Pythagoreans and EleaticsEdward HusseyPYTHAGORAS AND THE EARLY PYTHAGOREANSPythagoras, a native of Samos, emigrated to southern Italy around 520, and seemsto have established himself in the city of Croton. There he founded a society ofpeople sharing his beliefs and way of life. This spread through the Greek citiesof southern Italy and Sicily, acquiring political as well as intellectual influence.Some time after his death, the original society broke up and its continuity waslost; yet groups of self-styled ‘Pythagoreans’ appeared repeatedly thereafter.Palpably reliable evidence about early Pythagorean activities is so scanty thatsome initial scepticism is in order about Pythagoras as a philosopher, or as a‘natural philosopher’ in Ionian style.1Sporadic early reports depict Pythagoras asprimarily a magician and miracle-worker; and, on the theoretical side, a collectorand expositor in dogmatic style, rather than a creator or investigator. It is clearthat some doctrines later seen as ‘Pythagorean’ were already current, around thesame time, in the theological and cosmological poems attributed to Orpheus.Plato barely mentions Pythagoras by name; but incorporates into some of hismyths material which is likely to be genuinely Pythagorean. After Plato,philosophically-inspired reconstructions of Pythagoras begin to appear, in whichhe is represented as the head of a regular school, promoting research intophilosophy and the mathematical sciences; or as an enlightened statesman andinstructor for political life.2At best, even when based on good sources, thesefourth-century accounts (which themselves survive only in later reports) aremore or less anachronistic idealizations. Still less reliance can be placed in thegreat 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 interestswere then attributed to Pythagoras.The cosmology of Parmenides (see below)and the poems of Empedocles have a substratum of ideas that may be suspectedto be Pythagorean in inspiration. All in all, there is a body of general ideas,appearing by the mid fifth century and reasonably firmly associated withPythagoras, which was to be influential in a programmatic way throughout thecentury, and on Plato, above all, in the fourth. These ideas may be grouped underthe 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, andmore or less closely tied to it were various ideas, not necessarily clearlydistinguished by Pythagoras himself.(1) One was the radical redefinition of theselfwhich the doctrine involved: anew belief about what we human beings really are. The implication is that we arenot, as in traditional Greek belief, mortal beings, with at best a shadowy afterlifein Hades, but that we are truly immortal, perhaps fallen gods. Our real selveshave always existed and will always exist. And the heritage which they havelost, but may recover, is a divine, paradise-like existence.(2) Connected with that is the belief that we arenot at homein the body, infact that this life and all incarnations are really punishments, or at best periods ofrehabilitation. It follows that we are not hereprimarilyto enjoy ourselves. It doesnotnecessarily follow that the body is intrinsically evil, or that the ordinarykinds of enjoyment are bad in themselves. That extreme puritanical conclusionmay have been drawn by some, but probably not by Pythagoras himself.(3) Pythagoras saw the world as sharply and systematically polarized betweengood and evil. The real role of the self is to be a moral agent, to participate in themoral struggle; and it is rewarded and punished accordingly. A systematiccosmological dualism, associating all aspects of the world with the good—evilpolarity, seems also to have been characteristically Pythagorean. It may be thisdualism which accounts (psychologically at least) for the doctrine of the cyclicalrecurrence of events. Given a systematic dualism of good and evil, good musttriumph, but evil cannot be abolished. The simplest solution is to suppose thatthere 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, isobscure.3Mathematics and the Importance of Abstract StructureAnother leading idea of Pythagoras was that of the key importance ofmathematical structuresin the universe.Pythagoras himself was no creative mathematician; there is no reliableevidence that he proved any mathematical theorems at all (not even ‘Pythagoras’theorem’). The evidence suggests rather that the Pythagoreans’ focus was on aspeculative numerology applied to the cosmos. (For example, the dualism ofgood and evil was paralleled, and perhaps meant to be explained, by the dualismof 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 abstractstructure in things; and secondly that this abstract structure may be the key to theessential nature of things. It is through these ideas that Pythagoras became themidwife of pure mathematics, which began to develop from now on; and indeedthe founder of the whole mathematical side of scientific theory.PARMENIDESThe Poem of ParmenidesParmenides was a citizen of the Greek city of Elea in southern Italy. Hisphilosophical activity belongs to the first half of the fifth century. He expoundedhis thoughts in a poem, using Homeric hexameter verses. Verse for publicrecitation was then still a natural medium for diffusing ideas; yet the ‘naturalphilosophers’ of the sixth century had chosen prose, to show their rejection of theauthority of the poets, and their closeness to ordinary experience. Parmenides’choice of hexameter verse may imply in its turn a rejection of the naturalphilosophers.The poem begins with a first-person narrative of a journey. Accompanied bythe daughters of the Sun, the narrator rides in a chariot into remote regions, toreach ‘the gate of the paths of night and day’. Passing through, he is welcomedby a goddess, who promises that he is to ‘find out everything’. She goes on tofulfil the promise, in an exposition which constitutes the whole of the rest of thepoem.Over one hundred verses of the poem survive, including all of the introductorynarration and probably almost all of the first and fundamental part of thegoddess’s exposition. Together with comments of Plato, Aristotle and others, thisis a fine corpus of first-rate evidence, the survival of which is due principally toSimplicius, the sixth-century Neoplatonist commentator on Aristotle.4Yet controversy dogs almost every part of Parmenides’ thinking, for aconjunction of reasons. First, there are gaps in our information at certain crucialpoints. Next, Parmenides’ language is often obscure, in spite of his evidentstriving for maximal clarity. The constraints imposed, by the metre andvocabulary of epic verse, on the exposition of a subject-matter for which they werenever designed, are bad enough. Then there is the problem of supplyingwhatever, 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 manypoints.5The Promise of the GoddessCentral to the understanding of Parmenides is the promise made by the goddess:It is necessary that you find out everything: both the unmoving heart ofwell-rounded reality[alētheiē],and the opinions of mortals, in which thereis 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 goingthrough everything.(DK 28 B 1.28–32)The division of the objects of discovery into two determines the structure of therest 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, concernedwithalētheiē,will contain only certainties. The second part, of which the truthcannot 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 completesystem.6If we leave on one side, for the moment, the ‘opinions’ and what is here saidabout them,7the next fundamental question is the meaning of the wordalētheiē.In English, it is usually translated by ‘truth’, to which it seems to correspond inthe spread of its early usage. The adjectivealēthēs,from which it is formed, hasmuch the same spread as ‘true’ (covering the areas indicated by the words‘truthful’, ‘accurate’, ‘real’, and ‘genuine’; though not that of ‘faithful’). But inParmenides the translation ‘reality’ for Parmenides’alētheiēmust 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 truestatement isabout,and is guaranteed by: the underlying actual state of things, thereality. So, later on, the goddess marks the end of the first part by saying, ‘At thispoint I end for you my trusty tale and thought concerningalētheiē.’8While ‘reality’ may be taken as the closest word to the intended primarymeaning here, it is also true thatalētheiē,as in Homer, carries implications aboutthecertaintyof what is said, and of thecorrectnessandaccuracyof the methodby which it is found. Parmenides wants to insist on these points too; which hedoes, here and elsewhere, primarily by words indicating trustworthiness and itsguarantees(pistos, pistis, peitbō). The goddess promises not only insight intosome reality, but a guarantee of the truth of the insight.This reality is ‘well-rounded’, presumably because it forms a satisfactorilycoherent and closed system; and it has an ‘unmoving heart’, presumably becauseat least in essentials it is not subject to change. Both of these thoughts reappearsignificantly later.The Choice of WaysThe narrator’s ‘finding out’ of reality is represented as a matter of simplylistening to the goddess. Yet there are hints that his was an active pursuit of thetruth; it was his own desire that started him off. The metaphor of travel, and theimplication of active pursuit in ‘finding out’, are now carried further. There istalk 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 expositionenvisages an active rethinking, by the listener, of the course of Parmenides’thinking.Why theactiveparticipation of the listener is needed becomes clear from whatfollows. The exposition concerning reality is in the form of a deductiveargument, which one cannot properly follow and grasp, without recreating it inthe 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 themention of which it is, naturally enough, identified in the exposition. Rigorousdeductive arguments were possibly already in use in mathematics; but they musthave been novel to most of Parmenides’ contemporaries. Hence the effortsParmenides makes, using the metaphor of the ‘ways’, to keep the course of thearguments, their interrelation and their overall effect, absolutely clear.Come then, I will tell you (and you, listen and take in the story!), whichways of enquiry alone are to be thought: the one, that it is and cannot notbe, 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—thistrack, I tell you, isutterly unconvincing…(DK 28 B 2.1–6)This presents, as a starting-point, a choice between two such ways, which aremutually exclusive. Clearly, though, they are not jointly exhaustive, since theremight also be ways involving unrealized possibilities (‘it is, but might possiblynot be’, ‘it is not, but might possibly be’). In fact, in the sequel, Parmenides willpresent only one more way, the ‘way of mortals’, which, as stated, is evidentlyself-contradictory. The two named here are apparently the only ones that ‘are tobe thought’; and, of these, one is to be rejected as false.What is going on here seems to be as follows. Parmenides holds (on whatgrounds, remains to be examined) that to speak of unrealized possibilitiesinvolves 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 theless, 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 acontradiction, 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 onlyremaining possibility. Accordingly, it is accepted as true by elimination, and itsconsequences examined.What then is meant here by ‘it is’ and ‘it is not’? First, what is ‘it’? In theGreek, the verbestistands alone, as Greek verbs can, without even a pronoun tofunction as the grammatical subject. But unless Parmenides is making someradical and improbable departure from ordinary practice, an intended subject ofdiscourse, of which ‘is’ and ‘is not’ are here said, must have been meant to bereadily supplied from the context. Unfortunately for us, the original context isnow partly missing. Between the promise of the goddess and the statement of thetwo ways, some now lost stretch of text, probably not long, once stood. None theless, what remains is sufficient for near-certainty as to the intended subject.The ways are ‘ways of enquiry’. An enquiry, then, is presupposed as beingalready afoot. What that enquiry is concerned with, is likely to be what the firstpart of the goddess’s promise is concerned with: reality. It is true that the wordalētheiēnowhere appears subsequently in the subject place attached to the verbesti. In the exploration of the true way that says ‘it is’, the subject of ‘is’ appearssometimes, cloaked in the unspecific designation(to) eon,‘that which is’. Thisphrase, though, can be taken without artificiality as another, and metrically moreconvenient, way of referring toalētheiē. (So taken, it involves a metaphysicalpun: see below on the meanings of the verbeinai.)This conclusion, thatalētheiē,in the sense of ‘reality’, is the intended subject,is central to the interpretation of Parmenides to be presented here.9It has beenreached by a simple yet powerful argument. It has yet to be subjected, though, toa series of severe tests. A reconstruction of Parmenides deserves acceptance onlyif it makes convincing sense of thewholeof the surviving evidence.The first test arises immediately. Can one make sense of an initial choicebetween ‘necessarily, reality is’ and ‘necessarily, reality is not’? At this point, wemust also ask about the possible meanings of the verbeinai(‘be’).In general, it seems to make sense, whateverxmay be, if one is making anenquiry intox,to start by asking ‘is there any such thing asxor not?’ Thenormal usage of the verbeinaieasily covers such a sense of ‘is’. In launching anenquiry intoalētheiē,understood in extension from Homeric usage in a ‘summedsense’, as what would be jointly indicated by all true statements, Parmenides is ineffect asking, sceptically, ‘why do we have to suppose that there is any suchthing in the first place?’Since this entirely normal and familiar use ofeinaifits the context so well,there is no need at the outset to look for more exotic possibilities.10Later, though,when the subject of discourse is referred to as ‘that which is’(to eon),a differentuse of the verb bears the logical weight. Another common use ofeinaiis that inwhich it means (said of possible states of affairs) ‘obtain, be the case’. Ifalētheiēis thought of as a ‘summed state of affairs’, then to say that there actually existssuch 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 ofDescartes. Both start with a philosophical enquirer, an apparently isolated mind,trying to establish what it can know with absolute certainty. Parmenidesapproaches the problem via the concept ofalētheiē,the reality that would have tounderwrite any knowledge. What is next to be examined is his argument toestablish that there must be such a reality. It is here that his further initialpresuppositions, if any, are to be found. This is the argument that rejects the waythat says ‘it is not’.The Rejection of ‘it is not’The passage in which Parmenides justifies the rejection of the second way isprobably 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)…thistrack, I tell you, is utterly unconvincing [or: undiscoverable]; foryou would not recognize [or; become aware of] what is not (for that cannotbe 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 theargumentative weight is thrown, are, in the aorist forms used here, commonHomeric words for ‘recognize’ and ‘point out’; they are cognitive ‘successverbs’. Their objects can be either ordinary individuals or ‘that’-clauses. So it isnecessarily true that (1) ‘you would not recognize [to be the case: i.e. getknowledge of], or point out [as being the case: i.e. show, demonstrate], what isnot’.The natural way to expand (1) into a relevant argument is as follows. If thereis no such thing as reality, then no-one can recognize it, nor point it out. In thatcase 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 areavailable: (2) that knowledge involves or consists in awareness of reality, andcommunication of knowledge involves or consists in the pointing-out of reality;(3) that knowledge and its communication are possible.11Did Parmenides supply any support for (2) and (3)? As to (2), there is no wayof telling; maybe it was taken as following immediately from the meaning ofalētheiē,as that which truths areabout,and knowledge isof. As to (3), first of allsome evidence of Aristotle comes in here opportunely. Aristotle identifies, as anunderlying thesis of the Eleatics, that ‘some knowledge or understanding(phronēsis)is possible’ (Aristotle,On the HeavensIII. 1, 298b14–24). Thissupports 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 showthat enquiry is always successful. It will now be suggested that in fact theremaining pieces of text, dealing with the rejection of ‘it is not’, give thesupporting 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, butwhat 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 theyshould be treated, at least initially, as (part of) a separate argument from the onereconstructed above. Once again, Homeric usage is an important guide. ‘It is forbeing/thinking/saying’ represents an idiom familiar in Homer: ‘A is for x-ing’means either ‘there isAavailable to do some x-ing’ or ‘there isAavailable to bex-ed’.Much depends here on what sort of thing might be said to be ‘available for sayingand thinking’. In Homeric usage, the object of the verbs ‘say’ and ‘think’ isusually expressed by a ‘that’-clause. What the clause describes is the state ofaffairs, in virtue of which the saying or thinking is true or not.An interpretation is possible within these linguistic constraints. Parmenides isarguing for the thesis that what can be said and thought, must actually be thecase; i.e. that one can say and think only ‘things that are’, these being thought ofnot as true statements but as actual states of affairs.The argument has a very close affinity with one which troubled Plato invarious places, notably in theSophist(but he did not accept it as correct, in theSophistor elsewhere).12The ‘Platonic problem’ (as it may be called forconvenience) starts from the premiss (4), that saying and thinking must have, asobjects, a state of affairs, actual or not; i.e. a genuine case of saying or thinkingmust 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 mustexist. Now, (6) for a state of affairs, to exist is just to be actual. Hence (7) onlyactual 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 afar-reaching modal variation of it. What can be thought and said, must by (4) beat least a possible state of affairs (‘it is for being’). But (8) there can be nounrealized, ‘bare‘ possibilities. The argument to this effect is brought outeffectively by the idiomatic ‘is for being’. What ‘is for thinking’, also ‘is forbeing’, and therefore necessarily is. There can be nothing more to ‘being forbeing’, than just being. Anything that is not, cannot be in any sense, and socannot even ‘be for being’. Hence every possible state of affairs is actual, and soit 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 orsaying, from the strong modal principle (8), that there are no unrealizedpossibilities. They are, of course, akin; in both (7) and (8), there is a refusal tohave any philosophical truck whatever with any non-existent state of affairs. It isprinciple (8) that also supplies what is obviously needed: an explanation ofParmenides’ 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.13Inthe 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’ butin the constructing of fictions of some sort,apparentstatements. 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 overallreconstruction of the rejection of ‘it is not’. The overall effect of the rejection ofthe way ‘it is not’ is to establish that since there must be true thinking andsaying, there must be some objective reality. The first piece of text ((A) above) isa sketch of this overall argument, using premisses (1) (2) (3). In reply to thisargument, a sceptic might question premiss (3): granted that thinking and sayingoccur, why should it be that some thinking and saying must betrue?14SoParmenides engages with this objection in the further argument which terminatesin the second piece of text ((B) above). This argues that (7) there is no such thingas 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 hisfirst task is to show that knowledge is possible, his next problem, having shownthat, is of a Kantian kind: given that knowledge and correct thoughtmustbepossible, 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 asreality. There must be something for the knowledge to beabout,andof,which bybeing so guarantees it.15The ‘Way of Mortals’After rejecting the way that says ‘it is not’, the goddess mentions, asunacceptable, yet another way, not previously mentioned:Then again [I shut you out] from this [way], which ignorant mortals wanderalong [or. construct], two-headed (for it is helplessness that steers thewandering mind in their breasts); they drift along, deaf and blind, in adaze, confused tribes: they accept as their convention that to be and not tobe is the same and not the same [or:that the same thing and not the samething 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 themass. The ‘confused tribes’ can hardly be just a particular group of theorists.16Besides, the goddess associates this way with an unthinking interpretation of theevidence of the senses, which is due to ‘habit of much experience’ and thereforepresumably 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 lazyhabits in selecting and interpreting the information given by sense perception.The distinction had already been made by Heraclitus, who remarked: ‘Badwitnesses to people are eyes and ears, if [those people] have uncomprehendingsouls’ (DK 22 B 107). It is reason that must dictate how sense perception is to beunderstood, and not the other way round.On what grounds Parmenides took ordinary people to be enmeshed incontradiction about reality, is not yet clear. The reference to ‘the controversialtest which I have stated’ must include the rejection of ‘it is not’. Parmenides maysee people as accepting both ‘it is’ and ‘it is not’, because, while they see theneed to assume some kind of reality, they at once contradict that assumption, asParmenides believes, by allowing reality to contain features which are excludedby the test. For example, the existence of unrealized possibilities, and other thingswhich are yet to be expressly excluded. The ‘controversial test’ probablyincludes also the negative implications of what is yet to come: the examinationof 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 manysigns: 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 isall together now—one, coherent.(DK 28 B 8.1–6)The ‘signs’ are best taken as the proofs, to follow, of the properties announcedhere as belonging to ‘what is’(eon),i.e. reality. Evidently the deduction of theconsequences 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 itgrow? Nor will I let you say or think that [it did so] out of what isnot, for it is not sayable or thinkable that it is not. Besides, whatnecessity 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 thispart. For convenience, the subject (‘what is’ or ‘reality’) will be denoted byE.Suppose thatEdoes at some time come-to-be. Then Parmenides asks:out of whatdoes 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-evidentlytrue; it is plausible to connect it with other places in this argument where heseems to have some variety of the Principle of Sufficient Reason in mind.So, ifEcomes-to-be, it comes-to-be out ofF(say). Then forF,in turn, there arethe two possibilities:Fis, orFis not. Parmenides considers the secondpossibility, but not, apparently, the first one. This is a first problem.There is an extra twist to it. So far, we have considered Parmenides’ reasoningsabout ‘it is’ and ‘it is not’ without taking account of the ambiguities of thepresent tense. The rejection of the way ‘it is not’ does not call for these to beconsidered. But the Greek present tense is ambiguous in the same ways as theEnglish one; and where, as here, possible past and future events are beingdiscussed, it becomes necessary to distinguish the various uses. ‘It is’ and ‘it isnot’ may be timeless, or refer to the time of the coming-to-be, or to the time ofutterance, if that is different. Parmenides gives us no help at all on this point; butit is plausible to assume that he means the question ‘isFor is it not?’ to beunderstood 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:Ecomes-to-be out ofF;isF,or is it not, at the time ofE’scoming-to-be? First, ifFis, at that time, then at that time it is part ofE,sinceE(on the interpretation followed here) is the whole of reality. But nothing cancome-to-be out of a part of itself, since that does not count as coming-to-be atall. This point will account for Parmenides’ failure to examine the supposition ‘Fis’.Second, suppose then thatF,at the time ofE’scoming-to-be, is not. This isthe case which Parmenides examines. He gives two arguments.One argument is: ‘It is not sayable or thinkable that it[F]is not’. This invokesthe results of the rejection of the way that says ‘it is not’. By principle (7), ‘Fisnot’ is not sayable or thinkable, because it is not true.But why is ‘Fis not’ not true? By principle (i), if it were true, then F would notbe capable of being recognized or pointed out; so it could not figure intelligiblyin 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),nosentence of the form ‘Xis not’ can be true.It might be objected that this principle (11) (so far as has been shown) appliesonly to what is notat the time when the utterance is made;in other words that heretoo there is a crucial ambiguity in the present tense. What ifFis not, at the timeof coming-to-be, but is, at the time of speaking? In that case, it would seem to bepossible to recognize and point outF,and say of it intelligibly that it was not, atsome earlier time. Again, Parmenides seems unaware of this objection. Is there ahole in his proof? It is more charitable, and perhaps more plausible, to supposethat Parmenides tacitly applies a principle of tense-logic such as this: (12) for anytimet,and any statementS,if it is true now thatSwas (will be) true att,then itwas (will be) true attthatSis true then. By this means, Parmenides can transferthe force of principle (1) to the time of the supposed coming-to-be. ‘Fis not’cannot be (have been, be going to be) true atanytime, because, if so, it would betrue at the relevant time that ‘Fis not’; but, by principle (11), the truth of ‘Fisnot’ (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 whichinvolves reference to things that supposedlyat any timeare not.The other argument begins: ‘Besides, what necessity would have driven it onto 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 inwhich there is nothing that is, there could hardly be any way of grounding thenecessity. Even if there were, why did it operate ‘later or sooner’: at oneparticular time rather than at another?The rest of section (a) is occupied, on this interpretation, only withrecapitulation and summing up. Only the case of coming-to-be has beendiscussed; there is no parallel treatment of ceasing-to-be, presumably because thearguments 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 respectmore in any one place (which would obstruct it from holdingtogether) nor in any respect less: all is full of what is. Hence it is allcoherent, 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 dividesE. What could thatbe?By the fact thatEis ‘summed’, comprising all that is, anything other than E hasto be something that is not. By the same argument as before, it can never be trueto say thatEis divided by something that is not. HenceEis not divided byanything other than itself. This limb of the argument, though suppressed here asobvious, appears in the parallel passage at 8.44–8.What is here explored is the other possibility: thatEis divided by itself, i.e. byits own internal variations. The possibility of internal qualitative variations is notmentioned; presumably they would not count as creating divisions. What ismentioned 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 likemanner’. 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 fixedthere; for strong necessity holds it in the fetters of the limit, whichfences 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 todeduction. It must be read backwards from the end. The starting-point is thatreality (E) is complete or ‘not lacking’. Once again the strategy is the same, thatofreductio ad absurdum. SupposeEis lacking; then E must lack something.What is this something? It cannot be part ofE,for then it would not be lackingfrom E. Therefore it is not part ofE,and hence is not; but it is not true that it isnot, by the now familiar argument.Given thatEis not lacking, it is complete, and has a ‘limit’. The word used herehas no close English equivalent: Homer’s usage applies it to anything that marksor achieves any kind of completion. Here, the ‘limit’ functions as a constraint onreality: the need for completeness is a (logical) constraint. Completeness rules out,in particular, all change and movement, and enforces uniqueness: reality is ‘byitself’ or ‘on its own’. Why?Completeness has these consequences because it embodies the principle thatEcontains everything that is. This now enables Parmenides to get some grip on theproblem 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 thereality of the past (or future), does not here have to be decided. Either way, therecan be no such thing as change or movement of reality: for those would implythe previous existence and present non-existence of some part of reality. Eitherpast (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 inrelation to anything else. For ‘anything else’ that could be taken into account couldnot 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 strikingimage:But since there is an outermost limit, it is perfect from every direction, likethe 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 neitheris there what is not, which would obstruct it from holding together, nor isthere any way in which what is would be here more and here less thanwhat is; for it is all immune from harm. For, equal to itself from everydirection, 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’ (theuniformity of its manner of being) and its perfection, is deduced its symmetryabout a centre. What is surprising is not the symmetry, since that could be seenas a form of perfection, but the ‘middle’, a privileged central location, introducedwithout explanation (on this see the next section).The Nature of RealityHaving followed the proofs of the properties of reality, one may still be uncertainjust what those properties are. How strong, for instance, is the claim that realityis ‘one and coherent’ meant to be? Does ‘completion’ include spatial andtemporal boundedness, and in general does reality have any spatial or temporalproperties at all? How, if at all, is it related to the world apparently given inordinary experience?Is reality spatial or temporal or both?First, the question of spatial and temporal properties. Parmenides shows nohesitation in applying to reality words which would normally imply spatial andtemporal 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 impliesnothing about space or time; but it is also said that this limit ‘fences it about’ andis ‘outermost’. The simile of the ball might not be meant spatially, but what ofthe 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 locationor extension. This point chimes with another: if one supposes that reality isspatially extended, its spherical symmetry is problematic. The ‘limit’ cannotpossibly be meant as a spatial boundary, since for reality to be bounded in spacewould be for it to be incomplete. It must be right, then, to take the spatial termsmetaphorically. They must be aids to grasping how reality inhabits a kind of‘logical space’. This works out smoothly. The ‘spherical symmetry’ mustexpress metaphorically the point that reality is exactly the same, however it isviewed by the mind: it presents no different ‘aspects’. The ‘middle’, about whichit 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, notlogically plural, not self-contradictory. ‘What is lies close to what is’, in thesense that any internal variation between parts does not constitute an essentialdifference. ‘Staying the same in the same and by itself makes the point thatreality does not exist in relation to anything other than itself, and so not inrelation to any external temporal or spatial framework; it is unique, and providesits own frame of reference. The metaphorical understanding of these terms issupported, above all, by the nature of the proofs. As has been seen, these makeno 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 toconceive of a state of affairs as being in, and lasting through, time. Parmenidesargues against anychangein reality; but this is still consistent with the view thatreality is something which persists, without change,throughtime. Did he wish togo 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 andthrough time, time must itself be seen as an aspect of reality. The basic temporalphenomenon must be the temporal extension of reality. This already goes someway beyond the simple notion of a persisting reality.Second, the argument (section (c) above) to the conclusion that reality ischangeless 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, sinceit is all together now.’ If one cannot even say that ‘it was’ and ‘it will be’, thenone 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 ofbefore and after, and therefore no meaning in tensed statements.The metaphorical interpretation of these spatial and temporal terms, as appliedto reality, does not, of course, imply that for Parmenides the spatial and temporalproperties of ordinary objects areillusory. It still remains to be seen (in the nextsection) how Parmenides deals with the world of ordinary experience.17In what sense is reality one?There is no doubt that Parmenides was a monist of some kind; the comments ofPlato and Aristotle alone would prove it, even if the fragments were lacking.18The 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) showsinter aliathat there is nothing other than reality (it is ‘unique’ or ‘byitself’). 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 apassage to be discussed below) they decide to ‘name two forms, one of whichought not [to be named]; this is where they have gone astray’ (B 8.53–4). Thefundamental error of the ‘mortals’ of the cosmology is to allow there to be twodifferent subjects of (apparent) discourse, rather than just one.Parmenides is then committed at least to a logical monism: there is one andonly one subject about which anything is true. This seems also to be themaximum that needs to be claimed, and the maximum that is imputed byAristotle’s remark that ‘[Parmenides] seems to be getting at that which is one indefinition’ (MetaphysicsI.5, 986b18–19). The argument for unity (section (b))demands nothing more. In particular it does not exclude internal variation, nordoes it impose qualitative homogeneity. Reality consists of a set of facts true ofitself. It is not excluded that reality might be constituted by more than one suchfact; and after all many statements about reality are made by the goddess herselfin the course of the argument; it would be absurd to suppose that they are meantto be seen as identical. Even though one may talk (as even the goddess sometimesdoes) in a misleading conventional way, this ‘plurality’ of facts must not beunderstood as a genuine plurality: what we are really dealing with here isdifferent aspects of reality. Even when different parts of reality are distinguished,the correct formulation does not admit them as subjects in their own right, butspeaks only of ‘what is’: ‘what is comes close to what is’; ‘what is cannot behere more or here less than what is’.So when the goddess distinguishes ‘the unmoving heart’, and the ‘middle’, ofreality, implying that there is also a peripheral part, she must be understood asspeaking, in a conventional way, about a situation which could be describedmore correctly. What she means by it, has now to be considered.The Errors of ‘Mortals’ and the Place of Ordinary ExperienceIt remains to ask how this reality is supposed to be related to the world ofordinary experience.In Parmenides’ rejection of the ‘way of mortals’, it was seen that senseexperiencein itself did not seem to be blamed for their mistakes. It was mortals’habitual misinterpretation of sense experience which caused them to fall intoself-contradiction.After the exploration of the nature of reality, it is possible to specify thefundamental 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 notfind thinking apart from what is, in which it is made explicit. For nothingother is or will be outside what is, since that has been bound by fate to bewhole and unchanging. Hence it will all be [just] name, all the things thatmortals have laid down, trusting them to be real, as coming-into-being andperishing, 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 realchange. Such people are committed to the reality of what are in fact conventionalfictions or ‘names’, taken as putative objects of thinking and saying. The passagestarts 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 (withits various consequences) the only true thought. Because there is no saying andthinking something false, apparent false thought must be ‘mere names’. So too atthe beginning of the cosmology (see next section), where we shall see that even‘mere names’ (like other conventions, so long as intelligently made and properlyobserved) can have their uses.It is the subjects of ordinary discourse, the things that we normally identify asthe plural changing contents of the world, that are here denounced as just‘name’, conventional noises and nothing more. Since statements about themcannot be true, they are not capable of being genuinely spoken and thoughtabout.The self-contradictory ‘way of mortals’ is now explained. ‘Mortals’ recognizethe existence of an objective reality, and therefore say ‘it is’. But they also haveto say ‘it is not’, because they take reality to be something truly plural andchanging.19The denunciation of ‘mortals’ doesnotexclude the substantial reality of theordinary world of experience—provided a construction is put upon that worldwhich is radically different from the usual one, on the two key points of pluralityand change. The temporal dimension may be kept, so long as it is in effectspatialized, with becoming and change ruled out as an illusion. The multiplicityof things in both spatial and temporal dimensions may be kept, so long as it isseen 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 byreasoning, which alone are objects of knowledge. The outside is ‘what meets theeye’: the contingent snippets of reality as perceived by the senses. Senseperception,even when in fact veridical, presumably does not yield knowledgebecause of the possibility of deception.20What it reveals, not being part of the coreof 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 theuselessness of all other cognitive states. Far from it. He recognizes both thepossibility and the practical value of ‘opinions’ about the cosmos, whenorganized into a plausible and reliable system. Here, building on the ideas ofXenophanes, he turns out to be the first recognizable philosopher of science.21This is why the conclusion of the investigation of reality does not mark theend of the poem. There still remains the second half of the promise of thegoddess, which must now be recalled:It is necessary that you find out everything: both the unmoving heart ofwell-rounded reality[alētheiē],and the opinions of mortals, in which thereis 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 goingthrough everything.(DK 28 B 1.28–32)This promise of an exposition of ‘mortal opinions’ is taken up at the end of theexploration 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 orderingof my words… This world-ordering I reveal to you, plausible in all itsparts, 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 conclusionalready reached, that for him sense-perception cannot give knowledge. For he isat pains to emphasize that such a system has no ‘proper guarantee of truth’; andthat it is ‘deceptive’ (it purports to give knowledge, but does not). It appeals toempirical evidence for support, not to reason. So it lacks any claim to be anobject of knowledge. The deeper reason why it cannot be supported by appeal topure reason is presumably that it is concerned with ‘peripheral’, contingentaspects of reality.But there is still a problem. If conducted in the usual way, a cosmology must alsonecessarily be not so much false as meaningless verbiage, since it takes seriouslythe illusions of plurality and change, speaks as though they were real, and offersexplanations of such changes in terms of physical necessities. Parmenides’‘Opinions’ is such a cosmology. Why does he deliberately offer a system ofwhich he himself thinks, and indeed implicitly says (in calling it ‘deceptive’, andbasing 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, butliterally meaningless, statements could be at need translated back into the correctbut 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 statementsarepracticallyuseful. In some way they correspond to the way the worldpresents itself to us. The fictitious entities they mention correspond to thefictions we create on the basis of our misread ordinary experience. Thatexperience shows they may be usefully manipulated to give a practicallyworkable understanding of the phenomenal world.22(Cosmology so conceived islike science as seen by ‘operationalist’ philosophers of science; and likedivination and natural magic—a thought perhaps taken further byEmpedocles.23)Within such limits, cosmology may none the less be required to satisfy certainformal demands.24Parmenides sets out these demands explicitly, for the firsttime. The original promise of the goddess stresses that the cosmology to be toldis (1) reliable; (2) comprehensive. Both of these points are echoed in the laterpassage, (1) Reliability is echoed by ‘deceptive’ and ‘plausible’. The demands onthe cosmology are further that it be a ‘world-ordering’, not only (2)comprehensive but also (3) coherent and formally pleasing; and (4) the bestpossible of its kind. These last two points may also include economy or beautyof explanation. The Principle of Sufficient Reason, which is closely related to thedemand for economy, appears, as in the exploration of reality, so again in thecosmology, to yield a symmetry between the two cosmic components.In fact, Parmenides devises an elegant and economical basis for cosmology byfollowing a hint given by the ‘way of mortals’. Any conventional cosmology hasto tread that false way, and to say both ‘it is’ and ‘it is not’. The simplest way tocommit this error is to suppose initially not one logical subject but two: onewhich is, and one which is not.25The physical properties of the two subjects arethen a kind of cosmic parody or allegory of the logical properties of what is andwhat is not.Now, they have fixed their judgements to name two forms, one of whichshould not [be named]; this is where they have gone astray. They haveseparated their bodies as opposites, and laid down their signs apart fromone another: for the one form, heavenly flamingFire,gentle-minded, verylight, 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,unknowingNight,a dense and weighty body.(DK 28 B 8.53–9)Fire and Night are the physical embodiments of the two opposed principles. Thecosmology is dualistic, and there is reason to suspect that, as with thePythagoreans from which it may borrow, the dualism was a moral (and anepistemic) as well as a physical one. The two opposed ‘forms’ are associatedfrom the outset with knowledge and ignorance; perhaps also with good and evil.Traces of morally charged struggles and loves of ‘gods’ within the cosmosremain in the testimony.26A cycle of cosmic changes is the most likelyexplanation of a detached remark (DK B 5) about circular exposition.Not only is the basis economical, but there are overall formal demands on thetwo forms. They must jointly exhaust the contents of the cosmos; and there mustbe cosmic symmetry as between them (DK B 9),27Conclusion: the Trouble with ThinkingThis account of Parmenides must end with questions on which certainty seems tobe out of reach.The overarching question is this: is Parmenides’ ‘framework’, in which histheory of reality is embedded, itself meant to be grounded in that theory ofreality? By the ‘framework’ is meant roughly the following: the originalassumption about the actual existence of thinking and certain knowledge; thedistinction between knowledge and opinion; the application of logic in thediscovery of the nature of reality; and the assertion of the practical, empiricaleffectiveness of systematized ‘opinions’.Even if this question cannot ultimately be given any confident answer, itusefully focuses attention on one sub-problem, which has so far been kept to oneside. This is the problem about the relation between thinking and reality. Wehave seen that thinking, for Parmenides, can only be of truths, indeed ofnecessary 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 thatthinks must be (part of) reality. If so, one would think that it ought tobe deducible from the nature of reality that it thinks about itself. No suchdeduction appears in the text, though; and the thesis that true thinking occursseems to be (as Aristotle took it) an initial assumption which is taken asunquestionable, but not formally proved.On this point, there are two parts of the poem which might serve as some kindof a guide. One is the introductory narrative of the journey to the goddess.Another is the outline of ‘physical psychology’, a general theory of perceptionand thought, which is attested as part of the cosmology.The journey to the goddessThe chariot-ride of the narrator in the introduction (preserved in DK B 1) hasusually been taken as an allegory of Parmenides’ own intellectual odyssey, and ofthe framework with which he starts.28Its chronology and geography are elusiveand dreamlike. The individual beings mentioned, even the narrator and thegoddess herself, are but shadowy outlines. Only certaintechnological objects—the chariot wheels and axle, the gate and its key—stand out in relief. IsParmenides here proclaiming his advances in the technology of thinking, as themotive power in, and the key and gateway to, all that follows? The ‘paths ofNight and Day’ would then be the ways of ‘it is’ and ‘it is not’. The chariot, thehorses, the daughters of the Sun who act as guides, and Parmenides’ ownambition, would correspond to everything Parmenides needs to get him as far asthe choice of ways,—that is, to the ‘framework’. All too much, though, must beleft uncertain, even if such an approach looks plausible in general.The empirical psychologyThe psychology or ‘theory of mental functioning’ which was outlined inParmenides’ cosmology is, equally, not much more than a tantalizing hint.Theophrastus says that for Parmenides as for several others ‘sense perception isby what is similar’, and goes on:‘As for Parmenides, he goes into no detail at all, but just [says] that, therebeing two elements, cognition[gnōsis]is according to what predominates.For, as the hot or the cold predominates, the intellect[dianoia]alters, butthat [intellect] which is [determined] by the hot is in a better and purerstate, though even that kind needs a kind of proportioning. He says:According as the compounding of the wandering limbs is in eachcase, in such a way is mind present in people; for it isthe same thing in each and in all that the nature of the limbs has inmind: the more is the thought.For he talks of sense perception and mental apprehension as being thesame; which is why [he says that] memory and forgetting occur from these[constituents] by the [change of] compounding. But whether, if they are inequal 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 makessense perception [occur] for the other element [(the cold)] in itself, is clearfrom the passage where he says that the corpse does not perceive light andhot and noise, because of the lack of fire, but does perceive cold andsilence and the opposite things. And in general [he says that] everythingthat is has some kind of cognition. Thus it seems he tries to cut short, byhis dogmatic statement, the difficulties that arise from his theory.(TheophrastusOn the Senses3–4, citing DK 28 B 16)This would seem to be at least a two-tier theory. The lower tier is basic senseperception, available to everything that exists, and ‘by the similar’; i.e. what isfire can perceive fire, what is night can perceive night, and what is a mixture canperceive both. The higher tier is that of mind and thought, somehow due to a‘proportionate compounding’ in human (and other?) bodies.29Any inferences from these indications can be but tentative. Briefly, the generalshape of Parmenides’ theory of reality shows that any real thinking must be (partof) reality thinking about itself.30The account of Parmenides’ intellectualjourney 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 courseinfected with the fictitiousness of the whole cosmology; yet it is probably meantto correspond, somehow, to the truth about thinking. If one element (‘fire’) in thecosmology corresponds to reality, then the fact that it is fire that cognizes firereflects the truth that it is reality that thinks of reality. It is a pity we can know nomore of what Parmenides thought about thought.ZENOIntroductionZeno of Elea, fellow citizen and disciple of Parmenides, became famous as theauthor of a series of destructive arguments. There is no good evidence that he putforward any positive doctrines. Plato and Aristotle were deeply impressed by theoriginality and power of the arguments; such knowledge of Zeno as survives isdue principally to them and to the Neoplatonist scholar Simplicius.31The Arguments against Plurality(a)Plato on Zeno’s book and the structure of the argumentsPlato’s dialogueParmenidesdescribes a supposed meeting in Athens, around450 BC, of the young Socrates and others with two visitors from Elea,Parmenides and Zeno. Plato’s fictional narrator gives some biographical dataabout the two Eleatics, and recounts a conversation between ‘Socrates’ and‘Zeno’ which tells (127d6–128e4) of the genesis of Zeno’s arguments againstplurality, their structure and their aim. Even if based solely on Plato’s ownreading of Zeno’s book, this has to be taken seriously as testimony.32According to this testimony, there was a book by Zeno which consistedentirely of arguments directed against the thesis ‘there are many things’. Eachargument began by assuming the truth of this thesis, and proceeded to deduce apair of mutually contradictory conclusions from it, in order to make areductio adabsurdumof the original thesis. In support of this account, Simplicius theNeoplatonist gives verbatim quotations from two of the arguments, which can beseen to exemplify the pattern; Plato’s narrator himself gives the outline of another.(b)The aim of the argumentsThis account of the structure of Zeno’s arguments leads ‘Socrates’ in thedialogue 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 conclusionof ‘Socrates’ is not completely accepted by ‘Zeno’, who denies that the book wasa ‘serious’ attempt to establish Parmenidean monism, and goes on:actually this [book] is a way of coming to the aid of Parmenides’ theory, byattacking those who try to make fun of it [on the grounds] that, if there isone thing, then many ridiculous and self-contradictory consequences followfor the theory. Well, this book is a counter-attack against the pluralists; itpays them back in the same coin, and more; its aim is to show that theirthesis, that there are many things, would have even moreridiculous consequences than the thesis that there is one thing, if one wereto go into it sufficiently.(Parmenides128c6–d6)The natural way to read this is as saying that the arguments had anad hominemelement. ‘Zeno’ cannot be saying that Parmenides’ thesis really had ridiculousconsequences; ‘even more ridiculous consequences’ points to the employment byZeno of assumptions made by Parmenides’ opponents, but not accepted by Zenohimself.Yet ‘Socrates’, a little earlier, has said that the arguments give ‘very many,very strong grounds for belief (128b1–3) that pluralism, of any varietyincompatible with Parmenides, is false. In Plato’s opinion the arguments,however they originated, were usable against all varieties of anti-Parmenideanpluralism. Therefore Plato’s testimony on Zeno cannot be fully understoodunless we know how he interpreted Parmenides, which cannot be investigated inthis chapter.Provisionally, it is enough to note that there is no danger of contradiction inPlato’s testimony, provided we may assume that Zeno’s original opponentsmade, and Zeno himself used against them, only such assumptions as were eitherinconsistent with Parmenides; or plausibly seen as articulations of commonsense.33The question can be finally decided, if at all, only by analysis of thearguments themselves.One further piece of information is given at 135d7–e7: the arguments wereabout ‘visible things’, i.e. they addressed themselves to the question of pluralismin the ordinary world, using assumptions derived from experience.34(c)The argument by ‘like’ and ‘unlike’According to Plato (Parmenides127d6–e5), the argument (the first one in thebook) purported to show that ‘if there are many things, they must be both likeand unlike’. Nothing further is known.(d)The argument by ‘finitely many’ and ‘infinitely many’Simplicius preserves the entire text of this argument. The compressed, austerestyle is reminiscent of Parmenides.If there are many things, it must be that they are just as many as they areand neither more of them nor less. But if they are as many as they are, theywould be finite.If there are many things, the things that are are infinite. For there arealways other things between those that are, and again others betweenthose; and thus the things that are are infinite.(DK 29 B 3, SimpliciusPhysics140.27–34)The first limb insists on the implications of countability. If it is true to say ‘thereare many things’ and to deny that ‘there is one thing’, that implies that (a) thereis one correct way of counting things; (b) that that way of counting the things thatare leads to a definite result. But a definite result implies finitely many things: ifthere were infinitely many, counting them would lead to no result at all.The second limb invokes the relation ‘between’(metaxu). Any two distinctthings are spatially separate (the converse of Parmenides’ argument for theoneness of reality from its undividedness). But what separates them must itselfbe something that is, and distinct from either. From this principle, an infiniteprogression of new entities is constructed.Though this involves an appeal to spatial properties, it might easily berephrased in terms of logical ones. The principle would be: for any two distinctthings, there must be some third thing different from either which distinguishesthem 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 enoughinformation to recover the rest in outline.The first limb claimed that ‘if there are many things, they are so small as tohave no size’. The argument proceeded, according to Simplicius, ‘from the factthat each of the many things is the same as itself and one’ (Physics139, 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 upof securely unified ones. Then consider each of these units. The line may havebeen (compare Melissus DK B 9): what has size has parts; what has parts is notone. Hence each of the units must be without size.The second limb contradicted this in successively stronger ways. First, itclaimed to show that, in a plurality, what is must have size. Suppose somethingdoes not have size, then it cannot be:For if it were added to another thing that is, it would make it no larger: forif something is no size, and is added, it is not possible that there should beany increase in size. This already shows that what is added would benothing. But if when it is taken away the other thing will be no smaller, andagain when it is added [the other thing] will not increase, it is clear thatwhat was added was nothing, and so was what was taken away.(DK 29 B 2, SimpliciusPhysics139.11–15)This argument in terms of adding and taking away obviously makes essential useof the assumption ‘there are many things’; it could not, therefore, have beenturned 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 beheld by most mathematizing theorists of the time.35The 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 sizeand bulk, and that one part of it is at a distance from another. The sameaccount applies to the part in front: for that too will have size and a part ofit will be in front. Now, it is alike to say this once and to keep saying it allthe time: for no such part of it will be the endmost, nor will it be that [anysuch part] is not one part next to another. Thus if there are many things, itmust be that they are both small and large: so small as to have no size, solarge as to be infinite.(DK 29 B 1, SimpliciusPhysics141.2–8)One axiom used is that anything having size contains at least two partsthemselves having size. This clearly generates an unending series of parts havingsize. 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. Thereis some analogy with the ‘Stadium’ and ‘Achilles’ (see (c) below): just as therunner’s supposedly finite track turns out to contain an infinite series ofsubstretches, each of positive length, so here the object with supposedly finitesize turns out to contain an infinite series of parts, each having size. If we try torecompose the original thing out of the parts, we shall never finish, but always beadding to its size; and this, Zeno might plausibly claim, is just what is meantwhen we say something is infinite in size.36(f)Methods and assumptionsIn the light of the arguments themselves as preserved, the question of their aimsand methods can be taken up again.It is evident that some of the assumptions used by Zeno in these arguments arenot due to simple ‘common sense’. Common sense does not make postulatesabout the divisibilityad infinitumof things having size; nor suppose that ‘to be isto be something having a quantity’; nor insist on a single correct way of countingthings. Hence Zeno’s arguments are not directed against unreflecting ‘commonsense’. In fact, these are the kind of assumptions that are naturally and plausiblymade, 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 theconstructions of progressionsad infinitum,and the remark when one isconstructed: ‘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 outpossible to make precisely the same step yet again.37The Arguments about Motion(a)Aristotle’s evidenceThere is only one certain primary source for the content of Zeno’s argumentsabout motion: Aristotle, who states and discusses them inPhysicsVI and VIII (VI2, 233a21–30; VI 9, 239b5–240a18; VIII 8, 263a4–b9). Aristotle’s source is notknown; no book of Zeno that might have contained them is recorded. It isperfectly possible that they reached Aristotle by oral tradition. In any case, whilethere is no reason to doubt that they are substantially authentic, there is also noreason 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 contradictionsfrom the supposition that something moves. Three of them purport to show thatwhat moves, does not move. They are ‘dramatized’, in so far as they introduceparticular supposed moving things: a runner; two runners; an arrow; threemoving and stationary masses. Aristotle presents the arguments as designed to bemutually independent.38(b)The ‘Stadium’ and the ‘Achilles’Suppose a runner is to run along a running-track. The stretch to be traversed (callitS) may be considered as divided up into substretches in various ways. Given thestarting and finishing points we understand what is meant by ‘the first half ofS’,‘the third quarter ofS’ and so on. It seems that however short a substretch isspecified in this way, it will always have positive length and may be thought ofas divided into two halves.39Going on in this way we can specify a division ofSinto substretches whichwill be such that the runner runs through a well-ordered but infinite series ofsubstretches. 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 integern,atthe end of thenth substretch the runner has coveredofS,and thenth substretch is ½nof the whole length of the track. However large a finite numbernbecomes, thefraction is never equal to 1; there are infinitely many substretches.With such a division, the series of substretches is well-ordered, and the runnerwho traversesShas been throughallof the substretches in order: for every finitenumberN,the runner has traversed theNth substretch. Hence the runner hastraversed an infinite series of substretches, in a finite time; but this is impossible.This is an expansion of Aristotle’s formulations (PhysicsVI 2, 233a21–23; VI9, 239b11–14) of the ‘Stadium’ argument.40The ‘Achilles’ (PhysicsVI 9,239b14–29) makes the same point more dramatically, pitting a very fast runneragainst a very slow one. The slow runner is given a start. The stretch covered bythe faster runner is divided up in such a way that it appears the faster can nevercatch the slower within any finite time. This drives home the point that speed isirrelevant. No limit of speed is prescribed or needed by the argument; the speedof 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-stretchis 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 intwo places at once; so it must occupy ‘a space equal to itself (i.e. of the sameshape and size).2 The arrow must be at rest at this moment. There is no distance throughwhich it moves,ina moment; hence it does not moveata moment, so itmust be at rest at that moment.3 But the moment chosen was an arbitrary moment during the flight of thearrow. It follows that the arrow must be at rest atallmoments during itsflight.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 ofAristotle’s abbreviated report (PhysicsVI 9, 239b5–9 and 30–3).41Aristotlehimself is interested only in step (4), where he thinks to find the fallacy; he givesthe only briefest sketch of (1), (2) and (3).42(d)The ‘Moving Rows’Aristotle (PhysicsVI 9, 239b33–240a18) reports this argument in terms ofunspecified ‘masses’ on a racecourse; to make it easier for a modern reader, themasses may be thought of as railway trains.43Consider three railway trains of the same length, on three parallel tracks. Oneof the trains is moving in the ‘up’ direction, another is moving at the same speedin 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 itdoes to pass the other moving train.Just how Zeno derived a contradiction from this fact, is uncertain. Accordingto Aristotle, Zeno simply assumed that the passing-times must be equal, since thespeeds are equal and the two masses passed are equal in length. Then it followsthat the time is equal to twice itself. The assumption, though, has often beenthought too obviously false to be Zeno’s. It may simply be Aristotle’s attempt tofill a gap in the argument as it reached him.44Yet Aristotle himself thought it notobviously fallacious, and worthy of detailed refutation.(e)Method and purpose of the four argumentsAs Aristotle describes them, these four arguments are simply ‘the arguments ofZeno about motion which cause difficulties to those who try to solve them’: nosuggestion that they wereallof Zeno’s arguments on the subject. Aristotlepresents them as mutually independent, and in an order which is not dictated byhis own concerns, presumably that of his source.Once again, as with the arguments against plurality, some of the assumptionsare manifestly theorists’ initial assumptions, rather than those of simple‘common sense’; but they are close to common sense.45If one starts trying tothink systematically in an abstract way, analogous to mathematics, about thephenomena of motion and its relation to time and space, these are assumptionsthat it is natural to start with. It is natural to assume that both the time-stretch andthe track of the moving thing may be treated for theoretical purposes asgeometrical lines obeying Euclidean geometry. This means that they are divisiblead infinitum,and that points along them exist ‘anywhere’: i.e. at all placescorresponding to lengths constructible by Euclidean procedures. There was notheory equivalent to that of the real numbers available in Zeno’s time, but suchassumptions correspond to elementary theorems and constructions of planegeometry as it was beginning to be developed.The way of thinking about physical phenomena embedded in Zeno’sassumptions is therefore an abstracting, mathematizing physicist’s way. Zeno’soriginal opponents are likely to have been natural philosophers, very likely fromthe loose group of ‘Pythagoreans’ (see below), who were then taking the firststeps towards a mathematized theory of the natural world.46Just because Zeno’s assumptions are natural ones for any mathematisingtheorist to make, his arguments still arouse heated discussion amongphilosophers. The suggestion, still sometimes made, that Zeno’s arguments havebeen made obsolete by developments in modern mathematics (particularlydifferential calculus and the theory of infinite series), misses the point. The valueand interest of all Zeno’s arguments is just that they are challenges to thefoundations of any mathematics and any physics that uses infinites andindivisibles of any kind and applies them to the physical world.47Other ArgumentsReports 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 soad infinitum’ (PhysicsIV 1,209a23–5). Elsewhere he gives the problem in the form: ‘if a place is something,in what will it be?’ (PhysicsIV 3, 210b22–24). If this was originally oneargument, it constructed an infinite series out of the common assumption thateverything that is, is in a place, which is something other than itself: applying theassumption to places themselves, we shall have places of places, places of placesof places, and so on. Such a series could have figured in one of the argumentsagainst plurality. In any case, it would be a good parry to any attack onParmenides’ monism which sought to show that his ‘One’ must occupy a placeother than itself.Also from Aristotle (PhysicsVII 5, 250a19–22): Zeno argued that, if a heap ofgrain makes a noise when it falls, then a single grain and any fractional part of itmust make a noise too. One may conjecture that Zeno’s dilemma was: either itmakes a proportionately small noise, or none at all. If the latter, a natural andfundamental assumption of mathematizing physics is undermined: theassumption that the magnitudes of effects are in direct proportion to themagnitudes of their causes. But if the former, then why do we not hear theproportionately small noise? If it fails to affect our senses, the assumption ofproportionality breaks down somewhere else. Such an argument would obviouslyfit Zeno’s programme of attack on any possible mathematical physics.ConclusionExamination of the evidence for Zeno’s arguments leads to satisfyinglyconsistent results, and bears out the testimony of Plato.First, Zeno attacked principally certain commonly-held views involving thereality of plurality and change, but did not confine himself to those targets. Thisfits well with Parmenides, who saw the twin beliefs in the reality of essentialplurality and of change as the two marks of deluded ‘mortals’.48Second, 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 tobe ‘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 reducingeverything to mathematics (Metaphysics1.5, 985b23–986b8 and 987a9–27).49The only individual one, about whom something of tangible philosophicalinterest can be known, is Philolaus of Croton.50Five fragments which may be reasonably taken as genuine reveal a theory ofunderlying structure in the universe which is heavily influenced by thedevelopment of mathematics as an abstract study.51This theory is propounded, itseems, on the basis of an analysis of ordinary human knowledge and itspresuppositions.Philolaus’ starting-point isgnōsis,the everyday activity of cognitive‘grasping’ (individuation, identification, reidentification, reference) of ordinaryindividual things. This ‘cognizing’ implies that its objects ‘have number’, i.e. arein some sense measurable or countable. Quite generally, any cognizable objectmust be marked off from everything else by a sharp, definite boundary. Whetherthis boundary be spatial or temporal, the object within it will have somemeasurable quantity (volume, time-duration). Also, a cognizable collection ofobjects must have a number; indeed even a single object must be recognizable asa single object and not a plurality, which implies a definite and practicallyapplicable method of counting. These points are recognizably related to somearguments 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 unitsand hence boundaries round these units. That what is must be a unit and have aboundary is also argued in Parmenides (see above, p. 41).The concept of a ‘boundary’ is central here. Philolaus’ analysis of thepresuppositions of cognition leads him to a logical separation of the contents ofthe universe into ‘things which bound’ and ‘things unbounded’. Everything inthe cosmos, and that cosmos itself, is claimed manifestly to exhibit a structure‘fitted together’ from the two kinds of thing. This dualism is obviously closelyrelated to views which Aristotle attributes to ‘the people called Pythagoreans’.He reports that some of them set up two ‘columns of correlated opposites’, whichfeatured such items as limit/unlimited, odd/even, one/plurality, right/ left, male/female, etc. (MetaphysicsI. 5, 986a22–6).Philolaus’ careful attempt to build up a general ontology on the basis of ananalysis of ordinary cognition, guided by mathematics, leads him naturally in thedirection of Aristotelian ‘form’ and ‘matter’. Whatever stuff an individual isthought of as being ‘made of, is in itself not ‘bounded’; for it might be present inany 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 notfound, and it seems that Philolaus thought this question beyond the reach ofhuman 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 dualisticstructure. All that we can say is that even humble human cognition presupposessuch a structure of things in particular and in general; the first example, it hasbeen suggested, of a Kantian transcendental argument.52MELISSUSMelissus of Samos (active around the mid fifth century) is best grouped with thephilosophers of Elea, to whom he obviously owes much. In spite of thepreservation of ten fragments (plus a paraphrase of other arguments) bySimplicius, and a reasonable amount of supporting testimony (the most usefulfrom Aristotle), Melissus’ intentions are not obvious. Many of his argumentsseem obviously weaker and cruder than those of Parmenides; on these groundshe was dismissed with contempt by Aristotle.53Foundations of MonismAs found in the quotations by Simplicius, Melissus starts by considering‘whatever was’ (DK B 1). The emphatic use of thepasttense already signals adeparture from Parmenides. It may have been justified by an initial argument tothe effect that thinking and speaking require the existence of something thoughtor spoken about; it is impossible to think or speak ‘about nothing’.54By the timewe reflect on our own thinking and speaking, they are in the past, so what isguaranteed by the argument is that somethingwas.‘Whatever was’ is also apparently more non-committal than ‘reality’. Asquickly becomes clear, however, this entity is conceived of by Melissus asextended in space and in time. It is ‘the universe’ rather than ‘reality’. Variousthings are proved about it. First (B 1), it cannot have come into being, because itwould have to have done so out of nothing, which is impossible. Next (B 2), italways was and always will be; Melissus here assumes that ceasing-to-be is justas impossible as coming-to-be.Next (B 3), an obscure argument to show that the universe is spatiallyunbounded, perhaps intended to parallel the argument for no coming-to-be andno ceasing-to-be. The thought seems to be that a ‘beginning’ or ‘end’ in space isjust as inconceivable as one in time; in either case we should have to suppose thatthere was nothing beyond. But that is unacceptable, apparently. Why? Possiblyagain 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 couldnot be unbounded, but would have bounds with each other’. Why should internalboundary lines be ruled out? Perhaps (cf. Parmenides and Zeno) because eveninternal boundary lines involve what is not; they cannot be part of either of thecomponents they separate, so are either themselves components, and need furtherboundaries, or are ‘nothing’, which is again impossible. So there cannot be two ormore distinct components in the universe.A further vital point, proved we know not how, was that the universe ishomogeneous. It also has no ‘bulk’ or ‘body’, on the grounds that that wouldmean that it would have physical parts, and not be a unity (B 10).Arguments against Change, Void and MotionMelissus’ arguments against the possibility of any kind of change proceedbriskly but none too convincingly. First, qualitative change would imply lack ofinternal homogeneity in the universe, since it would have to be qualitativelydifferent at different times. Next comes ‘change ofkosmos’; apparently some moreessential type of change (change of internal structure?). The argument is thatsuch a change necessarily involves what has already been ruled out: e.g. increaseor partial perishing or qualitative change.There follows the at first sight bizarre corollary that the universe does notexperience pain or mental distress; since pain and distress imply change orinhomogeneity in various ways. To deny that would be pointless, unless theuniverse were at least possibly a sentient being. If Melissus, like Parmenides,began with the assumption that some mental activity occurred, that would for himhave 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’ andtherefore does not exist. Hence there must be a plenum, which cannot admitanything from outside into itself, and so there can be no movement, since nothingcan budge to make room for the moving thing. Two corollaries: first, no actualdividing 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 SensePerceptionWhere does Melissus’ monism leave common sense and sense perception? Themessages of sense perception cannot be true. Melissus bases his attack on thefact that sense perception tells us that change occurs. The argument is: ifsomething 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 waterhas heated up, they would be contradicting themselves. So either our senses donot really tell us anything; or there is no change, when again our senses havemisled us.The aim is clear: it is to undermine any common-sense objections to thepositive doctrine about the universe. It therefore has to be an independentargument. The central idea of this independent argument against change is thatnothing that is true can cease to be true, ‘for there is nothing stronger than whatis really so’. We need a conception of truth as unchanging; but then thedeliverances of sense perception need to be at least reinterpreted, for they give usonly time-bound truths. So we need to revise the common-sense notion thatsense perceptions are straightforwardly true.CONCLUSIONThis chapter began with Pythagoras, as the presumed source of some persistentlyinfluential thoughts. His influence on philosophy was diffuse and non-specific. Hisquestioning of ‘what we really are’, and his insistence that we are moral agents ina morally polarized world, prepared for the creation of moral philosophy bySocrates and Plato.55Above all, Pythagoras’ insistence on the relevance of mathematics andimportance of abstract structure links him to the Eleatics. For what seems to becommon to both Pythagoreans and Eleatics is that they take seriously the ideal ofmathematically exact knowledge, the constraining force of mathematicallyrigorous argument, and the cardinal role of abstract structure in the nature ofthings. (Pythagoras’ other main concerns—the nature and destiny of the self, andthe dualism of good and evil—surface in the Eleatics, if at all, only inParmenides’ cosmology.)The Eleatic philosophers, likewise, had an influence which reached far beyondtheir few actual followers, and is still active today. Higher standards of precisionin statement and rigour of argument are noticeable everywhere in the later fifthcentury. Metaphysical argument in the Eleatic style appears: in Melissus, and asan intellectual exercise or for sceptical purposes, as in the sophist Gorgias. Moresignificantly, Socrates’ step-by-step, mostly destructive argumentation is Eleaticin spirit; it developed into the philosophical method of Plato and Aristotle, bothof whom pay tribute to ‘father Parmenides’.In the philosophy of scientific theorizing, it was Zeno’s dazzling attacks onincipient mathematizing physics that, for a long time, stole the show. Their effectwas not wholly negative: they stimulated further investigations into thefoundations of mathematics, and its relation to the physical world, whichculminated in the work of Aristotle. The more constructive thinking ofParmenides and Philolaus about scientific theorizing has only very recentlybegun to be understood and appreciated.NOTES1 The classic study of Walter Burkert (Burkert [2.25]) supersedes all previousdiscussions of the evidence. It may go too far in the direction of scepticism aboutPythagoras as theoretician: see Kahn [4.2]. The (pre-Burkert) catalogue of sourcesin Guthrie [2.13] Vol I: 157–71 is still serviceable.2 Those of Aristoxenus, Dicaearchus and Heraclides Ponticus were the earliest andmost influential: see Burkert [2.25], 53–109.3 Certain animal foods were taboo, but a comprehensive ban on the slaughter andeating of animals is improbable and poorly attested for Pythagoras himself. Someunder Pythagorean or Orphic influence, such as Empedocles, did observe such aban. 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 standardedition for reference purposes; the most reliable and informed recent edition, onmatters 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. Amongminor sources are some other Neoplatonists (Plotinus, Iamblichus, Proclus), andSextus 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 verybrief indications of the spread of opinion on cardinal points; they do not try tooutline 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 Onalētheiēand related words in early Greek, scholarly discussion has been toooften darkened by philosophical prejudice. See the useful study of Heitsch [4.29];also Mourelatos [4.24], 63–7 and references there.Alētheiēin 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 ornecessary truth’.9 So Verdenius [4.30]. Allied to this view are those who take the intended subject tobe ‘what is’ in the sense of ‘what is the case’ (e.g. Mourelatos [4.24]). Otherleading 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 bespoken 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 subjectat all is intended, at least initially, is suggested in different ways by e.g. Calogero[4.18], Coxon [4.8].10 On the verbeinai‘be’ in early Greek, see items [4.31] to [4.34] in the Bibliography.The entirely straightforward Homeric usage (‘Xis’—‘there is such a thing asX’) isthe obvious first hypothesis for theestiandouk estipaths. Some, though, have putthe so-called ‘veridical’ uses (‘be’=‘be true’ or ‘be’=‘be so’, ‘be the case’) in theforefront (e.g. Jantzen [4.23], Kahn [4.42]); others make the use ofeinaiinpredication 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 thenormal uses.11 In fact premiss (2), even without (1) and (3), gives a reason to reject the way thatsays ‘it is not’. For this way says, about reality generally, that it doesn’t exist orobtain. 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 PlatoTheaetetus188c9–189b6,Sophist237b7-e7. On the versions of this argumentin 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 theirderivatives are used in ways that seem inconsistent with principle (7). (a) Thegoddess 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 werepossible to think its falsities, (c) She speaks of ‘[my] trusty account (“saying”) andthought about reality’ (B 8.50–1), as though it were possible to have un-trustythought. Of these passages, though, (b) and (c) are rhetorical flourishes, in no wayessential to the argument; while (a), which occurs before principle (7) has beenintroduced, need only mean thatat mostthose two ways can be thought.14 Xenophanes, for instance, would have questioned the ambition of establishing thetruth, rather than mapping out by enquirycoherent possibilities for well-basedopinion.15 Whether this reality isobjectiveor 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. Literalsphericity 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, PlatoSophist242d4–6;Parmenides12834–433; AristotleMetaphysics1.5, 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 itselfpresupposing that mistake: it is plural, and it implies the reality of death. By theirvery error, they condemn themselves to appear to themselves as plural andephemeral. Interesting parallels for this in early Brahmanical monism, e.g. in theKatha Upanishad:…Herein there’s no diversity at all.Death beyond death is all the lotOf 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 hadbeen 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 practicalusefulness, is not clear. There is a hint (‘ithad to bethat opinions should reputablybe’, B1.32) that Parmenides did envisage such a guarantee; and see below on thecosmology as formally parallel to the section dealing withalētheiē.Scholarly opinion has been much divided on the status and purpose of the sectionconcerned 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-studyin 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 bemeant 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 hiscosmology: Empedocles DK B 110 and 111.24 On the internal structure of the ‘opinions’, and the parallelism withAlētheiē,seeMourelatos [4.24], 222–63.25 This reading is supported by Aristotle’s testimony (Metaphysics1.5,26 ‘Love’ as a power: DK 613, cf. AristotleMetaphysics1.3, 984b20–31; struggles ofgods: PlatoSymposium195c, CiceroOn the Nature of the CodsI.II.28 (DK 28 A37). There is no need to be puzzled by the appearance of Hesiodic divinities here, ifParmenides, 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 mentalfunctioning, on which see pp. 150–1; see Guthrie [2.13] II: 57–70.28 But there is much disagreement about the details. An extended ancientallegorization is found in Sextus Empiricus (Adversus MathematicosVII.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 textand 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 aloneconstitutes reality, nor that reality is just what thinks itself. (It does follow thatreality is not ultimately ‘mind-independent’, in that it is necessarily thought byitself. In this rather special sense, Parmenides is an idealist, but not provably in anywider sense.)31 Zeno was ‘the Eleatic Palamedes’ (PlatoPhaedrus261d6), the ‘inventor ofdialectic’ (Diogenes LaertiusLivesVIII.57 (W.D.RossAristotelis FragmentaSelecta,Oxford, 1955:15).32 Plato’s evidence has not gone unchallenged. Zeno has sometimes been read asattacking Parmenides as well as his opponents, particularly by those who questionwhether Parmenides was a monist. The attempt of Solmsen [4.72] to underminePlato’s testimony was countered by Vlastos [4.73]; but even Vlastos doubts Plato’stestimony 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 wouldnot have tried to make fun of Parmenidesby arguments,as Zeno implies hisopponents did.34 This fits the earlier suggestions ofad hominemargumentation by Zeno. It does notimply that, in Plato’s opinion, Parmenides’ monism was a monism about the ordinaryworld.35 So Aristotle,MetaphysicsIII 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 manylater ancient writers.37 Other possible arguments of Zeno against plurality appear at: AristotleOnGeneration and Conception1.2, 316a14–317a12 (not attributed, and introduced inthe context of Democritus’ atomism); and SimpliciusPhysics139.24–140, 26,ThemistiusPhysics12.1–3, PhiloponusPhysics80.23–81.7 (attributed toParmenides or Zeno). On these as possibly Zeno’s: see Vlastos [4.64], 371–2 andMakin [4.66].38 On their possible interdependence, see section (e).39 Compare the assumption needed in (e) above, that anything having size can bedivided into two things each having size.40 Sometimes known as the ‘Dichotomy’. Aristotle’s own solution is atPhysicsVIII8, 263a4–b9.41 Aristotle’s phrase corresponding to ‘at a moment’ is ‘in the now’, i.e. ‘in the presentunderstood as an indivisible instant’. This excludes periods of time, evensupposedly indivisible ones. It is possible that Zeno’s argument somehow dependedcrucially on the instant’s being taken aspresent(as suggested by Lear [4.67]).42 Diogenes Laertius (LivesIX.72, DK 29 B 4), using a source independent ofAristotle, gives a summary of an argument which may possibly descend fromZeno’s formulation of step (2): ‘that which moves does not move either in the placein 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 asAristotle’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 exhaustingthe theoretical possibilities for pluralism. The idea goes back to the nineteenthcentury; 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 divisiblead 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 formulationsboth of the arguments and of the possibilities for meeting them: see especiallyGrünbaum [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). Theirauthenticity 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 particularlyto Nussbaum [4.79],51 See DK 44 B 1, 2, 4, 5, 6.52 Nussbaum [4.79], 102.53 AristotleMetaphysicsI.5, 986b25–7 (‘rather crude’);PhysicsI 2, 185a10–11 (‘lowgrade’).One purported source, the pseudo-Aristotelian essayOn MelissusXenophanes Gorgias (MXG),is an exercise in ‘philosophical reconstruction’, fromwhich it is not possible to disentangle with confidence any further informationabout Melissus.MXGis not drawn on here. The most noteworthy modern attemptto rehabilitate Melissus as a philosopher is that of Barnes [2.8], chs. 10, 11, 14.54 This is a conjectural interpretation of Simplicius’ paraphrase,Physics103.15: ‘ifnothing 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 periodinto Pythagoras.BIBLIOGRAPHYPythagoras and the Early PythagoreansTextsNo authentic writings survive. Collections of early Pythagoreanakousmataandsumbola,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 studies4.1 Burkert [2–25].4.2 Kahn, C.H. ‘Pythagorean philosophy before Plato’, in Mourelatos [2.19]: 161–85.‘Orphism’ and sixth- and fifth-century religion4.3 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 science4.7 Lloyd [1.7]. See also [2.27], [2.34], [2.38], [2.41].ParmenidesTexts with translation and commentary4.8 Coxon, A.H.The Fragments of Parmenides,Assen/Maastricht, Van Gorcum, 1986.4.9 Gallop, D.Parmenides of Elea, Phoenixsuppl. vol. 18, Toronto/Buffalo/ London,University of Toronto Press, 1984.4.10 Heitsch, E.Parmenides: Die Fragmente,2nd edn, Munich/Zürich, Artemis Verlag,1991.4.11 Hölscher, U.Parmenides: Vom Wesen des Seienden,Frankfurt, Suhrkamp Verlag,1969.4.12 O’Brien, D. and Frère, J. in P.Aubenque (ed.)Études sur Parménide,vol. I:Le Poèmede Parménide,Paris, J.Vrin, 1986.4.13 Tarán. L.Parmenides,Princeton, NJ, Princeton University Press, 1965.4.14 Untersteiner, M.Parménide: testimonianze e frammenti,Florence, La Nuova ItaliaEditrice, 1967.General studies and collections of essays4.15 Aubenque, P. (ed.)Études sur Parménide,vol. II:Problèmes d’interprétation,Paris,J.Vrin, 1986.4.16 Austin, S.Parmenides: Being, Bounds, and Logic,New Haven and London, YaleUniversity Press, 1986.4.17 Bormann, K.Parmenides: Untersuchungen zu den Fragmenten,Hamburg, FelixMeiner 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, YaleUniversity Press, 1970.4.25 Owens, J. (ed.)Parmenides Studies Today, Afonist62/1, 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 proem4.28 Burkert, W. ‘Das Proömium des Parmenides und die Katabasis desPythagoras’,Phronesis14 (1969): 1–30.Alētheiēin early Greek and in Parmenides:4.29 Heitsch, E. ‘Die nicht-philosophischealētheia’,Hermes90 (1962): 24–33.4.30 Verdenius, W.J. ‘Parmenides B 2, 3’,Mnemosyneser. 4, 15 (1962): 237.The verbeinai(‘be’) and the concept ofbeingin early Greekand in Parmenides:4.31 Hölscher, U.Der Sinn vom Sein in der älteren griechischen Philosophie,Sitzungsberichte der Heidelberger Akademie der Wissenschaften, philosophischhistorischeKlasse, Jahrgang 1976, 3. Abhandlung, Heidelberg, Carl WinterUniversitätsverlag, 1976.4.32 Kahn, C.H.The Verb ‘Be’ in Ancient Greek,Foundations of Language, suppl. series16, Dordrecht and Boston, Reidel, 1973.4.33—‘Why existence does not emerge as a distinct concept in Greek philosophy’,Archivfür Geschichte der Philosophie58 (1976): 323–34.4.34 Matthen, M. ‘Greek ontology and the “Is” of truth’,Phronesis28 (1983): 113–35.Methods of argument, and the nature of thinking4.35 Lesher, J. ‘Parmenides’ critique of thinking: thepoludéris elenchosof Fragment 7’,Oxford Studies in Ancient Philosophy2 (1984): 1–30.4.36 Mourelatos, A.P.D. ‘Mind’s commitment to the real: Parmenides B 8.34–41’, inAnton 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 inAncient Philosophy6 (1988): 39–67.4.38 Hintikka, J. ‘Parmenides’Cogitoargument’,Ancient Philosophy1 (1980): 5–16.The nature of what is4.39 Barnes, J. ‘Parmenides and the Eleatic One’,Archiv für Geschichte der Philosophie61 (1979): 1–21.4.40 Finkelberg, A. ‘Parmenides between material and logical monism’,Archiv fürGeschichte der Philosophie70 (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 Metaphysics22 (1969): 700–24.4.43 ——‘More on Parmenides’,Review of Metaphysics23 (1969): 333–40.4.44 Ketchum, R.J. ‘Parmenides on what there is’,Canadian Journal of Philosophy20(1990): 167–90.4.45 Malcolm, J. ‘On avoiding the void’,Oxford Studies in Ancient Philosophy9 (1991):75–94.4.46 Owen, G.E.L. ‘Eleatic questions’,Classical QuarterlyNS 10 (1960): 84–102; repr.in Allen and Furley [2.15], vol. 2; 48–81, and in M.Nussbaum (ed.)Logic, Scienceand Dialectic: Collected Papers in Greek Philosophy,London, Duckworth, 1986:3–26.4.47—— ‘Plato and Parmenides on the timeless present’,Monist50 (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 derPhilosophie52 (1970): 113–35.The ‘Platonic problem’ of not-being4.49 Denyer, N.Language, Thought and Falsehood in Ancient Greek Philosophy,Londonand New York, Routledge, 1991.4.50 Pelletier, F.J.Parmenides, Plato and the Semantics of Not-Being,Chicago andLondon, University of Chicago Press, 1990.4.51 Wiggins, D. ‘Sentence meaning, negation and Plato’s problem of non-being’, inG.Vlastos (ed.)Plato: A Collection of Critical Essays,I:Metaphysics andEpistemology,Garden City, NY, Doubleday, 1971:268–303.Cosmology (including psychology)4.52 Curd, P.K. ‘Deception and belief in Parmenides’Doxa’,Apeiron25 (1992): 109–33.4.53 Finkelberg, A. ‘The cosmology of Parmenides’,American Journal of Philology107(1986): 303–17.4.54 Laks, A. ‘The More’ and ‘The Full’: on the reconstruction of Parmenides‘ theory ofsensation in TheophrastusDe Sensibus3–4’,Oxford Studies in Ancient Philosophy8(1990): 1–18.4.55 Long, A.A. ‘The principles of Parmenides’ cosmology’,Phronesis8 (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 dieBegriffswelt der Vorsokratiker,Darmstadt, Wissenschaftliche Buchgesellschaft,1968:391–422.Miscellaneous4.57 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.ZenoTexts with translation and commentary4.59 Lee, H.D.P.Zeno of Elea,Cambridge, Cambridge University Press, 1936.4.60 Untersteiner, M.Zenone: testimonianze e frammenti,Florence, La Nuova ItaliaEditrice, 1963. Translations of the relevant parts of PlatoParmenidescan be found inCornford [4.19]. The testimony of Aristotle in thePhysicsis translated in:4.61 Barnes, J. (ed.)The Complete Works of Aristotle,Princeton, NJ, Princeton UniversityPress, 1984.General studies4.62 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 UniversityPress, 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 VlastosStudies in Greek Philosophy,ed. D.W. Graham, vol. 1,Princeton, NJ, Princeton University Press, 1995:241–63.The arguments against plurality4.65 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’,Phronesis27 (1982): 223–38.The arguments about motion4.67 Lear, J. ‘A note on Zeno’s arrow’,Phronesis26 (1981): 91–104.4.68 Owen, G.E.L. ‘Zeno and the mathematicians’,Proceedings of the AristotelianSociety58 (1957/8): 199–222; repr. in Allen and Furley [2.15], vol. 2:143–65; inSalmon [4.76]: 139–63 and in Nussbaum (see [4.46]): 45–61.4.69 Pickering, F.R. ‘Aristotle on Zeno and the Now’,Phronesis23 (1978): 253–7.4.70 Vlastos, G. ‘A note on Zeno’s arrow’,Phronesis11 (1966): 3–18; repr. in Allen andFurley [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 Philosophy4 (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 Zeno4.72 Solmsen, F. ‘The tradition about Zeno of Elea re-examined’,Phronesis16 (1971):116–41; repr. in Mourelatos [2.19]: 368–93.4.73 Vlastos, G. ‘Plato’s testimony concerning Zeno of Elea’,Journal of Hellenic Studies95 (1975): 136–62, repr. in Vlastos, ed. Graham (see [4.64]); vol. 1: 264–300.Zeno, Aristotle and modern philosophy4.74 Bostock, D. ‘Aristotle, Zeno and the potential infinite’,Proceedings of theAristotelian Society73 (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 commentary4.78 Huffman, C.A.Philolaus of Croton: Pythagorean and Presocratic,Cambridge,Cambridge University Press, 1993.Studies4.79 Nussbaum, M.C. ‘Eleatic conventionalism and Philolaus on the conditions ofthought’,Harvard Studies in Classical Philology83 (1979): 63–108. See also [4.1]and [4.2].MelissusText with translation and commentary4.80 Reale, G.Melisso: testimoniaze e frammenti,Florence, La Nuova Italia Edi trice,1970.