HERMODORUS OF SYRACUSE AND SEXTUS EMPIRICUS’ ‘PYTHAGOREANS’ ON CATEGORIES AND PRINCIPLES

Abstract Hermodorus of Syracuse, a Sicilian disciple of Plato, is reported by Simplicius to have set out a classification of beings, which is of a piece with an argument for principle monism (in Ph. 247.30–248.18 > F 5 IP2; 256.28–257.4 = F 6 IP2). A similar classification appears in Sextus Empiricus’ Aduersus mathematicos X (262–75), where it is officially ascribed to some ‘Pythagoreans’ (Πυθαγορικοί) or ‘children of the Pythagoreans’ (Πυθαγορικῶν παῖδες), but seems ultimately based on Early Academic material. Virtually all commentators have read these classifications conjointly. More radically, both have been taken to record Plato's oral teaching and to give essentially the same categorial scheme, which is regarded as the most developed instance of a so-called ‘Academic doctrine of the categories’. This article re-examines these texts and provides an alternative reading. Section 1 focusses on Hermodorus and defends three theses: (1) there was never such a thing as an ‘Academic doctrine of the categories’; (2) Hermodorus does not seem to recount what Plato said, but to propose an integrated interpretation and defence of aspects of his thought; (3) Hermodorus’ pronouncements about principles are incompatible with other testimonies on Plato's unwritten teaching, notably Aristotle's. Section 2 moves to Sextus and defends a fourth thesis: (4) despite their similarities, the classifications of Hermodorus and Sextus’ Pyrhagoreans are considerably different, though perhaps originated from the same debate.

INTRODUCTION 1 In Ph.A 9, Aristotle assesses the account of the principles of coming-to-be provided by Plato and other Academics.In fact, no philosopher is explicitly named in the chapter.The thinkers criticized are generically introduced as ἕτεροί τινες (191b35).But they are said to speak about the 'great-and-small' (192a7).So Aristotle should have at least Plato in mind. 2 The plurals (191b35 ἕτεροί τινες; 192a6-7 οἱ δέ; 192a9-10 προῆλθον; 192a11 ποιοῦσιν) might suggest that some of his followers are also included. 3But this alone is not decisive, for sometimes Aristotle uses the plural to refer to one person only. 4till, the τις at 192a11 is probably Plato,5 and his way of conceiving of the 'great-and-small' is partly distinguished from that of the οἱ δέ at 192a6-7.So the target seems to include both Plato and other Academics. 6Aristotle praises them for having touched upon the notion of an underlying nature, but criticizes them for having failed to distinguish between matter and privation.For, according to Aristotle, they identified matter both with the great-and-small and with not-being, and made this unique nature that from which things come to be.Thus, although they spoke of the material substrate as a two-pronged nature (great-and-small), in fact, by identifying it with not-being, they ended up with a dyadic account of the principles of coming-to-be, that is, matter(/ great-and-small/not-being), as a joint cause (192a13 συναιτία), 7 and form.Aristotle thinks this is mistaken and that the principles must be three: matter, privation and form.
Commenting on this Physics chapter, Simplicius replies that Plato did not make an ἀρχή of matter.To this end, he avails himself of two quotations from the Περὶ Πλάτωνος of Hermodorus of Syracuse, a Sicilian disciple of Plato (in Ph. 247.30-248.18> F 5 IP2 ; 256.28-257.4= F 6 IP 2 ). 8We know nearly nothing about Hermodorus.His Syracusan origin is mentioned by Philodemus (T 1 IP 2 Συρακόσιος).We learn that he was a disciple of Plato from the Suda (T 3 IP 2 ἀκροατής) and Simplicius (F 5 IP 2 ἑταῖρος).Presumably he met Plato during one of the latter's trips to Syracuse. 9Philodemus and the Suda add that Hermodorus brought Plato's writings to Sicily (T 1 IP 2 [μετ]αφέ|[ρω]ν ̣;10 T 3 IP 2 κομίζων).Perhaps, after meeting Plato in Sicily, he followed him to Athens, attended the Academy for a while and later returned to Sicily, taking the master's works with him. 11This seems the only ground for saying that Hermodorus was ever actually part of the Academy, 12 as opposed to being a mere follower of Plato.
In the first of Simplicius' quotations, Hermodorus sets out a classification of beings, which is of a piece with an argument for principle monism.A similar classification appears in Sextus Empiricus' Aduersus mathematicos X (262-75), where it is officially ascribed to some 'Pythagoreans' (Πυθαγορικοί) or 'children of the Pythagoreans' (Πυθαγορικῶν παῖδες), but seems ultimately based on Early Academic material (see n. 62 below).Virtually all commentators have read these classifications conjointly (see n. 60 below).More radically, both have been taken to record Plato's oral teaching and to give essentially the same categorial scheme, which is regarded as the most developed instance of the so-called 'Academic doctrine of the categories'.
This article re-examines these texts and provides an alternative reading.I begin in Section 1 with Hermodorus and defend three theses: (1) there was never such a thing as an 'Academic doctrine of the categories'; (2) Hermodorus does not seem to recount what Plato said, but to propose an integrated interpretation and defence of aspects of his thought; (3) Hermodorus' pronouncements about principles are incompatible with other testimonies on Plato's unwritten teaching, notably Aristotle's.In Section 2 I move to Sextus and defend a fourth thesis: (4) despite their similarities, the classifications of Hermodorus and Sextus' Pyrhagoreans are considerably different, though perhaps originated from the same debate.

HERMODORUS' CLASSIFICATION OF BEINGS AND PRINCIPLE MONISM
Simplicius takes Hermodorus' pronouncements to show that Aristotle is right to claim that Plato calls matter great-and-small, but wrong to contend that Plato made a principle of matter.Simplicius has access to Hermodorus' work in a doubly mediated way: he reads Porphyry, who refers to a passage of Dercyllides' On the Philosophy of Plato,13 which in turn quotes Hermodorus. 14Since Simplicius specifies that Dercyllides quotes (λέξιν παραγράφειν) Hermodorus, scholars regard this as a genuine fragment, despite the double mediation. 15xt.Here is the relevant excerpt from Simplicius: From this text it is clear that Plato hypothesized matter in terms of the unlimited and indefinite and explained matter on the basis of those things which take on the more and the less, which themselves also include the great and the small.For [Hermodorus] 19 says: '[Plato] says that of things which are, some, such as human and horse, are in themselves, some are in relation to other things, and of these, some are in relation to contraries, as good is to bad, others are in relation to something, and of these some are definite, others indefinite.'

He adds:
'and all those which are described as great in relation to small possess the more and less.For, in virtue of their being still greater and smaller, they are brought to the unlimited, and likewise too broader and narrower, heavier and lighter, and everything which is expressed this way will go on to the unlimited.But things which are described in the way the equal and the stationary and the tuned are do not possess the more and the less but rather the contraries of these do, since one unequal is more unequal than another, one moving thing is more moving than another, and one out-of-tune thing is more out-of-tune than another.Consequently, all of the pairs of both of these have received the more and the less, except the single element.Consequently, such a thing, being unstable and formless and unlimited and a not-being, is described by negation of being, but to such a thing neither principle nor substance is appropriate, but it is driven into confusion.For [Plato] makes clear that just as in one way what acts is the cause in the strict sense and distinctively, so too it is a principle, but matter is not a principle.Therefore, it was said also by Plato and his followers that there is only one principle.
Although κατηγορία is absent from the fragment, I use the term 'category' (and cognates), as a synonym of 'genus of being', following a standard practice.
that the two sections are not obviously on a continuum and this interruption may be significant.The first section contains a categorial classification which proceeds diairetically. 22(A) Beings are first split into those in themselves (καθ' αὑτά) and those relative to others (πρὸς ἕτερα), such as human and horse.(B) Relatives to others are in turn dichotomously divided into 'relative to contraries' (πρὸς ἐναντία) and 'relative to something' (πρός τι).(C) The third subdivision distinguishes two types of πρός τι, the definite relatives (ὡρισμένα) and the indefinite ones (ἀόριστα).The following scheme results: (A) Dichotomous divisions of beings comparable to the one made in the first step of S1 are not uncommon in the Early Academy. 23But both the terminology and the way of developing this binary scheme are inconsistent.Xenocrates (F 15 IP 2 ) and the div.arist.67M/32DL divide beings into καθ' αὑτά (or καθ' ἑαυτά) and πρός τι; but Aristotle's De bono fr. 2 opposes καθ' αὑτά to ἀντικείμενα; Hermodorus even locates πρός τι under a more general class of πρὸς ἕτερα (see below).The div. arist.32DL (but not 67M) explains the opposition between καθ' ἑαυτά and πρός τι through the notion of 'expression' (ἑρμηνεία), absent from all other instances of the bicategorial scheme.Aristotle and Hermodorus frame the bicategorial distinction with an account of the principle(s); Xenocrates and the div.arist.67M/32DL do not.And while Aristotle reports that for Plato there are two principles and each category depends on both, Hermodorus, as we shall see, crisply denies the existence of a second principle.So the only thing these texts 22 The τὰ μὲν … τὰ δέ construction also suggests that the division is into exhaustive and mutually exclusive categories (cf.M. Duncombe, 'Plato's absolute and relative categories at Sophist 255c14', AncPhil 32 [2012], 77-86, at 78).W.-R. Mann, The Discovery of Things [Princeton, 2000], 144-5 locates the whole 'distinction of Hermodorus and others … within the class of πρὸς ἕτερα beings' of a higher division into καθ' αὑτά and πρὸς ἕτερα, which he reads in Plut.Adu.Col. 1115F Pohlenz-Westman.Unfortunately, there is no evidence of this in Simplicius or anywhere else. 23 end up sharing is just the dichotomous character of their divisions of beings.And while Hermodorus develops this binary scheme further, by dividing up the second category into subcategories, nothing similar is attested for Xenocrates, the div.arist.67M/32DL, or Aristotle's De bono fr. 2. The only (possibly) Early Academic categorial classification with a complexity comparable to Hermodorus' is Sextus' (but see §2 below).Hence, they can hardly be expressions of a unique and consistent doctrine.In other words, there seems to have been nothing like an 'Academic doctrine of the categories' 24 -my thesis (1).The extant evidence attests just a series of attempts to classify beings into two main groups (variously named), occasionally connected (in various ways) to principles.
But how should we understand Hermodorus' distinction between καθ' αὑτά and πρὸς ἕτερα?These Academic classifications share another trait.They all seem ultimately based on Plato's distinction, at Soph. 255c13-14, between τὰ αὐτὰ καθ᾽ αὑτά and τὰ πρὸς ἄλλα (often dubbed absolutes and relatives). 25Regarding Xenocrates and the div.arist.67M/32DL, I have argued elsewhere that, for this and other reasons, contrary to appearances and to what many commentators believe, the absolutes-relatives contrast is not equivalent to the Aristotelian substances-accidents contrast. 26I think the same holds for Hermodorus, 27 despite the higher complexity of his categorial classification.This seems prima facie disproved by the examples 'human being' and 'horse', 28 typical Aristotelian (secondary) substances.But the class of καθ' αὑτά is said to exclude only things which admit a contrary and those which admit the more-and-less.So nothing prevents the καθ' αὑτά from including, for example, definite numerical attributes such as four, or geometrical attributes such as triangle 29 (obviously not substances for Aristotle).The distinction seems rather between things which naturally form a pair (of contraries or correlatives) with their counterpart; and things which instead are stand-alone.
(B) The second division splits τὰ πρὸς ἕτερα into τὰ πρός τι and τὰ πρὸς ἐναντία.This is puzzling, because πρός τι and πρὸς ἕτερα are often interchangeable phrases; 30 and, if anything, one would expect the latter to name a class subordinate to that named by the former, 31 since 'something' (τι) is a more extended concept than 'other' (ἕτερον).Instead, τὰ πρὸς ἕτερα refers here to entities characterized by a generic type of relativity encompassing both contrariety and a somewhat more specific type of relativity.The nature of such a generic type of relativity is unfortunately not spelled out.
Here is a hypothesis: as I said, in Aristotle's De bono fr. 2 the 'in themselves' (καθ' αὑτά) are contrasted with 'opposites' (ἀντικείμενα).In Cat. 10 and elsewhere τὰ ἀντικείμενα include, among other things, both contraries (ἐναντία) and relatives (πρός τι).32 Perhaps Hermodorus had something similar in mind. 33And in the De bono Aristotle claims to assess Plato's view, whereas in the Categories he sets out his own.So he does not seem to claim originality on the two-pronged subdivision of generic opposites.If I am right, Hermodorus might be distinguishing, among those things which naturally form a pair of mutually related opposites, between those which cannot coexist, so that the coming-into-being of one requires the removal of the other; and those which must coexist, so that they come-to-be and are removed together. 34C) The third step of the division is the most debated.As I read it, Hermodorus continues dividing dichotomously, by sticking to the same pattern of (A) and (B). 35hus, he divides τὰ πρός τι into definite (τὰ ὡρισμένα) and indefinite (τὰ ἀόριστα).The grammar clearly supports this reading. 36Just as at 248.3, καὶ τούτων referred to τὰ πρὸς ἕτερα, so too καὶ τούτων at 248.4 is most naturally read as picking up τὰ πρός τι.More generally, the rhythm and the phraseology of the sequence are consistent throughout: we begin with a genitive plural indicating the kind to divide (248.2 τῶν ὄντων; 248.3 τούτων; 248.4 τούτων), and the two subkinds resulting from the division are introduced by the τὰ μὲν … τὰ δέ clause (248.2-3,3-4, 5).Hermodorus is therefore here distinguishing between two subkinds of relatives, definite and indefinite.Metaph.Δ 15.1020b32-1021a11 has often been evoked as a parallel passage. 37This is helpful, but Aristotle restricts there the definite-indefinite distinction to numerical relatives.This hardly applies to Hermodorus, because it would leave unmapped in his categorial tree non-numerical indefinite correlatives like right-left, father-son and similar. 38o perhaps (C) distinguishes between relatives linked by a determinate numerical relation (for example double-half) and relatives bound by an indeterminate relation (numerical or not).
Most scholars, however, construe step (C) differently. 39They (unnaturally) take the demonstrative τούτων at 248.4 to refer to all τὰ πρὸς ἕτερα, not just to τὰ πρός τι.To see why, the part of the fragment following ἐπάγει at 248.5 must be scrutinized.It will emerge that the exegetical worries of these interpreters are sensible, but can be accommodated without doing violence to the grammar.
The impulse for Simplicius' quotation from Hermodorus was that Aristotle frequently mentions that Plato calls matter τὸ μέγα καὶ τὸ μικρόν, but no sooner than 248.5 do we read something about this notion.We are here told that 'all those which are described as great in relation to small possess the more and less', in that they can 'go on εἰς ἄπειρον'.The salient feature of these items is that they are susceptible to unlimited increase or decrease.By calling something 'great', one means that it is great in relation to something, but places no limit on how great it is with respect to it.So things described as great in relation to small are presumably indefinite relatives. 40Hermodorus is thus paving the way for the claim that Plato locates matter under the right branch of (C).
But starting from 248.9 the picture gets more complicated.Unlike things described as great in relation to small, 'things which are described in the way the equal and the stationary and the tuned are do not possess the more and the less but rather the contraries (ἐναντία) of these do'.Since the possession of the μᾶλλον-ἧττον has been explanatorily connected to indefiniteness, we must conclude that the definite-indefinite distinction applies not just to τὰ πρός τι but also to τὰ πρὸς ἐναντία, and that S1 should be modified accordingly: A cryptic interim conclusion follows: 'all of the pairs of both of these have received the more and the less, except the single element'.I take 'both of these' (τούτων ἀμφοτέρων) to refer to unequal-unequal, moving-moving and out-of-tune-out-of-tune.Each forms in turn a pair (συζυγία) of contraries respectively with the equal, the stationary and the tuned.What Hermodorus is therefore saying is that all these pairs of contraries have received the more-and-less.But, of course, not both components of each pair are subject to the more-and-less: for in each pair one branch is always susceptible to unlimited increase or decrease (for example unequal); the other branch is not (for example equal). 41ow, to make the two parts of Hermodorus' fragment align scholars have felt obliged, as I anticipated, to go one step further and take the demonstrative τούτων at 248.4 to refer to all τὰ πρὸς ἕτερα and not just to τὰ πρός τι. 42The price to pay is high, because we have seen that the grammar and the phraseology of the fragment's first part leave no room for doubt.On the other hand, the exegetical worry motivating such a radical move is sensible.For the mention of τὰ ἐναντία at 248.10 makes it hard to avoid the conclusion that a branch of the contraries is also subject to the more-and-less, and that, therefore, the definite-indefinite distinction applies to both subdivisions of πρὸς ἕτερα. 43We seem to be at a dead end.But there is an economical way out.We have seen that Simplicius' citation is clearly interrupted by ἐπάγει at 248.5.The two sections, then, may well not be on a continuum.It is possible that Simplicius or Porphyry or Dercyllides left out a portion of Hermodorus' text to get straight to the point relevant to the discussion about the status of matter.We can thus reasonably suppose that Hermodorus began his discussion by proposing a division like the one represented by S1, but then made further considerations-not reported by Simplicius-that enabled him to revise it so as to obtain S2.There is therefore no need to do Procrustean violence to the grammar of the fragment's first part: the doctrinal requirements posed by the second part can be accommodated by giving due weight to the clear break splitting the quotation.
One last point before turning to principles: did Hermodorus need to rely on Plato's oral teaching to draw this categorial tree?I think not.(A) is clearly based on Soph.255c and (B) and (C) can easily be seen as resulting from an interpretative combination of claims made in the Theaetetus and (especially) the Philebus. 44At Tht. 186a-b, good and bad are among entities standing in a relation of contrariety (ἐναντιότης), which are repeatedly called πρὸς ἄλληλα, like at Phlb. 25d11-e1.Yet again Phlb.24a-25c thematizes the connection, crucial for Hermodorus, between ἄπειρον and μᾶλλον-ἧττον, and distinguishes relatives which admit the more-and-less (for example hotter and colder) from those which do not (for example double and half).What emerges so far seems thus less a record of Plato's unwritten teaching than a formal attempt to provide a systematic composition of claims made in various passages scattered in Plato's later dialogues.This gives part of the evidence for my thesis (2).The next sub-section will bolster it further.
Hermodorus' argument for principle monism.Simplicius' goal was to argue, against Aristotle and through Hermodorus, that Plato did not conceive of matter as a principle.This conclusion is drawn in the rest of the fragment, starting from 248.13.Here Hermodorus affirms first that a thing subject to the more-and-less 45 is unstable, 42 Cf.n. 39 above. 43At Phlb.25a6-b3, relatives such as τὸ ἴσον and τὸ διπλάσιον are called ἐναντία of things admitting the more-and-less.And at 26d1-2 Plato calls the τὸ ἧττον an ἐναντίον of τὸ μᾶλλον.Some have therefore thought that Hermodorus borrows this loose sense of ἐναντίον and the whole second section of the fragment concerns the right branch of (C) alone, with no further subdivision of τὰ ἐναντία (cf.C. de Vogel, Rethinking Plato and Platonism [Leiden, 1986], 193, apparently revising de Vogel [n.16]).This would leave S1 untouched, but would also questionably make Hermodorus carelessly use the technical terminology he himself introduced.
44 Cherniss (n.18), 286-7 n. 192 thinks that Hermodorus might also be relying on Plt.283c-286c.This is possible but unnecessary, because the Philebus (terminologically closer to Hermodorus' fragment) already contains all the relevant elements Hermodorus needed from the Statesman.
formless, unlimited and a not-being, by negation of being (248.13-14).Two points on this: First, the thrust of Hermodorus' conclusion is that matter is fundamentally privative: since matter is said like the great-and-small, it is an indefinite relative; 46 thus, it is subject to the more-and-less; hence, being susceptible of unlimited increase or decrease, it has no determinate limit or stability.A crucial aspect of Hermodorus' strategy emerges here (and one which marks a substantial difference with Sextus): once he has drawn his categorial tree, Hermodorus can now use it as a framework for analysis, by precisely locating matter in it and drawing certain metaphysical consequences.
Second, Hermodorus did not need to rely on any oral teaching for this point either, and could draw from a number of textual cues in Plato's later dialogues, first and foremost the description in the Timaeus of the receptacle as ἄμορφον (50d7, 51a7) and subject to completely unstable and disorderly motion (52d-53c).Cherniss, however, has also contended that by calling matter a not-being, 'by negation of being' (κατὰ ἀπόφασιν τοῦ ὄντος), Hermodorus contradicts Soph.257b and 258e, where, for Cherniss, 'not-being as the negation of being Plato forcibly puts aside'. 47et in those passages the Eleatic Stranger by no means denies that not-being is the ἀπόφασις of being.He argues instead that an ἀπόφασις does not necessarily indicate contrariety, and that when a negation particle is prefixed to a name, it only indicates difference from what that name designates (257b9-c3)-hence, τὸ μὴ ὄν does not indicate the long declared off-limits contrary of being (258e6-8).Further, to contradict the Sophist, Hermodorus would have to be suggesting that matter is utterly not-existent.This would also clash with the Timaeus, where the receptacle, though formless and deprived of every attribute, is none the less certainly existent (cf., for example, 52a8); and also with the Philebus, which lists ἄπειρον as a genus of being (25a1, 26c9-d1).But this cannot be what Hermodorus means, because he started from a division of ὄντα.More plausibly, he is suggesting that matter is the negation of determinate being, 48 as it is purely passive and thereby ready to receive determinations, as the ensuing lines of the fragment suggest.Let us look at them.At 248.14-18 Hermodorus concludes that since matter is privative it cannot be a substance, let alone a principle.Like Aristotle, Hermodorus thinks that, since an indefinite relative cannot be a fundamental being (οὐσία), it cannot even be a principle 49 -but unlike Aristotle, he thinks matter is such an indefinite relative.This is further underpinned through the subsequent remark that matter is not just privative but also constitutively passive.Hermodorus evokes Plato's recurrent analysis of causation in terms of production (prominent in the Philebus) 50 and argues, consistently with the Timaeus (50b8-c2, 50d7-e1), that since matter is not what acts, but what is acted upon, it cannot be the true cause of anything.The closing lines accomplish this argument for principle monism: matter is not a principle (ἡ δὲ ὕλη οὐκ ἀρχή) and there is only one principle (μία, ὅτι ἡ ἀρχή).
Scholars have not taken this claim seriously enough-some of them no doubt aiming to find a uniform doctrine of principles professed by the Academics. 51Hermodorus does not say, as often believed, that there is a second principle somewhat subordinate to the first.He crisply denies the existence of a second principle.Since matter is an indefinite relative, it is also privative and passive and therefore neither a substance nor a principle. 52Hence there is only one principle.It is consequently perplexing to read such scholars as Dillon saying on the very same page that 'Hermodorus declares matter not to be a principle [and] asserts a certain degree of metaphysical monism for Plato', but also that 'we have here, attested by Hermodorus, independently of Aristotle, in the first generation after Plato's death, confirmation of the two supreme principles of Plato's so-called 'unwritten doctrines' ([n.10], 202, my emphasis).Even Isnardi Parente, adamantly sceptical about the existence of Plato's unwritten doctrines, rightly emphasizes Hermodorus' pronounced 'tendenza al monismo' ([n.23], 131), but then lets slip that for Hermodorus 'i due principi si pongono differentemente che non in una relazione di parità' (126). 53The absence of the phrase ἀόριστος δυάς, which may or may not be significant, 54 is not my worry here.The problem is philosophical: Hermodorus is a monist, and it is a mistake to conflate his denial of the existence of a second principle with the claim that the second principle is subordinate to the first.
Accordingly, Hermodorus' pronouncements on principles are incompatible with other testimonies on Plato's unwritten teaching-my thesis (3).Most of these testimonies, first and foremost Aristotle's, ascribe to Plato a two-principle theory. 55o, even assuming the historical existence of Plato's oral teaching, 56 according to these testimonies there exists a second principle and the real problem concerns its nature and relation to the One.As Gaiser said at the beginning of his magnum opus, 'the central factual and historical problem' of the doctrine of principles is 'how the two opposing principles ultimately relate to each other in Plato's view'. 57By contrast, I submit that this problem does not even arise for Hermodorus, because for him there simply is no second principle.Indeed, Hermodorus was probably trying to defend Plato, through an integrated interpretation of aspects of his thought, from criticisms comparable to those Aristotle sets out in Ph.A 9 or Metaph.N 1 against the Platonic account of principles. 58To speak of a second principle in relation to Hermodorus betrays therefore the core of his argument.
I argued that Hermodorus could set out both his classification of beings and his argument for principle monism by relying just on Plato's dialogues.I now add that, if critics insist that Hermodorus must have relied on Plato's oral teaching, they will also have to deal with the unwelcome consequence that the indirect evidence for this teaching is at least on this point inconsistent. 59I leave this problem to others, but one aspect of it will be relevant to my next section.

SEXTUS EMPIRICUS' 'PYTHAGOREANS' ON THE GENERA OF BEING AND THEIR PRINCIPLES
Received view.Starting from Heinze's Xenokrates commentators have generally found the content of Hermodorus' fragment exposed in more detail and precision by Sextus Empiricus in Aduersus mathematicos X (262-75). 60Here a classification of beings comparable to Hermodorus', paired with an account of principles, is exposed and officially ascribed to some Πυθαγορικοί or Πυθαγορικῶν παῖδες. 61Scholars have long suggested, however, that Sextus' account of the Pythagoreans, whatever its immediate source, is ultimately based on Early Academic material. 62Thus Hermodorus and Sextus have been taken to give virtually the same categorial scheme, regarded as expression of Plato's unwritten teaching.I shall devote this last section to show that this reading is unconvincing and that, despite their similarities, Hermodorus' and Sextus' accounts are considerably different, though perhaps originated from the same debate-my thesis (4).(For this passage from Sextus, see also Brennan [n.62] below.) Classification of beings.Sextus' Pythagoreans divide beings into three genera: 1) those 'conceived in virtue of a difference' (τὰ μὲν κατὰ διαφορὰν νοεῖται), which exist by themselves (καθ' ἑαυτά) and in virtue of their own individuality, for example a human being or a horse.For these exist absolutely (ἀπολύτως) and not according to its state in relation to another thing (οὐχ ὡς κατὰ τὴν πρὸς ἕτερον σχέσιν); 2) those conceived 'in virtue of contrariety' (τὰ δὲ κατ' ἐναντίωσιν), which 'are observed on the basis of a contrariety between one thing and another' (ἑτέρου πρὸς ἕτερον), for example good-bad; 3) those conceived 'in relation to something' (τὰ δὲ πρός τι), which are conceived in terms of their state in relation to another thing (κατὰ τὴν ὡς πρὸς ἕτερον σχέσιν), for example right-left, up-down.
Contraries and relatives differ in two respects: (i) contraries cannot coexist, as the coming-into-being of one requires the removal of the other; whereas relatives must coexist, as they are cogenerated and coremoved.(ii) Contraries do not have any intermediate; relatives do.The similarities with Hermodorus' classification are considerable, but at least two differences are apparent.The first is the use of the term of Stoic flavour κατὰ διαφοράν to describe things in themselves.But this might be determined by linguistic updates of the Hellenistic era.The second is that Sextus gives a threefold division. 63nlike Hermodorus, he does not locate contraries and relatives under a common genus.It may be objected that the difference is minimal and that the description of both contraries and relatives includes the relevant expression πρὸς ἕτερον.But the difference in the classificatory structure can hardly be explained away as an accident.We shall see that it was probably due to a precise strategy to account for the categories-principles relation.Note that the problem does not concern, as Isnardi Parente (n.23), 125 believes, the greater or lesser adherence to 'Platonic orthodoxy'.For while Plato seems to favour the dichotomous model, he admits divisions into three or more terms (cf.Plt.287c3-5; Phlb.16c10-d7).
Sextus' Pythagoreans' argument for principle dualism.The Pythagoreans' classification of beings, like Hermodorus', is combined with, and indeed functional to an account of the principles.But the strategies and the results are completely different. 64We have seen Hermodorus using his categorial tree as a framework for analysis, to locate matter within it and thereby set out an argument to deny that matter is a principle.By contrast, the account of Sextus' Pythagoreans is evoked as one of the ways they teach that there are two ultimate principles of beings, the One and the Indefinite Dyad, 65 neither of which is located within the categorial tree, but is instead the ultimate superordinate γένος under which each category, by participating in it, is located (X 262.1-5).This is achieved in two steps.Each category is first related back to one of three different genera (X 274.2-3): • the κατὰ διαφοράν to the one τὸ ἕν, because each of them is one and regarded as by itself; 63 See the insistence on the number three at Sext.Emp.Math.X 269.1 τῶν τριῶν ὄντων γενῶν. 64Cf.Menn (n.40). 65To be sure, at Sext.Emp.Math.X 261.4-5 the Indefinite Dyad is produced by the One 'added to itself, in virtue of otherness'.But the following lines insist on the duality of principles (262.1-2,276.1-3).
• the κατ' ἐναντίωσιν to the equal-unequal (ἴσον-ἄνισον), in which the nature (φύσις) of contrariety is best observed; • finally, the πρός τι relate back to excess-deficiency (ὑπεροχή-ἔλλειψις), since they are always susceptible to increase and decrease.
In the second step, the intermediate pair ἴσον-ἄνισον is bifurcated: the first member relates back to the One, which is 'pre-eminently equal to itself'; whereas the unequal to excess-deficiency, which in turn falls under the Indefinite Dyad, the principle of indetermination.This results in the following scheme (from Krämer [n.24], 286): We can now see the import of the threefold categorial scheme.At first blush the dichotomous scheme would have appeared more convenient to show that the totality of reality is governed by two principles.But the Pythagoreans were explicitly attached to the idea that each genus of being must in turn have its own principle: so all the categories, 'being genera, are found placed under other genera (X 274.2-3 γένη ὄντα, εὕρηται ἄλλοις γένεσιν ὑποταττόμενα)'.Thus, instead of making contraries and relatives subdivisions of a single category, they preferred to construct a hierarchy of principles culminating in the pair One-Indefinite Dyad.
Both the classification of beings and the account of principles of Sextus' Pythagoreans are therefore considerably different from Hermodorus'.It is possible (and indeed quite likely) that they originated in a common debate which began in the Academy and continued afterwards and even elsewhere.But their identification is, in my view, contradicted by the texts.

CONCLUSION
All four theses of my plan have now been defended: (1) there was never such a thing as an 'Academic doctrine of the categories'.(2) Hermodorus does not seem to recount what Plato said, but proposes an integrated interpretation and defence of aspects of his thought.(3) Hermodorus' pronouncements about principles are incompatible with other testimonies on Plato's unwritten teaching, notably Aristotle's.(4) Despite their similarities, Hermodorus' and Sextus' classifications are considerably different.
Overall, if we want to make progress in the understanding of Early Academic discussions about principles and of the historical background of Aristotle's doctrine of the categories, instead of trying at all costs to 'scholasticize' and normalize the sources, we need rigorously to explore the reasons for their differences.

We cannot know how much text, if any, has been left out. But we will see 18
F 6 IP 2 is virtually identical to part of F 5 IP 2 , so I do not analyse it separately.For Susemihl (n.16), 522 n. 671 the last two sentences are Simplicius'.Zeller, Heinze, Robin and Isnardi Parente apparently agree and exclude them from the fragment, as does H. Cherniss, Aristotle's Criticism of Plato and Academy [Baltimore, 1944], 171 n. 96, who however thinks they are more probably by Dercyllides.But the switch to the first-person plural at 248.18-19 suggests the return of Simplicius' authorial voice and corroborates Diels's inclusion of both sentences in the quotation from Hermodorus.It might be objected that Hermodorus is himself one of οἱ περὶ Πλάτωνα, but οἱ/αἱ περὶ + acc.nominis proprii is primarily a standard way to refer to certain individuals: cf.S.L. Radt, 'Οἱ (αἱ etc.) περὶ + acc.nominis proprii bei Strabon', ZPE 71 (1988), 35-40.19ForP.S. Horky, 'Persian cosmos and Greek philosophy: Plato's association with the Zoroastrian magoi', OSAPh 37 (2009), 47-103, at 86 n. 150, the subject here is Dercyllides, not Hermodorus (and in the next sentence Hermodorus, not Plato).But Simplicius introduces the quotation by saying that 'from [Hermodorus'] text it is clear that Plato hypothesized matter' in a certain way, a claim echoed at 248.18-20 and suggesting that the subject of the quotation is Plato himself.Cf. 256.35-257.4(< F 6 IP 2 ) with Festugière (n.16), 308.