Odd and even

Use in the book

A being that knows neither two and three nor odd and even and is altogether ignorant of number cannot give an account of what it only perceives and remembers (977c). The progression joins individual small numbers to a classification within number. Knowing odd and even is therefore more than being able to repeat the words “two” and “three”; it indicates an elementary grasp of numerical distinctions.

The passage is a negative test, not a complete positive curriculum. The Athenian does not assert that identifying one even and one odd number is already sufficient for wisdom. Nor does the example define reason exclusively as parity calculation. Numerical discrimination stands at the threshold of the accountable understanding that the argument requires, while the fuller account of why number is foundational is deferred.

The soul's later description includes memory and reckoning in odd and even changes (981c). The phrase connects the psychic capacity to the numerical vocabulary but is compressed; the dialogue does not explain a specific algorithm or identify a particular alternation of astronomical periods at that point. It should not become a platform for unmarked reconstruction.

The curriculum finally requires the whole generation and power of the odd and even and what it contributes to the nature of things (990c). Elementary discrimination has become systematic inquiry into numbers themselves. The development matters: the earlier threshold and later comprehensive study belong to different stages, and the latter helps explain why ordinary practical counting has not already answered the question of wisdom.

Places to read

1. 977cδύο καὶ τρία — Small numbers supply the first concrete illustration
2. 977cμηδὲ περιττὸν μηδὲ ἄρτιον — Odd and even are explicit objects of missing knowledge
3. 977cἀγνοοῖ δὲ τὸ παράπαν ἀριθμόν — The deficiency is completed as ignorance of number altogether
4. 981cἐν περιτταῖς τε καὶ ἀρτίαις ἅμα μεταβολαῖς — Odd and even changes enter the description of soul's reckoning
5. 990cτοῦ περιττοῦ τε καὶ ἀρτίου γενέσεώς τε καὶ δυνάμεως — The later study concerns their whole generation and power

Translation in this edition

The consistent “odd” and “even” make the three passages comparable while leaving their constructions different. “Power” in the curricular phrase requires attention to its mathematical setting; it should not be assumed to be identical with every earlier natural capacity called dynamis.

Elementary parity — ἄρτιος (artios)

ἄρτιος (artios)

Elementary parity. The even is the paired numerical category whose total ignorance accompanies the inability to give an account of perceived and remembered things. Attested: ἄρτιον (977c). Odd and even elementary discrimination and later study.

Elementary parity — περιττός (perittos)

περιττός (perittos)

Elementary parity. Knowing the odd is paired with knowing the even as a minimal numerical distinction missing from the being that cannot give an account. Attested: περιττὸν (977c). Odd and even elementary discrimination and later study.

Numerical knowledge — γιγνώσκειν (gignōskein)

γιγνώσκειν (gignōskein)

Numerical knowledge. The hypothetical being fails to know two and three or odd and even, and so cannot give an account beyond perceptions and memories. Attested: γιγνώσκοι (977c). Odd and even elementary discrimination and later study.

Extension of counting — τρεῖς (treis)

τρεῖς (treis)

Extension of counting. The neuter τρία appears among the small numbers the ignorant being lacks and among the further numbers inferred from repeated appearances. Attested: τρία (977c). Odd and even elementary discrimination and later study.

Use in the book

The arithmetic account distinguishes even numbers divided into equal parts from odd numbers divided into unequal parts with a middle. Another passage judges the odd more complete because it has beginning, end, and middle, though its ensuing geometrical explanation is obscure. Cleanthes’ hymn uses the paired terms in praise of the god who makes what is uneven even and brings order to disorder.

Places to read

1. Arith.6Odd and even are defined through division and a middle.
2. Arith.5bNumerical completeness is connected with boundary and ratio.
3. I.1.12bThe vocabulary supports a poetic account of divine ordering.

Translation in this edition

“Odd/even” serves arithmetic; “uneven/even” preserves the hymn’s wider force. The difficult geometry is not reconstructed without evidence.