Double and half

Use in the book

At 168c Socrates extends the greater/smaller test. Something double of other doubles and itself would have to be double of itself considered as half, because only a half can be the correlate of a double in that relation. The same item would occupy both proportional roles relative to itself.

The example strengthens the specificity of the relational reasoning. It is not enough to say that the object shares some general quality with the subject; it must possess precisely the role required by the subject's power. That thought is generalized at 168d when self-hearing requires sound and self-seeing colour.

One need not supply modern exceptional arithmetic cases to the text's ordinary ratios of quantities. Its concern is the structure of the comparisons its participants recognize, followed by the open question whether knowledge behaves similarly. The double example establishes a difficulty in a particular relational family, not a general theorem that everything related to itself is contradictory.

Places to read

1. 168cDouble and half exemplify incompatible reflexive ratio roles
2. 168dThe general object-condition drawn from these examples

Translation in this edition

The edition's “as of halves” makes explicit the role of the genitive hēmiseos. It should not be reduced to “double things are halves”, which loses the correlativity.

διπλάσιος

Attested: διπλάσιόν / διπλασίων / διπλάσιον (diplasion / diplasiōn / diplasion). Double or twice as much; the attempted double-of-itself relation would also require the same thing to be its own half. References: 168c. Lexical identification: lemma: διπλάσιος. See Double and half.

ἥμισυς

Attested: ἡμίσεος (hēmiseos). Half; the correlate that something double must be double of in the relational argument. References: 168c. Lexical identification: lemma: ἥμισυς. See Double and half.