Being, Appearing, and the Platonic Idea in Badiou and Plato

Abstract This essay considers the ambiguous sense in which Badiou is a Platonist. It alleviates this ambiguity by considering how two characteristics of Platonism are treated in the metaphysics of Being and Event: (1) the split between being/appearing, and (2) Platonic Ideas. It considers how in Badiou and Plato’s metaphysics the treatment of both these characteristics of Platonism is comparable. Accordingly, it compares both such characteristics in relation to Being and Event and the Theaetetus and Phaedo, and, using concepts from each philosopher, explores three possible ways in which being/appearing and Platonic Ideas may be interrelated, thereby constituting a philosophy of experience. These explorations necessitate parallel consideration of how the sense in which Badiou is a Platonist is determinable by the interrelation of Platonic Ideas with being/appearing in his metaphysics. It is determined that how being/appearing and Platonic Ideas interrelate in Badiou is markedly different from how they do so in Plato, making it questionable whether Badiou is a Platonist in this sense. However, it is indicated that an esoteric sense in which Badiou is a Platonist (the dialectic of the One and the multiple) has been considered throughout this essay, and that more such senses deserve consideration.

The second reason is that arguably Theaetetus et al, like Badiou, are concerned with the multiple, the concept of which is the most fundamental in Badiou's metaphysics. The multiple must be defined at the outset of this essay, then.
Theaetetus' proposition, and Protagoras' and Heraclitus' theses, will be interpreted in this ordering of these propositions of which their arguments are comprised: that (1.1) being and appearing are the same, that (1.2) being/appearing is coming to be, and that (1.3) coming to be is multiple.

Being and appearing are the same
Theaetetus' proposition that 'knowledge is perception'10 -which Socrates identifies with Protagoras and Heraclitus, whose theses Theaetetus then agrees with11 -is relevant to both epistemology and the philosophy of perception. Theaetetus asserts that what is known (an epistemological matter) is what is perceived (a matter for the philosophy of perception). But insofar as he is asserting what is the case over what merely is known or perceived to be so, his assertion is of the most fundamental relevance to ontology. And since this essay's primary concern is ontology, it will also give this aspect of his proposition the most weight. We will now justify why the proposition that 'knowledge is perception', and the theses of Protagoras' and Heraclitus' which Socrates identifies with it, may primarily be understood as part of an ontological philosophy of experience. Accordingly, we will see why in this part of the Theaetetus the ontological claim is made that being and appearing are the same.
Over the course of the dialogue -from 151d-172c -Socrates identifies these three propositions with Theaetetus' initial assertion: (i) what is is what we know, (ii) what we know is what we perceive, and (iii) what we perceive is multiple. Only the second -that what we know is what we perceive -can be derived from Theaetetus' initial proposition that 'knowledge is perception'. As we will see, the first comes from Protagoras and the second from Heraclitus.12 And since the first pertains to section 1.1 of this essay, and the second and third to sections 1.2 and 1.3, let us now proceed in the proper order by considering the first.
Socrates identifies Theaetetus' assertion with Protagoras' proposition (or aphorism), 'Man is the measure of all things: of the things which are, that they are, and of the things which are not, that they are not.'13 It is important to recognise that Protagoras does not assert 'of the things which are, that they are [for his knowledge or perception], and of the things which are not, that they are not [for his knowledge or perception].'14 Qualifiers, of the kind I have put in square brackets here, are not used by Protagoras. He asserts without qualification that 'of the things which are, that they are, and of the things which are not, that they are not. '15 So, in Protagoras' philosophy of experience there is no disjunction between what is (in the first instance), what one knows (in the second instance), and what one perceives (in the third instance): what is is what one knows, which is what one perceives; on the other hand, what is not is what one does not know, which is what one does not perceive. The first and second instances will now be addressed in sections 1.2 and 1.3.

Being/appearing is coming to be
Socrates explains why what is, for Heraclitus, is coming to be, arguing that nothing is one or any kind of thing. What is really true, is this: the things of which we naturally say that they 'are', are in process of coming to be, as the result of movement and change and blending with one another. We are wrong when we say they 'are', since nothing ever is, but everything is coming to be.16 10 Ibid., 151e. 11 Ibid., 151d-153a. 12 Socrates eventually asserts that all three positions are actually coextensive, saying, 'So we find the various theories have converged to the same thing: that of Homer and Heraclitus and all their tribe, that all things flow like streams; of Protagoras, wisest of men, that man is the measure of all things; and of Theaetetus that, these things being so, knowledge proves to be perception. '  A clear argument can be picked out here. The first premise is that no thing is qualitatively different from any other thing; rather, all things are only ever quantitatively different. The second premise is that all things are in a constant process of undergoing quantitative difference. These premises lead to the conclusion that since nothing is qualitatively different from anything else, and since the undergoing of quantitative difference is a constant process for all things, all things are always intermingling -or, as Socrates says, 'blending'. This argument defines what coming to be is.
In relation to the second premise, it is important to further define what the undergoing of quantitative difference means for all things. Socrates later argues that this process is constituted by 'two forms of motion, alteration and spatial movement.'17 Furthermore, 'all things are always in every kind of motion.'18 That is, all things are always moving in terms of both their composition (or 'alteration') and their orientation in space (or 'spatial movement').
When we next consider this concept of coming to be in relation to Badiou, we will see that what Heraclitus means by 'all things [being] always in every kind of motion' may be interpreted as all things being multiple. Badiou names 'multiple' what is most fundamentally presented by any thing whatsoever. In other words, a multiple is what all things present at their most fundamental level; whether those things are 'pigs, stars, gods'19 or anything else at all, what any of those things is qua its presentation is a multiplicity. A multiplicity is 'a multiple of multiples':20 that is, what a multiple is at its most fundamental level is not one. In his settheoretical ontology (which uses the ZFC axiomatisation of set-theory, or Zermelo-Fraenkel set-theory with the axiom of choice21), Badiou chooses to refer to sets as multiples in order to draw attention to the dialectic of the one and the multiple that this mathematical theory illuminates -for this reason, he also dubs settheory 'the mathematical theory of the pure multiple'.22

Coming to be is multiple
On the subject of the dialectic of the one and the multiple which is illuminated by set-theory, he says, 'The multiple […] is composed solely of multiplicities. There is no one. In other words, every multiple is a multiple of multiples.'23 Hence, a multiple is 'not one' in the sense it is made up of multiples; those multiples are made up of more multiples; and those multiples are made up of even more multiples. And so on not ad infinitum, but until the point is reached at which it is no longer possible to determine that there any more multiples: the void (∅).24 Badiou's non-standard use of this mereological term to denote the empty set in his set-theoretical ontology should be noted here.
To understand in relation to Heraclitus how Badiou's ontology uses set-theory in this way, it will be helpful to consider a particular example which Badiou gives of multiples: trees.25 He describes 'two plane trees caught in the headlights of [a] motorist in a hurry and those observed in minute detail by [a] daydreamer in [a] valley.'26 Let us imagine that the motorist in a hurry and the dreamer in the valley are the same person. One day she is driving by the two trees, and it appears they are exactly alike. Another day she walks right up to the two trees, and they appear as two quite distinct things with different features.
All she sees the first day, when she is driving by, are two trees α and β which have no more discernable features than their outlines. The second day when she walks right up to them, she discerns another feature of them both: they have light-brown scaling bark γ. This multiple γ, therefore, belongs to both α and β: This may be read, 'For all the light-brown scaling bark (γ) in the scene that the traveller observes, this bark belongs to the first tree (α) and the second tree (β).' Now, when she walks even closer to the trees, she notices that the outer bark on both has flaked off in places. On α the outer bark has flaked off to reveal white bark (δ), and on β to reveal yellow bark (λ): This may be read, 'For all the white bark (δ) in the scene that the traveller observes, this bark belongs to the light-brown scaling bark (γ), and γ belongs to the first tree (α)' and 'For all the yellow bark in the scene that the traveller observes, this bark belongs to the light-brown scaling bark (γ), and γ belongs to the second tree (β)'. On account of the following formula, we may also say that γ is included in, or is a subset of, α and β, and likewise that δ and λ are subsets of, or are included in, γ.
We may, then, formulate inclusion as follows. Take the first tree (α): the light-brown scaling bark (γ) belongs to it. And the white bark (δ) belongs to the brown bark (γ). It follows that This may be read, 'For all the white bark (δ) in the scene that the traveller observes, if δ belongs to the brown bark (γ) then δ also belongs to the first tree (α).' This formula is equivalent to the following: γ ⊂ α, meaning γ is 'included' in, or is a 'subset' of, α. In other words, every multiple that belongs to γ also belongs to α. To say that δ is included in γ (δ ⊂ γ), we must introduce another multiple π for which And the same, of course, applies for λ.
Before proceeding any further, we should acknowledge that it might be objected that our interpretation of Badiou here uses set-theory in a non-standard way. This objection would be that we have interpreted Badiou as arguing that sets can be members of (or belong to) concrete objects (such as the trees), when only sets themselves can have members (or belong to other sets). However, this objection would miss the point that we have interpreted Badiou as arguing not that sets can belong to concrete objects, but that sets can belong to concrete objects that are apprehended qua their being sets. That is, we know that for Badiou everything fundamentally is a set. And here, we are not apprehending one of the trees which the traveller sees, for example, qua its appearance in a world (and therefore as a concrete object), but qua its being (and therefore as a set). In other words, our concern here is not phenomenology (which would concern how the tree appears in a world as a concrete object), but ontology (which concerns what the tree is irrespective of its appearance in any world: a set). Our interpretation is in line with the distinction that Badiou makes in Logics of Worlds between any thing's phenomenological being-there and its ontological multiple-being.28 Returning to our example, in summary our traveller first apprehended two multiples α and β, both of which appeared as trees. When walking right up to α and β, she apprehended that another multiple γ (the light-brown scaling bark) belonged to both of them. When walking even closer, she apprehended that each of these sets -(γ ∈ α) & (γ ∈ β) -included the subsets δ (white bark) and λ (yellow bark), so that (δ ⊂ α) 27 See Badiou, "Technical note: the conventions of writing", in Being and Event, 49-51. 28 It might also be helpful to note that though Badiou presents the trees as an example of the former (when he summarises in his Second Manifesto a central point of Logics of Worlds), here we are presenting them as an example of the latter. Though in Being and Event he does not consider such mundane concrete objects, he does nonetheless consider determinate things that are there (in the sense of phenomenological being-there), and not simply indeterminate things that do not appear in any world (in the sense of ontological multiple-being). For example, he considers music (Shoenberg), political action (in the history of Marxism), and poetry (Hölderlin and Mallarmé). Crucially, he considers these things qua their being, not qua their appearing in a world. In this essay, we have chosen to be concerned with the former -that is, with how in Badiou's philosophy of experience what we experience, in any case whatsoever, is fundamentally not something that appears in the world in which we experience it. In other words, we have chosen to be concerned not with beings, but with being. Badiou turns his attention to beings in Logics of Worlds when he proposes 'to think the being [i.e. any given being] as localised, to include the "there" in the thinking of being -something that the mathematical (ontological) theory of the pure multiple […] does not allow.' Badiou, "Logics of Worlds", 99-102 (102). & (λ ⊂ β). In this way, the trees went from being quantitatively similar and almost equal to quantitatively dissimilar in a particular way.
All the while as she approached the two trees -first from afar, then close by, then even closer -they were in a process of coming to be for her. In Heraclitus' vernacular, they were 'in every kind of motion'.29 That is, they moved (or changed) their orientation in space (or 'spatial movement'30) as she moved towards them from afar. Their composition (or 'alteration'31) changed when she, in doing so, apprehended that their sets included subsets.
However, it is important to qualify that the two trees only changed in orientation and composition for her. But in Badiou's philosophy of experience, he wants to be able to explain not just what is coming to be for her (or in particular), but what is coming to be as such (or universally).
In Badiou's ontology the axiom of replacement (or substitution) states that a new set may be obtained by replacing (or substituting) the variables of a given set with new variables.32 For instance, in our example we may, according to this axiom, replace the trees (α & β) with bananas, the brown bark (γ) with yellow banana skin, the white bark (δ) with the brown flecks that show the banana is ripe, and the yellow bark (λ) with greenfly -or anything else at all.
Here, we have formulated the coming to be of any multiples whatsoever that are structured in this way; we can conceive of not just any multiple in particular, but multiplicity in general. Badiou does not consider the metaphysics of being and appearing from the standpoint of perception (i.e., a posteriori) but conception (i.e., a priori).33 In this way, his philosophy is wholeheartedly rationalist: he advocates determining the being of things by proving what can be conceived of them, and thereby what perceptions of them are possible. Thereby, he sharply subordinates the a posteriori (or empirical) to the a priori (or rational). He addresses not any particular situation (or 'presented multiplicity'), but the universal situation of ontology itself, proving what is conceivable not of particular appearance, but universal appearing.34 He thereby presents 'the presentation of presentation.'35 Because this universal dimension of what appears admits of only quantitative and not qualitative distinctions, the trees in our example were distinct because of the different multiples that belonged to them, not because of how any qualities they may have had appeared to the traveller -like the roughness of their bark, the swishing of their leaves in the wind, or the delicate play of light on their branches. Badiou is not concerned with how the trees were presented to her perceptually (in an empirical or a posteriori sense), but how this presentation itself is to be presented conceptually (in a rational or a priori sense). The latter abstracts from a particular situation to the universal situation: namely, ontology.
Returning to Plato, Badiou would, taking all this into account, say that, while these theses which incorporate Theaetetus, Protagoras, and Heraclitus are actually only concerned with appearing qua perception, he is genuinely concerned with appearing qua being.

Being and appearing are distinct. Being is One and appearing is multiple
This conclusion -that Theaetetus and Protagoras, understood in relation to Heraclitus, cannot infer what being is from their relativistic perspective -is the very one that Plato draws in the Theaetetus. We will now see in 2.1 why this conclusion may be drawn from both Badiou and Plato's perspectives. Accordingly, we will see why it is necessary to posit that being and appearing are distinct, leading Plato to the view that being is One and appearing is multiple in 2.2.

Being and appearing are distinct
It may already have been noticed that Protagoras' thesis is paradoxical. For how can it be universally true, when it itself proposes that the truth of being is perspectival? In other words, if all things are as anyone measures them to be, then why should one person's measurement -in this case Protagoras' -be favoured over anyone else's?
This paradox may be explained in terms of our previous example. Imagine our traveller a has been accompanied by someone b. To recap, a thinks that white bark (δ) belongs to the first tree (α). However, b does not see any of this white bark (δ) at all; b thinks that only brown bark (γ) belongs to α. So, a thinks that (δ ∈ α); b thinks that ~ (δ ∈ α). This is the conjunction of a proposition and its negation: (δ ∈ α) & ~ (δ ∈ α). Although it violates the law of non-contradiction, as Socrates himself points out,36 it is consistent with Protagoras' view that both a and b are right. However, because for Badiou ontology is mathematics, he must uphold such laws and not 'pay the price -in which all mathematics is abolished -of the incoherency of the language.'37 Yet Protagoras' thesis, when it is combined with Heraclitus', is short-circuited more quickly still. For if the objects of perception always are in flux, and knowledge is perception, then knowledge always is in flux too.38 And not only this, but the percipient him-or herself always is in flux, or discontinuous.39 There is no possibility for fixed knowledge; neither is it possible for the person to whom that knowledge would belong to be fixed, or continuous.40 This is a 'general ruin of thought as such',41 like the one Badiou identifies in the Parmenides.
Nevertheless, in the Theaetetus Socrates, Theaetetus, and Theodorus maintain that appearing is coming to be. Unlike Protagoras and Heraclitus, they distinguish this conception of appearing from being, in opposition to the theses that Socrates related to Theaetetus' proposition that 'knowledge is perception'.42 They aver that knowledge is not perception; rather, knowledge is of things which are neither in flux nor discontinuous: the Forms.43 A metaphysics of being and appearing is made possible by the continuity of things outside appearance: knowledge of these things can put such an ontology on sure footing, and thereby ground a philosophy of experience.
This move in the Theaetetus will best be understood by referring to the Phaedo, because the Forms are only evoked, but not fully elaborated upon, in the Theaetetus. And it will be necessary to understand what precisely is outside appearance for Plato, in order to appreciate Badiou's alternative solution.

Being is One and appearing is multiple
Socrates describes the coming to be or multiplicity of things with the following example: 'do not equal stones and sticks sometimes, while remaining the same,' he asks, 'appear to one to be equal and to another to be unequal?'44 He means that the equality of these stones and sticks is always coming to be, and is never fixed or continuous. Their changing (i) spatial orientation and (ii) composition, depending on how and by whom they are apprehended, means they merely partake in equality, and are not wholly Equal.
In order to ensure that these stones and sticks remain equal despite their changing forms apropos of (i) and (ii), thereby avoiding the paradoxes of Heraclitus' theory of flux (which were addressed in section 2.1), Socrates must posit there are Forms that determine the stones and sticks' equality. As regards Socrates' example, they might seem to one person to be unequal because of the angle and distance from which she views them (i). Likewise, the colours and textures etc. that they are composed of might seem unequal, because of a light that is cast on them (ii). However, to another person from a different perspective they might be viewed as equal in these very same respects.45 For Plato the forms (with a lower-case 'f') of the stones and sticks, which in themselves are unfixed or discontinuous, are given fixity and continuity when they are referred to the Forms (with a capital 'F') of which they, in a sense, are representations.46 The question here is, 'How are forms "referred" to Forms in this way?' They are referred to by the jolting of memory, since knowledge of Forms, Socrates argues, is innate.47 In this way, the inconsistent multiplicity of forms, which are things we apprehend in the 'visible' world, are given consistency by the 'invisible' Forms, of which we have innate knowledge.48 Our a posteriori experience of this world (appearing), which is unstable, is structured by our a priori knowledge of another world (being), which is stable. T herefore , the empirical is subordinate to the rational in Plato's philosophy of experience as well as in Badiou's . Though the stones and sticks 'appear' not wholly equal, we know they 'are' equal by referring to our innate knowledge of Equality. The forms are, so to speak, held up to the Forms and compared to them; in comparison to the Form of Equality, which is wholly Equal, forms -such as stones or sticks -may be said to be equal to an extent. The quantitative similarity of the stones and sticks is bolstered by the qualitative identity of the Equal to which they, as it were, aspire.
On the basis of all this Socrates says the following to Theaetetus: 'if [someone] is seeing any one thing, he must be seeing a thing which is. Or do you think that a "one" can be found among the things which are not?'49 Theaetetus replies, 'I certainly don't.'50 This brings to mind Leibniz's famous maxim which Badiou quotes in the first meditation of Being and Event: 'What is not a being is not a being'.51 Without knowledge of the Forms we would not apprehend 'any one thing' at all, but only a flux of things 'blending with one another'.52 But how is Plato justified in positing that there is such a One? As Badiou says, 'It is an Idea [Εἶδος or Form] which is not an idea [εἶδος or form], whilst being that on the basis of which the very ideality of the idea maintains its being'.53 It is 'not an idea' because it is not presented. And Badiou indicates that 'one cannot see how there could be an access to being outside all presentation.'54 But an Idea 'outside all presentation' is precisely what Plato is positing there is. He is, contrary to the 'wager' of Being and Event, proposing that 'ontology is not actually a situation.'55 45 We might consider this issue with more of an emphasis on conception than perception, as Plato does in the Symposium with regard to love, for instance (Plato,"Symposium",. 46 Plato, "Phaedo", 74a-75a. 47 Ibid., 75b-77a 48 Ibid., 79a. 49 Plato, "Theaetetus", 188e. 50 Ibid. 51 Badiou, Being and Event, 23. 52 Plato, "Theaetetus", 152d-e. 53 Badiou, Being and Event, 26. 54 Ibid., 23. 55 Nevertheless, it should be noted that he is not positing there is a One that is beyond being, as Parmenides and the Neoplatonists each did. For the Plato of the Phaedo, any given thing is equal because of its 'being', which is the Form of Equality. The Form of Equality, or any other Form, is therefore not beyond being. It is One in the sense that it is not coming to be multivocally (as things which appear are), but simply is univocally (as something which does not appear, but which nonetheless structures everything which does). To posit that the being of any given thing is One is therefore not to posit 'The One' in the Parmenidean sense of the doctrine that 'all things are One, and that this One stands still, itself within itself, having no place in which to move.' (Plato,"Theaetetus",180e) Nonetheless, it should also be noted that the very problematic of being and appearing, which this essay tackles, was bequeathed to Plato by Parmenides. Badiou attributes to Heidegger the insight that 'what Parmenides is supposed to have stated for the first time in perfect clarity is the connection of two irreconcilable differences. The first is the difference between being and nonbeing [.] […] The second is the difference between being and seeming. The insight [of Heidegger] is that the two delimited differences […] are strictly speaking constitutive of philosophy, for it is they that began philosophy.' (Badiou, "Heidegger's Parmenides", 27); Badiou, Being and Event, 26-27.

Appearing counts-as-one and being is multiple
In opposition these theses of Plato's, in section 3.1 it will be demonstrated through Badiou that it is unnecessary to propose that things are Equal (or One) in their being, because they themselves may be perfectly equal (or counted-as-one) in their appearing. Consequently, in section 3.2 the tables will be turned by proposing that, while appearing counts-as-one, being is multiple.

Appearing counts-as-one
We agreed with Plato that everything apprehensible either a posteriori or a priori is coming to be or multiple. That is, whatever way something is apprehended by perception or conception, that thing will come to be different; it will appear not as a one or unity of necessary features, but as a multiplicity or disunity of contingent features. But if any given thing reveals itself to be disunited (or multiple) on closer inspection, then why does that thing nevertheless seem to be united (or one), insofar as it appears as this or that thing which is distinguishable from other things? It is this seemingly obvious question -which is in fact profound -that will lead us to affirm with Badiou that, though things are multiple as regards their being, they nonetheless are counted-as-one as regards their appearing.
We will be led to this conclusion by first considering the problem of equality which was broached in relation to Plato. In Badiou's ontology equality is determined by the axiom of extensionality, which defines equal sets as being sets all of whose multiples belong to one-another.56 That is, It is critically important to understand that α and β are not equal because of anything intrinsic to themselves. Rather, they are equal because of the fact that extrinsically another multiple (γ) belongs to both of them. Their equality is predicated on the fact that this multiple belongs to both of them. The 'predicate' of belonging (∈), when it is combined with the variables γ and β, creates a proposition which, accordingly, is either true or false -(γ ∈ β).
Here, we must appreciate the fundamental importance of this predicate, which is a posited relation included within the axioms of set-theory. Philosophers like me, who have only a secondary-school understanding of mathematics, need only consult a basic guide to set-theory to remind themselves that sets 'are completely characterised by their elements', that in set-theory the 'only non-logical symbol is the binary relation symbol ∈', and that in set-theory it is 'not formally necessary' to use any symbol but ∈.58 For example, we know that (γ ∈ α) and (γ ∈ β) are equivalent (↔) only on the basis that all γ 'belong' to α and β. It is on this basis we propose α and β are equal, not on the basis of their being the same things in themselves.
Hence, as a retort to Plato, Badiou can say that equality is defined not substantively but relationally. He asserts that 'the pure essence of being-there, or appearing, consists not of a form of being but of forms of relation.'59 Things are not held up to the Form of equality, and defined as being qualitatively similar to it, such that they are deemed to be equal themselves. Rather, things are held up to each other, and defined as being quantitatively similar to one-another. It is shown that there is a multiple that belongs to each of them, and they are deemed to be equal on this basis. There is no Equality (with a capital 'E') as such, only equality (with a lower-case 'e') with regard to this or that multiple. Nevertheless, there is an idea (εἶδος) of equality that is not outside appearing.
It is for reasons such as this that Badiou affirms, 'I am a sophisticated Platonist, not a vulgar one.'60 For one thing, it may be the case he is an idealist Platonist. At this juncture we will hypothesise that Badiou is an idealist, and reexamine this in light of our findings at this essay's conclusion. We will make this hypothesis because he asserts, 'Parmenides is right: being and thinking are the Same.'61Against critiques of metaphysics Badiou calls 'critical, positive, dialectical, or hermeneutic', he maintains, 'Dogmatic metaphysics defends the rights of indeterminacy only within the bounds of a preliminary thesis which affirms that thought and the thinkable are homogenous to each other.'62 Accordingly, let us bear these assertions in mind, and consider the following argument. If ontology is mathematics, and what is is simply what is thinkable (as idealism affirms), then axioms -such as the axioms of replacement and extensionality (see sections 1.3 and 3.1 respectively) -are real and not merely formal. For this reason Peter Hallward suggests that Badiou's ontology might be described as 'an amalgam of the formalist and realist positions' of mathematics.63 Every axiom of his ontology posits an idea (εἶδος) which makes up the very fabric of reality.
And most fundamental to all these axioms is the posited relation of belonging (∈), which we now, therefore, understand to be the most fundamental idea (εἶδος) that makes up this fabric of reality, on Badiou's view. He calls 'belonging, ∈, the unique and supreme Idea of the presented-multiple'.64 For it is because of belonging that multiples are presented as one despite being multiple.65 Badiou calls this presentation of multiples as one counting-as-one.66 Multiples are counted-as-one by, as variables, being predicated on the binary operator of belonging.
The predicate ∈ itself is not one, because it is a 'binary operator' whose 'operation', therefore, is relational rather than substantive. It does not substantively propose a multiple is any one kind of thing -for example, that it is not (~), or is a tautology (⊤) or contradiction (⊥). In other words, its operation is not unary; it is not a unary operator. Rather, it relationally predicates a given multiple on its belonging to another. These multiples are 'counted-as-one', but are not one. There is no unary operation that establishes they are One, but only the binary operation ∈ that establishes their oneness. For example, the oneness of the sets denoted by the proposition (α ∈ β) is predicated on α belonging to β. However, α is not a one on its own, nor is β, and neither do α and β combine to make a one. Instead, they simply count-as-one in relation to each other.
This, therefore, is the most fundamental idea (εἶδος) for Badiou: though all things are always coming to be because they are never One, they nevertheless are counted-as-one. But this still would not seem to be enough to avoid Heraclitus' flux. For if the oneness of any thing is simply an operation, then why does that thing appear not as a temporary 'forming into one' of multiples which are, in Socrates' words, 'blending with one another',67 but as a thing that is one in itself?
To demonstrate how this paradox can be avoided, let us again consider our traveller (from section 1.3). The trees, to appear as things in themselves, must not simply appear as the belonging of the tree-outlines (α and β) to the brown bark (γ), and γ to the white (δ) and yellow (λ) bark, and so on to the trunk, the sap, the cell structure, and beyond. As Plato puts it in the Parmenides, 'If oneness isn't present in the others [i.e., multiples], the others are neither many nor one.'68 That is, if each tree, while nonetheless appearing multiple, does not appear as one multiple, then it will be inapprehensible. It must, rather, be possible to apprehend it as 'this multiple whose terms let themselves be numbered on the basis of the law that is structure (the count-as-one).'69 Hence, the coming to be of each of these sets must be stopped, and a more stable set must be produced from the subsets that are included in the tree-outlines. 70 This more stable set must be sufficient to present 69 This means that strictly speaking 'being is neither one (because only presentation itself is pertinent to the count-as-one), nor multiple (because the multiple is solely the regime of presentation).' (Badiou,Being and Event,28,24.) 70 Ibid., 93-95. trees in themselves, and not the temporary 'forming-into-one'71 of trees which then, in the blinking of any eye (à la perception) or the passing of a thought (à la conception), 'blend'72 and come to be indistinguishable.73 Nevertheless, the production of a new set from these subsets, which Badiou calls 'the count-as-one of subsets',74 is founded on belonging (∈). For we saw in section 1.3 that inclusion (⊂) is derived from belonging. Hence, without belonging there cannot be subsets to count-as-one, thereby producing a new set. Therefore, the count-as-one of subsets -which Badiou also calls 'metastructure' and 'the count of the count'75 -does not change the fact that the idea (εἶδος) which is most fundamental for him is belonging. He stresses that this 'does not introduce a special operation, nor any primitive relation other that of belonging.'76 By positing the count-as-one of subsets, he can propose that 'the being of presentation is inconsistent multiplicity, but despite this, it is never chaotic.'77 However, the flipside of this proposition is that though presentation is never chaotic, its being is still inconsistent multiplicity -the predicate of which is ∈. We will now enquire further into what this 'being of presentation' is on Badiou's view.

Being is multiple
Here, we must pay close attention once more to the fact that belonging (∈) is a posited relation included within the axioms of set-theory. This relation is therefore necessarily not the sort of thing that can be proven. In other words, an axiomatic decision to think in a certain way is not the same thing as a proof that something is the case. Although theorems about belonging -such as the theorem of the point of excess in Meditation Seven of Being and Event78 -can be proven, the axioms themselves in which belonging is included as a posited relation are by definition not proven, but postulated 'axiomatically'. For the following reasons, this means that the subsistence of belonging, or whether belonging 'is', is indeterminate.
Belonging is the operation through which multiples count-as-one in relation to each other. But this operation is not itself counted-as-one; it is simply posited. It was demonstrated in section 3.1 that for any thing to be thinkable, its coming to be must be stopped by counting-it-as-one. Accordingly, because the operation of belonging is not itself counted-as-one, the foundation of this operation, or the means by which it comes to be, is unthinkable. The means by which it comes to be is not thought (as a proof) but decided (in the axioms of set-theory) -like Parmenides, Badiou '[establishes] philosophy as a decision.'79 And because 'being and thinking are the Same',80 to be is to be thinkable and not to be is to be unthinkable. Hence, the count-as-one, whose axiomatic foundation is unthinkable, does not subsist. In other words, there is no being of the count-as-one. It is founded on nothing; 'the essence of structure is the void.'81 The void (∅) is the 'proper name of being',82 since it can only be named and not thought. For this reason Badiou asserts 'metaphysics is, and this is its shortest definition, that which makes a predicate of the impredicable.'83 The operation of belonging, which can be thought, is predicated on the void, which cannot be thought and therefore is 'impredicable' in this sense.
We learnt that for Badiou belonging is the most fundamental idea (εἶδος) that constitutes reality. Now we understand that the subsistence of this idea cannot be determined, because the basis on which multiples count-as-one, and therefore are thinkable, is not itself counted-as-one, and therefore is unthinkable. Therefore, the very fundament of reality is indeterminacy, on Badiou's view. Another way of saying this is as follows: being is pure multiplicity.84 Being is multiple because how to think (or present) it can be decided in multiple ways -however, without a prior decision about how being is to be presented, being is neither one nor multiple, but void (∅). Adrian Johnston is wrong to reproach Badiou because 'the ground, origin, or source of this operation [of belonging]', which is how Badiou decides to present being, 'remains mysteriously unspecified.'85 For it is necessary that this operation be one way among many in which how to present (or think) being can be decided, so that the choice to think this way in particular be a decision with subjective force -as Badiou outlines especially in Meditation Twenty of Being and Event, and which Daniel Sacilotto characterises as an 'ur-decision'.86 Regrettably, there is not space to address this aspect of Badiou's philosophy here.
Returning to Plato, we can now see that Badiou opposes his negative conception of being to Plato's positive conception. That is, while for Plato being is something in reality with positive existence, for Badiou being is inexistence. That is, it is what in-exists or, as it were, exists 'inside' multiples as what is always already deconstructing them. They are always already being deconstructed because they are structured by nothing (∅) -'there is no structure of being.'87 They are not, on the other hand, structured by Ideas as Plato posits them in the Phaedo.
We can now reappraise what Badiou means when he says, 'I am a sophisticated Platonist, not a vulgar one.'88 He is not a Platonist in the sense that he posits the Idea (Εἶδος) with a capital 'I', though he does posit the idea (εἶδος) with a lower-case 'i'. For Badiou reality is fundamentally constituted by an idea: the relation of belonging, whose subsistence is indeterminate. However, it is critically important to understand that because of the indeterminacy of this relation, it is not an 'Idea' in the sense that Plato gives to this concept in the Phaedo (i.e., it does not subsist). For in Logics of Worlds Badiou asserts, 'In Being and Event, I assumed the dissemination of the indifferent multiple as the ground of all that there is, and consequently affirmed the ontological non-being of relation.'89 Here, he affirms 'the ontological non-being of relation' in the sense that ontologically the relation of belonging, which constitutes reality as its fundamental idea, is indeterminate. In other words, non-being = indeterminacy.90 These findings shed more light on whether and in what senses Badiou is both (I) a Platonist and (II) an idealist.
I. The definition of Platonism that we have chosen to be concerned with, at this juncture, is the doctrine that abstract entities subsist. However, the abstract entity of belonging (∈) which Badiou posits, and the other abstract entities like equality (↔) which he derives from this entity, do not subsist. What subsists is the indeterminacy of belonging (or set-membership) as the basis on which all things are apprehensible as such. Badiou's philosophy of experience is therefore founded on a radical contingency. Moreover, his truly is a 'Platonic gesture',91 because for him apprehending things as abstract entities is only a contingent foundation for thought and not representative of the necessary way things are. Therefore, David and Ricardo L. Nirenberg are wrong to assert, 'Badiou is, throughout [Being and Event], asserting that sets and their properties are there, regardless of whether anyone thinks them or not.'92 Badiou is not 'a Platonist for whom the huge universe of set-theoretical objects is actually real.'93 On the contrary, Badiou, as Jason Barker argues, is an 'ontological anti-realist' in the sense that he 'denies the very existence of an objective, mind-independent universe of objects.'94 This being said, is Badiou an idealist?
II. Daniel Sacilotto argues that Badiou does not, like Hegel, posit an absolute idealism that 'render[s] the difference between thought and its object immanent to thinking'.95 That is, Badiou does not render immanent the difference between the two propositions that (i) things must, in thought, be apprehended as members of sets, and that (ii) things must, in actuality, be members of sets.96 Even if we accept that it is necessary to think of things as sets, it does not follow in Badiou's metaphysics that things in themselves are sets. As Kant asserts, 'The things that we intuit are not in themselves what we intuit them as being.'97 Of course, intuition is not at issue here. Nevertheless, does Badiou not posit a transcendental idealism whereby things qua their apprehensibility are determinate sets and things qua themselves are indeterminate and therefore inapprehensible?98 Ray Brassier points out it would appear that Badiou is merely reiterating a familiar transcendental distinction between the formal features of being as characterised relative to thought, or 'for us', and being as it is independently of thought, or 'in itself'; which would amount to the well-worn dualism of phenomenon and noumenon. 99 Sacilotto offers a solution to this problem: 'Badiou shows that for mathematical ontology the two terms [of being and thinking] remain merely indistinct, rendering both ontologically indifferent qua multiples.'100 That is, 'the same' is coextensive with 'undifferentiated'. For Badiou everything is the same (or undifferentiated) ontologically because the logic of being (or onto-logy) of every thing is pure multiplicity (i.e., pure quantitative and not qualitative difference). It is in this sense that being and thinking and the same. Brassier offers the same solution, arguing that 'the law of the count as condition for existence […] is ultimately indiscernible from the ontological inconsistency whose presentation it forecloses.'101 Nevertheless, how can Badiou account for the connection between how things 'are' (as ontologically inconsistent) and how we 'think' about them (as phenomenologically consistent)? This question concerns 'the connection between set-theory and the world'.102 It asks how a connection between the being of undifferentiated sets, and the appearing of differentiated phenomena, can be posited in a philosophy of experience without idealism. Becky Vartabedian asks this question too, arguing that 'there is something like an act of faith required in the assertion that this [ontological] apparatus tells something essential about the world or world(s).' 103 Brassier and Johnston also argue that this problem has not been solved.104 My contribution here has been to point out that if Badiou is an idealist, he is not one in the Platonist sense of positing the subsistence of abstract entities. Nevertheless, I can also point out that his metaphysics constitutes a novel approach to the philosophy of experience. That is, Badiou splits being and appearing, in conformity with orthodox Platonism. But he assigns Platonic Ideas not to being but to appearing, in a decidedly unorthodox move. However, Badiou's decision to keep the idea (εἶδος) centre-stage in this way arguably legitimises his claim to being a philosopher proudly in the tradition of Plato.
We must ask whether this conceptualisation of the idea (εἶδος) conforms with how Plato himself conceptualised it in the Parmenides, in spite of the fact it is not in conformity with conventional Platonism. For this question, it is this dialogue that must be our focus, because it is the one Badiou meditates on in the meditation on Plato in Being and Event. With regard to the Parmenides, we might be tempted to preempt our conclusion by wondering whether multiplicity itself is not the very Platonic Idea shared by both Badiou and Plato, and which therefore legitimises once and for all Badiou's claim to being a Platonist. This is indeed the conclusion that we must draw, while not being able to give the full attention to this matter -i.e., to the sophistication both of this dialogue and of Badiou's response to it -which could only be given by an entire essay on this subject alone. For Badiou writes, What Plato is endeavouring to think here [in the Parmenides], in a magnificent, dense text, is evidently inconsistent multiplicity, which is to say, pure presentation, anterior to any one-effect, or to any structure105 However, Badiou argues that, because Plato did not have the conceptual resources of set-theory, he was unable to conceptualise '[the] below-the-presentable that is multiple-presentation.'106 Therefore, he mistakenly believed that 'there is no form of object for thought which is capable of gathering together the pure multiple, the multiple-without-one, and making it consist'.107 The definition of the Idea which, Badiou argues, is common to all of Plato's dialogues is (in Badiou's words) 'the occurrence in beings of the thinkable.'108 However, for Badiou what is thinkable of beings (i.e., their being) can, contrary to what Plato argues in the Parmenides, be presented: namely, by set-theory.
Nonetheless, though Plato was not able to take this argument to its logical conclusion, he went far enough to leave his doctrine of the Idea (Εἶδος) -in the form in which he had presented it in the Phaedo and elsewhere -in question. For what the Parmenides establishes is in Badiou's words 'the non-being of the One.'109 It is on this basis that the 'Idea' -in the sense of a unitary abstract entity which, although it is not presentable, supervenes upon everything presented -cannot be.110 These concluding remarks notwithstanding, I would like to finish this essay by acknowledging that a comprehensive study of the sense in which Badiou is a Platonist, in all his sophistication, would need not just to discuss whether he is a Platonist ontologically, but also to discuss how his Platonism is amorous, poetical, and political -in correspondence with what he considers to be the other three conditions of philosophy. The polemical force of his 'Platonic gesture'111 'as a provocation or a banner',112 and his mission to '(re)turn'113 to its roots the very tradition that Plato inaugurated, must both be explained in light of all four such conditions. Badiou's mission must, furthermore, be explained in light of not just the place of abstract entities in his metaphysics of being and appearing, but the five theses that, in The (Re)turn of Philosophy Itself, define Platonism as the doctrine that (i) there are four philosophical conditions, (ii) philosophy is 105 Badiou, Being and Event, 33. 106 Ibid, 34. 107 Ibid. 108 Ibid., 36. 109 Ibid., 31. 110 We would, then, seem to be justified in supposing that Badiou is claiming here that Plato rejected the doctrine of the Idea as it is presented in the Phaedo and elsewhere, that in the Parmenides he sought 'the occurrence in beings of the thinkable' in another place. To justify this supposition, we would have to hold that in claiming to being Platonist, Badiou -or any other philosopher -must be committed to the view that Plato, after everything is said and done, had only one position on this doctrine, or indeed any other doctrine advanced in the dialogues. However, it would be in the spirit of this essay to suggest that there are perhaps multiple Platos, and that Badiou may simply have decided on the Plato that is most useful for his philosophy. essentially opposed to sophistry, (iii) philosophy produces no truths, (iv) philosophy is both 'a fiction of knowledge' and 'a fiction of art ', and (v) philosophy is 'a pinch of truths'. 114 Barker rightly points out that this 'rather unconventional brand of Platonism' is a 'critique of Platonism' but not of Plato, because it sets out to re-turn Platonism to the truths in Plato it has turned away from. 115 The most important such truth for Badiou, because of which he calls his philosophy a 'Platonism of the multiple',116 is the very dialectic of the One and the multiple which this essay has explored not via the Parmenides -as Badiou does in the second meditation of Being and Event -but via the Theaetetus and Phaedo. Last but not least, the Good is perhaps the idea that Plato shines through in Badiou117 -Nirenberg and Nirenberg therefore could not be more wrong that 'he is a Platonist […] without Good'.118 , 119