# 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: polansky@duq.edu, Joe Cimakasky: joecimakasky@gmail.com

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. S

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