Zeno's Second Argument against Plurality
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Zeno's Second Argument against Plurality SANDRA PETERSON 1. INTRODUCTION Proclus reports that Zeno gave forty arguments against plurality .' Plato reports that these were in support of the thesis of Zeno's colleague Parmenides that there is exactly one being. 2 I take it that we are to understand the Parmenidean thesis to be incompatible with Zeno's being distinct from Parmenides and Zeno's giving forty arguments against plurality. 3 2. A NEW RECONSTRUCTION Zeno's second argument against plurality attempts to reduce the assumption that there are many to the absurdity that there would be both finitely many and infinitely many. I offer a reconstruction of the part of the argument for the consequence that there would be finitely many. The text is: If there are many, it is necessary that they be as many as they are, neither more nor fewer. But if they are as many as they are, they must be finitely many. 4 "If there are many" is the assumption for reduction to absurdity. The "it is necessary " indicates an implication of there being many: whatever number x is, if x is greater than 1, it is exactly what it is, that is, x = x. "Neither more nor fewer" I take to be an explication of x's being as many as it is: for any number n, greater than 0, x :~ x + n and x ~ x - n. For the purpose of the reconstruction we Can use a single instance of the general claim for the result of addition: x :~ x + 1. "They must be finitely many" is the conclusion that the number of things there are is finite. My re-expression of Zeno's argument has made use of the word "number"; the Greek 6pt0poq does not occur in the text of the second argument against plurality, nor anywhere else in the Fragmente from Zeno. Use of "number" and of the variable x, The label "second argument against plurality" is from Encyclopedia of Philosophy, Edwards ed., s.v. "Zeno of Elea" (written by G. Vlastos). I H. Diels and W. Kranz, Die Fragmente der Vorsokratiker, 12th ed. (Dublin, 1966), 29a15. 2Prin. 128c-d. 3G. E. L. Owen ("Zeno and the Mathematicians," Proceedings of the Aristotelian Society 58 [1958] : 199) says that "Zeno certainly held . . . that there is only one thing in existence." H. Cherniss (Aristotle's Criticism of Presocratic Philosophy [New York, 1971], p. 145), commenting on one of Zeno's arguments, says, "This argument is a refutation of the doctrine that there can be more than a single Real Being; it is in accord with the purpose of Zeno's dialectic as given by Plato." ' Vlastos's translation ("Zeno," p. 371) of Diels and Kranz, 29b3, and G. S. Kirk and J. E. Raven, The Presocratic Philosophers (Cambridge, 1964), no. 366. "Finitely many" translates ~e~spaop~,vct."As many as there are" translates xot~tOxa... 6o~t ~,ax't. [261] 262 HISTORY OF PHILOSOPHY representing particular numbers of collections, shortens considerably the presentation of the argument. It is in no other way crucial to my reconstruction. The reconstruction could be rephrased without using "number." For example, instead of saying "for any number x, x ~: x + 1," one could say, "for any collection C, that collection C does not have just as many items in it as the collection consisting of C together with some new item i not in C. I stipulate that here the latter is all I mean by the former. The following is my reconstruction: 1. Suppose there are k, where k is more than 1. [Assumption for reduction to absurdity] 2. For any x, if x is greater than 1, x -- x. [Premise] 3. For any x, if x = x, then x ~ex + 1. [Premise] 4. For any x, if x r x + 1, then x is finite. [Premise] 5. k is finite. [Conclusion] The assumption might also be expressed: the number of things there are is k. Of the three premises, 2 and 3 are directly from the text. Premise 4 has no counterpart in the text: I have added it as, for me, the most natural means to bridge the inferential gap between 1, 2, 3...

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