TRANSLATION NOTES — Chapter 7: On Extension as Intelligible Matter¶
1. Philological & Conceptual Analyses¶
1.1. Materia Intelligibilis (Intelligible Matter)¶
Latin: Materia intelligibilis.
Greek Origin: ὕλη νοητή (hylē noētē; Aristotle, Metaphysics VII, 10, 1036a11; VII, 11, 1037a4; VIII, 6, 1045a34).
Thomistic Doctrine: In St. Thomas (ST I, q. 85, a. 1, ad 2; In Boeth. de Trin., q. 5, a. 3), intelligible matter is quantified substance considered under continuous or discrete quantity, abstracted from sensible qualities (heat, cold, color, hardness).
Hoenen’s Synthesis: Intelligible matter is the materia ex qua of mathematical construction. Continuous quantity is the matter of geometry; discrete quantity (multitude) is the matter of arithmetic. It is passive potency awaiting constructive actuation.
1.2. Genus Subiectum vs. Subiecta Secundaria vs. Propriae Passiones¶
Latin: Genus subiectum (subject genus); subiecta secundaria (secondary subjects); propriae passiones (proper passions/attributes).
Scholastic Framework: In Aristotle’s Analytica Posteriora (I, c. 10, 76b11–16) and St. Thomas (In I Anal. Post., lect. 2, no. 5; lect. 15):
Every science has a primary subject genus whose existence is presupposed (in geometry: magnitude / continuous extension).
Primary divisions of this subject produce boundaries (surfaces, lines, points), which are primary proper attributes (passiones).
These attributes then become secondary subjects (e.g., triangle, square, circle), whose own proper passions (e.g., having internal angles equal to two right angles) are demonstrated by construction.
Commentary Crux: In Footnote 1, Hoenen notes the famous textual difficulty in Anal. Post. I, c. 1 (71a14), where Aristotle treats “triangle” (trigonon) as a passio, resolved by J. Alvarez Laso (1952).
1.3. Demonstratio Quasi Operativa (Quasi-Operative Demonstration)¶
Latin: Demonstratio quasi operativa.
Source: St. Thomas, In I Anal. Post., lect. 2, no. 5.
Meaning: In geometry, demonstrating the mathematical existence of a figure is accomplished not by verbal syllogism alone, but by an operative procedure (e.g., Euclid I.1: constructing an equilateral triangle upon a given line using compass intersections).
Modern Anticipation: Hoenen highlights that the Scholastics anticipated P. W. Bridgman’s operational method (operational definition) by seven centuries, but with full metaphysical grounding.
1.4. Poiountes Gignōskousin (ποιοῦντες γιγνώσκουσιν) / Facientes Cognoscunt¶
Greek / Latin: ποιοῦντες γιγνώσκουσιν / facientes cognoscunt (“in making / acting they know”).
Source: Aristotle, Metaphysics IX (Θ), c. 9, 1051a21–33:
“It is by an activity also that geometrical constructions are discovered; for we discover them by dividing. If the figures had been already divided, the constructions would have been obvious; but as it is, they are present only in potency… It is by making [acting] that they know.”
Hoenen’s Vindication: Geometry is neither passive empiricism nor unconstrained mental invention. The geometer actualizes what is already present in potentia in extension.
1.5. Ta Ex Aphaireseōs (τὰ ἐξ ἀφαιρέσεως)¶
Greek: τὰ ἐξ ἀφαιρέσεως (“things from abstraction”).
Aristotelian Usage: Aristotle repeatedly designates mathematical entities as ta ex aphaireseōs (Phys. II, 2, 193b31; De Anima III, 7, 431b12; Metaph. XI, 3, 1061a29).
Hoenen’s Critique: Modern commentators who claimed that “Aristotelian abstraction cannot justify mathematical certainty” ignored that Aristotle specifically formulated the doctrine of formal abstraction to explain the certainty of mathematics.
1.6. Species Sunt Sicut Numeri¶
Latin: Species sunt sicut numeri (“Species are like numbers”).
Source: Aristotle, Metaphysics VIII (H), c. 3 (1043b33); St. Thomas, In VIII Metaph., lect. 3, no. 1722.
Philosophical Principle: Adding or subtracting a unit changes the species of a number; similarly, adding or subtracting a specific difference changes the essence of a being.
Scientific Realization: In § 3, no. 4, Hoenen identifies the dramatic physical realization of this adage: Mendeleev’s periodic table and Henry Moseley’s atomic numbers (:math:`Z = 1, 2, 3, dots, 92`), where chemical species are ordered strictly by the series of natural numbers.
1.7. Nisi Ipse Intellectus¶
Latin: Nihil est in intellectu quod non prius fuerit in sensu, nisi ipse intellectus.
Traditional Attribution: Popularly attributed to G. W. Leibniz (Nouveaux Essais sur l’entendement humain, Bk. II, ch. 1).
Thomistic Precedence: Hoenen demonstrates that St. Thomas explicitly stated this centuries earlier in Quaestiones Disputatae de Malo, q. 6, a. 1, ad 18: the intellect knows external objects through sense, but knows its own nature reflexively from its act.
2. Historical, Intertextual & Bibliographical Context¶
2.1. Aristotle & Aquinas vs. Bertrand Russell on Mathematical Existence¶
Source: Bertrand Russell, Principles of Mathematics (1903).
Russell’s Error: Russell assumed that every geometric curve, circle, and triangle already exists in act within physical or absolute space.
Hoenen’s Thomistic Correction: In unformed continuous extension, figures exist in potency only. There are no actual triangles or spheres pre-existing inside an uncarved block of marble. The geometer must draw the boundaries; the act of dividing actualizes the figure. Mathematical existence is the demonstrated possibility of actualization in intelligible matter.
2.2. Aristotle vs. Kant on “Constructive Science”¶
Source: Immanuel Kant, Kritik der reinen Vernunft (A 713 / B 741).
Kant’s Thesis: Kant claimed that philosophy is cognition through concepts, while mathematics is “cognition through the construction of concepts,” presenting this as an idealist breakthrough.
Hoenen’s Counter-Demonstration: Kant inherited, and subjectivized, the classical Aristotelian doctrine. St. Thomas (In Boeth. de Trin., q. 5, a. 1, ad 3; In I Anal. Post., lect. 2) explicitly defined geometry as an operative, constructive science whose demonstrations prove mathematical existence through physical or mental construction.
2.3. Émile Meyerson and Mathematical Physics¶
Source: Émile Meyerson, Identité et réalité (Paris: Alcan, 1908).
The Observation: « Le mathématique se détache du fond du réel » (“The mathematical detaches itself from the background of the real”).
Context: In pure mathematics, the mind abstracts intelligible matter directly and easily. In experimental physics, the physicist must use elaborate experimental artifice to isolate mathematical forms (laws) from messy, complex material phenomena. Experimental method is the empirical imitation of mathematical abstraction.
2.4. Dalton, Hyletropic Bodies, and the Emergence of Exactitude¶
Sources: John Dalton, A New System of Chemical Philosophy (Manchester, 1808); Wilhelm Ostwald, Klassiker der exakten Wissenschaften (Leipzig, 1902).
Context: Dalton discovered that chemical elements combine in stoichiometric ratios of small integers (\(1:1, 1:2, 2:3\)). Ostwald observed that this holds only if one first operationally defines “pure substances” (hyletropic bodies) that undergo phase changes without compositional alteration. Discrete arithmetic intelligible matter underlies the atomic constitution of the physical universe, validating the Aristotelian insight that nature is mathematically ordered.
3. Technical Glossary for Caput VII¶
Latin / Greek Term |
English Rendering |
Philological & Conceptual Definition |
|---|---|---|
|
intelligible matter |
Quantitative extension and multitude conceived without sensible qualities; the matter of mathematical construction. |
|
subject genus |
The primary subject of a science whose existence is presupposed (Anal. Post. I, 10). |
|
secondary subjects |
Geometric figures (triangle, circle) constructed out of boundaries in the primary subject. |
|
quasi-operative demonstration |
Demonstrative proof of mathematical existence through an operative procedure or construction. |
|
“in making they know” |
Aristotle’s principle (Metaph. IX, 9) that geometric truths are discovered by active division. |
|
things from abstraction |
Aristotle’s technical designation for mathematical entities (Phys. II, 2). |
|
species are like numbers |
Scholastic principle that essential forms are ordered like the discrete natural numbers. |
|
the intelligible is being |
The fundamental noetic axiom derived from mathematical intuition. |