TRANSLATION NOTES — Chapter 6: On the Fundamental Subject of Geometry

1. Philological & Conceptual Analyses

1.1. Subiectum Fundamentale (Fundamental Subject)

• Latin: Subiectum fundamentale geometriae.

• Thomistic Context: In Aristotelian epistemology (Anal. Post. I, c. 10), every science has a formal subject genus (subiectum proprium) whose proper attributes (passiones) it demonstrates.

• Hoenen’s Insight: Geometry’s subject is not the arbitrary figures constructed in imagination, nor is it “empty space.” Its subject is continuous extension as such (extensio ut sic). Because continuous extension in itself is isotropic and homogeneous (zero intrinsic Gaussian curvature), the Euclidean structure is the fundamental structure of extension.

1.2. Ens Rationis cum Fundamento in Re vs. Kant’s Unding

• Latin: Ens rationis cum fundamento in re (“a being of reason with a foundation in reality”).

• Scholastic Background: An ens rationis is an object of thought that cannot exist in extramental reality on its own. However, when it has a fundamentum in re, the intellect forms the concept to express real relations existing among actual beings (such as relations of mutual contact, distance, and topography).

• Kant’s *Unding*: Immanuel Kant referred to absolute empty space as an Unding (an “un-thing” or non-entity) because an empty container existing without bodies is an absurdity (Kritik der reinen Vernunft, A 39 / B 56). Hoenen agrees that absolute empty space is extramentally unreal, but rejects Kant’s idealist conclusion that space is merely a subjective form of human sensibility. For Hoenen, space is an objective mental system grounded directly in the mutual contacts of extended bodies.

1.3. Quod Factum Est Infectum Fieri Nequit (The Irreversibility of Time)

• Latin: Quod factum est infectum fieri nequit (“What is done cannot be undone”).

• Philosophical Tradition: An ancient philosophical adage cited by Aristotle (Nicomachean Ethics VI, 2, 1139b10: “For this alone is lacking even to God, to make undone things that have once been done”, quoting Agathon) and St. Thomas Aquinas (ST I, q. 25, a. 4).

• Application to Fluent Extension: Hoenen uses this principle to demonstrate the radical asymmetry between linear spatial extension and temporal duration:

  • In a spatial line, motion can proceed from \(A\) to \(B\) and return from \(B\) to \(A\) (the road between Athens and Thebes, Phys. III, 3).

  • In duration, motion is strictly irreversible. The flow from past to future is an intrinsic, necessary law of existential act.

1.4. Nullus Est Motus in Praedicamento “Quando”

• Latin: Non datur motus in praedicamento quando.

• Aristotelian Background: In Physica V, c. 2 (225b10), Aristotle establishes that motion (motus) occurs only in three categories: Quantity (quantitas = growth/diminution), Quality (qualitas = alteration), and Place (ubi = local motion). There is no motion in the category of Substance, nor in Relation, nor in Time (quando).

• Noetic Significance: Bodies change place, distance, and configuration in space; but the temporal order of events cannot be reordered (\(A, B, C\) cannot become \(A, C, B\)). The contact of durations is irrevocably fixed by the inception of their existential act.

1.5. Monstrum Metaphysicum (The Metaphysical Monster)

• Latin: Ponitur monstrum metaphysicum.

• Context: § 3, no. 2 C (p. 222).

• Target: The ontological hypostatization of Hermann Minkowski’s 1908 four-dimensional continuum (\(x, y, z, ict\) or \(ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2\)).

• The Core Error: Conflating essence and existence. Permanent 3D extension is an accident inhering in material essence; successive 1D duration is the flow of the act of existence (esse fluens). To collapse duration into a fourth static spatial dimension turns time into space, eliminates genuine becoming, freezes the universe into a static block, and treats existential act as if it were a geometric coordinate. Hoenen permits spacetime as a mathematical calculating device (ens rationis), but rejects it as an ontological cosmology.

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2. Historical, Intertextual & Bibliographical Context

2.1. Henri Poincaré: Refuting Spatial Conventionalism

• Source: Henri Poincaré, La science et l’hypothèse (Paris: Flammarion, 1902, pp. 80–84).

• Poincaré’s Claim: If we take a physical circle, measure its radius and circumference, and find \(C / 2r \neq \pi\), we have not performed an experiment on “space,” but on the physical matter of the ring and measuring rod. Poincaré concludes that geometric axioms are mere conventions regarding the relations of bodies to “space.”

• Hoenen’s Refutation:

  1. Poincaré’s Right Premise: Experiments indeed bear on physical bodies, not on “empty space.”

  2. Poincaré’s Fatal Blind Spot: Poincaré fails to distinguish between a physical property (e.g., thermal expansion, known empirically as a contingent fact) and a geometric property (divisibility, which flows necessarily from extension).

  3. The Operative Definition: The experiment of dividing a body reveals the intelligible nature of extension itself. The geometer does not need “space”; he abstracts the necessary properties of extended being directly from the phantasm.

2.2. Albert Einstein’s Realist Epistemology (Forum, 1930)

• Source: Albert Einstein, “Raum-, Äther- und Feld-Problem”, Forum Philosophicum 1 (1930), p. 173.

• The Citation:

  “Space presupposes the conception of the objective corporeal world. I can recognize bodies through their sensible characteristics without yet conceiving them as spatial. If the concept of bodies is formed in this way, sensible experience forces us to establish local relations between bodies, that is, relations of mutual contact. What we indicate as spatial relations between bodies is nothing else. Therefore: without the concept of bodies, no concept of spatial relations between bodies; and without the concept of spatial relations, no concept of space.”

• Thomistic Concordance: Hoenen highlights that Einstein, guided by physical common sense, independently recovered the classical Aristotelian-Thomistic doctrine:

  • Aristotle (Phys. IV, 1–4) and Aquinas (In IV Phys., lect. 1–6) taught that the notion of place (locus) arises entirely from the mutual contact of bodies in motion.

  • “Space” is neither an absolute Newtonian arena nor a subjective Kantian form; it is the mental abstraction of the network of mutual contacts between real extended bodies.

2.3. Aristotle and the Theory of Measurement (Metaphysics X)

• Source: Aristotle, Metaphysics X (I), c. 1 (1052b20–1053a25).

• The Measure (μέτρον): Aristotle defines the measure as that by which quantity is known (ᾧ τὸ ποσὸν γιγνώσκεται).

• Homogeneity / Connaturality (συγγενές): The measuring unit must belong to the exact same genus as the measured object (a length measures a length, a duration measures a duration).

• Discovery, Not Creation: Measuring does not create the quantitative species; it detects an already existing quantitative magnitude in the real object and expresses it numerically.

2.4. Permanent Extension vs. Successive Duration: Comparative Categories

Category	Permanent Extension (Extensio Permanens)	Successive Extension / Duration (Extensio Fluens)
Ontological Order	Essence / Quiddity (determination of material form)	Existence (actus essendi, fluent act of esse)
Dimensionality	Three-dimensional (surface = 2D boundary; line = 1D)	Strictly Unidimensional (multi-dimensional duration is absurd)
Directionality	Reversible (opposite directions on a line: left/right)	Strictly Irreversible (single direction: past \(\to\) future)
Sheaf of Directions	Continuum of Directions (forming angles, sheaf around point)	No Angles / No Sheaf (two durations cannot form an angle)
Coincidence / Contact	Requires Spatial Alignment (angle must be zero; contact varies)	Spontaneous Penetration (co-beginning implies total coincidence)
Modifiability	Contingent & Variable (bodies can be rearranged in space)	Irrevocable (quod factum est infectum fieri nequit)
Category	Underlies category Ubi (place) and Quantitas	Manifested in category Quando (time; motus non datur)

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3. Technical Glossary for Caput VI

Latin / Greek Term	English Rendering	Philological & Conceptual Definition
subiectum fundamentale	fundamental subject	The primary subject genus of a demonstrative science (Anal. Post. I, 10); for geometry, continuous extension.
ens rationis cum fundamento in re	being of reason with a foundation in reality	A mental construct grounded objectively in real relations among physical beings (e.g., relational space).
extensio permanens	permanent extension	Three-dimensional, static spatial extension belonging to corporeal essence.
extensio fluens	fluent extension / duration	One-dimensional, successive duration belonging to the existential act of motion (esse fluens).
monstrum metaphysicum	metaphysical monster	Hoenen’s term for the ontological hypostatization of 4D spacetime, which conflates essence with existence.
μέτρον (metron)	measure / unit	That by which quantity is known (Metaph. X, 1).
συγγενές (syngenes)	connatural / homogeneous	The requirement that the measuring unit belong to the same genus as the measured object.