Liber III.A — Physicist vs. Mathematician & the Potential Infinite — Critical Commentary & Notes¶
Historical-Mathematical (Heath) & Scholastic-Noetic (Hoenen) Apparatus · Bekker 193b22–194b15, 202b30–208a4
I. Mathematical-Historical Commentary (Secundum Heath & Euclidem)¶
Heath (Thirteen Books, Vol. I, pp. 199, 232–234; Mathematics in Aristotle, pp. 98–113) shows that Physics III.7 (207b27–34) directly explains the wording of Euclid’s Second Postulate! Notice that Euclid does not postulate an actually infinite straight line; instead, Postulate 2 reads: Καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ’ εὐθείας ἐκβαλεῖν (‘To produce a finite straight line continuously in a straight line’—i.e., to whatever finite length a construction requires). Euclid’s constructively extensible finite segment is the exact formalization of Aristotle’s ὅσην ἂν βούλωνται πεπερασμένην (207b30)!
Furthermore, in 206a16–29 and 206b3–12, Aristotle clearly distinguishes convergent geometric bisection (cutting off a constant ratio, such as $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$, which never exhausts the finite segment) from subtracting a fixed equal finite magnitude repeatedly, which necessarily exhausts and surpasses any finite whole (πᾶν γὰρ πεπερασμένον ἀναιρεῖ ὁτῳοῦν ὡρισμένῳ, 206b11–12)—an explicit formulation of the Axiom of Archimedes / Eudoxus (Elements V, Def. 4 & X.1).
II. Scholastic-Noetic Commentary (Secundum Hoenen, Thomam & Paciolum)¶
In De Noetica Geometriae (Caput I, § 3; Caput VI), Physics II.2 (193b31–35) provides the charter for Aristotelian-Thomistic formal abstraction (separatio in intellectu / abstractio formalis). Unlike the Platonists (‘those who speak of Ideas’, 193b36), who hypostatize physical attributes that cannot even be thought apart from flesh and motion, the geometer isolates continuous quantity (τὸ κοῖλον / τὸ καμπύλον), which genuinely possesses intrinsic intelligibility prior to sensible qualities (οὐδὲ γίγνεται ψεῦδος χωριζόντων — ‘nor does any falsity arise when they separate them’, 193b35, echoed in Scholasticism as abstrahentium non est mendacium).
Equally vital for Hoenen (Caput IV & VI) is Aristotle’s diagnosis at 203b22–25 of why mathematicians are tempted by an actual infinite: ‘because in thought (`ἐν τῇ νοήσει`) it never gives out.’ Aristotle answers at 208a14–19 that thinking something larger does not make it exist outside thought; geometric truth tracks real potential divisibility grounded in continuous extension, anticipating constructive mathematics’ critique of ungrounded cantorian actualities.
III. Locus-by-Locus Philological & Geometrical Notes¶
193b31–194a7 — ‘The Snub’ (To Simon) vs. ‘The Concave’ (To Koilon)
Aristotle’s celebrated contrast: ‘concave’ contains only pure spatial curvature in intelligible extension, whereas ‘snub’ embeds sensible matter (a nose) inside its very essence—mirroring geometry vs. physics.
206b33–207a14 — Inversion of Parmenides’ & Melissus’ Concept of Perfection
Against Melissus, Aristotle sides with Parmenides and the Pythagoreans: the whole and complete (teleion) is what possesses limit and form, whereas the apeiron is the indeterminate matter (ὕλη) of magnitude completion.
207b27–34 — Scale Invariance (Homothety) Replaces Actual Infinity
Aristotle proves with extraordinary geometric elegance why cosmic finitude never restricts Euclidean geometry: any ratio or construction requiring an ‘arbitrarily huge’ diagram can be executed homothetically by dividing an arbitrarily small segment in the exact same ratio (τεμεῖν γὰρ ἔστιν εἰς τὸν αὐτὸν λόγον τῷ μεγίστῳ μεγέθει ἕτερον πηλικονοῦν, 207b31–33).
IV. Sister-Corpus Citations Indexed in This Unit¶
Bekker Locus |
Treatise & Chapter |
Sister Repos |
Cited Locations (Volume, Page / Folio & Display) |
|---|---|---|---|
Physica II.2 |
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euclid (volume_I, I, p. 226): |
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Physica III.4 |
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euclid (volume_I, I, p. 351): |
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Physica III.5 |
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euclid (volume_I, I, p. 233): |
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Physica III.6 |
|
euclid (volume_I, I, p. 233): |
|
Physica III.7 |
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euclid (volume_II, VII, p. 280): |