Plato on the Mechanics of Koinōnia Formation

This paper argues that, in order to understand the unified relations that are commonly predicated of koinōnia in the ethical, political, and cosmological spheres respectively, one must first appreciate certain prerequisite “principles” or “rules” that are necessary for koinōnia formation. One principle which has been for long the subject of intense discussion in Platonic scholarship is proportionality. However, rather than stopping short at the unproblematically straightforward point of connection between proportionality —in the broadest possible sense— and well-ordered wholes, I suggest that we can get a much richer account from Plato’s preoccupation with proportionality and koinōnia formation by exploring the different proportional models he puts to work in different contexts. It will be argued that for Plato geometrical proportionality in particular is the binding principle par excellence for koinōnia formation as it is the fairest model of order and enables the most enduring complex wholes. Approaching the point from this angle may yet provide a further significant way of understanding the undeniably obvious political differences between the Republic and the Laws in terms of koinōnia formation.


Introduction
It is well-known that the term koinōnia is highly charged not only in the philosophy of Plato, but also in standard Greek thought generally. Broadly construed, the term seems to have signified some sort of shared or joint relation among a plurality of parts based on dynamic exchange, inclusiveness, and collectivism. Ancient physicians, politicians and natural philosophers alike were driven by the question of how to create well-functioning systems amidst complexity. Indeed, the Greek psychē was all too conscious of the potential threat that conflict and discord might pose to the complex structures (specifically the polis) that pervaded their reality. The guiding motivation for the ancients, then, was to conceive of systems that would mitigate such conflicts in order to ensure that the human body and the citizen-body functioned as optimally as possible, drawing on observations from the regularity of the cosmos as they did.
What might be said to be lacking in our reception of koinōnia, however, is a study on the very mechanics of how to form better a koinōnia-based relationship. That is to say, our understanding of koinōnia is only half complete until further substantive explanation is given to the antecedent "principles" or "rules" that go into making a particular instance of koinōnia. For the sake of clarity, let me briefly distinguish between two different aspects of koinōnia. There is, on one hand, what I take to be a basic (or non-normative) sense of koinōnia. Simple relations, interactions or transactions between two or more parties would suffice for a koinōnia of this type. Such relationships are ubiquitous, for we engage in all sorts of exchanges with our fellow citizens in everyday life. What is important for me to note here is that no prior planning or forethought is necessary-only that some sort of minimal or lower grade joint effort has taken place. This can, I think, then be contrasted with what I also take to be a more complex (or normative) sense of koinōnia, according to much deeper, richer, and long-lasting koinōnia relationships predicated on stronger pre-existing bonds. This latter is the type I shall be principally concerned with in this paper.
It is perhaps unconsciously taken for granted that koinōniai normatively understood simply exist. But the question we must ask, I contend, is exactly how these relationships come about in the first place. What are the necessary pre-requisites for koinōnia formation? This concern is not only of central interest for ancient theories of koinōnia in general, but even more so for Plato, who clearly puts special emphasis on koinōnia as an aim for political and human striving.
One of the first passages from antiquity to highlight the value of koinōnia in principally a political context comes from Thucydides' Histories (3.10). As Thucydides sets out to discuss the twin themes of justice and virtue, his speaker tells us that: (T1) "… neither friendship (philia) among individuals nor cooperation (koinōnia) among cities is secure unless it is undertaken between partners who show honour in their dealings with one another and who in general are of similar character (homoiotropoi); for where there is a discrepancy of minds differences in actions follow" 1 .
This excerpt clearly highlights one key feature that is paramount to koinōnia formation: likeness of customs or similarity of character (homoiotropoi). As I understand it, this is quite naturally one of the ways competing independent Greek states could see themselves as part of a much larger, connected community under the banner of the Greek speaking world. Despite geographic and local differences, Greek city-states could be bound by their collective customs and practices, which in turn gave them a distinctively "Greek" identity. It seems to me, therefore, that prioritising this key aspect can only be appropriate to Thucydides' purposes in the context of the Histories 2 . Notice, finally, the awareness in the latter part of the sentence how potential differences might breed conflict among people. As I alluded to at the outset, this premonition was always at the forefront of those pioneers who felt inspired to theorise about koinōnia-formation in society and is one we shall frequently return to in this paper.
One could, however, object that the passage still falls short of establishing how we get to thicker koinōnia-based relationships in the first place. True, likeness of customs might be an indispensable feature to facilitate civic togetherness and cooperation between cities. But we might still be justified to ask for a more technically robust and conceptually sophisticated explanation for universal koinōnia-formation. Do we not need something to determine how, mechanically, certain unified relations come to be held among a plurality of disparate parts? It is my intuition here that Plato was one of the first philosophers to address this concern. In his writings, I shall suggest, there is a deliberate attempt to explore the ideal "principles" or "rules" that are necessary for long- 1 The translation is taken from Rhodes, P. J., History, Warminster: Wiley-Blackwell, v. III, 1994. 2 We find similar musings on the importance of likeness of character in Aristotle's discussion of friendship in the NE, for which see 1157a10-12, 1162a9-15.
lasting koinōnia of the aforementioned normative type. My motivation might be reformulated thus: what, for Plato, are the bonds that best foster koinōnia relationships, creating a sense of long-lasting community amongst a plurality? Owing to the several different explanations one could take in response, here I would like to focus on just one aspect I take to be central to an appreciation of koinōnia in Plato: proportionality. The passages I have selected, including my analysis of them, must therefore be restricted to their bearing on the importance of proportionality to koinōnia formation in Plato.

The Origins of Proportionality
What exactly do we mean by such a manifestly broad concept as "proportionality"? Put simply, a proportionate relation is one where there is some similarity of terms in accord with a mathematically defined ratio or formula. The task of the person who is going to remedy conflict and create order from a pre-existing state of complexity is, then, to calculate the precise ratios, or equality of ratios, that are appropriate for fostering a certain relation among things in a given context. Pointedly, our ancient sources make it clear that already by the time of Plato's writing activity proportionality could be expressed by a number of specific ways. Consider the following from Archytas' Fragment 2: (T2) "There are three means in music: one is the arithmetic, the second geometric and the third sub-contrary [which they call 'harmonic']. The mean is arithmetic, whenever three terms are in proportion by exceeding one another in the following way: by that which the first exceeds the second, by this the second exceeds the third. And in this proportion (analogia) it turns out that the interval of the greater terms is smaller and that of the smaller greater. The mean is geometric, whenever they [the terms] are such that as the first is to the second so the second is to the third. Of these [terms] the greater and the lesser make an equal interval. The mean is subcontrary, which we call harmonic, whenever they [the terms] are such that, by which part of itself the first term exceeds the second, by this part of the third the middle exceeds the third. It turns out that, in this proportion, the interval of the greater terms is greater and that of the lesser is less" 3 . 3 The translation is taken from Huffman, C., Archytas of Tarentum Such an intellectual discovery of different proportional means, however illuminating, might make our task of interpreting proportionality in Plato even more challenging 4 . The question we must return to is whether, in certain contexts, Plato's engagement with proportionality is best understood as covering the general study of proportions 5 , or, conversely, whether we see any prioritisation of a specific model of proportionality for koinōnia-formation. While both possibilities still succeed in underscoring the value of proportionality for koinōnia-formation in Plato -which is after all my main purpose here-, I think we can get a much richer account from Plato's interest in proportionality by seeing him operate mostly with a specific model. That shall be a secondary aim of my purposes in what follows.

Geometrical Equality in the Gorgias
One place in the Platonic corpus where the link between proportionality and koinōnia is first explicitly adumbrated is the Gorgias. Our main authority for this is a well-known text which follows from the breakdown of Socrates' famous exchange with Callicles. After repeatedly dismissing Callicles' pleonectic lifestyle and stressing that the orderly man, and only he, can be supremely blessed and happy, Socrates then proceeds to draw some rather unfortunate implications for an advocate of the Calliclean lifestyle: (T3) (i) "For no other man would be a friend to such a man; nor would god. For he is incapable of community (koinōnein); and when there is no community (koinōnia) with a man, there can be no friendship with him". (ii) "Now the wise men say, Callicles, that heaven and earth, gods and men are bound by community (koinōnia) and friendship and order and temperance and justice; and that is why they call this whole universe the 'world-order', not 'disorder' or 'intemperance', my friend. But I think you don't heed them, though you're wise yourself. You haven't noticed that 4 I remain agnostic on the issue whether Plato was personally acquainted with Archytas or not (for which see Morrison, J. S., "The Origins of Plato's Philosopher Statesman", in: Classical Quarterly, v. LII (1958), p. 215 and fn. 21 below). However, I think it is quite plausible that Plato was at least familiar with the work of Archytas. Burnyeat (2000) quotes Republic 530d8 as evidence for such a connection, where he takes Plato to be referring to the Pythagoreans but especially to Archytas. See Burnyeat, M., "Plato on why mathematics is good for the soul", in: Smiley, T. J. (ed.), Mathematics and Necessity: Essays in the History of Philosophy, Oxford: Oxford University Press, 2000, 15ff. geometrical equality has great power among gods and men; you think you should practice taking more, because you are heedless of geometry" (507e3-508a8) 6 .
Let me divide this passage into two separate chunks. Relevant to our purposes, in the first (i) Socrates appears to highlight one pertinent outcome of his evaluation of the Calliclean lifestyle: that such a man is incapable of friendship and koinōnia with both his fellow citizens and with god. Now, why is it that the unrestrained, pleonectic individual cannot enter such positive relationships? The appearance of koinōnia might seem rather abrupt and unexpected, especially since the discussion of the dialogue so far has pertained to one's individualistic happiness irrespective of any positive relations with others. If anything, however, this serves to remind us that the happy life is all about good relations, at least in the context of the Gorgias. It is because of the unhappy disharmonious internal relations within Callicles' own soul that he is incapable of community with others. And this, in turn, is because Callicles' overbearing appetitive soul and desire to have more than others produces an imbalance the likes of which engenders discordant interpersonal relationships. What we need to have good relations with others, then, is neither excess on either side but rather a certain isomorphism between persons. And this can only be struck, according to this theoretical reconstruction, if both souls are well-ordered so that one neither overpowers nor is overpowered by the excess of the other's soul.
Apart from suggesting that Callicles is incapable of koinōnia with others, the text also points to relations between individuals and god (or the gods in the plural a few lines later). So, we must ask again: why can the intemperate man not form a friendship and koinōnia with the gods? The simple response is just to say that the pleonectic individual is on the wrong footing with the divine. But while interpersonal relationships largely depend on the moderation of each soul, our relationships with the divine might in one way be entirely unequal though in another way perfectly compatible with koinōnia. For instance, no one would dispute that the gods are infinitely stronger and more powerful than us, though the well-ordered person at least still forms a koinōnia with them. The problem with Callicles, then, is that he tries to match the strength of the gods with the excess of his own soul, but such an agreement or "fit" between mortals and the divine is simply ill-ordained in nature. Mortals need to be put into their proper 6 The translation is taken from Irwin, T., Plato: Gorgias, Oxford: Oxford University Press, 1979, cursives added. relations with the divine, but this is only achieved by respecting the superiority of the divine over the human world, and by keeping our own souls in check 7 .
Thankfully, the second half of our text from the Gorgias provides a significant clue how to remedy pleonexia and the incongruent relationships it creates both within ourselves and within our relationships with others. Socrates famously chides Callicles for failing to notice the great power that geometrical equality (ἡ ἰσότης ἡ γεωμετρικὴ) holds among gods and men. "Among gods and men" seems broadly unrestricted, and since this coupling was just paired with heaven and earth, we ought to assume that geometrical equality holds great sway over the entire universe. This means, in effect, that proportionate relationships secure all the koinōnia that exist within the order of the whole kosmos. Koinōnia, one might put it, requires geometrical equality. Now, equality itself was itself earlier associated with justice (dikaiosynē) by the masses -a point both Callicles and Socrates themselves proffer but whose truth-value both crucially disagree on-while pleonexia was said to be the cause of all injustice. It seems, then, that equality is being diametrically opposed to pleonexia. As Socrates puts it, "you think you should practice taking more (pleonexia), because you are heedless of geometry", where "geometry" here stands in place for the earlier geometrical equality in particular. Clearly, an understanding of geometry is at least a necessary condition for fostering koinōnia relationships. But at this stage, two complementary questions seem to present themselves: i) how would our lives and relationships with others be transformed to accommodate for koinōnia if we were well-versed in the power of geometrical equality, and what would this look like? ii) What is exactly encompassed in the term "geometrical equality" anyway? Since an answer to question ii) holds the key to understanding the more practical concerns of question i), I want to now turn to an examination of the key phrase "geometrical equality" and its place within the context of the dialogue.
As far as I can see, two competing interpretations in the literature have emerged from this edifying remark. First, the deflationary reading. Burkert 8 ,7 One might compare Eryximachus' eulogy to erōs in the Symposium, which contains much discussion on the universal causal force of erōs to bring all potentially hostile opposites into friendly relations. One relation that is especially highlighted is that between mortals and god, for which it is the province of the expert seer to work out the right sort of arrangement. Crucially, it is by recognising that the gods are stronger than us (tois kreittosin ēmōn theois, 188e1) that we are able to form a koinōnia with them. This unequal power dynamic between mortals and the divine may, I submit, be expressed in mathematical proportional terms. followed by Huffman 9 most recently in his monograph on Archytas, suggest that "geometrical equality" must be referring to something mathematically neutral. In other words, geōmetrikē in the broadest possible sense simply encompasses the general study of mathematical proportions, irrespective of its more specific instantiations (as exemplified in T2). Support for this interpretation is given by the fact that the only other mention of geometry in the Gorgias is broadly neutral as to its specific applications. Socrates had earlier vowed to compare technai with mere knacks in the way the geometricians (hoi geōmetrai) would: "as cosmetics is to gymnastics, so is sophistry is to legislation, and as cookery is to medicine, so is rhetoric to justice" (465c1-5). Here, Socrates clearly employs a simple mathematical formula, the sort of thing geometricians always get up to. They put things into all sorts of relations, notwithstanding their knowledge of more specific proportional means. Perhaps, then, what Callicles needs to combat his obsession with pleonexia is first and foremost a comprehensive knowledge of geometry. On this view, Callicles' pleonexia is so extreme that an appeal to "one specialised sort of equality is out of place" 10 .
Even a proponent of the deflationary reading such as Huffman, however, cannot avoid conceding that Plato does give special emphasis to the geometric mean here (see again T2) 11 . This is what we might call the standard interpretation, favoured by the likes of Dodds 12 in his seminal commentary on the dialogue and later by Vlastos 13 in several papers. This proportion is, we may note, the model that Plato will unqualifiedly end up with in the more mature stages of his thought, where in the Timaeus geometric proportion in the technical sense is the binding principle par excellence for the friendly relation that holds between the four elements of the world-body (32b-c). However, it is important to stress from now that putting special emphasis on geometrical proportionality is not to say that the general study of proportions does not form part of the ideal education advocated by Socrates. Certainly, Callicles will benefit from a general understanding of mathematical proportion. Only that 9 Huffman, C., Archytas of Tarentum: Pythagorean, Philosopher and Mathematician King, 2005, p. 209ff. 10 Ibid. 11 Ibid., p. 208. 12 Dodds, E. R., Plato: Gorgias, Oxford: Oxford University Press, 1945, pp. 339-340. 13  within this study there is specifically one proportion that is most conducive to combatting pleonexia and instantiating koinōnia.
To substantiate that claim, as previously mentioned two diverging ways of life have so far been offered in the dialogue. On the hand, the many were said to espouse the theory that justice is equality -where we can assume Plato has in mind pure democratic equality and right to participation regardless of personal differences (corresponding to simple arithmetic proportion)-while Callicles sets up his support for pleonexia in opposition to presumably this form of absolute democratic equality. But is the point not that a third, altogether preferable alternative, has completely escaped Callicles' notice? Arguably, Socrates can agree with the many that justice consists in equality without also having to endorse their specific version of democratic equality. If this was Plato's intention, then his remarks on democracy both in the Republic and elsewhere would surely be out of place. In the Republic, for instance, the many are wrongly said to give a sort of equality to both "equals and unequals alike", the converse of which gets us strikingly close to geometrical equality itself (558c1-4). I shall return to a fuller examination of the place of the Republic towards the end of this paper.
For now, while Callicles' model of pleonexia is neither the preferred option, I want to suggest that a version of that model will survive once it has undergone radical revision. For at the core of Callicles' theory is the straightforward idea that those who are superior ought to be given more. At one point, the better and superior man even becomes equated with the wiser man, to which it is then further agreed that the wise man ought to rule over the unwise even if they outnumber him by a thousand. In this way, it is agreed that the man who is better and wiser should rule over the lower men and have more than them (490a6-8).
As is well noted in the literature, this is hardly a point that Plato would find contentious 14 . In line with the supremacy of philosophical wisdom, the wiser man should indeed be given more of a right to rule over others. The untenability of Callicles' position is that he wants to endorse this largely non-controversial thesis on the one hand and at the same time maintain that everything should be concentrated in the hands of the superior, even if that means taking more 14 Irwin, T., Plato: Gorgias, Oxford: Oxford University Press, 1979, p. 226. Thus Harvey (1965 recognises the compatibility of Callicles' uncontroversial claim at 583e that "the greater should have more than the less" with geometrical proportion as Plato will present it, but fails to account for the general incompatibility of Callicles' allegiance to pleonexia with geometrical proportion. See Harvey, F. D., "Two kinds of equality", in: Classica et Mediaevalia, v. XVI (1965), pp. 127-128. than is due or right and encroaching on the demands of others. By force and unjustly if necessary, too. For Socrates, by contrast, the just rulership of the wise man does not entail that he should take all the benefits that are accrued from his leadership over the city. This is well-evidenced when he suggestively puts the question to Callicles "shouldn't he [i.e. the wise man] rather have more than some, and less than others?" (490c4-5). Notice that inequalities might still persist here -the wise may well have more than the unwise-but overall equality ensues since the wise deserve more by virtue of their merit when compared to others. This will not, crucially, exclude the less deserving from having some share, however small. Thus, overall equality will prevail since each is given only what is due, according to what is proportionate and fair. And this, I submit, is one way in which geometrical equality could in theory manifest in the state.
We are now in a position to attempt a response to question i), namely, how an appreciation of geometrical proportionality (in the preferred technical sense) translates into koinōnia-based relationships. We have already discussed the sorts of relationships one ought to cultivate with the gods. In the light of our recent examination of geometrical equality, this now becomes even more lucid. The gods, being superior to us, ought to be treated and revered as such, while humans, being inferior, ought to know their proper place and not overstep the boundaries of what is appropriate for them to do -similarly in our relationships with others in the political community. As long as we stay within the limits of what is deserving to each, friendly interaction is made possible between potentially unequal and disparate members of society. In this way, people can see themselves as part of a collective, shared community (koinōnia) based on mutual fellowship rather than grave or harsh inequalities.
One could finally speculate how geometrical proportionality might be expressed within the koinōnia of each individual soul comprising the community. For instance, Callicles was earlier contrasted with the initiated man who regulates his pleasures and desires rather than falling prey to them. According to this portrayal of the good man, he gives more to those pleasures that make him good thus allowing him to function well, and less to those that have the opposite effect. By putting the pleasures and desires of the soul into a sort of geometrical progression in this way, the good man may be free from internal conflict and discord. To that end, it is noteworthy that the uninitiated soul facing its judgement in the final myth of the Gorgias is described as being "full of disproportion and shamefulness" (525a5). The word for disproportion here is "asymmetria", a highly normative concept in Plato's later metaphysical vocabulary which became closely linked with geometrical proportionality in the Timaeus. It is even associated with koinōnia under the heading of taxis (order) in the Definitiones 15 . This final piece of evidence is what we need to make the link between proportionality (specifically the geometric type) and koinōnia inseverable.

Proportionality in the Political Dialogues
For the remainder of this paper, I would like to demonstrate that an awareness of proportionality (broadly construed) as a key explanatory principle for koinōnia formation runs through Plato's political dialogues. It does so, I suggest, because in these dialogues we find contexts that are principally concerned with the good binding, or koinōnia of the city, which is often made up of people who exhibit many different and sometimes conflicting qualities. If there is to be one expertise responsible for the overall organisation of the city, then, proportionality is an indispensable ingredient in ensuring that the whole is efficiently bound together and functions well.

The Statesman
How exactly the expert politician does this is one of the chief concerns of the dialogue named after the statesman. Indeed, the kingly art is said to concern a care of the "whole community" (koinōnia) together" with all its respective interrelations (276b7). But, as we shall see, in some important respects the skill of the politician is also on a par with the expertise of all those other craftsmen whose distinctive activity is to create a well-ordered whole from a pre-existing plurality. Within such an account, I would like to suggest, a preoccupation with due measure (to metrion) or simply "proportionality" is central. We might say from now that proportionality extends not just to the good relations existing within specifically the human koinōnia, but also to the koinōnia that holds different things within one complex whole of which each of the various different technai have their province. Let me, then, first begin with a closer examination of those passages which address the universal importance of due measure for technē, before fleshing out the more pertinent link between proportion and koinōnia, and finally what implications this has for the statesman's distinctive activity. Now, the Eleatic Stranger's investigation into the art of measurement first begins with a delineation between two different kinds: excess and deficiency, which are later given the more specific opposites "greater" and "lesser", "more" and "less" (283c3-6). At first glance, it seems that the stranger's purposes for introducing this division are self-reflexive. It is as if he is warning us against the length of his previous myth as well as speech in general. The salient point seems to be that the appropriate length of a speech depends on the subject matter at hand; any deviation from this norm in the direction of either excess or defect creates an unwanted disproportionality. Thankfully, there exists an art of measurement (metrētikē, 283d1) which serves to remind us not only of the appropriate length of speeches, but also of what is appropriate in the case of production generally where the material parts of a complex may sway towards either extreme. This art is then bifurcated into two further parts: (T4) "This way: one part will correspond to the sharing (koinōnian) by things in greatness and smallness [i)] in relation to each other, [ii)] the other to what producing things necessarily is" (283d7-9) 16 .
According to the preferred translation here, what is important to note is that the koinōnia of things that partake in the great and the small may be expressed in two different ways: either in relation to each other, or in relation to what production necessarily is. An alternative rendering of this second part of the division could be: to the necessary being (ousia) of production, implying that this second way of measuring has a higher ontological status than the first because of the objects it is disposed towards. The straightforward inference, it seems to me, is that were we to simply measure the koinōnia of an object in relation to its greater and lesser components, we would have no standard for ascertaining the appropriate measure for a given production. Thus, if I tailor a suit with reference to the largest and smallest size in disregard for your particular size, I would never be able to create a suit that is appropriate for you. While the second alternative is couched in cryptically vague language, we can at least say that what is absolutely essential for production is some distinct thing, separate from the greater and the lesser, but necessary for each individual case of production. This is the way the koinōnia of things partaking in the great and the small ought to be ideally regulated.
What the stranger has in mind is given further articulation just a few lines later, when he introduces the specific nature of due measure (to metrion). This is that standard, or paradigm that measurement looks towards rather than simply measuring things against the greater or the less. The koinōnia that holds between the parts of an object needs to be brought into a relation that remains faithful to due measure. Not doing so, we are told, would destroy all the various different technai and their productions, since they only truly operate well when they guard against the more and the less in relation to what is in due measure (to tou metriou, 284a5-b1). For it is only by preserving measure (here to metron) that they produce all that is fine and good, where I suspect to metron refers neutrally to "measure" as such, even though it is only the right measure (metron) that will create what is in due measure and thus metrion.
All of this might seem trivially uncontroversial. Of course, all craftsmen engage in measurement when they work on their creations. Yes, the stranger would concede, but not many realize that craftsmen are only successful to the degree that they measure with specific reference to to metrion. Indeed, there follows a similar criticism of certain sophisticated people (kompsoi) who also acknowledge that measurement is ubiquitous to creation, only to disregard exactly what type of measurement, and how measurement engenders stable, good, and long-lasting complex wholes. As the Stranger says: (T5) "For in a certain way it is the case that all those things that are the products of the various kinds of expertise share in measurement; but because of their not being accustomed to carrying on their investigations by dividing according to classes, the people in question both throw these things together at once, despite the degree of difference between them, thinking them alike, and correspondingly they do the opposite of this by dividing other things not according to parts, when the rule is that when one perceives first the community (koinōnian) of many things, one should not desist until one sees in it all those differences that are located in classes, and conversely, with the various unlikenesses, when they are seen in multitudes, one should be incapable of pulling a face and stopping before one has penned all the related things with one likeness and surrounded them in some real class" (285a3-b6) 17 .
Let me try to unpack this rather dense and complicated piece of text. Clearly, where the sophisticated have gone astray is in their ignorance of different relations pertaining to measurement. In fact, they muddle both types of measurements together, "thinking them alike". The way to separate the true art of measurement from the other, according to the Stranger's analysis, is to first begin by dividing things according to parts (kata merē). What this could mean is filtering out all the different components that make up one larger whole, taking in isolation each individual component and putting each into its proper place. As the Stranger says, one needs to perceive all those various differences and unlikenesses existing within one thing before gathering them into one overall likeness. Plausibly, this might well involve applying measure or proportion to the process; by putting different and unlike things into a relation based on due measure, the whole may be bound together as one likeness.
Even more important is that one first acknowledges the "koinōnia" of many things. Much like the earlier mention of koinōnia in the dialogue, this might mean that there are already certain relations existing within some complex items. But within such koinōniai, as the Stranger also says, one must notice things that are unlike and different. This means that while there may be certain pre-established relations within one koinōnia, it is still possible to conceive of even more sophisticated and integrated relations within one whole 18 . That is to say, when one takes a complex of parts one has to first consider those relations that already exist and all their differences, but by implementing due measure to the whole those relations can become even more tightly bound together.
While such important lessons in dialectics and divisions are helpful for the knowledge of all things (285d7), remember that the Stranger's digression into due measure is supposed to have implications for the statesman's own distinctive activity. How might we apply the technical vocabulary and important lessons to be drawn from the above passage to politics? Plainly, the task of the statesman is to take over a pre-existing community of people with all their respective interactions and interrelations before implementing an even higher degree of integration. In line with the Stranger's previous remarks, the statesman would first perceive the relations that he perceives in front of him; but within such a koinōnia, there are also several different parts which may be different and unlike; without stopping there, the statesman ought to take those differences into consideration and, by applying a certain measure or 18 This may have implications for the question whether there can be such a thing as imperfect mixtures or not, which Philebus 25e7 seems to problematise. While I contend that a mixture ought to be a good one, normatively understood as having a cause which imposes the right measure on a substrate, it seems to me that the Statesman at least leaves open the possibility that even those combinations of the greater and the lesser that are not informed by "due measure" are still nevertheless genuine combinations, albeit poorly created ones. Similarly, while koinōnia ought to denote good and orderly relations, it seems there can also be koinōniai that are less valuable, simply relations existing within one complex that can be improved for the betterment of the whole. See my distinction between two types of koinōnia on p. 2 of this paper.
proportion to the whole, create an even better koinōnia among the citizens by putting them all into one likeness. In this way, despite qualitative differences among the citizen, all can be made alike when due measure becomes a part of their newfound community.
In the same way that the weaver must take two very different materials and intertwine them into one cloth, the statesman thus takes people whose souls vary to the degree of their inclination towards the virtues of courage or moderation. Now, we might speculate whether these virtues can be put on two diametrically opposed ends of one scale 19 . What is vital for the statesman, then, is to find the right balance and mean among these extremes that allows each individual and society as a whole to flourish. On this account, certain souls may still predominate in either of the two virtues, but when proportion is applied they may obtain the right measure in their souls between courage and moderation, and so may society as a whole. The motivation, again, is to ensure that differences do not lead to large scale conflicts and wars. Proportionality mitigates against such risks by noticing the differences among the parts of one community and improving the relations within that community through one common measure for all.

The Republic
Let's move now to Plato's most cited political dialogue. While proportionality is not as explicitly emphasized as it is in the dialogues we have surveyed so far, I would like to show that proportionate relations are still at work in the koinōnia of the city in the Republic. We can also conjecture exactly what type of proportional equality is best suited for the tripartite political framework of the ideal state, and what implications this might have for situating the Republic within Plato's wider engagement with proportionality and koinōnia formation.
First, it is clear that the supreme motivating principle for the political project of the Republic is how to foster civic unity. When factions arise within the community and the citizens making it up are at war with each other, Plato warns, the city then becomes pluralised rather than unified. Since, however, we do see that society is composed of potentially hostile opposite groups of 19 The Stranger appears to criticise the sort of thing that is customarily said about the virtues: that all the parts of virtue are amicably disposed towards each other (306b12-c1). However, even if all the virtues are syggenes qua virtues (c5), courage and moderation in particular are said to be "extremely hostile to each other and occupy opposite positions in many things" (a9-10). Perhaps, then, these two virtues are the extremes the statesman takes as his material as he applies due measure in the souls of the citizens and in the city as a whole. people, the question is how to ensure that the whole becomes unified while also recognizing certain entrenched personal differences. The main response for Plato in the Republic is to suggest that each class only does what is naturally suited for it. This is what has traditionally come to be known as Plato's principle of "political justice", according to which each performs its own distinctive activity rather than impinging on the domain of another who is better suited for a different role 20 . Each may thus become "a single person rather than many people, and in this way the entire city may grow to be a single city rather than many cities" (423d4-6). We can, therefore, fairly proclaim from now that a minimum requirement for the unity of the city is that each part recognises and performs only its own unique function.
In fact, this specialization of functions is also necessary if there is going to be a genuine koinōnia in the city at all. As early as Book 2, Plato's principle of justice is said to allow different people to be 'associates and helpers' (koinōnous te kai boēthous) with each other (369c1-4), since each farmer or artisan makes his product "common (koinon) to all", "sharing" (koinōnounta) it with others (369e2-370a4). Socrates finally concludes that such exchanges within the city was the main reason for forming an association and establishing a city in the first place: (T6) "What about trade in the city itself? How will each group share its production with others? That after all was our reason for forming an association (koinōnian) and establishing a city" (371b4-6) 21 .
The desired koinōnia of the city is then encapsulated even more vividly in a famous proverb Plato alludes to at least twice in the dialogue, namely that everything should as much as possible be "shared among friends" (424a1-2, 449c8). Exactly how this maxim might become embodied in the political community is the task of Book 5 of the dialogue to work out 22 . Now, what is so distinctive about the Republic is the way in which it discusses several types of relations; from intra-familial and personal relations, to socio-economic relations within the three-tier class system, to the hierarchical relationship of ruler to ruled. If everything is going to be "shared among friends", then, clearly each of those relationships need to be organised in such a way so as to harmonise with that paradigmatic principle. What I would like to do now is go through each of these relations, before drawing some more specific conclusions on koinōnia in the context of the Republic.
Book 5 begins by first challenging the famous proverb in its generality. Merely saying that everything ought to be "shared among friends" is just as good as taking the easy route out, Socrates warns. If the maxim is going to be applied in the "right" (orthōs) way (449d6), then, it will require a "second major discussion" about the ideal state (450a7-8). Significantly, Socrates begins by reminding us of one initial premise for the koinōnia of the best constitution: that the men of kallipolis ought to assume the role of "guardians of the herd" (451c7-8). All the subsequent relations that follow within the city, therefore, must have some bearing to the ruling class which itself assumes control over the whole city. One relation that is especially discussed is the "taking of wives, marriage, and having children" (423e7). However, it is worth remembering that these relations are importantly restricted in their scope. For it is only true to say that the sharing of women, marriages, and children, are held in common within solely the guardian class. The philosopher ruler will regard the male children his sons, the female children his daughters, who in turn will call him father, and so forth with grandchildren. All the children born from parents around the same time will call themselves brothers and sisters, whereupon Socrates concludes: (T7) "There you are, Glaucon. That's what it is for women and children to be "in common" (koinōnia) among the guardians of your city.
With such familial ties among the inhabitants of the city dealt with, the next task for Socrates is to test whether what has been said is consistent with the best possible constitution sketched so far in the dialogue. Recall that the goodness of the city was predicated on its unity. The greatest evil for a city is said to be that which tears it apart and makes it many instead of one, while the greatest good is that which unites the city and makes it one (462a9-b2). Taking this as a preliminary guiding principle, Socrates then applies the same methodology as with intra-familial ties to feelings of pleasures and pains within the city. When, as far as possible, all the citizens are equally affected by pleasure, pain, gain and loss, then a community of feeling arises which makes the city one unity. This, in turn, will mitigate against any such individual variation in feeling that might lead to large-scale divisions. The well-regulated city, then, is one where the greatest number of people employ the phrase "mine" and "not mine" with reference to the same things. This thought is given further flesh through the analogy of a whole community and the human body. When, the analogy goes, someone has a pain in her finger, we say that the pain belongs to and is felt by the whole person. Likewise, with the human community. If one part of the community is hurting, then the whole, which is bound into one community through the organising power of the ruling soul, feels the pain (462c10-d5).
Immediately following the analogy, however, Socrates again shifts the focus onto specifically the guardian class. While the "common" people shall refer to the rulers as "saviours and defenders", the rulers by contrast shall refer to each other not just as "fellow-rulers" but, also as "relatives". Each individual guardian will call the other his brother, sister, father, mother, son, or daughter. Not merely as a verbal convention, but also as a matter of conscious behaviour and respect for one another. By thinking and speaking to one another in this way, all shall regard the same pleasures and pain as their own, and what will be called "mine" and "not mine" will also be held in common. Now, the reason why there can be such relations within the guardian class in the first place is because of the aforementioned intra-familial associations, according to which women and children are all held in common. This, above the general organisation of the city, is demarcated as the greatest good (megiston agathon) for the city (464b5-6).
But how is this at all consistent with Plato's clear remarks elsewhere that the unity of the city is a matter of the whole, each and every class included? Should the city not also be regarded as one whole family? Here, an appreciation of the function of the Noble Lie (414b-415d) might come to aid. I cannot give a full examination of the function of the myth here, but what I would like to note is that while the myth does reinforce tripartition, it is also intended to show that all citizens alike derive from one common autochthonous origin 24 . The consequence is that all the citizens, not just the guards, may see themselves as part of one family in which they can regard all others as their brothers and sisters. But this, it must still be conceded, is a mere far-cry from the much stronger familial relations that are predicated of solely the guardian class 25 . 24 Loraux, N., The divided city: On Memory and Forgetting in Ancient Athens, New York: Zone Books, 2002, p. 198. 25 One helpful solution to this problem might be that of Schofield (2006), who argues for "grades" or "modes" of brotherhood. See Schofield, M., Plato: Political Philosophy, 2006, p. 224 with fn. 91. According to this suggestion, the guards may have one mode of brotherhood with fellow guards, and a lower grade of brotherhood with the other class. This enables the city to be one whole family while also recognising a hierarchy of relations within that family.
As Socrates explicitly spells out, as long as factions do not arise within the guardian class, there can be no danger of the rest of the city being divided, of another class going against the guardians, or of the guardians fighting among themselves (465b8-10). This assumes a high degree of functional responsibility to the guardian class, who operate as the overall unifying principle for the whole city. In fact, when the ideal city does eventually break down, the fault ultimately lies with the guardian class. In their overseeing of breeding and birthcontrol, it is said that the guardians may at some point fail to apply calculation and observation correctly. The perfect geometrical number for the marriage between men and women, which is paramount to the koinōnia of the state, will escape their notice, resulting in offspring who lack the right nature for being guardians. They will first neglect education, and then even more importantly, they will lack the ability to discriminate between the different classes of the city: gold, silver, bronze and iron. When this happens, Socrates cautions us, "you will get unlikeness and discordant inequality (ἀνομοιότης ἐγγενήσεται καὶ ἀνωμαλία ἀνάρμοστος). And when you get those, wherever they occur, they always breed war and hostility" (547a2-4). This illustrates just how important class identification and separation is to the ideal state of kallipolis, which is regulated by the guardians who create the right conditions for the production of philosopher rulers through pre-planned marriage and breeding relations (460a8-10) 26 .
This provides an appropriate segway to discussing specifically inter-class relations within the Republic. If harmony, likeness and equality are going to prevail in the city, and if these are achieved by maintaining the separateness of the classes, other than the breeding relations just alluded to we ought to ask whether there are any specifically socio-economic conditions that maintain the class structure of kallipolis. The answer, it seems to me, will again involve an awareness of some sort of proportionate organisation. Admittedly, while proportionality is not explicitly mentioned as a binding principle, we can still 26 Note, however, that when Plato comes to speak of marriage relations in the Laws, he takes on an entirely different theoretical framework. Now, rather than the "eugenic manipulation" (Schofield, 2006) of only the best males and females in the city, Plato even applies the normative ideal of proportionality (as seen in the Statesman and fleshed out further in pp. 21-26) to marriage: for "in respect of excellence, what is evenly balanced and proportioned (symmetron) is infinitely superior to what is untempered" (773a6-7). Cf. Schofield, M., Plato: Political Philosophy, 2006, p. 225. In practice, this means that a man should always look to his opposite in marriage either in wealth or temperament. That this is now thought to be a good for the city shows that goodness in the city as a whole is being envisaged along somewhat nuanced lines, inextricably tied up with the notions of balance, measure and proportion. nevertheless tease out how an arrangement predicated on proportions might prefigure the tripartite class system of the Republic 27 . For instance, it is clear that if each class is going to perform its unique function, they will need access to only those things that contribute to them carrying out their function. This means, in essence, that distributions will be given out on a purely equal basis in accord with the principle of justice. Carpenters will be given access to only those tools that allow them to do carpentry well, and the auxiliaries will be given exactly what they need in their role of executing the orders of the rulers. That is to say, the carpenter will be denied access to what the auxiliaries receive, and vice versa, for each class only needs what each is by nature capable of performing (433e-434b). Nothing is said, however, on whether certain outstanding individuals within each class might receive more access to those benefits that assist their function if they do a better job. This is what we might have expected if Plato were operating with geometrical proportion, which we saw him propounding at least in outline in the Gorgias, and which he will finally favour in his later dialogues. But in the Republic Plato cold shoulders this model of proportion, and we shall perhaps see why in a moment.
Things are more complex when it comes to the distribution of benefits within the guardian class and the hierarchical relation that follows. Since philosophers do most for the city, we would naturally expect that they receive a much larger share of the city's rewards in turn. What we find instead is that the rights of the philosophers are severely curtailed. This means that unlike others in the community they should not own private houses, land, or property of any kind, and that they receive a modest living from the other citizens as compensation for their benevolent ruling (464c, cf. 416d-417b) 28 . Were they to become involved in real estate or finance, they would no longer be guardians but would assume some other role in the city, leaving them susceptible to betrayal and social upheaval. This arrangement, we might say, is again exactly what the guardian class needs to fulfil its function. For other rewards traditionally 27 As e.g. Sedley (2007) in fact does, when he compares the disproportionately equal distributions handed out by democracy at 558c with the distribution of rights at 433e-434b. See Sedley, D. N., "Philosophy, the Forms, and the art of ruling", in: Ferrari, G. (ed.), The Cambridge Companion to Plato's Republic, Cambridge: Cambridge University Press, 2007, p. 271 fn. 24. 28 Which leads to Adeimantus' charge at 419a that Socrates is making the guardians completely unhappy. Of course, Adeimantus has missed the larger import of civic happiness in the Republic, which does not come down to any one individual or class, but rather the happiness of the whole conceived as one functioning unity. associated with happiness such as wealth might corrupt the philosophic nature and derail the guardians from acting in the best interests of the city as a whole.
In one key respect, however, the powers of the guardian class do disproportionately outweigh and dominate the whole community. Despite the clear socio-economic equalities just mentioned, the political inequalities of kallipolis are severe. Patently, the right to participate in any of the functions of government are restricted to a few individuals preeminent in wisdom. And this is arguably why Plato disavows geometrical proportionality in the Republic, for that type of proportion involves giving more to those that are deserving and less to those who are less deserving 29 . By contrast, the guardians are simply given all of the share of government powers, without a consideration of what the lower classes might bring to the table. Now, turning back to our list of different proportional means in T2, it has been suggested that Plato may have singled out the harmonic proportion in the Republic since that type of progression recognizes merit to an even greater degree than the geometric one 30 . I cannot go into the precise mathematics here, but a simple harmonic progression will show that the differences between terms are even greater than in the geometric, securing even further the essential separation between the best and worst elements in the city. Indeed, the connection of harmony with music, both of which Plato clearly makes good use of in the Republic (esp. 432a-b), makes it more likely that he was working with specifically this model of proportion as opposed to the geometric. This, I nevertheless concede, can only be a matter of tentative speculation 31 . 29 Vlastos, G., "The theory of social justice in the polis in Plato's Republic", 1977, p. 24. 30 Morrison, J. S., "The Origins of Plato's Philosopher Stateman", 1958, p. 214. It is true that only the arithmetic and geometric proportions received an explicit political application in 5 th and 4 th century writers, while the harmonic proportion only seems to have received a political counterpart in later pseudo-Archytan writings, which are themselves probably spurious". However, Plato's tripartite class system is uniquely unparalleled in Greek literature, so perhaps those other writers were not able to make an obvious link between existing political constitutions and the harmonic proportion. While it is certainly possible that Plato may not have applied the harmonic proportion to politics in the Republic, we can at least be confident that he was well aware of this model of proportion when he applies it without question to cosmology in the Timaeus (see Iamblichus in Nic. 118.23). 31 Contra Harvey (1965), who suggests that Plato operates with geometrical proportionality in the Republic while nevertheless conceding that "Plato says nothing about it". Cf. Harvey, F. D., "Two kinds of equality", 1965, p. 109, fn. 34. But I cannot see how the rigid tripartite framework of the dialogue can accommodate for geometrical proportionality, especially with respect to the distribution of political rights between the different classes. Nor do I find Harvey's arguments to the effect that Plato disavows harmonic proportion in the Republic (1965, pp. 142-144) wholly convincing.
And yet, it is clear that Plato still thinks this hierarchical model is the key to achieving philia and koinōnia in the whole city. In a famous passage towards the end of Book 9 of the Republic, Plato makes Socrates say that the person whose reasoning element is weak ought to be regarded as the slave (doulon) of the best person. Not, however, according to Thrasymachus' notion of ruling, where the slave is sometimes harmed just for the benefit of the ruler. But rather as the following text demonstrates: (T8) "Ideally, he [i.e. the slave to the ruler] will have his own divine and wise element within himself, but failing that it will be imposed on him from outside, so that as far as possible we may all be equal (homoioi), and all friends (philoi), since we are all under the guidance of the same commander" (590d4-6) 32 .
What is more, Plato is at pains to stress elsewhere in the Republic that the subjugation of the lower classes will ultimately be consented to. It is not just that they are forced into a hierarchical relationship with the rulers; they willingly accept their rule, and may even have some sort of dim understanding that this is the best arrangement for them. Indeed, self-discipline (sōphrosynē) was differentiated from other virtues such as courage or wisdom insofar as it extends throughout the whole city and combines all the different parts to sing together in unison (431e10-432a9, cf. esp. 442c10-d1). This virtue just is the agreement on who should rule, so to the extent that the unwise grasp this truth they may yet have some sort of "popular" virtue.
In summary, while koinōnia is not especially flagged in the above passage, I think we can fairly assume that as well as being equal and friends (homoioi kai philoi), all the citizens of kallipolis will also be koinon when the wise element in the city rules over them and puts each class into its proper place. Each realizes that they are an important part of a collective whole and shared experience, and that what binds them together giving each its unique identity is the divine ruling element presiding over the whole city. But what is crucially still up for debate here is to what extent we find this koinōnia to be successful. For despite the strong impression the Republic gives of civic cohesion, as we have also seen, the koinōnia of the city is predicated on an inflexible class system which, regrettably, disenfranchises the lower parts of the city from having a right to political participation. One might protest that this is not consistent with equality at all, notwithstanding that equality and friendship for all is still the aim of the guardians. It is certainly no more consistent with geometrical equality which, we have seen, does admit more to the more deserving but still gives out some share of what is due to the less deserving. Finally, the koinōnia of interpersonal familial relations, of pleasures and pains, and of a sense of natural kinship to one another -which in the Laws will be said to be suitable only for "gods or children of gods" (739a-e)-seems solely restricted to relations within the guardian class 33 . This emphasis on the unity intrinsic to the guardian class and the accompanying responsibility given to it for the unity of the whole, combined with such grave inequalities in the political sphere, puts into doubt the very plausibility of koinōnia as the Republic envisages it.

The Laws
Plato's final political work, and the one to which I shall conclude this paper, provides a striking contrasting model of koinōnia. A model which, I shall suggest, both takes us back to Plato's musings on geometrical proportionality in the Gorgias while also building on the emphasis given to due measure in the Statesman.
One feature that immediately sets the Laws apart from any of Plato's other political dialogues is the positive attention given to freedom (eleutheria) as a principal aim (skopos) for the lawgiver. As early as Book 3, Plato has the Athenian say that a city "needs to be free, rational, and on friendly terms with itself" (693b3-5). What exactly this desired "freedom" could amount to as part of this threefold trinity of aims is perhaps best articulated in the ensuing examples of different historical constitutions from the past. Before a fuller examination of those examples, the Athenian first provides us with some preparatory guidance for understanding different political systems. Among them are two main types -monarchy and democracy-the former best exemplified by Persia and the latter by Athens. If there is to be freedom and friendship together with wisdom in any community, it is necessary to have a blend of both these systems (693d7-e1). Indeed, the Athenian cautiously warns against disregarding due measure (to metrion) between these constitutions by saying that if you give too much power to things that cannot take it -sails to ships, food to the body, or more importantly power to the human soul-, then excess takes over leading to injustice, the child of excess. Thus, the lawgiver needs to guard against 33 By contrast, as well noted by Schofield (2006, p. 223, n. 87), the communist agenda of the Republic is ultimately abandoned but, also in one way reworked in the Laws so as to apply to "the whole city" so far as that is possible (739b-c). See further Schofield, M., Plato: Political Philosophy, 2006, p. 223 and fn. 87. corrupting the human soul with too much power, and the way to remedy that is again by observing due proportion (691c1-d6).
It is said that our two main archetypes Persia and Athens have never been able to strike due proportion (to metrion) between these different extremes; Persia had traditionally been too attached to the principle of pure monarchy, while Athens was attached to pure freedom (693e5-694a1) 34 . There was however a time in Persia's history where they were closely led to to metrion between freedom and slavery under the reign of the king Cyrus. The consequences of this, according to the Athenian, were momentous. Just consider the following text: (T9) "This [i.e. to metrion between slavery and freedom] gave them, first, their personal freedom, and secondly, the mastery over many peoples. The rulers gave a share of freedom to those under their rule, putting them on an equal footing. This made the soldiers well disposed towards their generals, and they showed themselves eager to face danger. And further, if there was any among them with brains enough to offer good advice, the king was not one to resent this. He allowed freedom of speech, and promoted those whose advice was of some value. So someone like this could regard the benefit of his wisdom as belonging to everybody, and put it forward openly. So at that time all their affairs prospered as a result of their freedom, friendship and community of mind" (nou koinōnian, 694a3-b6) 35 .
The freedom alluded to here seems to encompass the ability of Cyrus' ruled subjects to participate in the functions of government. Were there too much freedom among them then the city might lack the direction it needs to make good decisions, and such excessive freedom might otherwise lead to licentiousness and corruption. It is not, however, that there should be no ruling principle, for then the Athenian would not have expressly said that "in any gathering or association (koinōniais)-for any activity whatever-it is right that in each and every case there should be a directive principle" (640a3-6). No, the issue at stake here is what sort of relationship between ruler and ruled best fosters a friendly community. And one thing that clearly contributes to such a healthy relationship is, as the end of the passage demonstrates, what I have translated as a certain "community of mind" (nou koinōnian). If there be someone wise (phronimos) enough among the soldiers then, rather than 34 Sparta and Crete, meanwhile, are said to be much closer approximations of the best constitution. For an explicit reference to the successes of Sparta in particular, see 692a-b. 35 The translation is taken from Schofield, M. & Griffith, T., Plato: The Laws, Cambridge: Cambridge University Press, 2016, cursives added. having their views unduly silenced, they are actively encouraged to contribute to the king's decision making. Clearly, giving other individuals the freedom to contribute to the decision-making process of government is a significant check on excessive kingly power. Such a pooling of ideas (nou koinōnian) among ruler and ruled, I suspect, is also intended to engender a more authentic and longlasting sense of koinōnia in society as a whole.
The time of Persia under the subsequent reign of king Darius provides an equally striking case of a constitution much closer to the normative ideal of due measure between monarchy (slavery) and democracy (unchecked freedom). Consider finally the following example: (T10) "And when it came to legislating, he decided the way to manage things was to introduce some degree of common equality (isotēta koinēn). He also gave the force of law to the distribution of tribute promised to the Persians by Cyrus, by this means creating friendship and community (philian kai koinōnian) among all the Persians, and winning over the common people of the Persians with money and gifts. The result was that the loyalty of his armed forces gained him as much territory again as Cyrus had bequeathed" (695c10-d6) 36 .
In the case of legislation, the passage suggests, Darius thought that to introduce some sort of "common equality" (isotēta koinēn) was the best for the Persians. This, along with certain distributions given to the Persians, created a strong sense of friendship and community (philian kai koinōnian) among the Persians. Now, we will not fail to notice that equality was also flagged in the previous Cyrus passage. By giving freedom to his subjects, Cyrus had put everyone "on an equal footing" (epi to ison agontes). Since both constitutions are by definition monarchies, however, equality is quite clearly being understood here in a much more nuanced sense. The point is not that the citizens are all made equal by having the exact same share of power. To the contrary, I take it that equality prevails when the rest of the populace is given some right to political participation, which may vary in degree depending on the value of each person's contribution.
Plato elaborates on equality and its inextricable connection with friendship even further in Book 6 of the dialogue. In an explicit reference back to the material from the Laws we have examined so far, he reinforces the message that a constitution should at all times maintain a middle position between monarchy and democracy. As Plato has the Athenian explicitly say, "slaves and 36 The translation is taken from Schofield, M. & Griffith, T., Plato: The Laws, 2016, cursives added. masters can never be friends" (757a1). This should already serve as a reminder that we are verging on quite different territory from the Republic. For while that dialogue emphasized both friendship and equality as desiderata for the rulers of the city, we also saw that such friendship was predicated on the lower members of the community assuming a position of quasi-slavery towards the best (T8). Moreover, this slavery coincided with the fact that the lower classes were obstructed from participating in the functions of government, which was exclusively restricted to the guardian class in accordance with the principle of justice such that each must only perform its own distinct function. Now if, as I have suggested, we are being urged to think of equality here along nuanced lines, and if equality bears a strong relationship to the statement "slaves and masters can never be friends", then we ought to ask what sort of equality Plato is operating within the Laws, and how this makes a significant difference to the resulting koinōnia of the city.
Plato does have the Athenian provide a crystal-clear response in the lines that follow (756e10-758a2). He begins by challenging the view of the majority that the lowly (phauloi) and the morally good (spoudaioi) are on an equal footing, since "equality between people who are not equal-and the absence of any proportionamounts to inequality" 37 . This, in turn, is said to breed civil unrest. Thus, while the old and true saying that "equality creates friendship" is uncontroversially straightforward, the precise type of equality in question is far from clear. The Athenian then helpfully contrasts two opposing forms of equality: (T11) "Equality comes in two forms, which, though they both have the same name, are really, in many respects, almost diametrically opposed. The first (i) is the equality of measurement, weights, and number, and applying it to the distribution of public honours is within the capacity of any city-or any lawgiver; they can use the drawing of lots to ensure equality. But (ii) the truest and best equality is not immediately obvious to everybody. It leaves the decision to Zeus, and its effect on mankind is always the same: it helps them but rarely, though whenever it does help either cities or individuals, it is the cause of all things good, since it allocates more to what is greater and less to what is lesser, and by giving each of them a measure (metria) related to its nature-in the case of public honours, greater honours to those whose endowment of human goodness is greater, and 37 Plato does articulate the same point in similar language in his critique of democracy in the Republic (558c1-4, cf. p. 8 above), but as I have argued the preferred converse model of equality (geometrical equality) plays no obvious role in the tripartite scheme of the dialogue. lesser honours to those whose endowment of goodness and education is the opposite-duly allocates to each class what is appropriate to it" 38 (757b1-c6).
The one employs measurement, weights, and numbers, and the lawgiver can use the drawing of lots to ensure this type of equality as indeed was the practice in contemporary Athens. We might call this "arithmetic" equality, corresponding to the first of our proportional models in T2. The "truest and best equality" which is not immediately obvious to the rest of us leaves the decision to Zeus, and is always the cause of what is good: for it "allocates more to what is greater and less to what is lesser, and giving each of them a measure related to its nature… duly allocates to each class what is appropriate to it". This equality clearly corresponds to the geometric mean we have encountered time and again, which takes into consideration personal differences and dishes out what is due in accord with merit. That is why people who are not equal should not receive equal treatment, but rather, since inequalities necessarily exist, what is truly equal is some sort of proportionate distribution-what the Athenian had earlier referred to as an "unequal but proportional yardstick" (τῷ ἀνίσῳ συμμέτρῳ, 744c3) 39 .
The Athenian finally stresses that this is precisely what statesmanship ought to be concerned with, so that there is neither "a handful of tyrants, or a single one, or some kind of popular control, but always justice", true justice being synonymous with the equality which gives out different equalities to unequal members of society. Notice that this equality is perfectly compatible with, indeed is necessitated by the Athenian's earlier remarks on freedom as a new desideratum for the lawgiver. Too much power in the hands of a few select individuals would necessarily hinder less capable individuals with some potentially important contributions from having a share in government, while too much freedom fails to consider personal differences and treats all alike. The right balance (to metrion) of monarchy and democracy thus creates room for 38 The translation is taken from Schofield, M. & Griffith, T., Plato: The Laws, 2016. 39 It should be noted that the reference here comes in the specific context of the distribution of property rights and the subsequent stratification of four different property classes in Magnesia. What is critical for my purposes is that the normative ideal of geometrical proportionalityaccording to which each is given a share of what is proportionate to their worth-is now even applied to social and property relations in the Laws (unlike the Republic) as well as political relations, further underscoring the importance of proportionality for the goodness and unity of the whole city. Proportionality in property relations, however, is also intended to "equalise opportunities in public life". geometrical equality, which duly allocates to each what is appropriate to it 40 . We should have no doubt that, as the earlier examples of Cyrus and Darius illustrate, this sort of proportional equality is the key to creating koinōnia in a society composed of unequal people varying in the degree of their character.
These considerations both take us back to the Gorgias where geometrical proportionality was first highlighted as the binding principle par excellence for well-ordered wholes, and build on the explanatory framework of the Statesman, where we saw that politics was strongly associated with creating the right proportion (to metrion) between different extremes. According to my overall reconstruction, however, it looks as though the rigid and inflexible class system of the Republic stands out as a sort of anomaly. But even there an appreciation of proportionality is still at work in the koinōnia of the city, albeit a proportional model that Plato will eventually abandon when he has his final say in the Laws. What I hope to have shown is nevertheless consistent throughout is the central place of proportionality to koinōnia formation in Plato's thought. I have argued that koinōnia, while being a normative condition of complex wholes already presupposing some level of functionality among different parts, needs to be supplemented with an investigation into the principles and necessary prerequisites that go into making a koinōnia good. As Plato freely experiments with different structural models for well-ordered wholes in different contexts, so does his application of the principle of proportionality to koinōnia formation. 40 One might protest that 1) due proportion (to metrion) or a balanced mixture between different constitutional extremes (monarchy and democracy) is an entirely different thing from 2) the specifically mathematical proportions we have been operating with and their own respective political applications (cf. Harvey, F. D., "Two kinds of equality", 1965, p. 105, fn. 12). But, if that were the case, why would Plato have introduced the two different equalities in T11 and show a clear preference for geometrical proportionality in the same context as he stresses that a constitution ought to have a "middle position" (meson, i.e. to metrion) between monarchy and democracy? With Morrow (1960), I am more convinced that the "mathematical conception of the mean [i.e. geometrical proportion] is present in Plato's thought of the political mean [i.e. to metrion between monarchy and democracy]". Thus Schofield (2013) rightly suggests that proportionate equality (geometrical proportion) is the key to achieving a "carefully considered blend" of monarchy and democracy. Cf. Morrow