Numbers themselves and the first mathematical study

Use in the book

Number had first appeared as the divine gift without which human beings could neither reason adequately nor possess the whole of virtue. The culminating course begins with a more particular educational demand: not numbers that have bodies, but numbers themselves, and the whole generation and power of odd and even as it contributes to the nature of what is (990c).

“Numbers that have bodies” contrasts the arithmetic of counted bodily items with the study of numerical relations considered in themselves. The expression does not require imagining bodily numerals physically written on a page. Nor does the opposite phrase, by itself, establish a separately existing realm of mathematical objects. The emphasis is on the object and generality of the inquiry: it must explain numerical nature rather than merely enumerate useful collections.

The odd-even distinction links this advanced curriculum to the basic cognitive deprivation imagined at 977c. A creature ignorant of two, three, odd, even, and number altogether cannot give an account of what it perceives and remembers. At the ending, what was a basic condition of reason becomes a systematic field to be understood in its full generation and power. The transition is from possessing elementary numerical discrimination to investigating the structures it makes available.

The sequence also explains why the next subjects do not abandon number. Geometry is introduced as a making alike of numbers in relation to plane figures, and stereometry extends the treatment into solid-dimensional relations. Musical means then disclose ratios useful in rhythm and harmony. A reader should not impose a rigid modern division between pure arithmetic, geometry, and physics before seeing how the text links them.

The phrase “greatest and first” describes the foundational place of this subject, while the course's goal remains the divine visible order. “First” is not a claim that memorizing elementary counting immediately confers the final wisdom. Preparation, progressive study, testing, and unity remain necessary. Similarly, the mention of numerical contribution to the nature of what is is a wide programmatic claim, not an already supplied mathematical derivation of everything that exists.

Places to read

1. 977cμηδὲ περιττὸν μηδὲ ἄρτιον — Ignorance of odd and even is part of the deprivation of number
2. 990cτὸ δὲ μέγιστόν τε καὶ πρῶτον — The first mathematical subject has foundational priority
3. 990cἀριθμῶν αὐτῶν ἀλλ’ οὐ σώματα ἐχόντων — Numbers themselves are distinguished from embodied counted numbers
4. 990cτοῦ περιττοῦ τε καὶ ἀρτίου — Odd and even return as objects of systematic study
5. 990cγενέσεώς τε καὶ δυνάμεως — Their generation and power are to be learned
6. 990cπρὸς τὴν τῶν ὄντων φύσιν — Numerical inquiry bears on the nature of what is

Translation in this edition

“Numbers themselves, but not of numbers that have bodies” retains the unusual Greek contrast. “Coming into being and power” avoids silently substituting a modern algebraic syllabus. The mathematical square-sense of dynamis becomes relevant in the following passage; it need not be forced upon every occurrence of “power” in the curriculum's opening sentence.

Generation of number — γένεσις (genesis)

γένεσις (genesis)

Generation of number. The generation of odd and even is part of pure-number study; it need not mean that numerals are bodily things produced in a workshop. Attested: γενέσεώς (990c). Numbers themselves and the first mathematical study.

Fundamental numerical class — ἄρτιος (artios)

ἄρτιος (artios)

Fundamental numerical class. Even complements odd in the first mathematical study, whose contribution to the nature of what is must be learned. Attested: ἀρτίου (990c). Numbers themselves and the first mathematical study.

Numbers considered in themselves — ἀριθμός (arithmos)

ἀριθμός (arithmos)

Numbers considered in themselves. The first study concerns numbers themselves rather than counted bodily collections; this is a methodological contrast, not by itself proof of separate mathematical existence. Attested: ἀριθμῶν αὐτῶν (990c). Numbers themselves and the first mathematical study.

Fundamental numerical class — περιττός (perittos)

περιττός (perittos)

Fundamental numerical class. The whole generation and power of odd is studied with even, extending the elementary distinction into a systematic subject. Attested: περιττοῦ (990c). Numbers themselves and the first mathematical study.

Numerical efficacy — δύναμις (dynamis)

δύναμις (dynamis)

Numerical efficacy. The capacity of odd and even, together with their generation, in relation to the nature of what is. Attested: δυνάμεως (990c). Numbers themselves and the first mathematical study.

Numbers themselves without bodily collections (contextual phrase)

ἀριθμῶν αὐτῶν ἀλλ’ οὐ σώματα ἐχόντων (990c)

The opposition directs study toward number and its relations considered in themselves rather than the counting of bodies. It does not independently prove a separate ontological realm or describe written numerals as the rejected bodily numbers. Numbers themselves and the first mathematical study.