, from the same root as German Zahl, ‘number’.
The ultimate units, therefore, we may say, being unique in themselves, are distinguished by iconic and narrative relations with one another. And this is just the case with the Gods, for the Gods are not primarily understood by us as operating this or that function, which arises from comparison, but primarily as incomparable units, and we will tend to distinguish them, in addition to their proper names, by referring to their familial relations with other Gods in their pantheon, or the other relations they display in the myths and iconography peculiar to them. The science of these relations, both those internal to the divine individual as well as those external, is then the true ‘theological arithmetic’, and a rich field for future inquiry. The significance of Plato’s remark that “the God eternally geometrizes”, therefore, is not to accord a naïve dignity to a science the limitations of which Plato is also well aware, but in the primary institution of possible relations and spaces of relation through the originary, mythic acts of the Gods, acts the meaning of which transcend categorization to inform all the sciences on a primordial level.
As I remarked earlier, it is very unlikely that Plato explicitly spoke of the Gods in his lecture on the Good. In a certain respect, this is because he didn’t need to. All around him temples hummed with the daily life of devotion, temples to the Gods of many nations. The noise and colors and smells of festivals were always in the air. That all of this could go away was inconceivable, even nonsensical. It is we who need to approach Plato’s thought from an explicitly polytheistic perspective, and by ‘we’ I by no means intend merely we polytheists, but we moderns generally. For whatever our intellectual project, it will run aground if the notion of the Idea remains trapped in the deadlock of form and matter, the legacy of the distortion consequent upon the monotheistic construct of a Platonism without the Gods.
In authentic Platonism, Ideas were always inseparable from the lives of souls, mortal souls like ours, but most importantly the immortal souls of the Gods, who do not suffer from our deficiencies. I have had little occasion to speak of Ideas in this essay, which would seem surprising under the conventional impression of Platonism as fundamentally concerned with these entities. In this, however, I merely follow Plato himself, who explains that as the Sun is to the things of the world, principle both of their very being and of their visibility, so is the Good to the Ideas, that by which they are and are known (Republic 509b). But the Good is the Unity of each thing, and so the study of the modes of unity enfolds and takes up into itself the inquiry into forms or ideas. The ultimate mode of unity, in turn, is the peculiar or unique; and the most unique of things are the Gods themselves. For the Platonist, this is how the priority of the Gods in and beyond the cosmos offers itself to our understanding.
* Lecture presented at the Polytheist Leadership Conference, Fishkill, NY, July 12, 2014.
 See, notably, Gerd Van Riel, Plato’s Gods (Farnham, Surrey: Ashgate, 2013).
 All translations from the Philebus and from the Sophist are by H. N. Fowler, occasionally modified.
 In accord with Homeric and other early poets’ usage of philon (see LSJ, φίλος, I.2.c.).
 The ambiguity of naming in the Cratylus should be compared to the account of naming in the Seventh Letter, in which the name of a thing, together with its definition, image, and the knowledge of it, form a single holistic totality in the soul (Epistle VII 342c) which at once exhibits the thing itself but also presents itself in tension, even opposition to it. The name has its privilege over the other elements of this system inasmuch as it corresponds to the ‘true name’ in denoting the object. This correspondence anchors the system by which a unique entity is joined to the totality of a language, but the ‘true name’ remains ‘ineffable’ simply on account of its irreducibility to this system.
 For an interpretation of the fragments of the historical Parmenides with which I am largely in agreement, though not necessarily with her reading of Plato, see Patricia Curd, The Legacy of Parmenides: Eleatic Monism and Later Presocratic Thought (Princeton: Princeton University Press, 1998).
 Mary Margaret McCabe, Plato’s Individuals (Princeton: Princeton University Press, 1994), pp. 4-5.
 Although in ordinary language not all ‘whats’ are also ‘whos’, the capacity to universalize the ‘who’ function, and thus the mode of unity it represents, is essential to the pantheistic position. I would argue that this analysis shows that pantheism, just like monolatry and henotheism, is best understood on a polytheistic basis.
 Something like this is at stake in the Stoic Chrysippus’s paradox about identical twins ‘Dion’ and ‘Theon’.
 Plato says in the Sophist (255e) that “each thing is different [heteron] from the rest not on account of its own nature, but through participating in the idea of the Different.” ‘Difference’ here is not the source of individuation, but of diacritical or differential identity, what we might term structural identity, which is necessarily holistic and in that respect a unity subordinating the individual ‘natures’ that are distinguished from one another by it.
 Translations from the Laws are by R. G. Bury.
 See the ten kinds of motion discussed at Laws 893b-894c. The formula for generating linear motion from the superimposition of circular motions was first clearly articulated by Nasir al-Din al-Tusi in the 13th c. CE (the ‘Tusi couple’), though anticipated in remarks by Proclus in his commentary on Euclid (In Eucl. 106).
 The two kinds of motion are described at Timaeus 40ab: the one, rotation, is “uniform motion in the same spot, whereby it conceives always identical thoughts about the same objects,” the other, revolution, is “a forward motion due to its being dominated by the revolution of the Same and Similar” (trans. R. G. Bury), the latter phrase referring to the motion of Identity which together with that of Difference is constitutive of the Soul as such (Tim. 36c).
 Trans. Paul Shorey
 Trans. Hippocrates G. Apostle.
 This is likely the first indication of the systematic distinction later Platonists would draw between ‘henads’ and ‘monads’, terms which seem synonymous in the Philebus, although the discussion there is clearly less technical than the discussions within the Academy it permits us to glimpse.
 It is noteworthy that one of the maxims attributed to Pythagoras explicitly juxtaposes numbers and names: “What is the wisest thing? Number; but second, the one who put names to things” (Diels-Kranz 58c4, quoted by Iamblichus, Vita Pythagorae 82).
 As reported by Plutarch, Quaes. Conv. 8.2.1.
 On this primary collective activity of the Gods, see in particular “The Second Intelligible Triad and the Intelligible-Intellective Gods,” Méthexis 23 (2010), pp. 137-157. [Reprinted in Essays on the Metaphysics of Polytheism in Proclus (New York City: Phaidra Editions, 2014).]