Pythagoreans and Eleatics

Pythagoreans and Eleatics Edward Hussey PYTHAGORAS AND THE EARLY PYTHAGOREANS Pythagoras, a native of Samos, emigrated to southern Italy around 520, and seems to have established himself in the city of Croton. There he founded a society of people sharing his beliefs and way of life. This spread through the Greek cities of southern Italy and Sicily, acquiring political as well as intellectual influence. Some time after his death, the original society broke up and its continuity was lost; yet groups of self-styled ‘Pythagoreans’ appeared repeatedly thereafter. Palpably reliable evidence about early Pythagorean activities is so scanty that some initial scepticism is in order about Pythagoras as a philosopher, or as a ‘natural philosopher’ in Ionian style.¹ Sporadic early reports depict Pythagoras as primarily a magician and miracle-worker; and, on the theoretical side, a collector and expositor in dogmatic style, rather than a creator or investigator. It is clear that some doctrines later seen as ‘Pythagorean’ were already current, around the same time, in the theological and cosmological poems attributed to Orpheus. Plato barely mentions Pythagoras by name; but incorporates into some of his myths material which is likely to be genuinely Pythagorean. After Plato, philosophically-inspired reconstructions of Pythagoras begin to appear, in which he is represented as the head of a regular school, promoting research into philosophy and the mathematical sciences; or as an enlightened statesman and instructor for political life.² At best, even when based on good sources, these fourth-century accounts (which themselves survive only in later reports) are more or less anachronistic idealizations. Still less reliance can be placed in the great mass of later statements about Pythagoras and his followers. Indirectly, the fact that certain later fifth-century thinkers were called ‘Pythagoreans’ (see below) gives some indication of what theoretical interests were then attributed to Pythagoras. The cosmology of Parmenides (see below) and the poems of Empedocles have a substratum of ideas that may be suspected to be Pythagorean in inspiration. All in all, there is a body of general ideas, appearing by the mid fifth century and reasonably firmly associated with Pythagoras, which was to be influential in a programmatic way throughout the century, and on Plato, above all, in the fourth. These ideas may be grouped under the headings of ‘metempsychosis’ and ‘mathematics’. ‘Metempsychosis’ and the Self ‘Metempsychosis’ was the doctrine of the repeated incarnations of an immortal ‘soul’ or self, in human or animal or plant bodies. Centred on that doctrine, and more or less closely tied to it were various ideas, not necessarily clearly distinguished by Pythagoras himself. (1) One was the radical redefinition of the self which the doctrine involved: a new belief about what we human beings really are. The implication is that we are not, as in traditional Greek belief, mortal beings, with at best a shadowy afterlife in Hades, but that we are truly immortal, perhaps fallen gods. Our real selves have always existed and will always exist. And the heritage which they have lost, but may recover, is a divine, paradise-like existence. (2) Connected with that is the belief that we are not at home in the body, in fact that this life and all incarnations are really punishments, or at best periods of rehabilitation. It follows that we are not here primarily to enjoy ourselves. It does not necessarily follow that the body is intrinsically evil, or that the ordinary kinds of enjoyment are bad in themselves. That extreme puritanical conclusion may have been drawn by some, but probably not by Pythagoras himself. (3) Pythagoras saw the world as sharply and systematically polarized between good and evil. The real role of the self is to be a moral agent, to participate in the moral struggle; and it is rewarded and punished accordingly. A systematic cosmological dualism, associating all aspects of the world with the good—evil polarity, seems also to have been characteristically Pythagorean. It may be this dualism which accounts (psychologically at least) for the doctrine of the cyclical recurrence of events. Given a systematic dualism of good and evil, good must triumph, but evil cannot be abolished. The simplest solution is to suppose that there are cosmic cycles: at the end of each cycle, all beings have been ‘saved’, and good triumphs; but then the moral fall starts all over again. (4) Yet another aspect is the kinship of all living things. What precepts, if any, Pythagoras deduced from this about human behaviour to other animals, is obscure.³ Mathematics and the Importance of Abstract Structure Another leading idea of Pythagoras was that of the key importance of mathematical structures in the universe. Pythagoras himself was no creative mathematician; there is no reliable evidence that he proved any mathematical theorems at all (not even ‘Pythagoras’ theorem’). The evidence suggests rather that the Pythagoreans’ focus was on a speculative numerology applied to the cosmos. (For example, the dualism of good and evil was paralleled, and perhaps meant to be explained, by the dualism of odd and even numbers; and so on.) But from here the thought emerges, first, that mathematics is not just a useful practical device; that it reveals an abstract structure in things; and secondly that this abstract structure may be the key to the essential nature of things. It is through these ideas that Pythagoras became the midwife of pure mathematics, which began to develop from now on; and indeed the founder of the whole mathematical side of scientific theory. PARMENIDES The Poem of Parmenides Parmenides was a citizen of the Greek city of Elea in southern Italy. His philosophical activity belongs to the first half of the fifth century. He expounded his thoughts in a poem, using Homeric hexameter verses. Verse for public recitation was then still a natural medium for diffusing ideas; yet the ‘natural philosophers’ of the sixth century had chosen prose, to show their rejection of the authority of the poets, and their closeness to ordinary experience. Parmenides’ choice of hexameter verse may imply in its turn a rejection of the natural philosophers. The poem begins with a first-person narrative of a journey. Accompanied by the daughters of the Sun, the narrator rides in a chariot into remote regions, to reach ‘the gate of the paths of night and day’. Passing through, he is welcomed by a goddess, who promises that he is to ‘find out everything’. She goes on to fulfil the promise, in an exposition which constitutes the whole of the rest of the poem. Over one hundred verses of the poem survive, including all of the introductory narration and probably almost all of the first and fundamental part of the goddess’s exposition. Together with comments of Plato, Aristotle and others, this is a fine corpus of first-rate evidence, the survival of which is due principally to Simplicius, the sixth-century Neoplatonist commentator on Aristotle.⁴ Yet controversy dogs almost every part of Parmenides’ thinking, for a conjunction of reasons. First, there are gaps in our information at certain crucial points. Next, Parmenides’ language is often obscure, in spite of his evident striving for maximal clarity. The constraints imposed, by the metre and vocabulary of epic verse, on the exposition of a subject-matter for which they were never designed, are bad enough. Then there is the problem of supplying whatever, in the course of his exposition, Parmenides left to be understood. Finally, his thought is itself novel and complex. Any translation, therefore, and any overall reconstruction of Parmenides, including the one now to be outlined, cannot but be highly controversial at many points.⁵ The Promise of the Goddess Central to the understanding of Parmenides is the promise made by the goddess: It is necessary that you find out everything: both the unmoving heart of well-rounded reality [alētheiē], and the opinions of mortals, in which there is no real guarantee of truth—but still, these things too you shall learn, how [or: since] it had to be that opinions should reputably be, all of them going through everything. (DK 28 B 1.28–32) The division of the objects of discovery into two determines the structure of the rest of the poem. It rests on the distinction (explicit since Xenophanes at least) between what can and what cannot be certainly known. The first pan, concerned with alētheiē, will contain only certainties. The second part, of which the truth cannot be guaranteed, will contain ‘opinions of mortals’. As with Xenophanes, there are better and worse opinions: those to be revealed are not any old opinions, but ones which enjoy the status of being ‘reputable’, and which form a complete system.⁶ If we leave on one side, for the moment, the ‘opinions’ and what is here said about them,⁷ the next fundamental question is the meaning of the word alētheiē. In English, it is usually translated by ‘truth’, to which it seems to correspond in the spread of its early usage. The adjective alēthēs, from which it is formed, has much the same spread as ‘true’ (covering the areas indicated by the words ‘truthful’, ‘accurate’, ‘real’, and ‘genuine’; though not that of ‘faithful’). But in Parmenides the translation ‘reality’ for Parmenides’ alētheiē must be insisted on, in order to bring out the essential point: what is referred to here is not anything (words, speech, thoughts) that is or makes a true statement; it is what the true statement is about, and is guaranteed by: the underlying actual state of things, the reality. So, later on, the goddess marks the end of the first part by saying, ‘At this point I end for you my trusty tale and thought concerning alētheiē.’⁸ While ‘reality’ may be taken as the closest word to the intended primary meaning here, it is also true that alētheiē, as in Homer, carries implications about the certainty of what is said, and of the correctness and accuracy of the method by which it is found. Parmenides wants to insist on these points too; which he does, here and elsewhere, primarily by words indicating trustworthiness and its guarantees (pistos, pistis, peitbō). The goddess promises not only insight into some reality, but a guarantee of the truth of the insight. This reality is ‘well-rounded’, presumably because it forms a satisfactorily coherent and closed system; and it has an ‘unmoving heart’, presumably because at least in essentials it is not subject to change. Both of these thoughts reappear significantly later. The Choice of Ways The narrator’s ‘finding out’ of reality is represented as a matter of simply listening to the goddess. Yet there are hints that his was an active pursuit of the truth; it was his own desire that started him off. The metaphor of travel, and the implication of active pursuit in ‘finding out’, are now carried further. There is talk of ‘ways of enquiry’; the listener is warned off from two of these ‘ways’, and told of ‘signs’ that appear in the course of the third. The exposition envisages an active rethinking, by the listener, of the course of Parmenides’ thinking. Why the active participation of the listener is needed becomes clear from what follows. The exposition concerning reality is in the form of a deductive argument, which one cannot properly follow and grasp, without recreating it in the movement of one’s own mind. The ‘ways of enquiry’ are ‘lines’ (as we say) of argument, each following deductively from its own initial premiss, by the mention of which it is, naturally enough, identified in the exposition. Rigorous deductive arguments were possibly already in use in mathematics; but they must have been novel to most of Parmenides’ contemporaries. Hence the efforts Parmenides makes, using the metaphor of the ‘ways’, to keep the course of the arguments, their interrelation and their overall effect, absolutely clear. Come then, I will tell you (and you, listen and take in the story!), which ways of enquiry alone are to be thought: the one, that it is and cannot not be, is the path of conviction, for it follows along after reality; the other, that it is not and that it is necessary that it is not—this track, I tell you, is utterly unconvincing… (DK 28 B 2.1–6) This presents, as a starting-point, a choice between two such ways, which are mutually exclusive. Clearly, though, they are not jointly exhaustive, since there might also be ways involving unrealized possibilities (‘it is, but might possibly not be’, ‘it is not, but might possibly be’). In fact, in the sequel, Parmenides will present only one more way, the ‘way of mortals’, which, as stated, is evidently self-contradictory. The two named here are apparently the only ones that ‘are to be thought’; and, of these, one is to be rejected as false. What is going on here seems to be as follows. Parmenides holds (on what grounds, remains to be examined) that to speak of unrealized possibilities involves a contradiction. Hence, taking ‘is it?’ as the basic question at issue, there can be only two premisses to be considered: ‘necessarily, it is’, and ‘necessarily, it is not’. The ‘way of mortals’, which says that ‘it is and it is not’, is self-evidently contradictory; it is therefore not ‘to be considered’. None the less, it is mentioned later, and the reader is expressly cautioned against it, because it is a popular and appealing way. Of the two ways worth consideration, the second, which says ‘necessarily, it is not’, also turns out to involve a contradiction, but this is not evident at the start; it has to be shown by argument. Once that has been done, the way that says ‘necessarily, it is’ is the only remaining possibility. Accordingly, it is accepted as true by elimination, and its consequences examined. What then is meant here by ‘it is’ and ‘it is not’? First, what is ‘it’? In the Greek, the verb esti stands alone, as Greek verbs can, without even a pronoun to function as the grammatical subject. But unless Parmenides is making some radical and improbable departure from ordinary practice, an intended subject of discourse, of which ‘is’ and ‘is not’ are here said, must have been meant to be readily supplied from the context. Unfortunately for us, the original context is now partly missing. Between the promise of the goddess and the statement of the two ways, some now lost stretch of text, probably not long, once stood. None the less, what remains is sufficient for near-certainty as to the intended subject. The ways are ‘ways of enquiry’. An enquiry, then, is presupposed as being already afoot. What that enquiry is concerned with, is likely to be what the first part of the goddess’s promise is concerned with: reality. It is true that the word alētheiē nowhere appears subsequently in the subject place attached to the verb esti. In the exploration of the true way that says ‘it is’, the subject of ‘is’ appears sometimes, cloaked in the unspecific designation (to) eon, ‘that which is’. This phrase, though, can be taken without artificiality as another, and metrically more convenient, way of referring to alētheiē. (So taken, it involves a metaphysical pun: see below on the meanings of the verb einai.) This conclusion, that alētheiē, in the sense of ‘reality’, is the intended subject, is central to the interpretation of Parmenides to be presented here.⁹ It has been reached by a simple yet powerful argument. It has yet to be subjected, though, to a series of severe tests. A reconstruction of Parmenides deserves acceptance only if it makes convincing sense of the whole of the surviving evidence. The first test arises immediately. Can one make sense of an initial choice between ‘necessarily, reality is’ and ‘necessarily, reality is not’? At this point, we must also ask about the possible meanings of the verb einai (‘be’). In general, it seems to make sense, whatever x may be, if one is making an enquiry into x, to start by asking ‘is there any such thing as x or not?’ The normal usage of the verb einai easily covers such a sense of ‘is’. In launching an enquiry into alētheiē, understood in extension from Homeric usage in a ‘summed sense’, as what would be jointly indicated by all true statements, Parmenides is in effect asking, sceptically, ‘why do we have to suppose that there is any such thing in the first place?’ Since this entirely normal and familiar use of einai fits the context so well, there is no need at the outset to look for more exotic possibilities.¹⁰ Later, though, when the subject of discourse is referred to as ‘that which is’ (to eon), a different use of the verb bears the logical weight. Another common use of einai is that in which it means (said of possible states of affairs) ‘obtain, be the case’. If alētheiē is thought of as a ‘summed state of affairs’, then to say that there actually exists such a thing is just the same as to say that it is the case. Parmenides’ philosophical starting-point looks, in this light, rather like that of Descartes. Both start with a philosophical enquirer, an apparently isolated mind, trying to establish what it can know with absolute certainty. Parmenides approaches the problem via the concept of alētheiē, the reality that would have to underwrite any knowledge. What is next to be examined is his argument to establish that there must be such a reality. It is here that his further initial presuppositions, if any, are to be found. This is the argument that rejects the way that says ‘it is not’. The Rejection of ‘it is not’ The passage in which Parmenides justifies the rejection of the second way is probably not preserved entire. There survive, in fact, only the beginning ((A) below) and the end, plus a single sentence presumably belonging closely with it ((B) below). (A)…this track, I tell you, is utterly unconvincing [or: undiscoverable]; for you would not recognize [or; become aware of] what is not (for that cannot be done), nor would you point it out. (DK 28 B 2.6–8) The claim is that the way ‘it is not’ must be rejected. The verbs on which the argumentative weight is thrown, are, in the aorist forms used here, common Homeric words for ‘recognize’ and ‘point out’; they are cognitive ‘success verbs’. Their objects can be either ordinary individuals or ‘that’-clauses. So it is necessarily true that (1) ‘you would not recognize [to be the case: i.e. get knowledge of], or point out [as being the case: i.e. show, demonstrate], what is not’. The natural way to expand (1) into a relevant argument is as follows. If there is no such thing as reality, then no-one can recognize it, nor point it out. In that case there can be no knowledge (if knowledge requires recognition of reality) and no communication of knowledge. This will suffice to reject ‘it is not’, provided two further premisses are available: (2) that knowledge involves or consists in awareness of reality, and communication of knowledge involves or consists in the pointing-out of reality; (3) that knowledge and its communication are possible.¹¹ Did Parmenides supply any support for (2) and (3)? As to (2), there is no way of telling; maybe it was taken as following immediately from the meaning of alētheiē, as that which truths are about, and knowledge is of. As to (3), first of all some evidence of Aristotle comes in here opportunely. Aristotle identifies, as an underlying thesis of the Eleatics, that ‘some knowledge or understanding (phronēsis) is possible’ (Aristotle, On the Heavens III. 1, 298b14–24). This supports the reconstruction; but does not tell what grounds if any were given for (3). It cannot be that this assumption is embodied in the initial acceptance of the ‘enquiry’, as something actually on foot, unless there was an argument to show that enquiry is always successful. It will now be suggested that in fact the remaining pieces of text, dealing with the rejection of ‘it is not’, give the supporting argument for (3). (B) For the same thing is for thinking and for being. (DK 28 B 3) It must be that what is for saying and for thinking, is; for it is for being, but what is not is not [for being]… (DK 28 B 6.1–2) These passages are part of the conclusion of the rejection of ‘it is not’. But they should be treated, at least initially, as (part of) a separate argument from the one reconstructed above. Once again, Homeric usage is an important guide. ‘It is for being/thinking/saying’ represents an idiom familiar in Homer: ‘A is for x-ing’ means either ‘there is A available to do some x-ing’ or ‘there is A available to be x-ed’. Much depends here on what sort of thing might be said to be ‘available for saying and thinking’. In Homeric usage, the object of the verbs ‘say’ and ‘think’ is usually expressed by a ‘that’-clause. What the clause describes is the state of affairs, in virtue of which the saying or thinking is true or not. An interpretation is possible within these linguistic constraints. Parmenides is arguing for the thesis that what can be said and thought, must actually be the case; i.e. that one can say and think only ‘things that are’, these being thought of not as true statements but as actual states of affairs. The argument has a very close affinity with one which troubled Plato in various places, notably in the Sophist (but he did not accept it as correct, in the Sophist or elsewhere).¹² The ‘Platonic problem’ (as it may be called for convenience) starts from the premiss (4), that saying and thinking must have, as objects, a state of affairs, actual or not; i.e. a genuine case of saying or thinking must be a case of saying or thinking that such and such is the case. But then, (5) if saying or thinking actually and not merely apparently occurs, its object must exist. Now, (6) for a state of affairs, to exist is just to be actual. Hence (7) only actual states of affairs are thought and said, i.e. all thinking and saying is true. In what is left of Parmenides’ text there appears, not quite this argument, but a far-reaching modal variation of it. What can be thought and said, must by (4) be at least a possible state of affairs (‘it is for being’). But (8) there can be no unrealized, ‘bare‘ possibilities. The argument to this effect is brought out effectively by the idiomatic ‘is for being’. What ‘is for thinking’, also ‘is for being’, and therefore necessarily is. There can be nothing more to ‘being for being’, than just being. Anything that is not, cannot be in any sense, and so cannot even ‘be for being’. Hence every possible state of affairs is actual, and so it must be that (7) what can be thought and said is true. We must, then, disentangle here the result (7), that there is no false thinking or saying, from the strong modal principle (8), that there are no unrealized possibilities. They are, of course, akin; in both (7) and (8), there is a refusal to have any philosophical truck whatever with any non-existent state of affairs. It is principle (8) that also supplies what is obviously needed: an explanation of Parmenides’ hitherto unjustified ruling-out of the ways ‘it is but might not be’ and ‘it is not but might be’. Both principles, (7) and (8), are important for the rest of the poem as well.¹³ In the deduction of consequences from ‘it is’, principle (8) will have a central role. Moreover, error, by principle (7), doesn’t consist in any ‘saying’ or ‘thinking’ but in the constructing of fictions of some sort, apparent statements. The effect of (7) is to force a new analysis of apparent falsehood, as will be seen later. These two partial reconstructions may now be put together to make an overall reconstruction of the rejection of ‘it is not’. The overall effect of the rejection of the way ‘it is not’ is to establish that since there must be true thinking and saying, there must be some objective reality. The first piece of text ((A) above) is a sketch of this overall argument, using premisses (1) (2) (3). In reply to this argument, a sceptic might question premiss (3): granted that thinking and saying occur, why should it be that some thinking and saying must be true?¹⁴ So Parmenides engages with this objection in the further argument which terminates in the second piece of text ((B) above). This argues that (7) there is no such thing as false thinking and saying, and (8) there are no unrealized possibilities either. On this reading, if Parmenides’ starting-point is like that of Descartes, and his first task is to show that knowledge is possible, his next problem, having shown that, is of a Kantian kind: given that knowledge and correct thought must be possible, what if anything follows about the nature of things? With the premisses (1), (2) and (3), he is able to show for a start that there must be such a thing as reality. There must be something for the knowledge to be about, and of, which by being so guarantees it.¹⁵ The ‘Way of Mortals’ After rejecting the way that says ‘it is not’, the goddess mentions, as unacceptable, yet another way, not previously mentioned: Then again [I shut you out] from this [way], which ignorant mortals wander along [or. construct], two-headed (for it is helplessness that steers the wandering mind in their breasts); they drift along, deaf and blind, in a daze, confused tribes: they accept as their convention that to be and not to be is the same and not the same [or: that the same thing and not the same thing both is and is not]; the path of all of them is back-turning. (DK 28 B 6.4–9) For surely it will never be forced that things that are not should be… (DK 28 B 7.1) There is no problem in understanding the rejection of a way that is clearly selfcontradictory. But why does Parmenides identify this way as the way of ‘mortals’; and are all human beings meant, or only some particular group? From the text, the ‘mortals’ seems to be ‘people’ generally, humanity in the mass. The ‘confused tribes’ can hardly be just a particular group of theorists.¹⁶ Besides, the goddess associates this way with an unthinking interpretation of the evidence of the senses, which is due to ‘habit of much experience’ and therefore presumably almost universal among adults: do not let the habit of much experience drive you along this way, exercising an unexamining eye, and a hearing and a tongue full of noise; but judge by reason the controversial test which I have stated. (DK 28 B 7.3–6) It seems to be, not sense perception itself which is at fault here, but people’s lazy habits in selecting and interpreting the information given by sense perception. The distinction had already been made by Heraclitus, who remarked: ‘Bad witnesses to people are eyes and ears, if [those people] have uncomprehending souls’ (DK 22 B 107). It is reason that must dictate how sense perception is to be understood, and not the other way round. On what grounds Parmenides took ordinary people to be enmeshed in contradiction about reality, is not yet clear. The reference to ‘the controversial test which I have stated’ must include the rejection of ‘it is not’. Parmenides may see people as accepting both ‘it is’ and ‘it is not’, because, while they see the need to assume some kind of reality, they at once contradict that assumption, as Parmenides believes, by allowing reality to contain features which are excluded by the test. For example, the existence of unrealized possibilities, and other things which are yet to be expressly excluded. The ‘controversial test’ probably includes also the negative implications of what is yet to come: the examination of the way that says ‘it is’. Consequences of ‘it is’ The other ways having been shown false, only the way that says ‘it is’ remains, so that this must be true. Only one story of a way is still left: that it is. On this [way] are very many signs: that what is cannot come-to-be nor cease-to-be; [that it is] whole, unique, unmoving and complete—nor was it ever nor will it be, since it is all together now—one, coherent. (DK 28 B 8.1–6) The ‘signs’ are best taken as the proofs, to follow, of the properties announced here as belonging to ‘what is’ (eon), i.e. reality. Evidently the deduction of the consequences of ‘it is’ constitutes, as expected, the journey along the way. (a) Reality cannot come-to-be nor cease-to-be (B 8.6–21) For, what origin will you seek for it? How and from where did it grow? Nor will I let you say or think that [it did so] out of what is not, for it is not sayable or thinkable that it is not. Besides, what necessity would have driven it on to come-to-be, later or sooner, starting from what is not? (DK 28 B 8.6–10) The first section of proof reveals the techniques of argument characteristic of this part. For convenience, the subject (‘what is’ or ‘reality’) will be denoted by E. Suppose that E does at some time come-to-be. Then Parmenides asks: out of what does it come-to-be? The implied premiss is: (10) whatever comes-to-be, comesto- be out of something. Parmenides seems to have taken (10) as self-evidently true; it is plausible to connect it with other places in this argument where he seems to have some variety of the Principle of Sufficient Reason in mind. So, if E comes-to-be, it comes-to-be out of F (say). Then for F, in turn, there are the two possibilities: F is, or F is not. Parmenides considers the second possibility, but not, apparently, the first one. This is a first problem. There is an extra twist to it. So far, we have considered Parmenides’ reasonings about ‘it is’ and ‘it is not’ without taking account of the ambiguities of the present tense. The rejection of the way ‘it is not’ does not call for these to be considered. But the Greek present tense is ambiguous in the same ways as the English one; and where, as here, possible past and future events are being discussed, it becomes necessary to distinguish the various uses. ‘It is’ and ‘it is not’ may be timeless, or refer to the time of the coming-to-be, or to the time of utterance, if that is different. Parmenides gives us no help at all on this point; but it is plausible to assume that he means the question ‘is F or is it not?’ to be understood as specialized (in line with ordinary usage) to the time of coming-tobe. This results, as will now be shown, in an intelligible argument. The question is then: E comes-to-be out of F; is F, or is it not, at the time of E’s coming-to-be? First, if F is, at that time, then at that time it is part of E, since E (on the interpretation followed here) is the whole of reality. But nothing can come-to-be out of a part of itself, since that does not count as coming-to-be at all. This point will account for Parmenides’ failure to examine the supposition ‘F is’. Second, suppose then that F, at the time of E’s coming-to-be, is not. This is the case which Parmenides examines. He gives two arguments. One argument is: ‘It is not sayable or thinkable that it [F] is not’. This invokes the results of the rejection of the way that says ‘it is not’. By principle (7), ‘F is not’ is not sayable or thinkable, because it is not true. But why is ‘F is not’ not true? By principle (i), if it were true, then F would not be capable of being recognized or pointed out; so it could not figure intelligibly in any sentence; so no sentence including it could be true, a contradiction. Parmenides in what follows will repeatedly appeal to the same consequence of (1): namely (11), no sentence of the form ‘X is not’ can be true. It might be objected that this principle (11) (so far as has been shown) applies only to what is not at the time when the utterance is made; in other words that here too there is a crucial ambiguity in the present tense. What if F is not, at the time of coming-to-be, but is, at the time of speaking? In that case, it would seem to be possible to recognize and point out F, and say of it intelligibly that it was not, at some earlier time. Again, Parmenides seems unaware of this objection. Is there a hole in his proof? It is more charitable, and perhaps more plausible, to suppose that Parmenides tacitly applies a principle of tense-logic such as this: (12) for any time t, and any statement S, if it is true now that S was (will be) true at t, then it was (will be) true at t that S is true then. By this means, Parmenides can transfer the force of principle (1) to the time of the supposed coming-to-be. ‘F is not’ cannot be (have been, be going to be) true at any time, because, if so, it would be true at the relevant time that ‘F is not’; but, by principle (11), the truth of ‘F is not’ (at any time) would involve a contradiction. The powerful general moral to be drawn, which will find further applications, is that, in assigning the properties of what is, none may be assigned which involves reference to things that supposedly at any time are not. The other argument begins: ‘Besides, what necessity would have driven it on to come-to-be, later or sooner, starting from what is not?’ The demand for a ‘necessity’, to explain what would have happened, implies, again, some variety of the Principle of Sufficient Reason. In an initial state in which there is nothing that is, there could hardly be any way of grounding the necessity. Even if there were, why did it operate ‘later or sooner’: at one particular time rather than at another? The rest of section (a) is occupied, on this interpretation, only with recapitulation and summing up. Only the case of coming-to-be has been discussed; there is no parallel treatment of ceasing-to-be, presumably because the arguments are intended to be exactly analogous. (b) Reality is undivided, coherent, one (B 8.22–25) Nor is it divided, since it is all in like manner, nor is it in any respect more in any one place (which would obstruct it from holding together) nor in any respect less: all is full of what is. Hence it is all coherent, for what is comes close to what is. (DK 28 B 8.22–5) The underlying strategy here is parallel to that of section (a). Suppose that reality (E) is divided. What that implies is that something divides E. What could that be? By the fact that E is ‘summed’, comprising all that is, anything other than E has to be something that is not. By the same argument as before, it can never be true to say that E is divided by something that is not. Hence E is not divided by anything other than itself. This limb of the argument, though suppressed here as obvious, appears in the parallel passage at 8.44–8. What is here explored is the other possibility: that E is divided by itself, i.e. by its own internal variations. The possibility of internal qualitative variations is not mentioned; presumably they would not count as creating divisions. What is mentioned is the possibility of variations of ‘more’ and ‘less’, i.e. in ‘quantity’ or ‘intensity’ of being. These are rejected, by the observation that ‘it is all in like manner’. Being admits of no degrees; anything either is or is not. (c) Reality is complete, unique, unchanging (B 8.26–33) The same, staying in the same, by itself it lies, and thus it stays fixed there; for strong necessity holds it in the fetters of the limit, which fences it about; since it is not right that what is should be incomplete, for it is not lacking—if it were, it would lack everything. (DK 28 B 8.29–33) This is a train of argument in which exposition runs the opposite way to deduction. It must be read backwards from the end. The starting-point is that reality (E) is complete or ‘not lacking’. Once again the strategy is the same, that of reductio ad absurdum. Suppose E is lacking; then E must lack something. What is this something? It cannot be part of E, for then it would not be lacking from E. Therefore it is not part of E, and hence is not; but it is not true that it is not, by the now familiar argument. Given that E is not lacking, it is complete, and has a ‘limit’. The word used here has no close English equivalent: Homer’s usage applies it to anything that marks or achieves any kind of completion. Here, the ‘limit’ functions as a constraint on reality: the need for completeness is a (logical) constraint. Completeness rules out, in particular, all change and movement, and enforces uniqueness: reality is ‘by itself’ or ‘on its own’. Why? Completeness has these consequences because it embodies the principle that E contains everything that is. This now enables Parmenides to get some grip on the problem of the past and the future. If past (or future) realities are still (or already) real, then they form part of reality; if not, not. The further question about the reality of the past (or future), does not here have to be decided. Either way, there can be no such thing as change or movement of reality: for those would imply the previous existence and present non-existence of some part of reality. Either past (future) and present coexist but differ, and then change is unreal; or the past (future) does not exist and then change is impossible. For similar reasons it must be ‘by itself, that is, unique, and not existing in relation to anything else. For ‘anything else’ that could be taken into account could not fail to be part of it. (d) Reality is spherically symmetrical (B 8.42–9) The grand finale of the way ‘it is’ combines points made earlier in a striking image: But since there is an outermost limit, it is perfect from every direction, like the mass of a well-rounded ball, in equipoise every way from the middle. For it must not be that it is any more or any less, here or there. For neither is there what is not, which would obstruct it from holding together, nor is there any way in which what is would be here more and here less than what is; for it is all immune from harm. For, equal to itself from every direction, it meets its limits uniformly. (DK 28 B 8.42–9) What is new in this section is spherical symmetry. From its ‘being all alike’ (the uniformity of its manner of being) and its perfection, is deduced its symmetry about a centre. What is surprising is not the symmetry, since that could be seen as a form of perfection, but the ‘middle’, a privileged central location, introduced without explanation (on this see the next section). The Nature of Reality Having followed the proofs of the properties of reality, one may still be uncertain just what those properties are. How strong, for instance, is the claim that reality is ‘one and coherent’ meant to be? Does ‘completion’ include spatial and temporal boundedness, and in general does reality have any spatial or temporal properties at all? How, if at all, is it related to the world apparently given in ordinary experience? Is reality spatial or temporal or both? First, the question of spatial and temporal properties. Parmenides shows no hesitation in applying to reality words which would normally imply spatial and temporal properties. It is ‘staying in the same thing’ and it ‘stays fixed there’; and ‘what is comes close to what is’. The word ‘limit’ (peiras) by itself implies nothing about space or time; but it is also said that this limit ‘fences it about’ and is ‘outermost’. The simile of the ball might not be meant spatially, but what of the statement that reality is ‘in equipoise every way from the middle’? Recall that reality has been interpreted to be a state of affairs. Such a thing, though it may persist or not through time, can hardly itself have a spatial location or extension. This point chimes with another: if one supposes that reality is spatially extended, its spherical symmetry is problematic. The ‘limit’ cannot possibly be meant as a spatial boundary, since for reality to be bounded in space would be for it to be incomplete. It must be right, then, to take the spatial terms metaphorically. They must be aids to grasping how reality inhabits a kind of ‘logical space’. This works out smoothly. The ‘spherical symmetry’ must express metaphorically the point that reality is exactly the same, however it is viewed by the mind: it presents no different ‘aspects’. The ‘middle’, about which it is symmetrical, can be identified with the ‘heart of well-rounded reality’ mentioned earlier; and be some kind of logical core (more on this later). Likewise the undividedness and coherence of reality mean that it is unified, not logically plural, not self-contradictory. ‘What is lies close to what is’, in the sense that any internal variation between parts does not constitute an essential difference. ‘Staying the same in the same and by itself makes the point that reality does not exist in relation to anything other than itself, and so not in relation to any external temporal or spatial framework; it is unique, and provides its own frame of reference. The metaphorical understanding of these terms is supported, above all, by the nature of the proofs. As has been seen, these make no appeal at all to properties of the space and time of experience. With respect to time, though, the situation is different. It is at least possible to conceive of a state of affairs as being in, and lasting through, time. Parmenides argues against any change in reality; but this is still consistent with the view that reality is something which persists, without change, through time. Did he wish to go further? There are good reasons for thinking so. First, the point made in connection with space, that reality exists ‘by itself, without relation to anything other than itself, means that, if reality exists in and through time, time must itself be seen as an aspect of reality. The basic temporal phenomenon must be the temporal extension of reality. This already goes some way beyond the simple notion of a persisting reality. Second, the argument (section (c) above) to the conclusion that reality is changeless and ‘by itself seems powerful enough to rule out, not merely change, but even mere time-lapse in relation to reality. Third, the initial list of conclusions states: ‘nor was it ever, nor will it be, since it is all together now.’ If one cannot even say that ‘it was’ and ‘it will be’, then one cannot say that it persists. Nor is it necessary to understand ‘now’ as a ‘now’ which implies a ‘time when’. It is much more plausibly a metaphorical ‘now’, indicating a single timeless state, in which there is no longer any distinction of before and after, and therefore no meaning in tensed statements. The metaphorical interpretation of these spatial and temporal terms, as applied to reality, does not, of course, imply that for Parmenides the spatial and temporal properties of ordinary objects are illusory. It still remains to be seen (in the next section) how Parmenides deals with the world of ordinary experience.¹⁷ In what sense is reality one? There is no doubt that Parmenides was a monist of some kind; the comments of Plato and Aristotle alone would prove it, even if the fragments were lacking.¹⁸ The relevant proofs are those given under (b) and (c) in the previous section. While argument (b) shows that reality is internally one (‘not divided’), argument (c) shows inter alia that there is nothing other than reality (it is ‘unique’ or ‘by itself’). Together these yield a monistic thesis: reality is both unified and unique, so there is but one thing. Just what this monism amounts to, may be seen by seeing what it excludes. The minimum that it must exclude is the error made by mortals when (in a passage to be discussed below) they decide to ‘name two forms, one of which ought not [to be named]; this is where they have gone astray’ (B 8.53–4). The fundamental error of the ‘mortals’ of the cosmology is to allow there to be two different subjects of (apparent) discourse, rather than just one. Parmenides is then committed at least to a logical monism: there is one and only one subject about which anything is true. This seems also to be the maximum that needs to be claimed, and the maximum that is imputed by Aristotle’s remark that ‘[Parmenides] seems to be getting at that which is one in definition’ (Metaphysics I.5, 986b18–19). The argument for unity (section (b)) demands nothing more. In particular it does not exclude internal variation, nor does it impose qualitative homogeneity. Reality consists of a set of facts true of itself. It is not excluded that reality might be constituted by more than one such fact; and after all many statements about reality are made by the goddess herself in the course of the argument; it would be absurd to suppose that they are meant to be seen as identical. Even though one may talk (as even the goddess sometimes does) in a misleading conventional way, this ‘plurality’ of facts must not be understood as a genuine plurality: what we are really dealing with here is different aspects of reality. Even when different parts of reality are distinguished, the correct formulation does not admit them as subjects in their own right, but speaks only of ‘what is’: ‘what is comes close to what is’; ‘what is cannot be here more or here less than what is’. So when the goddess distinguishes ‘the unmoving heart’, and the ‘middle’, of reality, implying that there is also a peripheral part, she must be understood as speaking, in a conventional way, about a situation which could be described more correctly. What she means by it, has now to be considered. The Errors of ‘Mortals’ and the Place of Ordinary Experience It remains to ask how this reality is supposed to be related to the world of ordinary experience. In Parmenides’ rejection of the ‘way of mortals’, it was seen that senseexperience in itself did not seem to be blamed for their mistakes. It was mortals’ habitual misinterpretation of sense experience which caused them to fall into self-contradiction. After the exploration of the nature of reality, it is possible to specify the fundamental mistake of ‘mortals’ more clearly, and Parmenides does so: The same thing is for thinking and [is] the thought that it is; for you will not find thinking apart from what is, in which it is made explicit. For nothing other is or will be outside what is, since that has been bound by fate to be whole and unchanging. Hence it will all be [just] name, all the things that mortals have laid down, trusting them to be real, as coming-into-being and perishing, being and not being, changing place and altering bright colour. (DK 28 B 8.34–41) ‘Mortals’, here too, includes all who accept a world of real plurality and real change. Such people are committed to the reality of what are in fact conventional fictions or ‘names’, taken as putative objects of thinking and saying. The passage starts with a reaffirmation of the principle derived from the ‘Platonic problem’ (see above): ‘what can be thought is just the thought that it is’, since this is (with its various consequences) the only true thought. Because there is no saying and thinking something false, apparent false thought must be ‘mere names’. So too at the beginning of the cosmology (see next section), where we shall see that even ‘mere names’ (like other conventions, so long as intelligently made and properly observed) can have their uses. It is the subjects of ordinary discourse, the things that we normally identify as the plural changing contents of the world, that are here denounced as just ‘name’, conventional noises and nothing more. Since statements about them cannot be true, they are not capable of being genuinely spoken and thought about. The self-contradictory ‘way of mortals’ is now explained. ‘Mortals’ recognize the existence of an objective reality, and therefore say ‘it is’. But they also have to say ‘it is not’, because they take reality to be something truly plural and changing.¹⁹ The denunciation of ‘mortals’ does not exclude the substantial reality of the ordinary world of experience—provided a construction is put upon that world which is radically different from the usual one, on the two key points of plurality and change. The temporal dimension may be kept, so long as it is in effect spatialized, with becoming and change ruled out as an illusion. The multiplicity of things in both spatial and temporal dimensions may be kept, so long as it is seen as non-essential qualitative variation within a single logical subject. Finally, if this is right, it yields a satisfactory sense for the mentions of the ‘unmoving heart’ of reality and of its ‘middle’, a core implying a periphery. The ‘heart’ or ‘middle’ is constituted by the necessary truths discovered by reasoning, which alone are objects of knowledge. The outside is ‘what meets the eye’: the contingent snippets of reality as perceived by the senses. Senseperception, even when in fact veridical, presumably does not yield knowledge because of the possibility of deception.²⁰ What it reveals, not being part of the core of reality, is non-essential and not demonstrable by reasoning. The Nature and Structure of Empirical Science: Cosmology as ‘Opinions of Mortals’ Parmenides’ stringently exclusive conception of knowledge does not entail the uselessness of all other cognitive states. Far from it. He recognizes both the possibility and the practical value of ‘opinions’ about the cosmos, when organized into a plausible and reliable system. Here, building on the ideas of Xenophanes, he turns out to be the first recognizable philosopher of science.²¹ This is why the conclusion of the investigation of reality does not mark the end of the poem. There still remains the second half of the promise of the goddess, which must now be recalled: It is necessary that you find out everything: both the unmoving heart of well-rounded reality [alētheiē], and the opinions of mortals, in which there is no real guarantee of truth—but still, these things too you shall learn, how [or: since] it had to be that opinions should reputably be, all of them going through everything. (DK 28 B 1.28–32) This promise of an exposition of ‘mortal opinions’ is taken up at the end of the exploration of reality: At this point I cease for you my trusty tale and thought concerning reality; from now on, learn the opinions of mortals, hearing the deceptive ordering of my words… This world-ordering I reveal to you, plausible in all its parts, so that surely no judgement of mortals shall ever overtake you. (DK 28 B 8.50–1 and 60–1) What Parmenides says about his system of ‘opinions’ confirms the conclusion already reached, that for him sense-perception cannot give knowledge. For he is at pains to emphasize that such a system has no ‘proper guarantee of truth’; and that it is ‘deceptive’ (it purports to give knowledge, but does not). It appeals to empirical evidence for support, not to reason. So it lacks any claim to be an object of knowledge. The deeper reason why it cannot be supported by appeal to pure reason is presumably that it is concerned with ‘peripheral’, contingent aspects of reality. But there is still a problem. If conducted in the usual way, a cosmology must also necessarily be not so much false as meaningless verbiage, since it takes seriously the illusions of plurality and change, speaks as though they were real, and offers explanations of such changes in terms of physical necessities. Parmenides’ ‘Opinions’ is such a cosmology. Why does he deliberately offer a system of which he himself thinks, and indeed implicitly says (in calling it ‘deceptive’, and basing it on an ‘error’), that it is not merely not certain, but, taken literally, meaningless all through? One possible answer is that Parmenides thought that his convenient, but literally meaningless, statements could be at need translated back into the correct but cumbersome language of timelessness and logical monism. Unfortunately, there is no indication in the text that it is merely a question of words. He does at least seem to reassure us that, meaningless or not, these statements are practically useful. In some way they correspond to the way the world presents itself to us. The fictitious entities they mention correspond to the fictions we create on the basis of our misread ordinary experience. That experience shows they may be usefully manipulated to give a practically workable understanding of the phenomenal world.²²(Cosmology so conceived is like science as seen by ‘operationalist’ philosophers of science; and like divination and natural magic—a thought perhaps taken further by Empedocles.²³) Within such limits, cosmology may none the less be required to satisfy certain formal demands.²⁴ Parmenides sets out these demands explicitly, for the first time. The original promise of the goddess stresses that the cosmology to be told is (1) reliable; (2) comprehensive. Both of these points are echoed in the later passage, (1) Reliability is echoed by ‘deceptive’ and ‘plausible’. The demands on the cosmology are further that it be a ‘world-ordering’, not only (2) comprehensive but also (3) coherent and formally pleasing; and (4) the best possible of its kind. These last two points may also include economy or beauty of explanation. The Principle of Sufficient Reason, which is closely related to the demand for economy, appears, as in the exploration of reality, so again in the cosmology, to yield a symmetry between the two cosmic components. In fact, Parmenides devises an elegant and economical basis for cosmology by following a hint given by the ‘way of mortals’. Any conventional cosmology has to tread that false way, and to say both ‘it is’ and ‘it is not’. The simplest way to commit this error is to suppose initially not one logical subject but two: one which is, and one which is not.²⁵ The physical properties of the two subjects are then a kind of cosmic parody or allegory of the logical properties of what is and what is not. Now, they have fixed their judgements to name two forms, one of which should not [be named]; this is where they have gone astray. They have separated their bodies as opposites, and laid down their signs apart from one another: for the one form, heavenly flaming Fire, gentle-minded, very light, the same as itself in every direction, but different from the other one; but that [other] one too by itself [they have laid down] as opposite, unknowing Night, a dense and weighty body. (DK 28 B 8.53–9) Fire and Night are the physical embodiments of the two opposed principles. The cosmology is dualistic, and there is reason to suspect that, as with the Pythagoreans from which it may borrow, the dualism was a moral (and an epistemic) as well as a physical one. The two opposed ‘forms’ are associated from the outset with knowledge and ignorance; perhaps also with good and evil. Traces of morally charged struggles and loves of ‘gods’ within the cosmos remain in the testimony.²⁶ A cycle of cosmic changes is the most likely explanation of a detached remark (DK B 5) about circular exposition. Not only is the basis economical, but there are overall formal demands on the two forms. They must jointly exhaust the contents of the cosmos; and there must be cosmic symmetry as between them (DK B 9),²⁷ Conclusion: the Trouble with Thinking This account of Parmenides must end with questions on which certainty seems to be out of reach. The overarching question is this: is Parmenides’ ‘framework’, in which his theory of reality is embedded, itself meant to be grounded in that theory of reality? By the ‘framework’ is meant roughly the following: the original assumption about the actual existence of thinking and certain knowledge; the distinction between knowledge and opinion; the application of logic in the discovery of the nature of reality; and the assertion of the practical, empirical effectiveness of systematized ‘opinions’. Even if this question cannot ultimately be given any confident answer, it usefully focuses attention on one sub-problem, which has so far been kept to one side. This is the problem about the relation between thinking and reality. We have seen that thinking, for Parmenides, can only be of truths, indeed of necessary truths, about reality. Is it a necessary truth that thinking occurs? If so, that truth itself is of course a necessary truth about reality; and whatever it is that thinks must be (part of) reality. If so, one would think that it ought to be deducible from the nature of reality that it thinks about itself. No such deduction appears in the text, though; and the thesis that true thinking occurs seems to be (as Aristotle took it) an initial assumption which is taken as unquestionable, but not formally proved. On this point, there are two parts of the poem which might serve as some kind of a guide. One is the introductory narrative of the journey to the goddess. Another is the outline of ‘physical psychology’, a general theory of perception and thought, which is attested as part of the cosmology. The journey to the goddess The chariot-ride of the narrator in the introduction (preserved in DK B 1) has usually been taken as an allegory of Parmenides’ own intellectual odyssey, and of the framework with which he starts.²⁸ Its chronology and geography are elusive and dreamlike. The individual beings mentioned, even the narrator and the goddess herself, are but shadowy outlines. Only certain technological objects— the chariot wheels and axle, the gate and its key—stand out in relief. Is Parmenides here proclaiming his advances in the technology of thinking, as the motive power in, and the key and gateway to, all that follows? The ‘paths of Night and Day’ would then be the ways of ‘it is’ and ‘it is not’. The chariot, the horses, the daughters of the Sun who act as guides, and Parmenides’ own ambition, would correspond to everything Parmenides needs to get him as far as the choice of ways,—that is, to the ‘framework’. All too much, though, must be left uncertain, even if such an approach looks plausible in general. The empirical psychology The psychology or ‘theory of mental functioning’ which was outlined in Parmenides’ cosmology is, equally, not much more than a tantalizing hint. Theophrastus says that for Parmenides as for several others ‘sense perception is by what is similar’, and goes on: ‘As for Parmenides, he goes into no detail at all, but just [says] that, there being two elements, cognition [gnōsis] is according to what predominates. For, as the hot or the cold predominates, the intellect [dianoia] alters, but that [intellect] which is [determined] by the hot is in a better and purer state, though even that kind needs a kind of proportioning. He says: According as the compounding of the wandering limbs is in each case, in such a way is mind present in people; for it is the same thing in each and in all that the nature of the limbs has in mind: the more is the thought. For he talks of sense perception and mental apprehension as being the same; which is why [he says that] memory and forgetting occur from these [constituents] by the [change of] compounding. But whether, if they are in equal quantities in the mixture, there will be mental apprehension or not, and what state this is, he does not go on to make clear. That he also makes sense perception [occur] for the other element [(the cold)] in itself, is clear from the passage where he says that the corpse does not perceive light and hot and noise, because of the lack of fire, but does perceive cold and silence and the opposite things. And in general [he says that] everything that is has some kind of cognition. Thus it seems he tries to cut short, by his dogmatic statement, the difficulties that arise from his theory. (Theophrastus On the Senses 3–4, citing DK 28 B 16) This would seem to be at least a two-tier theory. The lower tier is basic sense perception, available to everything that exists, and ‘by the similar’; i.e. what is fire can perceive fire, what is night can perceive night, and what is a mixture can perceive both. The higher tier is that of mind and thought, somehow due to a ‘proportionate compounding’ in human (and other?) bodies.²⁹ Any inferences from these indications can be but tentative. Briefly, the general shape of Parmenides’ theory of reality shows that any real thinking must be (part of) reality thinking about itself.³⁰ The account of Parmenides’ intellectual journey may be taken as acknowledging the need for starting-points for thinking —for a ‘framework’. The theory of mental activity in the cosmology is of course infected with the fictitiousness of the whole cosmology; yet it is probably meant to correspond, somehow, to the truth about thinking. If one element (‘fire’) in the cosmology corresponds to reality, then the fact that it is fire that cognizes fire reflects the truth that it is reality that thinks of reality. It is a pity we can know no more of what Parmenides thought about thought. ZENO Introduction Zeno of Elea, fellow citizen and disciple of Parmenides, became famous as the author of a series of destructive arguments. There is no good evidence that he put forward any positive doctrines. Plato and Aristotle were deeply impressed by the originality and power of the arguments; such knowledge of Zeno as survives is due principally to them and to the Neoplatonist scholar Simplicius.³¹ The Arguments against Plurality (a) Plato on Zeno’s book and the structure of the arguments Plato’s dialogue Parmenides describes a supposed meeting in Athens, around 450 BC, of the young Socrates and others with two visitors from Elea, Parmenides and Zeno. Plato’s fictional narrator gives some biographical data about the two Eleatics, and recounts a conversation between ‘Socrates’ and ‘Zeno’ which tells (127d6–128e4) of the genesis of Zeno’s arguments against plurality, their structure and their aim. Even if based solely on Plato’s own reading of Zeno’s book, this has to be taken seriously as testimony.³² According to this testimony, there was a book by Zeno which consisted entirely of arguments directed against the thesis ‘there are many things’. Each argument began by assuming the truth of this thesis, and proceeded to deduce a pair of mutually contradictory conclusions from it, in order to make a reductio ad absurdum of the original thesis. In support of this account, Simplicius the Neoplatonist gives verbatim quotations from two of the arguments, which can be seen to exemplify the pattern; Plato’s narrator himself gives the outline of another. (b) The aim of the arguments This account of the structure of Zeno’s arguments leads ‘Socrates’ in the dialogue to the view that Zeno’s aim was simply to refute the thesis of pluralism (‘that there are many things’), in any sense incompatible with Parmenides’ theory, and thereby to establish Parmenidean monism. However, this conclusion of ‘Socrates’ is not completely accepted by ‘Zeno’, who denies that the book was a ‘serious’ attempt to establish Parmenidean monism, and goes on: actually this [book] is a way of coming to the aid of Parmenides’ theory, by attacking those who try to make fun of it [on the grounds] that, if there is one thing, then many ridiculous and self-contradictory consequences follow for the theory. Well, this book is a counter-attack against the pluralists; it pays them back in the same coin, and more; its aim is to show that their thesis, that there are many things, would have even more ridiculous consequences than the thesis that there is one thing, if one were to go into it sufficiently. (Parmenides 128c6–d6) The natural way to read this is as saying that the arguments had an ad hominem element. ‘Zeno’ cannot be saying that Parmenides’ thesis really had ridiculous consequences; ‘even more ridiculous consequences’ points to the employment by Zeno of assumptions made by Parmenides’ opponents, but not accepted by Zeno himself. Yet ‘Socrates’, a little earlier, has said that the arguments give ‘very many, very strong grounds for belief (128b1–3) that pluralism, of any variety incompatible with Parmenides, is false. In Plato’s opinion the arguments, however they originated, were usable against all varieties of anti-Parmenidean pluralism. Therefore Plato’s testimony on Zeno cannot be fully understood unless we know how he interpreted Parmenides, which cannot be investigated in this chapter. Provisionally, it is enough to note that there is no danger of contradiction in Plato’s testimony, provided we may assume that Zeno’s original opponents made, and Zeno himself used against them, only such assumptions as were either inconsistent with Parmenides; or plausibly seen as articulations of commonsense. ³³ The question can be finally decided, if at all, only by analysis of the arguments themselves. One further piece of information is given at 135d7–e7: the arguments were about ‘visible things’, i.e. they addressed themselves to the question of pluralism in the ordinary world, using assumptions derived from experience.³⁴ (c) The argument by ‘like’ and ‘unlike’ According to Plato (Parmenides 127d6–e5), the argument (the first one in the book) purported to show that ‘if there are many things, they must be both like and unlike’. Nothing further is known. (d) The argument by ‘finitely many’ and ‘infinitely many’ Simplicius preserves the entire text of this argument. The compressed, austere style is reminiscent of Parmenides. If there are many things, it must be that they are just as many as they are and neither more of them nor less. But if they are as many as they are, they would be finite. If there are many things, the things that are are infinite. For there are always other things between those that are, and again others between those; and thus the things that are are infinite. (DK 29 B 3, Simplicius Physics 140.27–34) The first limb insists on the implications of countability. If it is true to say ‘there are many things’ and to deny that ‘there is one thing’, that implies that (a) there is one correct way of counting things; (b) that that way of counting the things that are leads to a definite result. But a definite result implies finitely many things: if there were infinitely many, counting them would lead to no result at all. The second limb invokes the relation ‘between’ (metaxu). Any two distinct things are spatially separate (the converse of Parmenides’ argument for the oneness of reality from its undividedness). But what separates them must itself be something that is, and distinct from either. From this principle, an infinite progression of new entities is constructed. Though this involves an appeal to spatial properties, it might easily be rephrased in terms of logical ones. The principle would be: for any two distinct things, there must be some third thing different from either which distinguishes them from one another; and so on. (e) The argument by ‘sizeless’ and ‘of infinite size’ Again Simplicius is our source. He quotes two chunks of the text, and enough information to recover the rest in outline. The first limb claimed that ‘if there are many things, they are so small as to have no size’. The argument proceeded, according to Simplicius, ‘from the fact that each of the many things is the same as itself and one’ (Physics 139, 18–19). It is not difficult to make a plausible reconstruction here. First, to speak of a ‘many’ implies, as in (d), a correct way of counting. The many must be made up of securely unified ones. Then consider each of these units. The line may have been (compare Melissus DK B 9): what has size has parts; what has parts is not one. Hence each of the units must be without size. The second limb contradicted this in successively stronger ways. First, it claimed to show that, in a plurality, what is must have size. Suppose something does not have size, then it cannot be: For if it were added to another thing that is, it would make it no larger: for if something is no size, and is added, it is not possible that there should be any increase in size. This already shows that what is added would be nothing. But if when it is taken away the other thing will be no smaller, and again when it is added [the other thing] will not increase, it is clear that what was added was nothing, and so was what was taken away. (DK 29 B 2, Simplicius Physics 139.11–15) This argument in terms of adding and taking away obviously makes essential use of the assumption ‘there are many things’; it could not, therefore, have been turned against Parmenides. It also needs some principle such as ‘to be is to be (something having) a quantity’: not a ‘commonsense’ axiom, but one likely to be held by most mathematizing theorists of the time.³⁵ The next and final step proceeds from size to infinite size: But if each [of the many things] is, then it is necessary that it has some size and bulk, and that one part of it is at a distance from another. The same account applies to the part in front: for that too will have size and a part of it will be in front. Now, it is alike to say this once and to keep saying it all the time: for no such part of it will be the endmost, nor will it be that [any such part] is not one part next to another. Thus if there are many things, it must be that they are both small and large: so small as to have no size, so large as to be infinite. (DK 29 B 1, Simplicius Physics 141.2–8) One axiom used is that anything having size contains at least two parts themselves having size. This clearly generates an unending series of parts having size. Less clear is the final step from ‘having infinitely many parts with size’ to ‘infinite (in size)’, which apparently was taken with no further argument. There is some analogy with the ‘Stadium’ and ‘Achilles’ (see (c) below): just as the runner’s supposedly finite track turns out to contain an infinite series of substretches, each of positive length, so here the object with supposedly finite size turns out to contain an infinite series of parts, each having size. If we try to recompose the original thing out of the parts, we shall never finish, but always be adding to its size; and this, Zeno might plausibly claim, is just what is meant when we say something is infinite in size.³⁶ (f) Methods and assumptions In the light of the arguments themselves as preserved, the question of their aims and methods can be taken up again. It is evident that some of the assumptions used by Zeno in these arguments are not due to simple ‘common sense’. Common sense does not make postulates about the divisibility ad infinitum of things having size; nor suppose that ‘to be is to be something having a quantity’; nor insist on a single correct way of counting things. Hence Zeno’s arguments are not directed against unreflecting ‘common sense’. In fact, these are the kind of assumptions that are naturally and plausibly made, when one sets about theorizing, in an abstract and mathematical spirit, about the physical world. The methods and the style of proof are also mathematical. Note-worthy are the constructions of progressions ad infinitum, and the remark when one is constructed: ‘it is alike to say this once and to keep saying it all the time’. However many times the operation is repeated, that is, it will always turn out possible to make precisely the same step yet again.³⁷ The Arguments about Motion (a) Aristotle’s evidence There is only one certain primary source for the content of Zeno’s arguments about motion: Aristotle, who states and discusses them in Physics VI and VIII (VI 2, 233a21–30; VI 9, 239b5–240a18; VIII 8, 263a4–b9). Aristotle’s source is not known; no book of Zeno that might have contained them is recorded. It is perfectly possible that they reached Aristotle by oral tradition. In any case, while there is no reason to doubt that they are substantially authentic, there is also no reason to suppose that Zeno’s own formulations have been faithfully preserved. (A source possibly independent of Aristotle is mentioned in (d) below.) The four individual arguments, as Aristotle reports them, derive contradictions from the supposition that something moves. Three of them purport to show that what moves, does not move. They are ‘dramatized’, in so far as they introduce particular supposed moving things: a runner; two runners; an arrow; three moving and stationary masses. Aristotle presents the arguments as designed to be mutually independent.³⁸ (b) The ‘Stadium’ and the ‘Achilles’ Suppose a runner is to run along a running-track. The stretch to be traversed (call it S) may be considered as divided up into substretches in various ways. Given the starting and finishing points we understand what is meant by ‘the first half of S’, ‘the third quarter of S’ and so on. It seems that however short a substretch is specified in this way, it will always have positive length and may be thought of as divided into two halves.³⁹ Going on in this way we can specify a division of S into substretches which will be such that the runner runs through a well-ordered but infinite series of substretches. First the runner traverses the first half, then half of what remains, then half of what remains, and so on. In this way, for any positive integer n, at the end of the nth substretch the runner has covered of S, and the nth substretch is ½n of the whole length of the track. However large a finite number n becomes, the fraction is never equal to 1; there are infinitely many substretches. With such a division, the series of substretches is well-ordered, and the runner who traverses S has been through all of the substretches in order: for every finite number N, the runner has traversed the Nth substretch. Hence the runner has traversed an infinite series of substretches, in a finite time; but this is impossible. This is an expansion of Aristotle’s formulations (Physics VI 2, 233a21–23; VI 9, 239b11–14) of the ‘Stadium’ argument.⁴⁰ The ‘Achilles’ (Physics VI 9, 239b14–29) makes the same point more dramatically, pitting a very fast runner against a very slow one. The slow runner is given a start. The stretch covered by the faster runner is divided up in such a way that it appears the faster can never catch the slower within any finite time. This drives home the point that speed is irrelevant. No limit of speed is prescribed or needed by the argument; the speed of the fast runner could increase without limit without removing the problem. (c) The ‘Arrow’ Another way of looking at things supposedly in motion throughout a time-stretch is to select any one moment during that stretch. Say an arrow is in flight. 1 At any one moment the arrow must be ‘in one place’. No part of it can be in two places at once; so it must occupy ‘a space equal to itself (i.e. of the same shape and size). 2 The arrow must be at rest at this moment. There is no distance through which it moves, in a moment; hence it does not move at a moment, so it must be at rest at that moment. 3 But the moment chosen was an arbitrary moment during the flight of the arrow. It follows that the arrow must be at rest at all moments during its flight. 4 Hence, since the arrow during its flight is never not at a moment of its flight, the arrow is always at rest during its flight; so it never moves during its flight. The above argument cannot claim to be more than a plausible filling-out of Aristotle’s abbreviated report (Physics VI 9, 239b5–9 and 30–3).⁴¹ Aristotle himself is interested only in step (4), where he thinks to find the fallacy; he gives the only briefest sketch of (1), (2) and (3).⁴² (d) The ‘Moving Rows’ Aristotle (Physics VI 9, 239b33–240a18) reports this argument in terms of unspecified ‘masses’ on a racecourse; to make it easier for a modern reader, the masses may be thought of as railway trains.⁴³ Consider three railway trains of the same length, on three parallel tracks. One of the trains is moving in the ‘up’ direction, another is moving at the same speed in the ‘down’ direction, and the third is stationary. As may be easily verified, either of the moving trains takes twice as long to pass the stationary train as it does to pass the other moving train. Just how Zeno derived a contradiction from this fact, is uncertain. According to Aristotle, Zeno simply assumed that the passing-times must be equal, since the speeds are equal and the two masses passed are equal in length. Then it follows that the time is equal to twice itself. The assumption, though, has often been thought too obviously false to be Zeno’s. It may simply be Aristotle’s attempt to fill a gap in the argument as it reached him.⁴⁴ Yet Aristotle himself thought it not obviously fallacious, and worthy of detailed refutation. (e) Method and purpose of the four arguments As Aristotle describes them, these four arguments are simply ‘the arguments of Zeno about motion which cause difficulties to those who try to solve them’: no suggestion that they were all of Zeno’s arguments on the subject. Aristotle presents them as mutually independent, and in an order which is not dictated by his own concerns, presumably that of his source. Once again, as with the arguments against plurality, some of the assumptions are manifestly theorists’ initial assumptions, rather than those of simple ‘common sense’; but they are close to common sense.⁴⁵ If one starts trying to think systematically in an abstract way, analogous to mathematics, about the phenomena of motion and its relation to time and space, these are assumptions that it is natural to start with. It is natural to assume that both the time-stretch and the track of the moving thing may be treated for theoretical purposes as geometrical lines obeying Euclidean geometry. This means that they are divisible ad infinitum, and that points along them exist ‘anywhere’: i.e. at all places corresponding to lengths constructible by Euclidean procedures. There was no theory equivalent to that of the real numbers available in Zeno’s time, but such assumptions correspond to elementary theorems and constructions of plane geometry as it was beginning to be developed. The way of thinking about physical phenomena embedded in Zeno’s assumptions is therefore an abstracting, mathematizing physicist’s way. Zeno’s original opponents are likely to have been natural philosophers, very likely from the loose group of ‘Pythagoreans’ (see below), who were then taking the first steps towards a mathematized theory of the natural world.⁴⁶ Just because Zeno’s assumptions are natural ones for any mathematising theorist to make, his arguments still arouse heated discussion among philosophers. The suggestion, still sometimes made, that Zeno’s arguments have been made obsolete by developments in modern mathematics (particularly differential calculus and the theory of infinite series), misses the point. The value and interest of all Zeno’s arguments is just that they are challenges to the foundations of any mathematics and any physics that uses infinites and indivisibles of any kind and applies them to the physical world.⁴⁷ Other Arguments Reports of yet other arguments by Zeno survive. Aristotle records a problem about place: ‘if everything that is is in a place, clearly there will be a place of the place too, and so ad infinitum’ (Physics IV 1, 209a23–5). Elsewhere he gives the problem in the form: ‘if a place is something, in what will it be?’ (Physics IV 3, 210b22–24). If this was originally one argument, it constructed an infinite series out of the common assumption that everything that is, is in a place, which is something other than itself: applying the assumption to places themselves, we shall have places of places, places of places of places, and so on. Such a series could have figured in one of the arguments against plurality. In any case, it would be a good parry to any attack on Parmenides’ monism which sought to show that his ‘One’ must occupy a place other than itself. Also from Aristotle (Physics VII 5, 250a19–22): Zeno argued that, if a heap of grain makes a noise when it falls, then a single grain and any fractional part of it must make a noise too. One may conjecture that Zeno’s dilemma was: either it makes a proportionately small noise, or none at all. If the latter, a natural and fundamental assumption of mathematizing physics is undermined: the assumption that the magnitudes of effects are in direct proportion to the magnitudes of their causes. But if the former, then why do we not hear the proportionately small noise? If it fails to affect our senses, the assumption of proportionality breaks down somewhere else. Such an argument would obviously fit Zeno’s programme of attack on any possible mathematical physics. Conclusion Examination of the evidence for Zeno’s arguments leads to satisfyingly consistent results, and bears out the testimony of Plato. First, Zeno attacked principally certain commonly-held views involving the reality of plurality and change, but did not confine himself to those targets. This fits well with Parmenides, who saw the twin beliefs in the reality of essential plurality and of change as the two marks of deluded ‘mortals’.⁴⁸ Second, Zeno’s argumentative assumptions are taken from his opponents. They may be characterized as those of theoretical physics in its infancy, of ‘mathematicized common sense’. PHILOLAUS AND ‘THE PEOPLE CALLED PYTHAGOREANS’ In the mid to late fifth century, there were various people and groups claiming to be ‘Pythagorean’; they were found principally in the west of the Greek world (Sicily and southern Italy). Aristotle, our most reliable source, tells of certain ‘Italians’ or ‘people called Pythagoreans’ who had a programme of reducing everything to mathematics (Metaphysics 1.5, 985b23–986b8 and 987a9–27).⁴⁹ The only individual one, about whom something of tangible philosophical interest can be known, is Philolaus of Croton.⁵⁰ Five fragments which may be reasonably taken as genuine reveal a theory of underlying structure in the universe which is heavily influenced by the development of mathematics as an abstract study.⁵¹ This theory is propounded, it seems, on the basis of an analysis of ordinary human knowledge and its presuppositions. Philolaus’ starting-point is gnōsis, the everyday activity of cognitive ‘grasping’ (individuation, identification, reidentification, reference) of ordinary individual things. This ‘cognizing’ implies that its objects ‘have number’, i.e. are in some sense measurable or countable. Quite generally, any cognizable object must be marked off from everything else by a sharp, definite boundary. Whether this boundary be spatial or temporal, the object within it will have some measurable quantity (volume, time-duration). Also, a cognizable collection of objects must have a number; indeed even a single object must be recognizable as a single object and not a plurality, which implies a definite and practically applicable method of counting. These points are recognizably related to some arguments of Parmenides and Zeno. Zeno (see above, pp. 153–4) argues that a ‘many’ implies a definite number; but also that it implies definite, distinct units and hence boundaries round these units. That what is must be a unit and have a boundary is also argued in Parmenides (see above, p. 41). The concept of a ‘boundary’ is central here. Philolaus’ analysis of the presuppositions of cognition leads him to a logical separation of the contents of the universe into ‘things which bound’ and ‘things unbounded’. Everything in the cosmos, and that cosmos itself, is claimed manifestly to exhibit a structure ‘fitted together’ from the two kinds of thing. This dualism is obviously closely related to views which Aristotle attributes to ‘the people called Pythagoreans’. He reports that some of them set up two ‘columns of correlated opposites’, which featured such items as limit/unlimited, odd/even, one/plurality, right/ left, male/ female, etc. (Metaphysics I. 5, 986a22–6). Philolaus’ careful attempt to build up a general ontology on the basis of an analysis of ordinary cognition, guided by mathematics, leads him naturally in the direction of Aristotelian ‘form’ and ‘matter’. Whatever stuff an individual is thought of as being ‘made of, is in itself not ‘bounded’; for it might be present in any quantity. But for there to be an individual, there must be also a ‘bound’. Further explication of just what is involved in this ‘fitting together’ is not found, and it seems that Philolaus thought this question beyond the reach of human knowledge. That conclusion is in conformity with his method. The ‘everlasting being’ of things, or ‘nature itself, is the subject of ‘divine cognition’ only. The ‘fitting together’ is achieved ‘in some way or other’. Mathematics, clearly, cannot help; for it too exemplifies, rather than explains, the dualistic structure. All that we can say is that even humble human cognition presupposes such a structure of things in particular and in general; the first example, it has been suggested, of a Kantian transcendental argument.⁵² MELISSUS Melissus of Samos (active around the mid fifth century) is best grouped with the philosophers of Elea, to whom he obviously owes much. In spite of the preservation of ten fragments (plus a paraphrase of other arguments) by Simplicius, and a reasonable amount of supporting testimony (the most useful from Aristotle), Melissus’ intentions are not obvious. Many of his arguments seem obviously weaker and cruder than those of Parmenides; on these grounds he was dismissed with contempt by Aristotle.⁵³ Foundations of Monism As found in the quotations by Simplicius, Melissus starts by considering ‘whatever was’ (DK B 1). The emphatic use of the past tense already signals a departure from Parmenides. It may have been justified by an initial argument to the effect that thinking and speaking require the existence of something thought or spoken about; it is impossible to think or speak ‘about nothing’.⁵⁴ By the time we reflect on our own thinking and speaking, they are in the past, so what is guaranteed by the argument is that something was. ‘Whatever was’ is also apparently more non-committal than ‘reality’. As quickly becomes clear, however, this entity is conceived of by Melissus as extended in space and in time. It is ‘the universe’ rather than ‘reality’. Various things are proved about it. First (B 1), it cannot have come into being, because it would have to have done so out of nothing, which is impossible. Next (B 2), it always was and always will be; Melissus here assumes that ceasing-to-be is just as impossible as coming-to-be. Next (B 3), an obscure argument to show that the universe is spatially unbounded, perhaps intended to parallel the argument for no coming-to-be and no ceasing-to-be. The thought seems to be that a ‘beginning’ or ‘end’ in space is just as inconceivable as one in time; in either case we should have to suppose that there was nothing beyond. But that is unacceptable, apparently. Why? Possibly again for the reason that a statement ostensibly ‘about nothing’ (i.e. where ‘nothing’ appears to refer to what the statement is about) is not a statement at all. Finally, an argument for the unity of the universe: ‘If it were two, they could not be unbounded, but would have bounds with each other’. Why should internal boundary lines be ruled out? Perhaps (cf. Parmenides and Zeno) because even internal boundary lines involve what is not; they cannot be part of either of the components they separate, so are either themselves components, and need further boundaries, or are ‘nothing’, which is again impossible. So there cannot be two or more distinct components in the universe. A further vital point, proved we know not how, was that the universe is homogeneous. It also has no ‘bulk’ or ‘body’, on the grounds that that would mean that it would have physical parts, and not be a unity (B 10). Arguments against Change, Void and Motion Melissus’ arguments against the possibility of any kind of change proceed briskly but none too convincingly. First, qualitative change would imply lack of internal homogeneity in the universe, since it would have to be qualitatively different at different times. Next comes ‘change of kosmos’; apparently some more essential type of change (change of internal structure?). The argument is that such a change necessarily involves what has already been ruled out: e.g. increase or partial perishing or qualitative change. There follows the at first sight bizarre corollary that the universe does not experience pain or mental distress; since pain and distress imply change or inhomogeneity in various ways. To deny that would be pointless, unless the universe were at least possibly a sentient being. If Melissus, like Parmenides, began with the assumption that some mental activity occurred, that would for him have the consequence that the universe has mental activity and so is sentient. Next, there can be no such thing as void, which would be ‘nothing’ and therefore does not exist. Hence there must be a plenum, which cannot admit anything from outside into itself, and so there can be no movement, since nothing can budge to make room for the moving thing. Two corollaries: first, no actual dividing of the undivided universe is possible, since that implies movement. Second, there can be no inner variation in respect of density, since ‘less dense’ can be understood only as meaning ‘having more void’. The Relation to Ordinary Experience and the Attack on Sense Perception Where does Melissus’ monism leave common sense and sense perception? The messages of sense perception cannot be true. Melissus bases his attack on the fact that sense perception tells us that change occurs. The argument is: if something is really so rather than so, it cannot cease to be true that this is so. Hence, if our senses tell us that, e.g. this water is cold, and then that this water has heated up, they would be contradicting themselves. So either our senses do not really tell us anything; or there is no change, when again our senses have misled us. The aim is clear: it is to undermine any common-sense objections to the positive doctrine about the universe. It therefore has to be an independent argument. The central idea of this independent argument against change is that nothing that is true can cease to be true, ‘for there is nothing stronger than what is really so’. We need a conception of truth as unchanging; but then the deliverances of sense perception need to be at least reinterpreted, for they give us only time-bound truths. So we need to revise the common-sense notion that sense perceptions are straightforwardly true. CONCLUSION This chapter began with Pythagoras, as the presumed source of some persistently influential thoughts. His influence on philosophy was diffuse and non-specific. His questioning of ‘what we really are’, and his insistence that we are moral agents in a morally polarized world, prepared for the creation of moral philosophy by Socrates and Plato.⁵⁵ Above all, Pythagoras’ insistence on the relevance of mathematics and importance of abstract structure links him to the Eleatics. For what seems to be common to both Pythagoreans and Eleatics is that they take seriously the ideal of mathematically exact knowledge, the constraining force of mathematically rigorous argument, and the cardinal role of abstract structure in the nature of things. (Pythagoras’ other main concerns—the nature and destiny of the self, and the dualism of good and evil—surface in the Eleatics, if at all, only in Parmenides’ cosmology.) The Eleatic philosophers, likewise, had an influence which reached far beyond their few actual followers, and is still active today. Higher standards of precision in statement and rigour of argument are noticeable everywhere in the later fifth century. Metaphysical argument in the Eleatic style appears: in Melissus, and as an intellectual exercise or for sceptical purposes, as in the sophist Gorgias. More significantly, Socrates’ step-by-step, mostly destructive argumentation is Eleatic in spirit; it developed into the philosophical method of Plato and Aristotle, both of whom pay tribute to ‘father Parmenides’. In the philosophy of scientific theorizing, it was Zeno’s dazzling attacks on incipient mathematizing physics that, for a long time, stole the show. Their effect was not wholly negative: they stimulated further investigations into the foundations of mathematics, and its relation to the physical world, which culminated in the work of Aristotle. The more constructive thinking of Parmenides and Philolaus about scientific theorizing has only very recently begun to be understood and appreciated. NOTES 1 The classic study of Walter Burkert (Burkert [2.25]) supersedes all previous discussions of the evidence. It may go too far in the direction of scepticism about Pythagoras as theoretician: see Kahn [4.2]. The (pre-Burkert) catalogue of sources in Guthrie [2.13] Vol I: 157–71 is still serviceable. 2 Those of Aristoxenus, Dicaearchus and Heraclides Ponticus were the earliest and most influential: see Burkert [2.25], 53–109. 3 Certain animal foods were taboo, but a comprehensive ban on the slaughter and eating of animals is improbable and poorly attested for Pythagoras himself. Some under Pythagorean or Orphic influence, such as Empedocles, did observe such a ban. On the whole subject of the taboo-prescriptions and mystical maxims (akousmata, sumbola) of the early Pythagoreans, see Burkert [2.25], 166–92. 4 The fragments of Parmenides have been edited many times. DK is the standard edition for reference purposes; the most reliable and informed recent edition, on matters of Greek linguistic usage and of textual history, is that of Coxon [4.8], which also gives much the fullest collection of secondary ancient evidence. Among minor sources are some other Neoplatonists (Plotinus, Iamblichus, Proclus), and Sextus Empiricus the Sceptic. 5 The scholarly literature is extensive. A small selection is given in the bibliography; the monograph of Mourelatos [4.24] can be particularly recommended for clarity, fullness of information and breadth of approach. The footnotes below offer very brief indications of the spread of opinion on cardinal points; they do not try to outline the arguments needed to justify the reading given in the text. 6 On Xenophanes and his relevance here, Hussey [2.35], 17–32. 7 On the ‘opinions of mortals’ see below pp. 147–9. 8 On alētheiē and related words in early Greek, scholarly discussion has been too often darkened by philosophical prejudice. See the useful study of Heitsch [4.29]; also Mourelatos [4.24], 63–7 and references there. Alētheiē in Parmenides is taken as ‘reality’ by Verdenius [4.30], Mourelatos [4. 24], 63–7, Coxon [4.8], 168. Others understand it as ‘truth’ or ‘manifest or necessary truth’. 9 So Verdenius [4.30]. Allied to this view are those who take the intended subject to be ‘what is’ in the sense of ‘what is the case’ (e.g. Mourelatos [4.24]). Other leading candidates for the role of subject of discourse: ‘that which is’ (so e.g. Cornford [4.19], Verdenius [4.27], Hölscher [4.22], O’Brien [4.12]); ‘what can be spoken and thought of (Owen [4.46]), ‘whatever may be the object of enquiry’ (Barnes [2.8]). That a wholly indefinite subject (‘something’) or no specific subject at all is intended, at least initially, is suggested in different ways by e.g. Calogero [4.18], Coxon [4.8]. 10 On the verb einai ‘be’ in early Greek, see items [4.31] to [4.34] in the Bibliography. The entirely straightforward Homeric usage (‘X is’—‘there is such a thing as X’) is the obvious first hypothesis for the esti and ouk esti paths. Some, though, have put the so-called ‘veridical’ uses (‘be’=‘be true’ or ‘be’=‘be so’, ‘be the case’) in the forefront (e.g. Jantzen [4.23], Kahn [4.42]); others make the use of einai in predication central (e.g. Mourelatos [4.24]); yet others (Calogero [4.18], Furth [4. 41]) have suggested that in Parmenides this verb is a ‘fusion’ of two or more of the normal uses. 11 In fact premiss (2), even without (1) and (3), gives a reason to reject the way that says ‘it is not’. For this way says, about reality generally, that it doesn’t exist or obtain. So by its own account it can’t state any truth, since truth presupposes reality. But there is nothing to show that Parmenides took this short cut. 12 Plato Theaetetus 188c9–189b6, Sophist 237b7-e7. On the versions of this argument in Plato, see e.g. items [4.49] to [4.51] in the Bibliography. 13 It is true that in places the words ‘say’ (legein, phasthai), ‘think’ (noein) and their derivatives are used in ways that seem inconsistent with principle (7). (a) The goddess describes (at least) two ways as those ‘which alone are to be thought’ (B 2. 2), including (at least) one false one. (b) She warns Parmenides against a false way: ‘fence off your thought from this way of enquiry’ (B 7.2), as though it were possible to think its falsities, (c) She speaks of ‘[my] trusty account (“saying”) and thought about reality’ (B 8.50–1), as though it were possible to have un-trusty thought. Of these passages, though, (b) and (c) are rhetorical flourishes, in no way essential to the argument; while (a), which occurs before principle (7) has been introduced, need only mean that at most those two ways can be thought. 14 Xenophanes, for instance, would have questioned the ambition of establishing the truth, rather than mapping out by enquiry coherent possibilities for well-based opinion. 15 Whether this reality is objective or not, is not here at issue. On this question, see ‘Conclusion; the Trouble with Thinking’. 16 Though verbal echoes suggest that Parmenides (not surprisingly) had Heraclitus, with his aggressive use of (?apparent) contradictions, particularly in mind. 17 Some have taken the spatial and temporal ways of speaking literally. Literal sphericity and centre: e.g. Cornford [4.19], Barnes [2.8]; against this, e.g. Owen [4. 46], 61–8. Persistence through time: e.g. Fränkel [4.20], sect 6; Schofield [4.48]; against this, Owen [4.47] The tense-logical principle ascribed to Parmenides at p. 140 above would not commit him to the reality of time in any sense. 18 For example, Plato Sophist 242d4–6; Parmenides 12834–433; Aristotle Metaphysics 1.5, 986b10–19. Recent views on just what the monism amounts to, and of the reliability of Plato’s testimony, have differed widely; Barnes [4.39] maintains that Parmenides is not a monist at all. 19 The contemptuous term ‘mortals’ may itself hint at their double mistake, by itself presupposing that mistake: it is plural, and it implies the reality of death. By their very error, they condemn themselves to appear to themselves as plural and ephemeral. Interesting parallels for this in early Brahmanical monism, e.g. in the Katha Upanishad: …Herein there’s no diversity at all. Death beyond death is all the lot Of him who sees in this what seems to be diverse. (R.C.Zaehner, Hindu Scriptures (Everyman’s Library: London and New York, Dent/Dutton, 1966); 178) 20 That the bare possibility of deception suffices to destroy a claim to knowledge had been pointed out by Xenophanes (DK 21 B 34). 21 On Xenophanes see the section ‘The Promise of the Goddess’. 22 But what it is (if anything), in the nature of reality, that underwrites this practical usefulness, is not clear. There is a hint (‘it had to be that opinions should reputably be’, B1.32) that Parmenides did envisage such a guarantee; and see below on the cosmology as formally parallel to the section dealing with alētheiē. Scholarly opinion has been much divided on the status and purpose of the section concerned with the ‘opinions of mortals’. They have been taken, for example, as a ‘dialectical’ refutation by analysis of the presuppositions of ordinary mortals (Owen [4.46]), a ‘history of the genesis of illusion’ (Hölscher [4.22]), a ‘case-study in self-deception’ (Mourelatos [4.24]); or as reportage of the latest (Pythagorean) fashion in cosmology (Cornford [4.19]). Or, as here, they have been taken to be meant seriously as empirical science (and philosophy of science); so e.g. Calogero [4.18], Verdenius [4.27], Fränkel [4.20]. 23 Empedocles promises magical powers to the disciple who meditates on his cosmology: Empedocles DK B 110 and 111. 24 On the internal structure of the ‘opinions’, and the parallelism with Alētheiē, see Mourelatos [4.24], 222–63. 25 This reading is supported by Aristotle’s testimony (Metaphysics 1.5, 26 ‘Love’ as a power: DK 613, cf. Aristotle Metaphysics 1.3, 984b20–31; struggles of gods: Plato Symposium 195c, Cicero On the Nature of the Cods I.II.28 (DK 28 A 37). There is no need to be puzzled by the appearance of Hesiodic divinities here, if Parmenides, as suggested, is taking an ‘operationalist’ view of what he is doing. 27 On details of the cosmology not discussed here (except for the theory of mental functioning, on which see pp. 150–1; see Guthrie [2.13] II: 57–70. 28 But there is much disagreement about the details. An extended ancient allegorization is found in Sextus Empiricus (Adversus Mathematicos VII.111–14). For the important parallels in Homer, Hesiod and Orphic writings, see Burkert [4. 28]. 29 On the theory of mental functioning, Fränkel [4.20], sect. 3; Laks [4.54]. Both text and meaning of the lines of Parmenides here quoted by Theophrastus are, unfortunately, uncertain at vital points. 30 Of course it does not follow from this that reality’s thinking is what alone constitutes reality, nor that reality is just what thinks itself. (It does follow that reality is not ultimately ‘mind-independent’, in that it is necessarily thought by itself. In this rather special sense, Parmenides is an idealist, but not provably in any wider sense.) 31 Zeno was ‘the Eleatic Palamedes’ (Plato Phaedrus 261d6), the ‘inventor of dialectic’ (Diogenes Laertius Lives VIII.57 (W.D.Ross Aristotelis Fragmenta Selecta, Oxford, 1955:15). 32 Plato’s evidence has not gone unchallenged. Zeno has sometimes been read as attacking Parmenides as well as his opponents, particularly by those who question whether Parmenides was a monist. The attempt of Solmsen [4.72] to undermine Plato’s testimony was countered by Vlastos [4.73]; but even Vlastos doubts Plato’s testimony that all the arguments in the book were directed against plurality. 33 Closeness to common sense is also suggested by the knockabout flavour of ‘making fun’ (kōmōidein). (The phrase ‘as against all the things that are said’ (127d9–10) is too vague to be of use.) But mere unreflecting common sense would not have tried to make fun of Parmenides by arguments, as Zeno implies his opponents did. 34 This fits the earlier suggestions of ad hominem argumentation by Zeno. It does not imply that, in Plato’s opinion, Parmenides’ monism was a monism about the ordinary world. 35 So Aristotle, Metaphysics III 4, 1001b7–16, who calls the argument ‘crude’ because of this assumption. 36 Vlastos ([4.64], 371) points out that the step made here was taken as valid by many later ancient writers. 37 Other possible arguments of Zeno against plurality appear at: Aristotle On Generation and Conception 1.2, 316a14–317a12 (not attributed, and introduced in the context of Democritus’ atomism); and Simplicius Physics 139.24–140, 26, Themistius Physics 12.1–3, Philoponus Physics 80.23–81.7 (attributed to Parmenides or Zeno). On these as possibly Zeno’s: see Vlastos [4.64], 371–2 and Makin [4.66]. 38 On their possible interdependence, see section (e). 39 Compare the assumption needed in (e) above, that anything having size can be divided into two things each having size. 40 Sometimes known as the ‘Dichotomy’. Aristotle’s own solution is at Physics VIII 8, 263a4–b9. 41 Aristotle’s phrase corresponding to ‘at a moment’ is ‘in the now’, i.e. ‘in the present understood as an indivisible instant’. This excludes periods of time, even supposedly indivisible ones. It is possible that Zeno’s argument somehow depended crucially on the instant’s being taken as present (as suggested by Lear [4.67]). 42 Diogenes Laertius (Lives IX.72, DK 29 B 4), using a source independent of Aristotle, gives a summary of an argument which may possibly descend from Zeno’s formulation of step (2): ‘that which moves does not move either in the place in which it is, or in the place in which it is not’. 43 The long illustrative example (240a4–17), implying a lettered diagram, is given as Aristotle’s own contribution; there is no reason to attribute it to Zeno. 44 Attempts to reconstruct a more satisfactory argument include those of Furley [4.63] and Owen [4.68]. 45 In some interpretations, the arguments have been seen as systematically exhausting the theoretical possibilities for pluralism. The idea goes back to the nineteenth century; notable in this connection is the theory of Owen [4.68]. On such a view, time and the track of the moving thing are considered in the ‘Stadium’ and the ‘Achilles’ as divisible ad infinitum; but in the ‘Arrow’ and the ‘Moving Rows’ as ‘atomized’, i.e. as consisting ultimately of indivisible units of extension. 46 On the indications connecting Zeno’s arguments with ‘Pythagoreans’ see Caveing [4.62], 163–80. 47 This is not to deny that modern mathematics enables us to give sharper formulations both of the arguments and of the possibilities for meeting them: see especially Grünbaum [4.75]. 48 See above pp. 145–7. 49 On Aristotle’s description and criticism of this programme, see Huffman [4.78], 57– 64; and Kahn [4.2]. 50 The surviving fragments attributed to Philolaus are due to various late sources (Diogenes Laertius, some Neoplatonists, and the anthology of Stobaeus). Their authenticity is controversial; on this question, see Burkert [2.25], 238–68; [4.78], The reading of Philolaus given here is indebted to Burkert [2.25] and particularly to Nussbaum [4.79], 51 See DK 44 B 1, 2, 4, 5, 6. 52 Nussbaum [4.79], 102. 53 Aristotle Metaphysics I.5, 986b25–7 (‘rather crude’); Physics I 2, 185a10–11 (‘lowgrade’). One purported source, the pseudo-Aristotelian essay On Melissus Xenophanes Gorgias (MXG), is an exercise in ‘philosophical reconstruction’, from which it is not possible to disentangle with confidence any further information about Melissus. MXG is not drawn on here. The most noteworthy modern attempt to rehabilitate Melissus as a philosopher is that of Barnes [2.8], chs. 10, 11, 14. 54 This is a conjectural interpretation of Simplicius’ paraphrase, Physics 103.15: ‘if nothing is, what would one say about it as though it were something?’ 55 It is not safe, though, to read back the mind-body dualism of Plato’s middle period into Pythagoras. BIBLIOGRAPHY Pythagoras and the Early Pythagoreans Texts No authentic writings survive. Collections of early Pythagorean akousmata and sumbola, and other later testimony about Pythagoras and the early Pythagoreans, are in DK [2. 2]: I, 446–80. On the surviving fragments of ‘Orphic’ writings, see West [4.5]. General studies 4.1 Burkert [2–25]. 4.2 Kahn, C.H. ‘Pythagorean philosophy before Plato’, in Mourelatos [2.19]: 161–85. ‘Orphism’ and sixth- and fifth-century religion 4.3 Burkert [1.43],290-304. 4.4 Dodds [2.28], ch. 5. 4.5 West, M.L. The Orphic Poems, Oxford, Oxford University Press, 1983. 4.6 Parker, R. ‘Early Orphism’, in A.Powell (see [2.36]): 483–510. Early Greek mathematics and science 4.7 Lloyd [1.7]. See also [2.27], [2.34], [2.38], [2.41]. Parmenides Texts with translation and commentary 4.8 Coxon, A.H. The Fragments of Parmenides, Assen/Maastricht, Van Gorcum, 1986. 4.9 Gallop, D. 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II: Problèmes d’interprétation, Paris, J.Vrin, 1986. 4.16 Austin, S.Parmenides: Being, Bounds, and Logic, New Haven and London, Yale University Press, 1986. 4.17 Bormann, K.Parmenides: Untersuchungen zu den Fragmenten, Hamburg, Felix Meiner Verlag, 1971. 4.18 Calogero, G.Studi sull’eleatismo, new edn, Florence, La Nuova Italia Editrice, 1977. 4.19 Cornford, F.M.Plato and Parmenides, London, Kegan Paul, 1939. 4.20 Fränkel, H. ‘Studies in Parmenides’, in Alien and Furley [2.15], vol. 2:1–47. 4.21 Heidegger, M.Parmenides, Bloomington and Indianapolis, Indiana University Press, 1992. 4.22 Hölscher, U.Anfängliches Fragen, Göttingen, Vandenhoeck and Ruprecht, 1968. 4.23 Jantzen, J.Parmenides zum Verhältnis von Sprache und Wirklichkeit, Munich, C.H.Beck, 1976. 4.24 Mourelatos, A.P.D. The Route of Parmenides, New Haven and London, Yale University Press, 1970. 4.25 Owens, J. (ed.) Parmenides Studies Today, Afonist 62/1, 1979. 4.26 Reinhardt, K.Parmenides und die Geschichte der griechischen Philosophie, 2nd edn, Vittorio Klostermann, Frankfurt, 1959. 4.27 Verdenius, W.J.Parmenides: Some Comments on his Poem, Groningen, J.B. Walters, 1942. The proem 4.28 Burkert, W. ‘Das Proömium des Parmenides und die Katabasis des Pythagoras’,Phronesis 14 (1969): 1–30. Alētheiē in early Greek and in Parmenides: 4.29 Heitsch, E. ‘Die nicht-philosophische alētheia’, Hermes 90 (1962): 24–33. 4.30 Verdenius, W.J. ‘Parmenides B 2, 3’, Mnemosyne ser. 4, 15 (1962): 237. The verb einai (‘be’) and the concept of being in early Greek and in Parmenides: 4.31 Hölscher, U.Der Sinn vom Sein in der älteren griechischen Philosophie, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, philosophischhistorische Klasse, Jahrgang 1976, 3. Abhandlung, Heidelberg, Carl Winter Universitätsverlag, 1976. 4.32 Kahn, C.H. The Verb ‘Be’ in Ancient Greek, Foundations of Language, suppl. series 16, Dordrecht and Boston, Reidel, 1973. 4.33—‘Why existence does not emerge as a distinct concept in Greek philosophy’, Archiv für Geschichte der Philosophie 58 (1976): 323–34. 4.34 Matthen, M. ‘Greek ontology and the “Is” of truth’, Phronesis 28 (1983): 113–35. Methods of argument, and the nature of thinking 4.35 Lesher, J. ‘Parmenides’ critique of thinking: the poludéris elenchos of Fragment 7’, Oxford Studies in Ancient Philosophy 2 (1984): 1–30. 4.36 Mourelatos, A.P.D. ‘Mind’s commitment to the real: Parmenides B 8.34–41’, in Anton and Kustas [2.16]: 59–80. The choice of ways and the rejection of ‘is not’ 4.37 Finkelberg, A. ‘Parmenides’ foundation of the way of Truth’, Oxford Studies in Ancient Philosophy 6 (1988): 39–67. 4.38 Hintikka, J. ‘Parmenides’ Cogito argument’, Ancient Philosophy 1 (1980): 5–16. The nature of what is 4.39 Barnes, J. ‘Parmenides and the Eleatic One’, Archiv für Geschichte der Philosophie 61 (1979): 1–21. 4.40 Finkelberg, A. ‘Parmenides between material and logical monism’, Archiv für Geschichte der Philosophie 70 (1988): 1–14. 4.41 Furth, M. ‘Elements of Eleatic ontology’, in Mourelatos [2.19]: 241–70. 4.42 Kahn, C.H. ‘The thesis of Parmenides’, Review of Metaphysics 22 (1969): 700–24. 4.43 ——‘More on Parmenides’, Review of Metaphysics 23 (1969): 333–40. 4.44 Ketchum, R.J. ‘Parmenides on what there is’, Canadian Journal of Philosophy 20 (1990): 167–90. 4.45 Malcolm, J. ‘On avoiding the void’, Oxford Studies in Ancient Philosophy 9 (1991): 75–94. 4.46 Owen, G.E.L. ‘Eleatic questions’, Classical Quarterly NS 10 (1960): 84–102; repr. in Allen and Furley [2.15], vol. 2; 48–81, and in M.Nussbaum (ed.) Logic, Science and Dialectic: Collected Papers in Greek Philosophy, London, Duckworth, 1986: 3–26. 4.47—— ‘Plato and Parmenides on the timeless present’, Monist 50 (1966): 317–40; repr. in Mourelatos [2.19]; 271–92, and in Nussbaum. (see [4.46]): 27–44. 4.48 Schofield, M. ‘Did Parmenides discover eternity?’, Archiv für Geschichte der Philosophie 52 (1970): 113–35. The ‘Platonic problem’ of not-being 4.49 Denyer, N. Language, Thought and Falsehood in Ancient Greek Philosophy, London and New York, Routledge, 1991. 4.50 Pelletier, F.J. Parmenides, Plato and the Semantics of Not-Being, Chicago and London, University of Chicago Press, 1990. 4.51 Wiggins, D. ‘Sentence meaning, negation and Plato’s problem of non-being’, in G.Vlastos (ed.) Plato: A Collection of Critical Essays, I: Metaphysics and Epistemology, Garden City, NY, Doubleday, 1971:268–303. Cosmology (including psychology) 4.52 Curd, P.K. ‘Deception and belief in Parmenides’ Doxa’, Apeiron 25 (1992): 109–33. 4.53 Finkelberg, A. ‘The cosmology of Parmenides’, American Journal of Philology 107 (1986): 303–17. 4.54 Laks, A. ‘The More’ and ‘The Full’: on the reconstruction of Parmenides‘ theory of sensation in Theophrastus De Sensibus 3–4’, Oxford Studies in Ancient Philosophy 8 (1990): 1–18. 4.55 Long, A.A. ‘The principles of Parmenides’ cosmology’, Phronesis 8 (1963): 90–107, reprinted in Allen and Furley [2.15], vol. 2:82–101. 4.56 Schwabl, H. ‘Sein und Doxa bei Parmenides’, in H.-G.Gadamer (ed.) Um die Begriffswelt der Vorsokratiker, Darmstadt, Wissenschaftliche Buchgesellschaft, 1968:391–422. Miscellaneous 4.57 Furley, D.J. ‘Notes on Parmenides’, in Lee, Mourelatos and Rorty (see [3.43]): 1–15. 4.58 Gadamer, H.-G. ‘Zur Vorgeschichte der Metaphysik’, in Gadamer (see [4–56]): 364–90. Zeno Texts with translation and commentary 4.59 Lee, H.D.P. Zeno of Elea, Cambridge, Cambridge University Press, 1936. 4.60 Untersteiner, M.Zenone: testimonianze e frammenti, Florence, La Nuova Italia Editrice, 1963. Translations of the relevant parts of Plato Parmenides can be found in Cornford [4.19]. The testimony of Aristotle in the Physics is translated in: 4.61 Barnes, J. (ed.) The Complete Works of Aristotle, Princeton, NJ, Princeton University Press, 1984. General studies 4.62 Caveing, M. Zénon d’Élée: prolégomènes aux doctrines du continu, Paris, J. Vrin, 1982. 4.63 Furley, D.J. Two Studies in the Greek Atomists, Princeton, NJ, Princeton University Press, 1967:63–78. 4.64 Vlastos, G. ‘Zeno of Elea’, in P. Edwards (ed.) The Encyclopedia of Philosophy, vol. 8, New York, The Macmillan Company and The Free Press, 1967: 369–79. Reprinted in Vlastos Studies in Greek Philosophy, ed. D.W. Graham, vol. 1, Princeton, NJ, Princeton University Press, 1995:241–63. The arguments against plurality 4.65 Fränkel, H. ‘Zeno of Elea’s attacks on plurality’, in Allen and Furley [2.15], vol. 2: 102–42. 4.66 Makin, S. ‘Zeno on plurality’, Phronesis 27 (1982): 223–38. The arguments about motion 4.67 Lear, J. ‘A note on Zeno’s arrow’, Phronesis 26 (1981): 91–104. 4.68 Owen, G.E.L. ‘Zeno and the mathematicians’, Proceedings of the Aristotelian Society 58 (1957/8): 199–222; repr. in Allen and Furley [2.15], vol. 2:143–65; in Salmon [4.76]: 139–63 and in Nussbaum (see [4.46]): 45–61. 4.69 Pickering, F.R. ‘Aristotle on Zeno and the Now’, Phronesis 23 (1978): 253–7. 4.70 Vlastos, G. ‘A note on Zeno’s arrow’, Phronesis 11 (1966): 3–18; repr. in Allen and Furley [2.15], vol. 2:184–200, and in Vlastos (ed. Graham) (see [4.64]); vol. 1: 205–18. 4.71 ——‘Zeno’s racecourse’, Journal of the History of Philosophy 4 (1966): 95–108; repr. in Allen and Furley [2.15], vol. 2:201–20, and in Vlastos, ed. Graham (see [4. 64]), vol. 1:189–204. Plato’s testimony on Zeno 4.72 Solmsen, F. ‘The tradition about Zeno of Elea re-examined’, Phronesis 16 (1971): 116–41; repr. in Mourelatos [2.19]: 368–93. 4.73 Vlastos, G. ‘Plato’s testimony concerning Zeno of Elea’, Journal of Hellenic Studies 95 (1975): 136–62, repr. in Vlastos, ed. Graham (see [4.64]); vol. 1: 264–300. Zeno, Aristotle and modern philosophy 4.74 Bostock, D. ‘Aristotle, Zeno and the potential infinite’, Proceedings of the Aristotelian Society 73 (1972/3): 37–51. 4.75 Grünbaum, A. Modern Science and Zeno’s Paradoxes, London, Allen and Unwin , 1968. 4.76 Salmon, W.C. (ed.) Zeno’s Paradoxes, Indianapolis and New York, BobbsMerrill, 1970. 4.77 Sorabji, R. Time, Creation and the Continuum, London, Duckworth, 1983. Philolaus and ‘The People Called Pythagoreans’ Text with commentary 4.78 Huffman, C.A. Philolaus of Croton: Pythagorean and Presocratic, Cambridge, Cambridge University Press, 1993. Studies 4.79 Nussbaum, M.C. ‘Eleatic conventionalism and Philolaus on the conditions of thought’, Harvard Studies in Classical Philology 83 (1979): 63–108. See also [4.1] and [4.2]. Melissus Text with translation and commentary 4.80 Reale, G. Melisso: testimoniaze e frammenti, Florence, La Nuova Italia Edi trice, 1970.