Unit I — Philosophy, the Quadrivium, the Odd Sieve & Perfect Numbers — Structural Digest

Argumentative Architecture & Interactive Diagram Guide · Book I, Chs. 1–7, 13, 16

Nicomachus I.3 — Ontological Division of Quantity & the Four Mathemata Boethius's Quadrivium ("fourfold path" from sense-perception to intelligible wisdom) BEING / QUANTITY (τὸ ποσόν) Objects of Wisdom (σοφία) MULTITUDE (τὸ πλῆθος) Discrete · Rooted in Monad, advancing infinitely MAGNITUDE (τὸ μέγεθος) Continuous · Bounded whole, infinitely divisible I · καθ’ ἑαυτό (Per se) ARITHMETIC Ἀριθμητική Even, Odd, Prime, Perfect Sovereign Mother (I.4–5) II · πρὸς ἄλλο (Ad aliud) MUSIC (HARMONICS) Μουσική 2:1 · 3:2 · 4:3 · 9:8 Relative Ratio III · ἐν μονῇ (At rest) GEOMETRY Γεωμετρία Triangle, Square, Cube Stationary Form IV · ἐν κινήσει (In motion) ASTRONOMY Σφαιρική Ordered Revolutions Celestial Harmony Ontological Priority (συναναιρεῖ μὲν, οὐ συναναιρεῖται δέ): Arithmetic precedes Music, Geometry, and Sphaerics
Book I, Chs. 3–5: Ontological tree of Quantity dividing into discrete Multitude (Arithmetic, Music) and continuous Magnitude (Geometry, Astronomy).

Architectural Thesis

Unit I establishes the philosophical charter of the Western mathematical curriculum before classifying odd primes and even perfect numbers:

  1. Immaterial Reality & the Dichotomy of Quantity (I.1–2): Following Pythagoras and Plato's Timaeus (plato:timaeus:27d-36d), true wisdom studies eternal immaterial realities rather than bodily flux. Quantity bifurcates into discontinuous Multitude (poson, bounded below by the monad and increasing without limit) and continuous Magnitude (pelikon, bounded as a whole and infinitely divisible).

  2. The Four Sister Sciences & Asymmetric Priority (I.3–6): Quoting Archytas (archytas:dk47b1), Nicomachus maps absolute vs. relative multitude to Arithmetic and Music, and stationary vs. revolving magnitude to Geometry and Astronomy. By asymmetric co-removal, Arithmetic precedes all three sister sciences and exists pre-eminently as the cosmic exemplar in the Mind of the Creator (I.6.1).

  3. The Odd Sieve of Eratosthenes & Perfect Numbers (I.7, 13, 16): Operating strictly on odd numbers \(\ge 3\), Eratosthenes's Sieve steps by \(p\) positions along the odd sequence (\(2p\) in integers) to separate primes from odd composites, while Euclid's formula \(2^{p-1}(2^p - 1)\) generates the perfect numbers \(6, 28, 496, 8128\).

Chapter Conspectus

Chapter

Core Doctrine

Hoche / D’Ooge

Boethius & Canon Parallel

I.1–2

Philosophy as love of wisdom; eternal being vs. bodily flux; Multitude (poson) vs. Magnitude (pēlikon)

  1. 1–5 / D. 181–183

Boethius Arith. I.1; Plato Timaeus 27d–28a

I.3

Four Mathemata (Quadrivium) as bridges/ladders; Archytas DK 47 B 1 & Epinomis 991d

  1. 5–9 / D. 183–187

Boethius Arith. I.1 (quadruvium); Proclus In Eucl. 35–37

I.4–6

Arithmetic as mother (mētēr) via co-removal (synanairein) & archetypal number in the Divine Mind

  1. 9–12 / D. 187–189

Aristotle Post. An. I.27; Aquinas In Boeth. de Trin. q. 5

I.7, 13

Monad, Even vs. Odd; prime, composite, and coprime odds via Eratosthenes’s Odd Sieve (koskinon)

  1. 13–14, 29–36 / D. 190–207

Euclid Elements VII Defs. 1–14; Boethius Arith. I.17

I.16

Ethical triad of superabundant, deficient, and perfect numbers (\(6, 28, 496, 8128\))

  1. 39–44 / D. 209–212

Euclid Elements IX.36; Boethius Arith. I.19–20