Counting the Hypotheses in Plato’s Parmenides

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  • Ron Polansky and Joe Cimakasky

    Counting the Hypotheses inPlatos ParmenidesAbstract: Parmenides exercise assists Socrates who is perplexed about formsand participation. The exercise assumes the one is and is not, and traces con-sequences for the one with respect to itself and the others and for the otherswith respect to themselves and the one. There appear to be eight or ninehypotheses. Counting the third makes all the odd-numbered hypotheses drawneither nor conclusions, while the even-numbered draw both and conclusions. Odd and even thus link with limit and unlimited principles, so thethird hypothesis on the instant clarifies forms and all beings. We also cast lighton the Presocratic origin of the theory of forms.

    Keywords: Plato Parmenides Hypotheses, Participation, Forms, Principles, Ana-xagoras, Plato, Parmenides, hypotheses

    Prof. Ron Polansky: Duquesne University, Pittsburgh, Pennsylvania 15282, United States;E-mail:, Joe Cimakasky:

    In Platos Parmenides, the youthful Socrates boldly counters Zenos paradoxi-cal argumentation by introducing the forms and participation. Parmenidesthen subjects Socrates to withering elenctic argumentation. Fortunately, Par-menides is prepared to assist the befuddled Socrates by providing a helpfulexercise. This exercise has Parmenides making the hypothesis that a one isand tracing what follows with respect to itself and with respect to the others.Moreover, he further assumes that the one is not and again determines theconsequences for the one and the others. This might lead to four series of de-ductions, but perhaps surprisingly there turn out to be eight or nine series ofdeductions.1 All commentators comment upon the number of hypotheses. Yetwe believe that we have a rather obvious, though so far as we know, unno-ticed observation to make about the counting of the hypotheses in the second

    1 Many commentators, e.g., Cornford 1957, 194; Meinwald 1991, 124129; Miller 1986, 251n53;Sayre 1996, 240241; Scolnicov 2003, 134; Turnbull 1998, 112, count eight hypotheses. Allen1983, 261 counts four hypotheses with two deductions each, except for the first hypothesis withthree deductions. The Neoplatonists, e.g., Proclus 1987, 402403, typically count nine hypoth-eses, with their numbering relating to their theory of emanation.

    DOI 10.1515/apeiron-2013-0014 apeiron 2013; 46(3): 229243

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  • part of this dialogue. This observation, along with our interpretation of thehypothesis on the instant, clarifies the numbering of the hypotheses and theposition of the third hypothesis. By way of this attention to numbering andordering of the hypotheses, we believe that we arrive at a comprehensive inter-pretation of the entire exercise.

    The assumption that the one is gives rise to at least four series of deduc-tions rather than merely two. Two deductions develop the consequences forthe one itself, and two further deductions produce the consequences for theothers. And the assumption that the one is not gives rise to four deductions ofconsequences, two with respect to the one itself and two with respect to theothers. Moreover, there is an additional line of deductions that is third in order(155e157b). Because this deduction seems unlike the others, yet looks to bebringing the first two together somehow, there is disagreement about whetherthere are eight or nine hypotheses (or series of deductions). Most modern com-mentators refer to this third deduction on the instant as an appendix, coda,auxiliary inquiry, corollary, insert hypothesis, or such, and they may num-ber it 2a.2

    Parmenides seems quite aware of the difficulty with the numbering becausethis third set of deductions is in fact the only one that gets explicitly numberedwith an ordinal numeral.3 Parmenides clearly announces that this is his thirdset of deductions: Lets speak of it yet a third () time (155e4).4 Thus thisdeduction that commentators dispute about counting in relation to the schemeof hypotheses is the only one that Parmenides actually counts. Some commen-tators point out that this is the only deduction explicitly numbered, but we donot find anyone attempting to explain why Parmenides counts only this one.This we clarify and show as a key to the entire exercise.

    2 Cornford 1957, 194 argues that the third hypothesis has no claim to the status, which manyassign to it, of a ninth independent Hypothesis. That would destroy the symmetry of the wholeset of Hypotheses. We reject Cornfords view and will show that far from destroying symmetry,counting nine hypotheses enhances the symmetry and clarifies the general structure of the ex-ercise.3 Sayre 1996, 240241 claims that the deduction that ensues, accordingly, should be under-stood as a further step in a series of inferences already underway, and not as the beginning ofa fresh set of inferences. Miller 1986, 251n53 believes: Platos reference to the third timeannounces not a new, third hypothesis, but rather, as the Greek proverb has it, a third timefor the savior, that is, an insight which will rescue the inquiry by getting beyond the seemingmanifold absurdity of hypotheses I and II.4 We use the Gill and Ryan translation in Cooper ed. 1997.

    230 Ron Polansky and Joe Cimakasky

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  • I Counting the Third Hypothesis

    A reason for counting just this third deduction is to call attention to the countand the importance of the position of this deduction, which, as many have re-cognized, is intended somehow to bring the first two hypotheses together, aswell as indicating how the subsequent pairs of hypotheses can be brought to-gether. The first two series of deductions assume that the one is. But the first ofthese immediately urges that the one that is, since it is one, cannot be many inany way (137c45). Consequently, the one cannot have any further attributes orplurality, which would make it more than one. Thus the assumption that theone is simply one leads to many neither nor conclusions concerning theone, such as that it is neither like nor unlike itself or the others.5 And thereforeit turns out that the one can neither have any being nor even be one(141e9142a1). In contrast to this result from assuming that a one is, the secondset of deductions allows that the one that is, since it has being, can be manyand have all sorts of attributes. Along this line, there develop both and conclusions, such as that the one is both like and unlike itself and the others.Whereas the first hypothesis then results in neither nor conclusions, thesecond hypothesis produces both and conclusions.

    Looking toward the remaining sets of deductions, when we count the thirdas third, we find that the fourth, sixth, and eighth sets lead to both and conclusions, while the fifth, seventh, and ninth produce neither nor conclu-sions. Hence, through inserting the third hypothesis, Parmenides arranges forall the odd-numbered hypotheses to have neither nor conclusions, while allof the even-numbered hypotheses have both and conclusions.6 The follow-ing diagram displays how inserting the third hypothesis creates clear order andsymmetry:

    Hypothesis ConclusionsOne Neither Nor Two Both And Three Neither Nor Four Both And Five Neither Nor Six Both And

    5 These many denials about the one, as denials, should not be supposed to give plurality to it.6 No previous interpreter to our knowledge has recognized that insertion of the third hypothe-sis has the consequence of lining up the odd-numbered hypotheses as neither nor and theeven-numbered hypotheses as both and deductions. Of course interpreters have not thenattempted to explain this. We need not raise the issue whether one is in fact a number for theancients since what we go on to say accords with viewing one as the principle of number.

    Counting the Hypotheses in Platos Parmenides 231

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  • Seven Neither Nor Eight Both And Nine Neither Nor

    What might be the significance of this arrangement of odd- and even-numberedhypotheses?

    II The Function of the Third Hypothesis

    The first hypothesis of a one with no manyness leads to neither nor conclu-sions such that it is not even one. This initial deduction responds to Socratessupposition in the first part of the dialogue that each form is merely the oneform that it is. Socrates there insists that he will wonder exceedingly if it can beshown that the forms themselves admit opposites and mix with each other (see129ce). The first deduction of the exercise provided by Parmenides discloses toSocrates how problematic it is to keep his principles, the forms, from havingany association with each other. Viewing any form as solely one with no many-ness about it whatsoever prevents it from even being what it is. Socrates wastherefore mistaken in trying to have each form be an isolated island unto itself.Forms, to be what they are, must be in some association with other forms. Forexample, each form must be the same as itself and other than all other forms.Thus in order for a given form to be the same as itself it must participate insameness, while being different from other forms requires it to share in other-ness. Evidently, contrary to Socrates supposition, the forms must associate witheach other.

    It should be the case, then, that understanding the way perceptible thingsparticipate in the forms necessitates an understanding of the way the forms, asthe principles of perceptible things, participate in each other. Consideration ofparticipation has to be raised up to the level of the forms themselves. Subse-quently, the second hypothesis of the exercise with the both and conclu-sions shows how any one form may participate in all sorts of other forms. Surelyall these both and conclusions are as dizzying, however, as the neither nor conclusions of the first hypothesis. Can the third hypothesis offer clarifi-cation?

    The discussion of the third hypothesis about how to speak of the one beginsthis way:

    Lets speak of it yet a third time. If the one is as we have described it ( -) being both one and many and neither one nor many, and partaking of time( ) must it not, because

    232 Ron Polansky and Joe Cimakasky

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  • it is one, sometimes partake of being, and in turn because it is not, sometimes not partakeof being? ( , , , ; 155e48)

    This summarizes the results of the first two series of deductions: the one is bothone and many, and the one is neither one nor many, and it is temporal. Parme-nides rather strangely adds partaking of time, but he leaves out that the onethat is neither one nor many, i.e., the one of the first hypothesis, lacks any tem-porality or relation to time at all (see 141de). By leaving this out he may makeit seem that participation in temporality explains how the one both partakes ofbeing and does not partake of being, or both is one and is not one. Accordingly,he says that because it is one, it sometimes () partakes of being, and be-cause it is not [one], it sometimes does not partake of being, as if there is atemporal shift from the one condition to the other, occasioning no violation ofthe principle of non-contradiction. But this can hardly be taken straightfor-wardly if the one of the first hypothesis never has anything to do with time.

    We propose alternatively that when Parmenides here says sometimes, hereally refers to just those times when he and his interlocutor Aristotle werespeaking of the one. Sometimes they were saying that the one partakes of being,and sometimes they were saying that it does not partake of being. Consequentlythe phrase , as we have described it (in 155e45) harmo-nizes with this interpretation insofar as it refers to the different times when theywere saying one thing or the other about the one.7

    If our interpretation regarding sometimes is correct, what Parmenides saysin the third deduction does not have to do with a shift in the one itself but intheir reflections upon the one. Of course once Parmenides puts it this way, andAristotle makes no complaints or raises no queries, Parmenides can shift moreand more to the view that it is the one itself that is undergoing these changes.But the shifts have in actuality rather been in their accounts of the one becausethe one allows for such shifting perspectives.

    Thus Parmenides is initially talking about the way they have been speakingabout the one: he is really saying that when they were saying that the one is,then surely it must partake of being, and when they were saying that it is not, itwas not partaking of being. He goes on to get agreement that when the oneparticipates in being or when they were saying that it participates it is un-able not to; and when it does not participate in being or when they were say-ing that it does not participate it cannot participate in it (155e810). Therethen seem to be different times, a time when the one participates in being and

    7 We have not found this interpretation in the literature.

    Counting the Hypotheses in Platos Parmenides 233

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  • another when it does not (155e1011). In fact this is most plausibly read as wesuggest such that Parmenides has the different times be the different times inwhich they were making the two different deductions.

    Parmenides subsequently inquires,

    Isnt there, then, a definite time when it gets a share of being ( )and when it parts from it ()? Or how can it at one time have and at anothertime not have the same thing, if it never gets and releases it? In no way. Dont youin fact call getting a share of being coming-to-be ( )? I do. And parting from being ceasing-to-be ()? Most certainly. Indeed the one, as it seems, when it gets and releases being, comes tobe and ceases to be. (156a1b1)

    If we are right that Parmenides should really be speaking about the way theyspoke differently of the one at different times, then we may wonder whether theones getting or losing a share of being has to do with its coming into being orceasing to be. But perhaps either Parmenides comments generally here aboutwhat occurs in time, or Aristotle usefully misunderstands what he says. What-ever the case, the one seems able not only to be and not to be, but also there-fore to become and to perish. It surely appears that the becoming and ceasingto be of the one is its change to being and not being respectively, i.e., it goesbetween the one of the first and second hypotheses. Yet Parmenides quicklyrevises our thinking about this, which might not be so surprising if we recognizethat he has actually been referring to how they were speaking of the one ra...


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