2 Speusippus and the Search for an Adequate System of Principles

Plato was succeeded as head of the Academy, as we have seen, by his nephew Speusippus, son of his sister Potone and a certain Eurymedon.¹ The circumstances of this succession are unfortunately quite obscure, since such ancient sources as we have—Philodemus' History of the Academy  ² and Diogenes Laertius—treat the event as unproblematical. Both Philodemus and Diogenes simply say ‘he succeeded’ (diedexato), without giving any indication of the procedure involved. Plato's last will and testament, preserved by Diogenes Laertius (III 41–3), really throws no light on the matter, since Speusippus, although he is named (along with six others) as an executor (epitropos), is not bequeathed anything therein, the only named beneficiary being Adeimantus (presumably Plato's nephew, or even grand‐nephew, rather than his older half‐brother). It is not even clear whether the school was the sort of entity which one could bequeath to anyone; certainly it receives no mention in the will.³

Sadly, all we have in the way of ‘biographical’ information on Speusippus (chiefly from Diogenes Laertius) is a farrago of unreliably attested anecdotes, most of which do not even sound plausible, and none of which would be of much importance in any case. By way of illustration of this, I append a sample:

He set up statues of the Graces in the shrine of the Muses erected by Plato in the Academy. He adhered faithfully to Plato's doctrines. In character, however, he was unlike him, being prone to anger and easily overcome by pleasures. At any rate, there is a story that in a fit of passion he flung his favourite dog into a well, and that pleasure was the sole motive for his journey to Macedonia to be present at the wedding‐feast of Cassander.⁴

The various elements in this have been well analysed by Leonardo Tarán.⁵ There is no reason to doubt Speusippus' contribution of the statues of the Graces to the shrine to the Muses established by Plato, but it would be precarious to draw any deeply significant conclusions from that. Rumours of his bad temper and proneness to pleasure are plainly malicious, and the latter allegation at least probably has something to do with Speusippus' known doctrine on pleasure, which we will examine in due course; the allegation of bad temper, however, might have some substance—the odd story about throwing his dog down a well has at least the virtue of a degree of specificity, though we have no idea of its provenance—but it conflicts with other evidence to be quoted below. As for his alleged motives for attending Cassander's wedding, the whole story must be dismissed on chronological grounds, since the only known wedding of the Macedonian regent Cassander was to Thessalonike, daughter of Philip II, in 316, long after Speusippus' death. Some fact may lurk beneath the muddy waters of this anecdote, but it is not obvious what it could be, or whether it ever really concerned Speusippus in the first place.

The same, we must regretfully conclude, goes for such snippets as the allegation that he had an affair with one of the lady pupils of the Academy, Lastheneia of Mantinea; that he suffered badly from arthritis in his latter years (not intrinsically improbable); or that he committed suicide out of depression (highly improbable).

There is some reasonably good evidence, on the other hand, of his being on good terms with Plato's friend and disciple, Dion of Syracuse, whom he got to know when Dion was living in exile in Athens in the 360s. I quote from Plutarch's Life of Dion, ch. 17:

Dion lived in the city with Callippus, one of his acquaintances, but for diversion he bought a country place, and afterwards, when he sailed to Sicily,⁶ he gave this to Speusippus, who was his most intimate friend at Athens. For Plato wished that Dion's disposition should be tempered and sweetened by association with men of charming presence who indulged seasonably in graceful pleasantries. And such a man was Speusippus. (trans. Perrin)⁷

This tells us something, not only about the relationship between Speusippus and Dion, but, more importantly, about his relationship with his uncle Plato. There are a number of other accounts of relations between the two, of varying degrees of reliability. One, relayed by Plutarch,⁸ alleges that Plato

reclaimed his nephew Speusippus from great self‐indulgence and debauchery, not by either saying or doing to him anything that would cause him pain, but when the young man was avoiding his parents, who were always showing him to be in the wrong and upbraiding him, Plato showed himself friendly and free from anger to Speusippus and so brought about in him great respect and admiration for himself and for philosophy. Yet many of Plato's friends used to rebuke him for not admonishing the youth, but Plato would say that he was indeed admonishing him: by his own, the philosopher's, manner of life, showing him a way to distinguish the difference between what is shameful and what is honourable.

While one must feel duly cautious about such unattributed anecdotes as this, it is perfectly possible that there is a kernel of truth in it. After all, the fact remains that Plato's nephew did follow him into the philosophic life when he might have done many other things, and this approach to the moral training of the young does, as Tarán points out,⁹ accord with Plato's own recommendation at Laws V 729b–c. The fact that there is a vaguely similar story told about Xenocrates' rehabilitation of Polemo, as we shall see presently, is really neither here nor there.

Another odd, but popular, little story¹⁰ tells of Plato becoming angry with one of his slaves, and, precisely because he was angry, asking Speusippus to beat the slave for him. Unfortunately for the reliability of this, however, there are fully seven other sources listed¹¹ which give Xenocrates in this role instead of Speusippus. It is in any case not of much significance.

Other than these, there are the details mentioned in the previous chapter: the famous fragment of Epicrates, depicting the students of the Academy analysing the pumpkin; and the story about Aristotle giving the aged Plato a hard time during Xenocrates' absence, when Speusippus was ill and unable to defend him. In either case, what we can deduce is that already in Plato's lifetime Speusippus was acknowledged as one of the key figures in the Academy, if not already the designated successor. But that is, after all, no more than what one would expect.

From the period of Speusippus' own short reign as head of the Academy (347–339), we have only the information that Heraclides of Pontus came first to study with him, before moving on to Aristotle (DL V 86),¹² and the tale of a confrontation with Diogenes the Cynic (DL IV 3), which would not be worth mentioning were it not for the fact that it may attest to the fact that he was afflicted with arthritis in his declining years (he was, after all, over sixty when he assumed the headship).

One other incident from this period, however, may be included, the historicity of which has long been under a cloud, but, I am now convinced, without adequate cause, and that is the matter of Speusippus' Letter to King Philip, which would, if genuine, have to have been composed in 343 or 342.¹³ It is a document of some interest, though not one which sheds a very creditable light on Speusippus, trying as it does to ingratiate him and the Academy with Philip, through adducing various mythical, and equally doubtful historical, justifications for the legitimacy of Philip's conquests, while attempting to blacken the reputation of the aged Isocrates and his students, in particular the historian Theopompus, who were basking in Philip's favour at the time. It does, however, constitute an example of the sort of politicking in which a head of school felt that he had to indulge in the period of the growth of Macedonian power. It does not, on the other hand, merit much attention in a study of his philosophy.

As regards Speusippus' works, we have a list in Diogenes Laertius' Life (IV 4–5), comprising twenty‐seven¹⁴ separate items, some involving multiple books (e.g. ten books of Homoia, or ‘Resemblances’), which Diogenes himself does not claim to be exhaustive (although he gives at the end a total of lines, which may be authoritative: 43,475). Diogenes describes them as comprising both treatises (hypomnēmata) and dialogues, and this seems to be borne out by the list of titles.

We have first a group of apparently ethical works, beginning with a dialogue Aristippus, no doubt attacking the hedonistic doctrines, ostensibly of Aristippus of Cyrene (with whom we may, perhaps, imagine Socrates to be in conversation), but primarily directed against Speusippus' colleague in the Academy, Eudoxus of Cnidus, who maintained also a hedonistic position.¹⁵ There follow treatises On Wealth, On Pleasure, On Justice, On Philosophy,  ¹⁶ and On Friendship. As we shall see, Xenocrates also wrote treatises on all these subjects, as on many other subjects treated by Speusippus, either in support or in emulation.

We find then a treatise On the Gods, followed by (perhaps) a series of dialogues, The Philosopher, Cephalus, Cleinomachus or Lysias, and The Citizen. The use of the names Cephalus and Lysias leads one to suppose that Speusippus is dramatizing the well‐known orator (whom he would have known) and his father (portrayed delightfully in Book I of Plato's Republic), but what these dialogues were about escapes us entirely. As for Cleinomachus, it has been suggested that he may be an eristic philosopher who was a contemporary of Eucleides of Megara, but this does not help us very much, except to suggest that Speusippus should not have approved of him. Both Philippus of Opus and Xenocrates wrote treatises On the Gods. The Citizen (Politēs) is a curious title, presumably of political content, not certainly a dialogue. but odd as the title of a treatise.

We find then a treatise On the Soul, a basic philosophical topic on which we also find treatises by Xenocrates (in two books), and Aristotle (in three), as well as single books by Theophrastus and Heraclides of Pontus. This no doubt developed Speusippus' distinctive definition of the soul, which will be discussed further presently. A treatise To  Gryllus follows, presumably addressed to Xenophon's son of that name, who died in the battle of Mantinea in 362, and whom Speusippus may well have known. Aristotle tells us (ap. DL II 65 = Fr. 68 Rose) that many people wrote encomia in his honour, and Aristotle himself composed one, as did the rhetorician Isocrates.

There follow a number of rather mysterious titles: Tekhnōn Elenchos probably means ‘A critical examination of rhetorical treatises’, since tekhnē can have that meaning; Hypomnēmatikoi Dialogoi could mean ‘memoirs (hypomnēmata) in dialogue form’, but more probably ‘dialogues for mnemonic purposes’, i.e. to help memorizing; and Tekhnikon probably has something to do with rhetorical treatises as well. So we may have here a little nest of treatises concerned, critically or otherwise, with the practice and theory of rhetoric—which would not be surprising: Platonic philosophers continued to take an interest in rhetoric to the end of antiquity, despite—or perhaps because of—Plato's strictures in the Gorgias, the Phaedrus, and elsewhere.

Next we have the well‐known Homoia, or On Similar Things, in ten books, though preceded by the puzzling rubric dialogoi, since it is highly unlikely that such a work was cast in the form of a dialogue. The rubric is best assumed to be out of place. It could have described a number of the previous works in the list, most probably the sequence from Philosophos to Pros Gryllon. The Homoia is followed by Diaireseis kai pros ta Homoia Hypotheseis (‘Divisions, and Hypotheses relating to the Homoia’), which sounds as though it is closely related to the Homoia itself, perhaps on a more theoretical level, if the Homoia was concerned, as on the evidence of the surviving fragments preserved by Athenaeus (Frs. 6–27 Tarán) it seems to have been, with the actual assembling of genera and species. There is a further work Peri genōn kai eidōn paradeigmatōn (‘On examples ofGenera and Species’—taking paradeigmatōn as governing the other two nouns, but the title is peculiar), which seems to relate to the same topic. Aristotle doubtless learned a good deal from Speusippus in this area, though he is loth to admit it. It is probable¹⁷ that when he refers in the Parts of Animals, as he does on two occasions (642^b12 and 643^a36), to ‘the written divisions’ he is actually referring to one or other of these works by Speusippus, especially since the context is one of trenchant criticism of dichotomous divisions.

We come next to a mysterious title, Pros ton amarturon, ‘In relation to the unwitnessed (one)’. This may, however, as Tarán proposes, be explained as yet another rhetorical work, a contribution to a controversy which involved the rhetorician Isocrates, who, shortly after the restoration of the democracy in 403, composed a speech on behalf of one Nicias against a man called Euthynous. Nicias, conceiving himself to be in danger during the rule of the Thirty Tyrants, had deposited a sum of three talents with Euthynous, who was a cousin of his, neglecting to bring anyone with him to witness this transaction. When he came to ask for the return of his money, Euthynous tried to fob him off with only two talents. Nicias had to wait till after the fall of the Thirty to bring suit against Euthynous, and, because of the circumstances, Isocrates was forced to compose a speech based solely on circumstantial evidence and probabilities—a considerable challenge for a forensic orator.¹⁸ Hence the title, Pros Euthynoun, amarturos. This speech, which was presumably successful, plainly acquired some degree of notoriety as an exemplum, becoming known simply as ‘the Amarturos’. Diogenes Laertius tells us (VI 15) that Socrates' follower Antisthenes composed a reply to it. This, we may assume, at some later stage Speusippus also did, perhaps as part of an ongoing feud which the Academy and Isocrates' school waged with each other during the middle years of the century.

Following on this, we have some more perspicuous items. First, the Encomium of Plato, or, as Diogenes terms it earlier, in his Life of Plato (III 2), The Funeral Banquet (Perideipnon) of Plato,  ¹⁹ in which we are told that Speusippus actually made the claim that his uncle was in fact the son of Apollo, relating a ‘miraculous birth’ story distinctly uncomplimentary to Plato's father Ariston:²⁰ ‘The story went that Ariston tried to force his attentions on Perictione, who was in her youthful bloom, and failed of his purpose; and then when he ceased from his violence, Apollo appeared to him in a vision; as a result of which he left her unmolested until her child was born.’

This extraordinary story is hard to evaluate, but we must reflect that it is told in the context of a funeral oration, and may therefore best be viewed as a rather sophisticated conceit (surely nobody in the room, in Athens in 347 bc, can have taken it literally?), making use of the ‘divine birth’ motif borrowed from similar stories already current about the birth of Pythagoras. It would thus be Speusippus' way—he was himself an enthusiastic Pythagorean—of assimilating his uncle to Pythagoras.²¹ To make sense of the narrative, I think, we must presume Ariston, rather than indulging in rape, already to have married Perictione, and to be portrayed as pressing his marital rights rather roughly on his timid young virgin wife (who has in fact, unbeknownst to him, already been visited by Apollo!).²²

Following on this, we have a trio of letters, addressed respectively to Dion, Dionysius (presumably Dionysius II), and Philip (of Macedon). Letters answering to these descriptions survive among the large collection of Greek letters. The first two are probably spurious, but the third, as I have suggested above, may well be genuine. It is certainly hard to see why anyone would have bothered to forge it.

Then there is a treatise On Legislation. Then one entitled Mathematikos (The Mathematician), which may have contained the remarks on the nature of mathematical postulates discussed below, pp. 82–6. Then what sounds like another dialogue, the Mandrobolos, or Mandroboulos, about which we know only what we can deduce from a passage of Aristotle's Sophistical Refutations (174^b19–27), where Aristotle tells us that a certain Cleophon (presumably not the extreme democrat leader who met his end in 404) in this work resorts to the device of claiming that his opponent's argument, though valid against one sense of a word he is using, is not valid against the sense of it that he intends.

There follow a further dialogue, the Lysias, which I presume to be identical with the Cleinomachus, or Lysias mentioned above; Horoi (Definitions), presumably not identical with the surviving pseudo‐Platonic Definitions, since they are full of Peripatetic and Stoic material, though that work may perhaps be a later reworking of a document of his; and lastly a curious title, Taxeis hypomnēmatōn, ‘Arrangements, or lists, of commentaries, or memoirs’, which can hardly in any case be a title of a treatise of any sort, but rather, if anything, a catalogue‐style list of treatises.

There follows a rather garbled numeral giving the total of lines for all the treatises, generally read as 43,475, but which Ritschl, followed by Tarán, reads as 224,075, which certainly seems rather high for the total of works included in Diogenes' list. But Diogenes is not claiming that this is complete, and we know of at least one important work omitted from it—in fact the only work of Speusippus (apart from the Letter to King Philip) of which we have any verbatim remains. That is the little treatise²³  On Pythagorean Numbers, quoted by the author of the pseudo‐Iamblichean Theology of Arithmetic (pp. 82, 10–85, 23 De Falco = Fr. 28 Tarán), which will be discussed below, pp. 59–64.

This is a reasonably impressive œuvre (though not by any means as extensive as that of Xenocrates, as we shall see), and it is a solemn thought that barely a line of it, apart from the last‐mentioned work, and (in my view) the Letter to King Philip discussed above, survives to us. If, as I shall argue below, we can add to that the substance of chapter 4 of Iamblichus' De communi mathematica scientia (DCMS), then the loss becomes a little less complete, but this is a controversial claim, and it is not easy to see in which of the above‐listed works Speusippus might have discussed the basic features of his metaphysics as set out in DCMS 4. At any rate, let us now turn to a consideration of his philosophical position.

In embarking on an evaluation of the doctrines of Speusippus, the first unhappy truth that confronts us is that we cannot evade the evidence of Aristotle. I have referred already²⁴ to the enormous importance which that evidence, biased and allusive as it is, takes on for the Old Academy as a whole—or at least for the Academy prior to 322 bc—but in the case of Speusippus it takes on particular importance, by reason of the fact that we have very little other evidence on his doctrine, and of that evidence that part which is potentially of most importance is unfortunately of rather doubtful status, as we shall see.

First, then, let us attempt to uncover Speusippus' doctrine of first principles, and specifically his theory of the derivation of other levels of being from his first principles of One and Multiplicity. Speusippus, naturally, takes his start from what was at least the later doctrine of Plato, itself much influenced by Pythagorean speculation, which postulates as first principles of all things a One and an Indefinite Dyad (for Plato, if we may credit Aristotle,²⁵ ‘the great‐and‐small’, or ‘the greater‐and‐smaller’). From the action of the first of these on the second, there derives the system of Forms, conceived as mathematical entities of some sort. Plato himself is not recorded as being very specific as to how this comes about, nor about the relation of forms to numbers (though the first ten numbers, the ‘decad’, seem to have a privileged status of some sort), or to the ‘mathematicals’, whatever they were.²⁶ However, Speusippus seems to have provided somewhat more positive proposals on the mechanics of the derivation of the various levels of being, and this gives Aristotle occasion for astringent criticism. This he delivers in Met. Z 1028^b21–4, and again at various points in Books M and N.

We need to look closely at these passages,²⁷ since they provide good illustrations, I think, of Aristotle's strategies in criticizing those he did not approve of, while also, if used judiciously, affording adequate clues as to Speusippus' true position. I should state at the outset that I believe that Speusippus does have a coherent metaphysical theory, and that it can be derived primarily from the valuable passage preserved by Iamblichus in De communi mathematica scientia, ch. 4,²⁸ with some help from a verbatim extract from his treatise On Pythagorean Numbers, preserved in the pseudo‐Iamblichean Theology  of Arithmetic,²⁹ and even (despite himself) from Aristotle. The scenario that Aristotle wishes to present to us, and that Tarán seems largely prepared to accept, is of a thinker so inconsequential that he should not have been let out on the streets without a nurse (to borrow a thought from Thrasymachus' abuse of Socrates in Republic I). It is not unreasonable, therefore, in the case of a man who shows various signs of having been a philosopher of considerable subtlety, to ask whether one cannot, with a little sympathetic imagination, put together a reasonably coherent theory.

The theory that emerges from DCMS 4, and our other sources of evidence, is as follows. Speusippus accepted the doctrine of two opposite principles propounded by Plato (at least in his later years), according to which all things are to be derived from a One and a principle of indefinite multiplicity (termed by Plato the Indefinite Dyad, and perhaps also the ‘great‐and small’, but by Speusippus himself plēthos, ‘multiplicity’), but he altered it in an interesting direction. Laying particular emphasis on the status of the first principles as ‘seeds’ or ‘potencies’ of all things, he argued that what is itself the cause of some quality in other things cannot have that quality in the same way, so that if the One is the cause of goodness and being for all other things, it cannot itself properly be termed good or even existent (Frr. 42–3 Tarán), any more than an acorn is itself an oak tree. This led him to an interpretation of Plato's first principle which placed it at an extreme of transcendence.³⁰

Let us first look at Aristotle's remarks at Met. N 1092^a11–17 (= Fr. 43 Tarán), where he is criticizing the view that what is more developed derives its existence from what is less developed. In this connection, he takes a dig at Speusippus (referred to anonymously, as usual):

Nor is someone correct who compares the principles of the universe (hai tou holou arkhai) to that of living things and plants, on the grounds that the more complete always comes from what is indefinite and incomplete (this being his reason for saying that this applies to the primary principles too, so that the One itself would not even be an existing thing).³¹ For even in this case the principles from which these things come are complete; it is a man that produces a man, and it is not true that the sperm is primary.

I think we may reasonably suspect Aristotle of being tendentious here. It is possible that Speusippus did somewhere (though not in the following passage) make a comparison between seeds of natural things and the One, but if he did, it can only have been in respect of the seed's (apparent) simplicity in comparison to the animals or plants which develop from it, as an analogue to the One's (actual) simplicity with respect to everything else in the universe; there could be no implication of incompleteness or imperfection in the case of the One.

We should now look at the beginning of DCMS (p. 15, 5ff. Festa):

Of mathematical numbers³² one must postulate two primary and highest principles, the One (which one should not even call Being (on), by reason of its simplicity and its position as principle of everything else, a principle being properly not yet that of which it is the principle); and another principle, that of Multiplicity (plēthos), which is able of itself to facilitate³³division (diairesis), and which, if we are able to describe its nature most suitably, we would liken to a completely fluid and pliable matter (hyle).³⁴

Out of these two—coming together, Speusippus says vaguely, ‘by reason of a certain persuasive necessity’ (a phrase, I think, intentionally recalling the ‘persuasion’ of Necessity by Reason in Timaeus 48a)³⁵—is generated Number, plēthos providing the principle of infinite divisibility, and the One imposing limit and quality, to produce the first principle of Number.

It is here, however, that difficulties begin to accumulate. The first product of these two, as I say, is Number. Speusippus appears to have portrayed this as the number One, which gives Aristotle a handle for much malicious criticism. It is a notable fact that neither Plato nor Aristotle appears to have considered ‘one’ as being properly a number, but rather the root or basis of number, whereas Speusippus, as Tarán shows (1981: 35ff.), took it—or at least a certain sort of one—as the first odd number. However, this should not lead us (as it unfortunately leads Tarán) to confuse this mathematical ‘one’ either with Speusippus' absolute first principle, or even with the first principle of his second level of reality, which is that of numbers. What Speusippus seems to have postulated is a first principle of number, which, as the representative of the One at the next level of being, is necessarily unitary, and may be termed ‘one’,³⁶uniting with the principle of multiplicity once again, this time to produce the series of natural numbers—or perhaps, in the first instance, just the sequence of numbers up to ten—since he exhibits, in his treatise On Pythagorean Numbers (= Fr. 28 Tarán), as we shall see, a special reverence for the Decad as, in a way, the summation of number.

However, in some way not clear from the evidence, this union of Number and Multiplicity also generates the first principle of Figure,³⁷ which is represented (again as the representative of the primal One) as a point (stigmē), this ‘point’ serving as the unitary first principle of all geometrical entities, whether two‐ or three‐dimensional, which unites in turn with a corresponding form of Multiplicity—which he terms thesis kai diastasis topōn, ‘position and spatial interval’ (p. 17, 16)³⁸—to generate.

The whole range of geometrical entities are thus generated, but there is also produced an essential geometrical entity which, once again, unites with the principle of Multiplicity to serve as the first principle of the realm of Soul, which, as we know,³⁹ Speusippus defined as ‘the form of the omni‐dimensionally extended’ (idea tou pantēi diastatou). Soul in turn generates all the souls in the universe, but also, by the same means, serves as the first principle of the physical world, or body.

So much, I think, we can discern, on the basis of the data presented in DCMS 4, about Speusippus' strategy for generating the totality of existence.⁴⁰ It is certainly tortuous and complex, and by no means without difficulties, and it plainly irritated Aristotle extremely, but I think that it can be shown to be not without some philosophical justification, and based on a creative analysis of difficulties within the Platonic metaphysical scheme that he had inherited.

First of all, there is a problem facing all monistic cosmological systems, particularly those which, like Plato's, postulate a totally simple and uniform first principle, acting on a totally undifferentiated material principle, and that is (as is pointed out at DCMS 4, pp. 16, 18ff.) that, if both elements involved are quite undifferentiated, their union should logically produce, if anything at all, only one thing, or one level of being, not a whole series of such levels, such as we seem to observe in the cosmos as we have it. Therefore, it seemed to Speusippus, one needs to postulate also some mechanism to explain this latter phenomenon. The best he could come up with—and it is this that Aristotle takes delight in satirizing as his episodic universe⁴¹—is the theory that the (logically) first product of the union of the two ultimate principles should then become a principle in its turn, mating, so to speak, in an incestuous union, with its mother (which Speusippus has been careful to characterize, as we have seen, as ‘a totally fluid and pliable matter’), and producing the next level of being.

That some such process is being envisaged by Speusippus seems indicated by further remarks of Aristotle in Book M of the Metaphysics. First of all, at 1085^a31–^b4 (= Fr. 51 Tarán), we find the following (he has just been criticising Plato):

These thinkers, then, generate magnitudes from this kind of matter, but others [sc. Speusippus] from the point (stigmē)—they regard the point as being, not one, but like (hoion) the one—and another material principle (hylē) which is like Multiplicity (plēthos), but not Multiplicity; yet in the case of these principles nonetheless we get into the same difficulties. For if the matter is one, then line, plane, and solid will be the same; because the product of the same elements must be one and the same.⁴² If on the other hand there is more than one matter—one of the line, another of the plane, and another of the solid—either the kinds are associated with (akolouthousin) one another, or they are not. Thus the same result will follow in this case also; for either the plane will not contain a line, or it will be a line.

This critique, while seeking to ridicule Speusippus' position, also gives us some insight into what it was. What Speusippus does indeed wish to postulate is a sort of akolouthia obtaining between the various levels of plēthos, or hylē, however he may have expressed this. Nor, it seems, was this notion entirely original to him. Somewhat earlier in this book (M 9, 1085^a8ff.), Aristotle criticizes those Platonists (hardly identical with Speusippus, though their identity is uncertain) who propose to derive lines, planes, and solids from what Aristotle chooses to term ‘species’ (eidē) of the ‘Great and Small’—‘the Long and Short’, the ‘Broad and Narrow’, and the ‘Deep and Shallow’—and the first principles (arkhai) corresponding to these. Speusippus was not therefore, it would seem, entirely on his own in worrying about the problem of the mechanics of deriving a multi‐layered universe from a pair of totally simple first principles.

Aristotle's polemical strategy here, as so often elsewhere, is to assume that his opponents are using the terms they use in precisely the sense that he would use them himself, and thus reduce their position to absurdity. In the section of Book M immediately following the passage quoted above (1085^b4–34 = Frs. 40 and rest of 51 Tarán), he does just that, in particular understanding the concepts ‘one,’ ‘number’, and ‘multiplicity’ in terms of his own system, in a way which renders Speusippus' position quite incoherent. What he does not allow for is that Speusippus is postulating first principles of unity, multiplicity, and number which are not subject to Aristotelian definitions. For instance, for Aristotle, a multiplicity is a multiplicity of units (monades), and ‘number’ is also made up of units, whereas in Speusippus' system there are such entities as primal Multiplicity and the first principle of number of which this does not hold true. It is therefore, as I say, hardly fair of Aristotle to employ arguments making these assumptions. Nevertheless, such a passage as this is most useful for discerning something of Speusippus' theory, if read with due caution and discernment.

The second aspect of Platonic doctrine that plainly caused Speusippus some trouble was the notorious Theory of Forms. Aristotle testifies⁴³ to the fact that the difficulties in the theory, even in the final form which it reached in Plato's thought, were what led Speusippus to abandon the Forms as such, and postulate instead simply a system of numbers, and then of geometrical figures, as the proper objects of knowledge (for which he fully accepted the need). What were these difficulties? They seem to have been for Speusippus very much the ones that have been brought against the theory by critics ever since, beginning with Aristotle. Not only is it very difficult to decide of what things there are forms, and what degree of ‘similarity’ they may have to the particulars which they ‘inform’, but also how exactly the forms, and especially forms of different degrees of generality (in particular genera and species) relate to each other.⁴⁴ In face of these problems, Speusippus plainly thought that the theory was not worth preserving in anything like its traditional Platonic form, but yet was concerned by the problem which had led Plato to postulate it in the first place, the need for there to be eternal and unchanging objects of knowledge.⁴⁵ He found adequate candidates for these, however, in the systems of numbers and geometrical figures, which did not have the same problems of diversity and hierarchy to which the traditional forms were subject.⁴⁶ This does not necessarily mean, however, as Aristotle would have us believe, that he simply gave up on the Forms. There was still a paradigmatic and creative function to be performed in the universe which could not be reasonably attributed to purely mathematical entities (as Aristotle takes pleasure in pointing out). Speusippus was quite cognizant of this, however, and seems to have attributed this function to the World‐Soul, as I shall explain presently. He is not to be regarded, therefore, as having simply abandoned the Forms; rather, he restructured and rationalized them.

If we examine Aristotle's polemical strategy in response to these innovations of Speusippus, we can see him, first of all, endeavouring to score points by emphasizing the apparent series of separate first principles to the exclusion of the linking process that I have postulated for Speusippus. Let us consider first Met. Z 1028^b15ff. (= Fr. 29a Tarán):

Some people consider the limits of bodies, such as the surface and line and point and monad, to be substances (ousiai), and indeed more so than body and the solid. Further, some [sc. the Pythagoreans] hold that nothing such as this exists independently of perceptible objects, while others think that there are a multiplicity of such entities, and that they are actually more real, since they are eternal, as for instance Plato holds that there are forms and mathematicals, as two distinct levels of substance, and then as a third the substance that consists of perceptible bodies, while Speusippus postulates even more substances, beginning with the One, and then first principles (arkhai) of each level of substance, one for numbers, another for volumes (megethē), and then that of the soul; and in this way he strings out (epekteinei) the levels of being.

As I say, this is cleverly laid out. First Plato is presented as producing the ‘mathematicals’ as a separate level of being between forms and physical particulars,⁴⁷ and then Speusippus is brought on as developing this tendency to absurd lengths—the verb epekteinei I would take as pointedly satirical.

This odd propensity of Speusippus is returned to later, in Book M 8, 1083^a20ff. (= Fr. 34 Tarán), in the context of a refutation of the Platonist concept of idea‐numbers (that is, numbers incomparable with one another). Aristotle's particular complaint against Speusippus here is that, since he abandoned the idea‐numbers, and retained only mathematical numbers, he cannot then declare ‘One’ to be a sort of idea of number, and not postulate ideas of all the other natural numbers as well:

But again, what certain others have to say about numbers is not well said either. These are those who do not accept the theory of Forms either as such or as some sort of numbers, but believe in the existence of mathematicals, and hold that numbers are first among beings, and that the first principle of them is the One itself (auto to hen). For it is bizarre (atopon) that there should be a One that is first among the ones, as those persons say, and not a Two for the twos, nor a Three for the threes; for they are all in the same logical situation. If, then, this is the situation as regards number, and one postulates the existence of mathematical number alone, then the One is not a principle (for then such a One would have to be of a different nature from all the other ones; and if this were the case, then there would have to be a primal Two different from the twos, and similarly for all the other numbers in turn). But if the One is a first principle, it would have to be rather as Plato used to maintain was the case in relation to numbers, that there was also a primal Two and a Three, and these numbers would not be commensurable (symblētous) with one another. But again, if one makes that postulate, as has been said, many impossibilities result. And yet it is necessary that either the latter or the former situation obtain, as, if neither is the case, there is no possibility that number be separable.

If my reconstruction of Speusippus' doctrine is even approximately correct, we can see Aristotle here being wickedly misleading about that doctrine, in order to set up Plato and his nephew as holding two equally implausible positions. I am not concerned here to defend Plato's theory of idea‐numbers; my concern is only with Speusippus. What Aristotle is doing in his case is, first, to present him as abandoning the Forms in favour of entities which he seeks to identify with Plato's ‘mathematicals’; and then to suggest that Speusippus tries to elevate the ordinary mathematical number ‘one’ to the status of a first principle, not only of numbers, but of everything.

We can see from what has emerged already, I hope, how misleading this is. Speusippus may indeed have involved himself in some terminological difficulties, but it seems clear enough that he is postulating three distinct entities: a supreme ‘One’, or Unity, the first principle of all things, a secondary ‘One’, or Unit, which is the immediate product of the primal One and Multiplicity, and serves in turn as the first principle of Number—and thirdly, the purely mathematical ‘one’, which (unlike both Plato and Aristotle) Speusippus regarded as the first odd number, and which is on the same metaphysical and logical level as all the other numbers.

As for abandoning the Forms (whether as numbers or haplōs), this, as I have said, would seem to be a further gross over‐simplification. Certainly, Speusippus seems to have unhitched the concept of Form in its causative aspect from either the mathematical or the geometrical levels of reality, but he does this only to establish it firmly at the level of Soul, which he (following, as I think, his non‐literal interpretation of the Timaeus) takes as the immediate transmitter of form to the physical universe.⁴⁸ Soul, let us recall, he defined as the ‘form of the omni‐dimensionally extended (idea tou pantēi diastatou)’. If we give due weight to that admittedly very compressed and elliptical definition, we must, I think, interpret it as the ‘executive aspect’, so to speak, of the Essential Living Being, or Paradigm, of the Timaeus(which in itself, Speusippus, as emerges from the surviving portion of his treatise On Pythagorean Numbers (= Fr. 28 Tarán), wished to identify with the Decad—and with the geometrical equivalent of the Decad). This system of numbers, projected already at the next level of being as geometrical entities, is finally set in motion by the Soul, and is projected in turn upon the ‘Receptacle’ of Matter in the form of the basic triangles and basic geometrical figures described (rather schematically and poetically) in the Timaeus.  ⁴⁹

Forms, then, in the strict sense, manifest themselves only in the World‐Soul (which is itself a sort of super‐form), not at any higher level. What they do, presumably, is to constitute that three‐dimensionality which Aristotle condemns as the ‘Pythagorean’ version of the basic substance of things,⁵⁰ and for which his own doctrine of ‘matter’ is designed as a substitute. Aristotle's denunciation of geometricals as invisible and insubstantial (being rather quantities, if anything) would, I think, leave Speusippus unmoved. For him, it is precisely the projection of these entities, in the form of combinations of the five ‘Platonic’ regular solids (which, we learn, he discussed in the first part of his treatise On Pythagorean Numbers)⁵¹ onto the field of force described by Plato in the Timaeus as ‘the Receptacle’ that produces what appears to us as the physical world. It is as ‘solid’ as we are, but our bodies too, it must be remembered, are made up of immaterial triangles. It is as the source, or matrix, of all these combinations, then, that the World‐Soul is denominated the idea of three‐dimensionality.⁵²

It is only at the level of soul, after all, Speusippus wishes to assert (and this is another assertion for which Aristotle satirizes him⁵³), that ‘goodness’ may properly be said to manifest itself (DCMS 4, p. 18, 2ff.).⁵⁴ The full significance of this remarkable doctrine needs some thought to elucidate. Speusippus wished to deny ‘goodness’ both to the primal One, and even to the mathematical and geometrical levels of reality, not because they were bad, but simply because he felt that the term had no real meaning at those levels.⁵⁵ So what, then, would be the ‘real’ meaning of ‘good’ at the cosmic level for Speusippus? It seems to me that it is closely allied to the creative, or ‘demiurgic’, activity of the World‐Soul, and in this connection I think that the well‐known characterization of the Demiurge as ‘good’ by Timaeus at 29e (agathos ēn . . .)⁵⁶ is profoundly relevant. In Speusippus' ‘deconstruction’ of the myth of the Timaeus, the function of the Demiurge breaks down fairly naturally into (a) an archetypal aspect, which is transcendent Intellect (the contents of which, in Speusippus' terms, would be the system of numbers and geometricals⁵⁷); and (b) an ‘executive’ function, which would be most naturally transferred to the World‐Soul. It is this latter that would most properly be described as agathos, as being the agent of all order and tendency towards perfection in the physical universe, and this, it seems to me, is the rationale behind identifying Soul as the first level of being that can properly be characterized as ‘good’.

Even as nothing is said of the soul in DCMS 4 (since such a topic is not relevant to Iamblichus' purpose there), so nothing is said of any lower level of being than soul⁵⁸—except an incidental remark at the end of the extract (p. 18, 9–12 Festa), à propos the origin and nature of evil in the universe. Speusippus declares that there is nothing either ugly or bad (oude aiskhron oude kakon) in the higher reaches of reality—the realm of the One, of Number, or of Figure, ‘but only at the lowest level, among the fourths and fifths, which are combined from the lowest elements, does evil come into being—and even then not principally (proēgoumenōs), but as a result of a falling‐away from and failure to control what is in accordance with nature.’

This, brief and elliptical as it is, provides much food for thought. However we may distinguish the ‘fourths and fifths’, we seem to discern here a five‐level universe, two more levels being postulated below the level of Figure. Since there is no other mention in this document of Soul, one might feel that we must identify the fourth level with that, but there are difficulties. I cannot see that Speusippus (or anyone else) would have described Soul as ‘combined from the lowest elements', and, though one might allow for the incidental arising of evil at the level of theembodied soul, one would hardly contemplate it arising at the level of pure soul. So the ’fourths and fifths' must, I think, be regarded as lower levels of being than pure soul, and this in turn would entail that Speusippus is not counting the One and Multiplicity themselves among the levels of being—a somewhat paradoxical, but not entirely surprising conclusion, in light of his view of the One as ‘not even yet a being’ and, as a principle, ‘not being such as are the things of which it is a principle’.⁵⁹

Soul, then, may be taken as occupying the third rank in Speusippus' universe. What can we take the ‘fourths and fifths’ to be? They must be bodies of some sort, since they are composites of elements, so there is no question of anything like prime matter being regarded as a level of being, any more than is the One. This would not in any case be coherent with Speusippus' doctrine, since he identifies a ‘material’ principle at every level of being, and, as I have suggested above, has no use for Aristotelian‐style matter, ‘prime’ or otherwise. Nor, I think, can we see here a distinction between the heavenly and sublunar levels of being, since one could hardly describe the heavenly bodies as being composed of ‘the lowest elements’. On the whole, I would suggest that the distinction here being made is between animate and inanimate physical beings, both of which are so composed, but the former of which are informed by soul, while the latter are informed merely by a lower offshoot of soul which could be termed a principle of cohesion—something like an ancestor of the Stoic hexis. Only at these levels do badness and ugliness manifest themselves, as deviations from nature, and instances of the imperfect dominance of matter by form.

If this is so, it presents an attitude to evil in accord with the doctrine of the Timaeus, where the ‘errant cause’ inherent in the Receptacle ensures that nothing in the physical world can go quite right all the time. However, we are left with the problem of how the system that we have seen unfold in the first three levels is continued on into the fourth and fifth. I can only suggest that Soul, as formal principle, unites with Multiplicity once again to produce the class of individual embodied souls, and also a lower formal principle—‘Nature’ (physis), as Plotinus would later term it—which is a kind of lower soul, and this in turn, uniting with Multiplicity, produces, finally, the inanimate creation.⁶⁰

I produce this scenario simply in an effort to grant coherence to the system which we find presented in DCMS 4, but since Aristotle does not anywhere in his writings dignify this level of the Speusippan universe with even a mention, there is not much to go on.

Some further light is thrown on Speusippus' metaphysical scheme by a curious little notice preserved in Proclus' Commentary on the Parmenides (VII, pp. 38, 32–40, 7 Klibansky).⁶¹ Let us look at this testimonium in its context:

For if the first One participated in Being in some way, although it is higher than Being and produces it, it would be a one which took over the mode of reality which belongs to Being. But it is not a one, and is the cause not just of Being but of everything, though of Being before the rest. And if everything must participate in its cause, there must be a ‘one’ other than the simply One, in which Being participates; and this ‘one’ is the principle of beings. This is also how Speusippus understands the situation (presenting his views as the doctrines of the ancients).⁶² What does he say?

‘For they held that the One is higher than Being and is the source of Being; and they delivered it even from the status of a principle. For they held that, given the One, in itself, conceived as separated and alone without other things,  ⁶³  with no additional  element, nothing else would come into existence. And so they introduced the Indefinite Dyad as the principle of beings.’

So he too testifies that this was the opinion of the ancients about the One: it is snatched up⁶⁴ beyond existence, and next after it comes the Indefinite Dyad. Here too, then, Plato proves this One to be beyond the existent and beyond the unity that is in the existent and beyond the whole One Being.

This passage is of interest, not just for Speusippus' doctrine of the role of the Indefinite Dyad, or Multiplicity, in generating the other levels of the universe, but also, perhaps, for a suggestion that it might provide as to how Speusippus may have interpreted the second part of Plato's Parmenides, and specifically the relation between the first and second hypotheses. The possibility that an ‘ontological’ (as opposed to a logical) interpretation of the hypotheses of the Parmenides might go back all the way to the Old Academy, and specifically to Speusippus, instead of arising in Neopythagorean circles some time in the late first century ad,⁶⁵ is a dangerously radical one, but I think that we must entertain it as a possibility in the light of this passage.⁶⁶

We must ask ourselves, after all, why Speusippus should be brought into this discussion by Proclus at all. What relevance has the Indefinite Dyad to the ‘one’ of the second hypothesis? A possible explanation is that Proclus found in some source to which he had access (possibly the commentary of Porphyry or of Iamblichus on the same dialogue) a reference to Speusippus' doctrine of first principles arising from his exegesis of the first two hypotheses of the Parmenides. The scenario would be the following: the ‘one’ of the first hypothesis is indeed so hedged about by negativities as to be, by itself, incapable of generating anything further. In the second hypothesis, on the other hand, we find a process set out in which a ‘one’ undergoes a process of self‐division, initially into ‘one’ and ‘being’, which generates, first, such basic categories as ‘sameness’ and ‘otherness’, and then the whole realm of numbers (cf. especially 143a–144a). The passage is worth quoting:

Now suppose we take a selection of these terms, say ‘being’ and ‘different’, or ‘being’ and ‘one’, or ‘one’ and ‘different’; in each case we are selecting a pair which may be spoken of as ‘both’. I mean: we can speak of ‘being’, and again of ‘one’. We have thus named each member of a pair. And when I say ‘being and one’ or ‘being and different’ or ‘different and one’, and so on in every possible combination, I am in each case speaking of ‘both’. And a pair that can properly be called ‘both’ must be two. And if a pair of things are two, each of them must be one.

This applies to our terms: since each set forms a couple, each term must be one. And if so, then, when any one is added to any pair, the sum will be three. And three is odd, two even. Now if there are two, there must also be twice times, if three, three times, since two is twice times one and three is three times one. And if there are two and twice times, three and three times, there must be twice times two and three times three. And, if there are three which occur twice and two which occur three times, there must be twice times three and three times two. Thus there will be even multiples of even sets, odd multiples of odd sets, odd multiples of even sets, and even multiples of odd sets. That being so, there is no number left, which must not necessarily be.

Therefore, if a one is, there must also be number. (trans. Cornford)

One may surely see in this passage the derivation of the set of natural numbers from the action of the Dyad on the One. I would suggest, then, that Speusippus (whether or not based on authorization from his uncle I forbear to speculate) interpreted the subject of the second hypothesis of the Parmenides as being, not the Indefinite Dyad as such, but rather the portrayal of the results of the action of the Indefinite Dyad on the original One.⁶⁷ The role of the Dyad is to initiate a process of ‘splitting’, by which, initially, unity is distinguished from being, and, then, by the adducing of the principles of ‘sameness’ and ‘otherness’, there is generated the whole system of numbers. What ontological values, if any, Speusippus may have assigned to the remaining hypotheses we can have no real idea, but, since he seems to postulate a five‐level universe (if we may draw this conclusion from the mention of the ‘fourths and fifths’ in DCMS 4), one might hazard a guess that he was prepared to identify the first five hypotheses (taking 155e–157e as the third, as was done by all later ancient Platonists) with the five levels, or at least basic features, taking these as: (1) the One by itself; (2) Multiplicity, acting on the One to generate the whole mathematical realm, figures as well as numbers; (3) Soul; (4) animate beings; (5) inanimate beings. This would imply that the last four hypotheses produce purely negative conclusions, as is implied by their format. But we are no doubt straying too far into the realm of pure speculation here. We do not even know, after all, how Plotinus identified the hypotheses after the first three; and yet he certainly assigned a value at least to those.

At any rate, we seem here to catch a glimpse of how Speusippus envisaged at least the initial cosmogonic process getting under way. Let us turn now, finally, by way of rounding off the examination of his metaphysics, to his treatise On Pythagorean Numbers,⁶⁸ for the light that it might cast both on his metaphysical scheme in general, and on his theory of mathematics in particular.

The form in which what we have of this treatise is preserved, as mentioned above, is rather troublesome. The Theologoumena Arithmetikēs is an anonymous work,⁶⁹ of the late antique or possibly early Byzantine period, largely made up of a series of extracts from arithmological works by Nicomachus of Gerasa and Anatolius of Laodicea. The extract from Speusippus occurs in the course of a passage apparently borrowed from Nicomachus,⁷⁰ which gives it a provenance in Neopythagorean circles of the second century ad, precisely the milieu in which, if anywhere, we might expect to find some surviving acquaintance, however minimal, with Speusippus' work. Nicomachus (if it is he) first gives a summary in his own words of the first half of the work—which he describes (a trifle patronizingly, perhaps!) as a ‘well‐turned’ or ‘subtle little book’ (biblidion glaphyron, p. 82, 13–14⁷¹)—and then presents an extract from the second part, which, he tells us, was exclusively concerned with the Decad, which Speusippus regarded as the summit of all number.

On his account, the first half of the book comprised the following:⁷² ‘In the first half of the book, he very elegantly expounds linear numbers, polygonal and all sorts of plane numbers,⁷³ solid numbers, and the five figures which are assigned to the elements of the universe, discussing both their individual attributes and their shared features, and their analogousness and correspondence.’

The item of chief interest here is the five regular solids. It is not at all clear from this whether Speusippus is remaining true to Plato in the Timaeus (47e–57c) in assigning the fifth regular solid, the dodecahedron, to ‘the universe as a whole’, or whether, like his colleagues Philippus of Opus and Xenocrates (though in their different ways)⁷⁴ he is postulating a fifth element, aether, to which to assign it. Tarán (1981: 265) tries to get round the problem by suggesting that this phrase ‘may be a parenthetical remark of the author of the excerpt and need not represent anything in Speusippus' book at all,’ but this is surely an unjustified assumption. On the whole, it seems most natural to conclude that Speusippus too postulated a fifth, celestial substance to answer to the dodecahedron, and, what is more, that he was prepared to attribute all this, quite anachronistically, to ‘the Pythagoreans’.

The other problem is whether we are to take the last phrase, which speaks of individual attributes (idiotēs) and shared features (koinotēs), analogousness (analogia) and correspondence (antakolouthia),⁷⁵ as referring to the five regular solids (which would be more natural), or to the various types of number mentioned before them—or both. On the whole, it would seem most natural to take them as referring to both. After all, Speusippus could well have discussed such topics as the idiotēs of the triangles which make up the cube (cf. Tim. 55b–c), which render earth non‐exchangeable with the other elements, as well as the analogies and correspondences betwen the various classes of numbers and the Platonic figures. We may assume, then, that every sort of property of and relationship between the numbers from one to nine, and the figures corresponding to them, were discussed in this part of the work.

The description of the contents continues as follows:

Next, in the remaining half of the book, he goes straight on to deal with the Decad, which he shows to be the most natural (physikōtatē) and perfective (telestikōtatē) of existent things, because it is, in itself, and not based on our conceptions or because we postulate that it happens to be so, a sort of productive form (eidos ti tekhnikon) of the finished products (telesmata) in the world, and set before the god who created the universe as a completely perfect paradigm.

There are many interesting points here, both of terminology and of doctrine. As becomes progressively more obvious, what is being described here is the Paradigm of Plato's Timaeus, though in terms distinctive of Speusippus. But let us attend first to details of terminology. First of all, the adjective physikōtatē would seem to denote that the decad is the sum‐total or quintessence of all natural things, while the rare adjective telestikōtatē indicates, as does tekhnikon below, that it is the agent responsible for bringing all things to realization.⁷⁶ As for the use of the rather loaded Platonic term eidos, it is not clear if it is to be taken in the fully technical sense of ‘form’, or simply in the sense of ‘sort’ or ‘type’, but I see no compelling reason not to take it in its technical sense. If so, however, we must observe that this eidos is given an active, demiurgic role in the universe, and thus the description of it as being ‘set before the god who is the creator of the universe’ must be taken as figurative language based on the Timaeus, which, as we know, Speusippus did not in any case take literally.

It is at first sight somewhat confusing, certainly, to find Speusippus making use of the machinery of creator god and paradigm, especially since he appears to wish to father all this doctrine on ‘the Pythagoreans’. If anything, the demiurgic role in Speusippus' universe should, as I have suggested above, be fulfilled by the World‐Soul, with the help of the psychic analogues of the content of the geometrical realm of existence. However, if we bear in mind that it was Speusippus' understanding that Plato himself was speaking figuratively in the Timaeus, he may well have felt entitled to adopt this same figurative language for himself.

After all, it is highly unlikely that Speusippus, any more than Plato, would have wished to excise ‘God’ (ho theos) from his system; the only problem was to decide just where to situate him. On the whole, the evidence, such as it is, seems to favour the World‐Soul as the most properly ‘divine’ element in the Speusippan universe (even as it appears to be, after all, in Book X of Plato's Laws, which would have been Plato's own last word on the subject). We have a snippet from Cicero's De Natura Deorum (I 13, 32 = Fr. 56a Tarán), admittedly from the recklessly polemical mouth of the Epicurean Velleius, which points in this direction. Having just disposed of Plato himself. Xenophon, and Antisthenes, he continues: ‘Very similarly Speusippus, following his uncle Plato, and speaking of a certain force that governs all things, and is endowed with soul (vim quandam dicens qua omnia regantur, eamque animalem), does his best to root out the notion of deity from our minds altogether.’ This vis animalis is most reasonably interpreted, I think, as being a garbled reference to the World‐Soul, in which case the connection being made with Plato is not entirely unjustified, at least with reference to the Laws.

Another straw in the wind may be a doxographic reference from Aetius (Placita I 7, 20 Diels = Fr. 58 Tarán): ‘Speusippus [declares God to be] Intellect, which is not identical with the One or the Good, but has a nature peculiar to itself (idiophyēs).’ This distinguishes ‘God’ (and Intellect) from Speusippus' highest principle (inaccurately given also the Platonic title of ‘the Good’), but it does not clearly identify it with any other level of Speusippus' universe. However, if we put two and two together, we may reflect that, in the Timaeus (the mythological trappings of which, as we recall, Speusippus would deconstruct) the Demiurge is at 47e clearly identified with Nous, and both at 30b and at 46e the principle is laid down (in a way very confusing for those who persist in taking the Timaeus literally) that ‘nous cannot be present to anything without soul’, or ‘the one and only existent thing which has the property of acquiring nous is soul’. So it would seem very probable that ‘God’ is to be identified with the World‐Soul in its rational, demiurgic aspect—and it is this that contemplates the Decad, which is in turn best seen as the sum‐total of the realm of Number, reproduced analogically, first as the realm of Figure, and then, with the principle of motion added, as the contents of Soul itself.

The actual verbatim quotation from Speusippus, while very good to have, does not actually advance our knowledge of his doctrine very significantly. In it, Speusippus is concerned to exhibit the perfection of the Decad in as many ways as he can, and this becomes an exercise in arithmology rather than mathematical theory in any modern sense. However, he does in the process illustrate what he means by analogia and antakolouthia as between the first four numbers and the geometrical figures corresponding to them, at one point (p. 84, 11–12) describing the point, line, triangle, and pyramid as ‘primary and first principles of the classes of entity proper to each’ (prōta kai arkhai tōn kath' hekasta homogenōn)—an important reminder of how Speusippus viewed the role of these primary figures. Strictly speaking, the point must be regarded as the first principle of the whole geometrical realm, even as the monad is of the realm of number, but plainly all four basic figures have an archetypal role, analogous to that of the first four numbers (the tetraktys) in the realm above.

We have now surveyed, as sympathetically as we can, what can be known or conjectured of the structure of Speusippus' universe. From metaphysics, or ‘physics’, let us now turn to ethics. Once again, the evidence is fairly exiguous, and once again, we have to contend with tendentious testimony from Aristotle.

The basic text on Speusippus' ethical doctrine is a passage of Clement of Alexandria's Stromateis,  ⁷⁷ which we may reasonably quote in full, since we have little enough else to go on in reconstructing his position:

Speusippus, the nephew of Plato, declares that happiness (eudaimonia) is a perfect state in the area of what is natural, or the state of [possession of]⁷⁸ goods, which is a state for which all men have a [natural] impulse, while the good aim at freedom from disturbance (aokhlēsia). It would be the virtues that are creative of happiness.

We have here a very summary (but, we hope, basically accurate) sketch of Speusippus' doctrine. He recognizes that men in general strive for happiness, but seems to introduce as a qualification of this that the good aim at ‘freedom from disturbance’. There must be some ellipse, though, here, I think, arising from Clement's compression of his source. The state (katastasis) for which all men have a natural impulse may be eudaimonia, but Speusippus may well have gone on to say that most men believe, foolishly, that this lies in the acquisition and enjoyment of sources of pleasure, while only the wise understand that it really resides in attaining ‘freedom from disturbance’. In that case, Speusippus would grant this much to the proponents of hedonism (such as Aristippus, founder of the Cyrenaic school, or his own colleague Eudoxus of Cnidos), that pleasure is indeed a natural object of striving for all living creatures, but he would deny that this proves what it is claimed to prove: that pleasure is the good for man. Speusippus wishes to maintain that for man, as a rational being, pleasure as an end must be transcended, with the aid of dialectic, and supplanted by a rational striving for aokhlēsia.

Speusippus appears to have used a number of arguments to support his position (which seems to have been developed in opposition to that of his colleague Eudoxus),⁷⁹ one of which in particular comes in for trenchant criticism from Aristotle.⁸⁰ Speusippus wished to argue that, just because pain is admittedly an evil, it does not follow that pleasure, as its opposite, is a good, since both may be evils, opposed (as extremes) to some third thing (sc. a mean between them), which is a good—even as the greater may be opposed both to the less and to the equal. Aristotle cannot very well object to this argument as such, since it is integral to his own doctrine of virtue as a mean between extremes, but he tries to pick holes in it on the specific ground that, if something is an evil, it is to be avoided (pheukton), and men do not in fact seek to avoid pleasure—quite the contrary—while they do seek to avoid pain.

This is not, however, an argument which Speusippus need feel defeated by. His response could well be that pleasure is indeed a thing to be avoided, even though men in general do not recognize this. What men are seeking is in fact happiness, and they wrongly imagine this to be attainable through pleasure, or even to consist in pleasure. What all too few of them realize, but which all should recognize, is that the state which constitutes true happiness is that in which the soul has reached the perfect balance between pleasure and pain, which is attained by the scientific application of limit (peras) to the unlimited spectrum of more and less which runs from extreme pain to extreme pleasure, either condition constituting a grave disturbance of the organism, and of the soul which presides over it. It is this median state which Speusippus denominates ‘freedom from disturbance’ (aokhlēsia)—and which, it could be argued, such otherwise opposed later philosophers as Zeno and Epicurus both recognized as an ideal as well, Zeno with his concept of ‘joy’ (khara), as opposed to ‘pleasure’, and Epicurus with his concept of ‘katastematic’, as opposed to ‘kinetic’, pleasure.

To a certain extent, this whole argument might seem to turn on a semantic quibble as to what constitutes pleasure, but this is not necessarily the case. Speusippus would not wish to deny, I think, that his preferred state of aokhlēsia is in some sense pleasant; his objection to ‘pleasure’, in its usual connotation in Greek, is rather that it essentially describes a process, and an open‐ended, disorderly one at that, while what he is aiming at, and recommending, is a steady state (hexis). He is to some extent, perhaps, constricted by a lack of suitable terminology, and this leads him to fasten on this rather negative‐sounding term for his ideal state, though without, I am sure, intending anything merely negative. Either of the later terminological distinctions mentioned above, Stoic or Epicurean, would have pleased him, I believe, if they had been brought to his attention.⁸¹

This brings us to a very vexed issue, the identity of ‘the enemies of Philebus’ in Phlb. 44a–d. A great many scholars have taken up positions on this question, many in favour of the identification with Speusippus, notably Wilamowitz,⁸² A. E. Taylor,⁸³ I. Düring,⁸⁴ H.‐J. Krämer,⁸⁵ and most recently Malcolm Schofield;⁸⁶ on the other hand, Leonardo Tarán, in his introduction to the fragments,⁸⁷ proposes to dismiss the identification as based on misunderstandings and inadequate evidence. He begins, however, with what seems to me to be a misconception of his own:

Because of his extreme anti‐hedonism, several scholars have ascribed to Speusippus the doctrine espoused by Philebus' enemies in Plato's homonymous dialogue (44b–d), according to which pleasure is merely the cessation of pain. This conception of pleasure as something purely negative  ⁸⁸ can hardly have been part of Speusippus' doctrine, however, since for him the neutral state between pleasure and pain coincides with the good. This neutral state he must have identified with the freedom from disturbance (aokhlēsia) which for him is the necessary condition of virtue and happiness. Hence, according to Speusippus the virtuous man must free himself from both pain and pleasure; and if pleasure were nothing but the cessation of pain, he would not have considered it an evil, nor would he have thought that the virtuous man must avoid it as such.

It seems to me that Tarán's mistake here is to take Plato too literally, even as he frequently seems to me to take Aristotle too literally in his attacks on Speusippus.⁸⁹ There is certainly nothing negative about Speusippus' doctrine on pleasure, but that does not mean that his uncle's satirical presentation of it might not seek to present it as negative—even as he is doubtless distorting in various ways the hedonist arguments of Eudoxus.

The dispute between Speusippus and Plato, after all, might be viewed as at least partly a semantic one. The mere cessation of pain could, certainly, be presented as a type of pleasure; but it need not be, and Speusippus chose not to present it as such. He wishes to reserve the term ‘pleasure’ to describe one of the equally reprehensible extremes which flank his ideal state, whereas Plato, it would seem, wishes to extend the notion of pleasure to take in also the state which Speusippus sees as transcending both pleasure and pain, and even to give it the title of ‘true’ or ‘pure’ pleasure.⁹⁰

This would not be the only area, we may note, in which Speusippus ventures to oppose what he feels to be a looseness in terminology on his uncle's part. On the other hand, he himself was left with a problem of terminology. It is unfortunate that Speusippus felt constrained to employ a negative term, aokhlēsia, to describe his ideal state, instead of, perhaps, the term to hileōn that Plato himself comes up with in Laws VII,⁹¹ or the later Stoic term khara, but that does not mean that he intended anything negative by it. Indeed, he undoubtedly wished to claim that this state was productive of profound satisfaction, not to say joy. He may, however, for want of a better adjective, have been constrained to describe it as hēdy, ‘pleasant’, though denying that it involved any form of hēdonē, and this gives Plato the chance to score a point off him, which he does not fail to take.⁹²

Let us now, in this connection, take a close look at a crucial passage of the Philebus, and see if we can divine just what sort of position is being criticized there. We may begin at 43c. Socrates has just secured Protarchus' agreement that not all changes in our constitution produce pleasure or pain, but only fairly considerable ones. He continues (43^c8–^d10):⁹³

socr.

: In that case, the form of life I mentioned just now would become a possibility again.

prot.

: What one is that?

socr.

: The one we said would be without distress or enjoyments (kharmonai).

prot.

: Certainly it would.

socr.

: To sum up, then, let us posit three forms of life, one of pleasure, one of distress, and one of neither. What would you say on the subject?

prot.

: Just what you have said, that there are these three lives.

socr.

: Now not being in distress would hardly be the same as enjoying oneself (khairein), would it?

prot.

: Of course not.

socr.

: So when you hear people say that the pleasantest (hēdiston) thing is to live all one's life free from distress, what do you think they are saying?

prot.

: They seem to be saying that not being in distress is pleasant (hēdy).

Let us pause here for a moment. Somebody, or some class of person, is being accused of claiming that the life without pain is the most pleasant (hēdy) of all. If this is Speusippus, then one would like to think that he is being deliberately misrepresented; otherwise he is being notably careless in his language. Speusippus does, admittedly, have a problem, as I have suggested above. He needs some positive value word to characterize his ideal psychic state, but he must be wary of employing words of the same root as hēdonē for that purpose. The fact, however, that Plato also uses khairein repeatedly⁹⁴ in this passage to describe this state might indicate that this was in fact the preferred term of the supporters of this position, who would thus be anticipating the Stoics in their technical use of this term. In that case, though, one might ask why they did not go all the way, and employ an adjective derived from the same root to characterize the preferred state, instead of falling back on hēdys. All one can suggest is that the available adjective, khartos, is somewhat rare and poetical,⁹⁵ and hēdys is much readier to hand.

Let us continue, however (43^e1–44^b3):

socr.

: Take any three things now, say one gold, one silver, one neither, just to have fine names.

prot.

: All right.

socr.

: Could the one that is neither possibly become either gold or silver?'

prot.

: How on earth could it?'

socr.

: Similarly, it would be a mistake for anyone to believe and therefore

to say that the midway life was either pleasant (hēdys) or distressing—at least if we are to be strict.

prot.

: How could it be?

socr.

: Yet we find people who say and believe these things.

prot.

: Certainly.

socr.

: Do they then think that they are enjoying themselves (khairein) on the occasions when they are not in a distressed condition?

prot.

: That's what they say, at any rate.

socr.

: So they believe they are then enjoying themselves (khairein), or they wouldn't say it.

prot.

: Certainly.

socr.

: Yet they are making a false judgement about enjoyment (khairein)—if, that is, enjoyment and lack of distress are two quite different things.

prot.

: But they turned out to be quite different things.

socr.

: We have a choice, then. We could hold, as we did just now, that there are three alternatives, or that there are only two, first distress, which we could say was an evil for men, and secondly release from distress, which, being no more nor less than good,⁹⁶ we should call pleasurable (hēdy).

I think we can see various instances of unfair argumentation in this passage. First of all, to produce gold and silver as examples of two extremes between which there is to be a third thing which is neither of them seems profoundly tendentious. Gold and silver are not, after all, opposites; and they are selected in such a way that any other item in the same class (presumably, of metals) is going to be worse than they are—unless, perhaps, we are to think of the fabled orichalc!

Further, Socrates is attempting to convict the protagonists of this position of wishing both to condemn pleasure and to commend their chosen intermediate state as something pleasant, which it would certainly be inconsistent for them to do, without allowing that they may have wished to claim that their intermediate state was characterized by a condition of mind far superior to pleasure, precisely as being the result of the imposition of limit and order on the disorderly and unlimited spectrum of sensations of which pleasure is, so to speak, one ‘wing’. They may have had difficulty in finding an adjective to describe this (and so may have incautiously fallen back on hēdys), but it rather looks, on Plato's own evidence, as if their preferred verb/noun for it may in fact have been khairein/khara.

This, after all, would by no means be the only time that Plato's Socrates was less than fair to a position of which he disapproved. But let us press on. Socrates now (44^b4–^c2) gets round to identifying a group which he describes as ‘the enemies of Philebus’, who are characterized as being ‘experts in natural science’ (deinoi ta peri physin) and being afflicted with a certain ‘crankiness arising from a not ignoble nature’ (tis dyskhereia physeōs ouk agennous):⁹⁷

prot.

: Why are we raising this question at this stage, Socrates? I don't see what you are getting at.

socr.

: Don't you know the real enemies of Philebus here?

prot.

: Who are you referring to?

socr.

: People with a considerable reputation as experts in the science of nature,⁹⁸ who deny any real existence to pleasures.⁹⁹

prot.

: How do they do that?

Socr.

: According to them, what Philebus and his friends call pleasures are nothing but cases of release from distress (lypōn apophygai).

The question to be addressed is whether the position stated here is in any way compatible with what appears from our other evidence to have been Speusippus' doctrine. On the face of it, one would be compelled to doubt it. After all, Speusippus is not attested as having denied the existence of pleasures; rather, he declared that they were just as much of an evil as pains, which would seem to recognize their existence at least to the same extent.

But then, one asks oneself, could anyone in his right mind have denied the existence of pleasures? Surely what we have here from Plato is the conclusion of a polemical argument against pleasure. The argument, I submit, would go something like this (and a complementary argument could in theory be employed against the existence of pain, except that there was no need to wean people from an irrational attachment to pain!). Any pleasure you care to name, if you analyse it, can be discerned as the result of freeing the organism from some form of distress or other, resulting from a disequilibrium in one direction. What is called ‘pleasure’ is simply the tilting of the balance in the other direction. So pleasures do not have a substantial existence, in that their manifestation is always a by‐product of the removal of pain. You do not experience a sensation of pleasure except in the context of the relief of some painful organic imbalance or other.¹⁰⁰

Now this is a pretty tendentious argument, but at least it is not manifestly absurd, as an outright denial of the existence of pleasure would be; and I think that it is a position that Speusippus could have taken up, consistently with his known views. The great problem that it leaves him with—and this is where, perhaps, his ‘crankiness’ (dyskhereia) comes in—is that he is forced to deny that what Plato would wish to term ‘pure pleasures’ (51aff.), such as those of smell or hearing, where there has been no previous distress, or indeed most pleasures of the mind, are to be described as pleasures at all.

But of course this is just what Speusippus wishes to deny, and it is here that his semantic dispute with his uncle becomes acute. Plato, at Phlb. 52^c1–^d1, actually makes very much the distinction that Speusippus must have made, but he makes it between ‘impure’ and ‘pure’ pleasures, whereas Speusippus would have made it between ‘pleasures’, which he would regard as intrinsically unmeasured and disorderly, and sensations arising out of his ideal state of equilibrium:

So now we have an orderly sorting out of the purified pleasures from what should be called unpurified cases. As a further point we ought to attribute to violent pleasures disorderliness (ametria), and to those that are not, orderliness (emmetria). Those that admit of great degrees and intensity, whether becoming such commonly or only rarely, we should put in the category mentioned earlier of that which is indeterminate (to apeiron genos) and which in varying degrees of more and less permeates both body and soul; the others we should put in the category of ordered things.

Socrates goes on (52^d5 ff.) to argue that the disorderly and indeterminate class of pleasures can properly be declared ‘false’, and only the ordered ones ‘true’, for very much the same reasons that ‘the cranks’ back in 44b–cwere criticized for denying the substantial reality of pleasure. Even there, we may note, the position of the cranks is not absolutely rejected; they are rather treated, ironically, as ‘inspired prophets’ (manteis, 44^c5), who have grasped an intimation of the truth through a certain natural talent, but have not worked it out dialectically. This, it seems to me, is a very suitable form of put‐down for Plato to use when dealing with a bumptious nephew. In fact, their two positions are not that far apart; it is just that Plato does not see the sense of denying the title of ‘pleasure’ to those states of the human organism which he has identified as ‘pure’ pleasures, whereas for Speusippus the essential distinction between these states and what he wants to designate ‘pleasures’ is precisely that they are states, and the others are processes, or motions, admitting of indefiniteness and of ‘more and less’.

Plato's contention is that the position of the ‘cranks’, while doing them credit in a way, has not been properly thought out. It seems to me, on the other hand, that there is a certain amount to be said for it, even if it presents difficulties. After all, if we bear in mind Speusippus' Pythagoreanizing world‐view (which is not, indeed, very different from that presented in the Philebus), it is important that, in the sphere of ethics, one postulates the existence of perfect states, representing the imposition of peras on the disorderly substratum of apeiria, and that it is such states that generate eudaimonia.  ¹⁰¹ The problem of how to describe the by‐products of such states, which may include such experiences as the enjoyment of the smell of roses, the sound of birdsong, or of a string quartet of Beethoven, as well as of philosophizing, is something that he may not have quite solved, though it seems possible, as we have seen, that he made some use of the verb khairein in this connection.

To complete our exegesis of Phlb. 44b–d, however: we may derive from it (or rather from its continuation, 44e–45d) a further polemical argument against pleasure which may well have been advanced by Speusippus. Socrates, at any rate, with his usual irony, thanks the ‘cranks’ warmly for supplying him with it.

The argument goes as follows. If we want to get a clear view of the nature of a given thing, we should look at it in its most extreme or unadulterated form. In the case of hardness, for example, we should examine the hardest things we can find, if we wish to acquire an accurate idea of what hardness is. In the case of pleasure, then, we look for the most extreme forms of pleasure, if we wish properly to comprehend the nature of pleasure. But then we are led to admit that people suffering from illness, either physical or mental—the argument glosses over the question whether all illnesses qualify in this regard, or only certain ones—experience more pleasure in more extreme forms (as a relief from their various pains or distresses) than healthy people. Thus it is argued that it is the most unbalanced natures that experience the most extreme pleasures. And so it is indicated (I hesitate to say proved) that pleasure is essentially connected with imbalance in the organism, and is thus the antithesis of a desirable state.

I see no reason not to accept that Plato is in fact borrowing this argument from Speusippus, and using it for his own purposes. He makes Socrates thank the ‘cranks’ for it, as I say. If anything, it brings home to us how close their positions really were. This does not, however, mean that Plato was any the less displeased with his nephew for taking up a position on his ‘right’, so to speak, on such a sensitive question as the status of pleasure. Mainly, though, he seems to have regarded this super‐austerity of Speusippus as evidence of deficiency in logical rigour. The sensations of the virtuous and self‐controlled have enough in common with those of the dissolute and intemperate to merit being given the same generic name. A more correct way of evaluating them is to make the distinction that he works out in the latter part of the dialogue between ‘adulterated’ and ‘pure’, ‘false’ and ‘true’ pleasures, rather than being left with a class of sensations for which no proper name at all had been developed. This, after all, is just the role that the ‘heavenly tradition’ set out in Phlb. 16–19 is meant to fulfil, making the correct distinctions within the previously vague and amorphous concept denominated ‘pleasure’.

It may after all turn out, though, that Speusippus has the best of this argument. The Stoics were second to none in logical rigour, and they, it seems to me, sided in retrospect with Speusippus. The Stoic theory of eupatheiai, or ‘equable states’, after all, is precisely developed to provide a set of sensations, including pleasurable ones, which are appropriate to a wise man who is quite free from passions, and that is the sort of Pythagorean sage that Speusippus seems to be envisaging. For the Stoics, the experiences of the sage are not to be assimilated to those of the vulgar, despite superficial resemblances between them, and that is surely very much the position of Speusippus before them.

There is not much other evidence remaining as to the ethical position of Speusippus, but a passage of Plutarch, On Common Conceptions, Against the Stoics 1065a (= Fr. 79 Tarán), provides us with the information that Speusippus (and Xenocrates after him) accepted that such things as health and wealth were not ‘indifferent’ in the Stoic sense, but could be ‘goods’ (even though minor goods, compared to those of the soul) if properly employed.¹⁰² We may reasonably conclude, I think, that both Speusippus and Xenocrates would have assented to Plato's views as expressed in Laws II 661a–c (cf. also V 728d–729a), to the effect that those things popularly regarded as goods, such as health, beauty, and wealth, are goods for the good man, but potentially great evils for the rest of mankind; and this presumably remained the official position of the Old Academy. Again, the difference with the Stoics is very largely a semantic one; once one recognizes, as did at least the later Stoics, that, among the ‘indifferents’, there are some to be ‘preferred’ and others to be ‘dispreferred’, one is very much back at the Old Academic position. But such terminological niceties are the very stuff of inter‐school disputes.

Let us turn to look now at some aspects of Speusippus' epistemology, and then at what we know of his contributions to logic.

We may take our start from a report of Sextus Empiricus (Adv. Math. VII 145–6 = Fr. 75 Tarán) as to Speusippus' doctrine of the criterion of truth, or of knowledge.¹⁰³ This is sandwiched between a report of Plato's doctrine, and of that of Xenocrates (to which we will come in due course), and is most interesting. Its provenance I would discern (despite the characteristically sceptical strictures of Tarán 1981: 431–2) as being in all probability a work of Antiochus of Ascalon,¹⁰⁴ and this will colour the technical terminology employed, but there is little reason to doubt the basic accuracy of the account.¹⁰⁵ What we find is an account of what Speusippus terms ‘cognitive sense‐perception’ (epistēmonikē aisthēsis), which is quite distinctive, and deserves some discussion. But first let us look at the passage in full:

Speusippus' view was that, since there are things which are sense‐perceptible and others which are intelligible, of those that are intelligible the criterion is cognitive reason (epistēmonikos logos), while of sensible things it is cognitive sense‐perception. And cognitive sense‐perception he conceived to be that which participates in the truth which accords with reason. To take an example: the fingers of the flute‐player or harper possess an artistic activity (tekhnikē energeia), which is, however, not brought to fruition primarily (proēgoumenōs)¹⁰⁶ through the fingers themselves, but is fully developed as a result of training under the co‐operative guidance of reasoning (logismos); and the sense‐perception of the musician, while it possesses an activity capable of grasping the harmonious and the non‐harmonious, nevertheless is not self‐produced (autophyēs) but is acquired by reason. Even so, cognitive sense‐perception naturally derives from reason the cognitive experience which it shares, and which leads to the unerring discrimination (aplanēs diagnōsis) of its proper objects (hypokeimena).

We need to probe carefully the theory being developed here, particularly as regards the status being accorded to the rational cognition of the physical world, a topic with which Speusippus is plainly much concerned. If we think of the hands of the skilled pianist, for example, flashing across the keys, we can say that there is a purely physical facility in the fingers, built up by fairly mechanical practice, which is certainly necessary for the skill of the pianist to be realized (and which the onset of arthritis, say, would impede or destroy). This would presumably answer to the basic physical efficiency of the sense‐organs in receiving their proper objects. But what makes the movement of the fingers purposeful and truly artistic is the intellectual power of the pianist, directing them in obedience to his conception of the form of the composition. Even so, our perception of the physical world, our discernment and identification of objects (pragmata), groups of objects, and situations within it, is a function, not of ‘raw’ aisthēsis, but rather of logos directing aisthēsis—that is to say, epistēmonikē aisthēsis.  ¹⁰⁷

Not unconnected with this doctrine, I think, is Speusippus' remarkable claim that knowledge of any given physical object requires knowledge of its differentiae in respect of everything else. This doctrine is communicated to us, critically, by Aristotle at An. Post. 97^a6–22 (= Fr. 63 Tarán), but as usual he criticizes Speusippus from the perspective of his own philosophical system, not from that of Speusippus:

There is no necessity for someone practising definition and division to know the totality of existents. And yet some people declare that it is impossible to know the differentiae between each thing and the rest without knowing each thing severally. [The argument goes that] without knowledge of the differentiae it is not possible to know each thing; for if something is not differentiated¹⁰⁸ from something else, it is the same as that thing, and if it is differentiated from it, it is other than it.

First of all, this is false; for a thing is not ‘other’ in respect of every differentia. There are many differentiae, after all, which belong to things within the same species, but not essentially nor in themselves. And then, when one fastens on a pair of contraries and the differentia which distinguishes them, and assumes that everything falls on this side or that [of the dichotomy], and assumes that what is being sought is on one side or other, and that this is the object of one's knowledge, it makes no difference whether one knows or doesn't know how many other things the differentiae are predicated of. For it is obvious that if, proceeding along these lines, one comes to what is no longer distinguished by a differentia, one will then have the account of the essence (logos tēs ousias  ¹⁰⁹).

Here Aristotle does seem to be on strong ground in his criticisms, but, once again, it is worthwhile, I think, to try to unravel what may have been Speusippus' motivation in making such a remarkable assertion.¹¹⁰ First, I think, we must recognize that Speusippus' purposes in practising definition are significantly different from those of Aristotle. Aristotle has a biologist's interest in classifying objects in the physical world into their natural kinds; furthermore, for him it is the individual combination of form and matter that represents primary substance, and only essential differentiae are necessary to establish this. For Speusippus, the true realities are numbers and geometrical figures, and it is only these that are truly knowable. Physical essences he seems to have regarded as analogous to monads or points, being merely the focus of all the relations that make each of them different from everything else. When he declares that, to attain knowledge of any one of these entities, one would have to be able to give a full account of the relations (of both similarity and difference) in which it stands to every other physical object, it seems best to suppose that he is just stating an impractical ideal, to make clear why it is impossible in practice to have epistēmē of physical objects. In the case of the number seven, after all, or of the equilateral triangle, it is not difficult to enumerate the totality of relationships of similarity and difference in which it stands with other entities of its class;¹¹¹ but in the case of the average plant, animal, or other natural object the task is much more complex. Yet Speusippus is plainly not advancing this difficulty in order to argue for complete scepticism as regards the physical world. We know, in fact, that he was a most industrious classifier of natural kinds, author of two works on the subject, the Homoia (‘Similarities’), in ten books, and Divisions and Hypotheses in  relation to the Similarities, and (mainly thanks to Athenaeus) we have a number of brief reports of classifications that he made, chiefly in Book II of the Homoia.  ¹¹²

Aristotle plainly studied this work fairly closely, though mainly to disagree with it. He criticizes Speusippus (a) for employing a rigid dichotomic scheme in his divisions even when this is not suitable; (b) for using privative terms (such as ‘wingless’, ‘bloodless’) in the course of these dichotomies, since these cannot properly provide a positive characterization; and (c) for making inapposite distinctions between natural species on the basis of such non‐essential features as habitat (as when he links water‐birds with fish on the basis that both are ‘water‐animals’, as opposed to ‘land‐animals’, under which he included land‐based birds).¹¹³

Speusippus' procedure, as I say, does seem on the evidence available to us to have been considerably less scientific than that of Aristotle, and the rigid practice of dichotomy does have obvious drawbacks, but we have to reflect, as I say, that Speusippus' purpose in engaging in this classificatory activity was probably very different from that of Aristotle. The actual title of his main work, Homoia, as well as what Aristotle tells us about his procedure, that he was enquiring in each case into ‘in what respects a given thing is similar, and in what respects it is different’ (ti tauton kai ti heteron),¹¹⁴ gives us some indication, I think, as to what his real purpose may have been.¹¹⁵ We may recall that, in the Timaeus (37a–c), where the function of the soul in cognizing the physical world (as well as the intelligible world) is being described, its elements of Identity and Difference are brought into play, and when its ‘circle of otherness’ (ho tou thaterou kyklos) is functioning correctly (orthos ōn), it conveys back accurate reports to the whole soul, and this results in ‘opinions and beliefs that are firm and true’ (doxai kai pisteis bebaioi kai alētheis). It seems to me that all of Speusippus' considerable efforts in the area of classification of similarities and differences had as their true purpose the purification of the soul, by helping its ‘circle of otherness’ to be orthos, and not the sort of scientific sorting‐out of natural kinds which was so important to Aristotle; and his remark about the impossibility of ever arriving at a full enumeration of the diaphorai which distinguished any physical object from all others, which so incensed Aristotle, was simply a reassertion of what we are repeatedly reminded of by Timaeus in his account, that the study of the physical world can only ever be an eikōs logos.  ¹¹⁶

A similar issue concerning the relations between the cognition of the physical and intelligible worlds seems to arise out of a view of Speusippus', reported by Proclus in his Commentary on the First Book of Euclid's Elements,  ¹¹⁷ on the distinction between a problem and a theorem, in connection with the theory of mathematics, and specifically of geometry. The basis of his complaint is that he takes the term problēma to connote ‘the bringing into being of something not previously existing’, whereas theōrēma is more proper to the study of objects which exist eternally. It is worth quoting the passage in full:

Again, the propositions that follow from the first principles he [sc. Euclid] divides into problems and theorems, the former including the construction of figures, the division of them into sections, subtractions from and additions to them, and in general the characters that result from such procedures, and the latter concerned with demonstrating inherent properties belonging to each figure. Just as the productive sciences have some theory in them, so the theoretical ones take on problems in a way analogous to production.

Some of the ancients, however, such as Speusippus and Amphinomus¹¹⁸ and their followers, insisted on calling all propositions ‘theorems’, considering ‘theorems’ to be a more appropriate designation than ‘problems’ for the objects of the theoretical sciences, specifically because these sciences deal with eternal things. There is no coming to be among eternals, and hence a problem has no place here, proposing as it does to bring into being or to make something not previously existing—such as to construct an equilateral triangle, or to describe a square when a straight line is given, or to place a straight line through a given point. Thus it is better, according to them, to say that all these objects exist and that we look on our constructions of them not as making, but as understanding them, taking eternal things as if they were in the process of coming to be. Hence we can say that all propositions have a theoretical and not a practical import.

This position of Speusippus seems to have something of a polemical edge to it. After all, it is not at all clear that the term problēma must denote something that is being postulated as being brought into being. Plato, indeed, is quite prepared to use it in Republic VII (530b) to refer the theorems of geometry and mathematical astronomy. The truth seems to be, as Proclus lets us know just below, that Speusippus' contemporary Menaechmus, a pupil of Eudoxus, a distinguished mathematician and associate (at least) of the Academy, took the position that all mathematical enquiries may be described as problēmata, and that Speusippus is here indulging in some intra‐school oneupmanship with a follower of his former rival—as well as once more ‘correcting’ his uncle.

Nevertheless, there is possibly a serious philosophical point behind this terminological squabble. As we have seen, Speusippus is insistent on the non‐literal, non‐temporal interpretation of the Timaeus, and in that connection his point is that the apparent description of temporal creation is to be regarded as analogous to the diagrams of the geometers, who appear to generate, say, a figure from a line, but only do this for sake of ‘clarity of exposition’.¹¹⁹ This would be the general principle of which that is a particular application.

Another significant passage from Proclus' commentary (p. 179, 12–22 = Fr. 73 Tarán) reports Speusippus making a distinction between our knowledge of principles (arkhai) and of those propositions which derive from them. First principles are characterized by simplicity (haplotēs), indemonstrability (to anapodeikton), and self‐evidence (to autopiston),¹²⁰ and the mode of their comprehension is thus to be distinguished from that of what follows from them. Once again, the passage is worth quoting:

Principles must always be superior to their consequences in being simple, indemonstrable, and self‐evident. In general, says Speusippus, in the hunt¹²¹ for knowledge in which our understanding is engaged we put forward some things and prepare them for use in later enquiry without having made any elaborate excursion, and our mind has a clearer contact with them than sight has with visual objects; but others it is unable to grasp immediately and therefore advances on them step by step and endeavours to capture them by their consequences. For example, drawing a straight line from a point to a point is something our thought grasps as obvious and easy, for by following the uniform flowing of the point and by proceeding without deviation more to one side than to another, it reaches the other point.¹²² Again, if one of the two ends of a straight line is stationary, the other end moving around it describes a circle without difficulty.¹²³ But if we should wish to draw a one‐turn spiral (helix monostrophos), we need a rather complicated device, for the spiral is generated by a complex of motions; and to construct an equilateral triangle will also require a special method for constructing a triangle. Geometrical intelligence will tell me that, if I think of a straight line one end of which is fixed and the other revolving about it, while a point is moving along it from the stationary end, I describe a monostrophic spiral; for when the end of the line which describes a circle has reached its starting point at the same time as the point completes its movement along the line, they coincide and make such a spiral. And again, if I describe two equal circles and join their point of intersection with the centres of the circles and draw a straight line from one centre to the other, I shall have an equilateral triangle. It is far from true, therefore, that these things can be done at first glance and by simple reflection; we should be content to follow the procedures by which the figures are constructed.

In what work Speusippus may have discussed this topic we cannot be certain, but the Mathematikos seems a good candidate. One can observe here a striking similarity in subject matter to the last chapter of Aristotle's Posterior Analytics (II 19), where Aristotle raises the question of how we can come to know first principles (arkhai), since they cannot be arrived at by demonstration (apodeixis), such as leads to scientific knowledge (epistēmē). Aristotle concludes that they can only be known by immediate intuition (nous). He doubtless had Speusippus' discussion in mind.

Lastly, in the area of logic, we may take note of an area wherein Aristotle has no great quarrel with Speusippus, but where he seems to be less than straightforward in acknowledging his debt to him, preferring rather to attack him on relatively minor points of dispute. This is on the subject of the division of names (onomata). Aristotle's own distinction at the beginning of the Categories between homonyms and synonyms is well known—though this is in fact a distinction between types of thing rather than of words¹²⁴—but it is less well known that he is here adapting a series of distinctions made already by Speusippus. We know this from Simplicius' commentary ad loc.,¹²⁵ who is himself reporting the testimony of the earlier Peripatetic commentator Boethus (1st cent. bc). According to Boethus, Speusippus divided names dichotomously, first into tautōnyma and heterōnyma (‘identical words’ and ‘different words’); then tautōnyma into homōnyma and synōnyma, by which he means respectively identical words which have different and unrelated meanings (e.g kuōn, standing for ‘dog’, ‘dog‐fish’, and even ‘dog‐star’), and identical words with identical meanings;¹²⁶ and then heterōnyma into idiōs heterōnyma (‘strictly different’, that is to say, quite unconnected, such as ‘grammar’, ‘man’, and ‘wood’) and polyōnyma (different words with the same meaning, such as aor, xiphos, phasganon, all meaning ‘sword’¹²⁷). Lastly (though here Boethus or Simplicius seems to have slipped up slightly in his reporting), he seems to have divided polyōnyma into idiōs polyōnyma and parōnyma (that is to say, words different but related to each other by derivation or other contiguity).

Aristotle does not explicitly criticize Speusippus' division of names (though he does so implicitly at the beginning of the Categories, by giving the designation of synōnyma to members of different species, such as ‘man’ and ‘ox’, which have the same generic name, ‘animal’), but he does attack a distinction which Speusippus seems to have made on the basis of this division. Speusippus took an interest in the analysis of the sources of ambiguity.¹²⁸ Speusippus, we can gather from Aristotle, Soph. El. 170^b12–171^b2 (= Fr. 69a Tarán),¹²⁹ divided all arguments into those addressed to the word (pros to onoma) and those addressed to the concept (pros tēn dianoian),¹³⁰ and in connection with the former kind of arguments he maintained that all sophistical refutations are based upon ambiguity (para to ditton). He therefore devoted himself to investigating ambiguity as a means of avoiding and refuting fallacies.

Aristotle (ibid.) rejects this classification of Speusippus' (cf. also SE 177^b7–9 = Fr. 69b Tarán), and denies that all sophistical refutations are based upon ambiguity:

No real distinction, such as some people propose, exists between arguments used against the word and those used against the concept; for it is absurd to suppose that some arguments are used against the word and others against the concept, and not the same in both cases. For what is failure to use the argument against the concept except what happens when a man does not apply the term in the meaning about which the man questioned thought that he was being questioned when he made the concession? And this is equivalent to using it against the word; whereas to use it against the concept is to apply it to the sense about which the man was thinking when he made the concession. If, then, when one word has more than one meaning, both the questioner and the man questioned were to think that it had only one meaning—as, for example, ‘being’ or ‘one’ have several meanings, but both the answerer answers and the questioner puts his question on the supposition that there is only one meaning and that the argument is that ‘all things are one’—will the argument have been directed against the word and not rather against the concept of the man questioned? (trans. Forster, adapted).

Of the examples mentioned here, ‘one’ is shrewdly chosen, if Speusippus is being aimed at, since, as we have seen, he had various uses for the word, and thus could be accused of ambiguity in that connection. In general, though, as Tarán very reasonably argues, Aristotle is engaged in point‐scoring by disregarding the nature of Speusippus' distinction between arguing ‘against the word’ and ‘against the concept’, and addressing it on his own terms. Speusippus seems really to have been concerned with a distinction between a disputation where both participants have gained an accurate conception of what is being argued about, and one in which verbal ambiguities are being exploited. He would not have been oblivious to the fact that there were various kinds of ambiguity, but he wants to claim that all sophistical refutations are based on ambiguity of one sort or another. The distinctions that Aristotle makes are doubtless more sophisticated than those of Speusippus, but at the same time he obscures the very considerable amount that he learned from his senior colleague.

This survey has, I hope, revealed a philosopher of some idiosyncrasy of viewpoint, perhaps, but by no means lacking in coherence or breadth of vision. Speusippus was certainly not the equal of Aristotle as an original mind, but neither was he an entirely unworthy heir to Plato's physical and intellectual establishment. He is, as we have seen, by no means a slavish follower of the doctrines of his uncle; indeed a number of his innovations in doctrine were not to find an echo in the Platonic tradition before the Neopythagoreans of the first and second centuries ad. It is only with his successor, indeed, that Platonism begins to settle down to some sort of ‘orthodoxy’—although even Xenocrates, as we shall see, is by no means devoid of original thought.

If one had to select the most original, and ultimately the most influential, aspect of Speusippus' theorizing, the choice would have to fall on his doctrine of the nature of the first principle—and that did not come into its own again (and even then on rather different premisses) until the time of Plotinus. But other aspects of his thought, such as his doctrine on happiness and the nature of pleasure, his theory of the nature and role of the World‐Soul, and a number of his logical formulations, seem to have proved stimulating both to his rival Aristotle and, later, to the founders of Stoicism.