Μονάς and ψυχή in the Phaedo

Sophia Stone

doi:10.14195/2183-4105_18_5

Abstract

The paper analyzes the final proof with Greek mathematics and the possibility of intermediates in the Phaedo. The final proof in Plato’s Phaedo depends on a claim at 105c6, that μονάς, ‘unit’, generates περιττός ‘odd’ in number. So, ψυχή ‘soul’ generates ζωή ‘life’ in a body, at 105c10-11. Yet commentators disagree how to understand these mathematical terms and their relation to the soul in Plato’s arguments. The Greek mathematicians understood odd numbers in one of two ways: either that which is not divisible into two equal parts, or that which differs from an even number by a unit. (Euclid VII.7) Plato uses the second way in the final proof. This paper argues that a proper understanding of these mathematical terms within Greek mathematics shows that the argument for the final proof is better than previously thought. Such an interpretation of the final proof lends credence to Platonic intermediates.

Highlights

FIRST PASSAGEIn the first passage under discussion, Socrates distinguishes the Forms Odd and Even from their instantiations
At the end of ‘Equals and Intermediates in Plato’, John Rist concludes that the attempts to show Plato held a doctrine of intermedi‐ ates in the dialogues should be suspect, yet admits that in some passages, ‘Plato appears to discuss a plurality of non‐sensible μονάδες.’[1]
The Greek mathematicians understood odd numbers in one of two ways: either that which is not divisible into two equal parts, or that which differs from an even number by a unit. (Euclid VII.7) Plato in the Phaedo at 105c6 is clearly using the latter definition

Summary

FIRST PASSAGE

In the first passage under discussion, Socrates distinguishes the Forms Odd and Even from their instantiations. The objects Socrates discusses, what we call odd, τὸ περιττόν, are not the Form, yet he points out that we always call them odd on ac‐ count of their nature, διὰ τὸ οὕτω πεφυκέναι, because the odd never leaves them Such is the nature of the triad, quintet, and the half of the whole multitude; ἡ τριὰς, ἡ πεμπτὰς καὶ ὁ ἥμισυς τοῦ ἀριθμοῦ ἅπας, 104a7‐b1. What is more probable and what I offer here is that the ‘by nature’ designation for περιττός in the first line of ἀριθμός means to establish a neces‐ sary relation with an object and its essential characteristic, and that the leaving out ‘by nature’ may mean an accidental relation, not necessary to the objects themselves.[14] If this is right, Socrates is not distin‐ guishing the nature of odd and the even, two aspects of arithmos, rather, Socrates is distin‐ guishing the nature of two kinds of number: whether they are themselves by themselves or whether they are connected to a composite, to a body. In other words, when you have a solid or when you have a bundle of objects, you only discover an odd or an even number when you further limit the multitudes into those which can be divided and those that cannot

SECOND PASSAGE

THIRD PASSAGE

CONCLUSION