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THE EPISTEMOLOGICAL SIGNIFICANCE OF PLATO'S THEORY OF IDEAL NUMBERS / על משמעותה האפיסטמולוגית של תורת המספרים האידיאליים לאפלטון

שמואל שקולניקוב and SAMUEL SCOLNICOV
Iyyun: The Jerusalem Philosophical Quarterly / עיון: רבעון פילוסופי
כרך כ‎', חוברת א'/ד‎' (טבת תשכ"ט-תשרי תש"ל), pp. 186-211
Stable URL: http://www.jstor.org/stable/23342189
Page Count: 26
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THE EPISTEMOLOGICAL SIGNIFICANCE OF PLATO'S THEORY OF IDEAL NUMBERS / על משמעותה האפיסטמולוגית של תורת המספרים האידיאליים לאפלטון
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Abstract

The breakdown of the position of the "Aristotelian misunderstanding", in the wake of the work of Stenzel and Jaeger, forces upon us a reconsideration of Plato's later thought as it was handed down to us by Aristotle and his disciples. This problem is especially acute in what refers to the so-called Theory of Ideal Numbers. As this theory is not openly discussed in Plato's written work, the question arises of the relation between the written theory of Ideas and the oral theory of Ideal Numbers. The present paper is an inquiry on the epistemological import of Plato's contention that ideas are numbers. In it, an attempt is made to show that the positing of the idea as a "one over many", i. e. as a synthetic unity, already implies the view that the idea is midway between the absolute One and absolute multiplicity, both of which are irrational and ineffable. If the ideas were many and were simple unities, and were different from one another, the difference between them would be irrational, as Melissus had shown. The Sophist brings us paradoxically to the conclusion that this irrationality can only be overcome by letting plurality into the idea itself. Now the problem of the one and the many makes its appearance already within the world of ideas. But, in contradistinction to the world of the senses, the multiplicity within the idea is not sheer multiplicity, but, as in the example of the letters in Philebus 17a, it is "between the one and the indefinite", it is a definite multiplicity. This multiplicity within the idea is made definite by virtue of the distinctive diairetic structure of each idea, according to the order of prior and posterior (Arist. Met. Δ 11. 1019al), the very same structure found in Plato's derivation of the (ideal) numbers, as Stenzel has shown at length. In a closer examination of the relation between ideas and numbers, Ross' and Wilpert's interpretations are discussed, and the conclusion is drawn that, while direct textual evidence would not be sufficient to confirm their contention that ideas and numbers were identified, indirect evidence stemming from the above interpretation of Plato's written doctrine would rather incline one to the view that numbers and ideas were regarded as prior and posterior, or as universals and their specifications. Further support to this view is adduced from an examination of the Eudoxian-Platonic concept of number as a "definite multiplicity" as against the Aristotelian concept of it as a "sum of unities". It is shown that Plato's broader concept of number enables him to see number primarily as structure and relation, and only secondarily as quantity. Insofar as the diairesis gives us the sequence of conditions and conditioned, it is "numerically" ordered, even though it does not necessarily deal with quantities. It seems thus that Ideal Number is number insofar as it represents rationality, viz. structured (= "limited") plurality. The last paragraph of the paper calls attention to the dual aspect of the idea as being at the same time under the category of relation and under the category of substance. This duality is well expressed in the parallel duality of the categorial status of number.

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