TRANSLATION NOTES — Chapter 2: On the Problem of Necessity¶
1. Philological & Conceptual Analyses¶
1.1. Constatatio (Empirical Finding / Direct Verification)¶
Latin: Constatatio (plural: constatationes).
Etymology & Translation: A Neo-Scholastic Latinization of the French philosophical term constatation (from constater, to observe, ascertain, or establish a direct fact). We render this throughout as “direct verification,” “ascertainment,” or “empirical finding.”
Epistemological Nuance: Hoenen uses constatatio to designate a direct, indisputable datum of consciousness or phenomenological inspection. It is not an abstract theoretical deduction, but an observational baseline that any attentive mind can personally verify.
1.2. Passio Propria and Proprium (Proper Attribute)¶
Latin: Passio propria, proprium.
Greek Equivalent: ἴδιον (idion).
Scholastic Framework: In Aristotle (Topics I, 5, 102a18; Posterior Analytics I, 4, 73a34–b5) and St. Thomas (In I Anal. Post., lect. 10), a proprium is a property that is not part of the essential definition (quid est) of a subject, but nevertheless flows universally and necessarily from that essence (predication quarto modo dicendi per se).
Application in Caput II:
Divisibility is a passio propria of quantitative extension.
Addibility and order are passiones propriae of multitude.
Unlike common accidents (accidentia communia, such as color), a proper attribute cannot be separated from its subject even in thought.
1.3. Nexus Formalis vs. Nexus Materialis¶
Latin: Nexus formalis; nexus materialis.
Translation: “Formal connection / nexus” vs. “material connection / nexus.”
Thomistic Lineage: St. Thomas Aquinas, In III Sent., d. 11, q. 1, a. 4, ad 6:
« Praedicatum inest subiecto ratione formae importatae per subiectum » (“The predicate belongs to the subject by reason of the form imported by the subject.”)
Conceptual Distinction:
Nexus materialis: The subject and predicate happen to coexist in the same concrete individual (suppositum) accidentally (per accidens), as in “This apple is sweet and red.”
Nexus formalis: The predicate belongs to the subject intrinsically in virtue of the formal nature signified by the subject (per se), as in “An extended line is divisible.”
1.4. Materia Intelligibilis (Intelligible Matter)¶
Latin: Materia intelligibilis.
Greek Origin: ὕλη νοητή (hylē noētē; Aristotle, Metaphysics VII, 10, 1036a11; VIII, 6, 1045a34).
Thomistic Definition: St. Thomas, Summa Theologiae I, q. 85, a. 1, ad 2:
« Materia intelligibilis dicitur substantia secundum quod subiacet quantitati. » (“Intelligible matter is substance insofar as it is subject to quantity.”)
Translation Significance: In modern philosophy, “matter” is almost exclusively associated with physical mass or sensible matter (materia sensibilis). For Hoenen and the Scholastics, materia intelligibilis is quantitative continuous extension and discrete multitude considered apart from sensible qualities (color, temperature, density). It is the proper object of mathematics.
1.5. In Actu Exercito vs. In Actu Signato¶
Latin: In actu exercito (in the exercised act) vs. in actu signato (in the signified act).
Philosophical Tradition: A classical Scholastic distinction formulated in medieval logic and grammar:
In actu exercito: Knowledge or operation actively functioning and operative in living consciousness, without being reflexively conceptualized or verbally formulated.
In actu signato: Knowledge explicitly codified as a formal thesis, theorem, or proposition.
Translational Context: Hoenen shows that when a person counts or reasons geometrically, foundational principles operate in actu exercito; the task of epistemology is to elevate this implicit operational knowledge into reflexive actu signato.
1.6. Motivum seu Determinativum (Motive and Determinant)¶
Latin: Motivum seu determinativum.
Source: Cardinal Cajetan (Thomas de Vio), In De Ente et Essentia, c. 1.
Translational Nuance: Refers to the role played by the sensible phantasm in intellectual intuition. The sensory image does not create the necessary relation; rather, it provides the determinate concrete content (materia) which specifies the judgment, while the intellect intuits its formal necessity.
1.7. Figurabile (Figurable / Capable of Shape)¶
Latin: Figurabile, corpus est figurabile quia extensum.
Source: René Descartes, Principia Philosophiae II, § 64.
Translational Context: Descartes established extension as capable of figure (figurabile). Hoenen adopts this precise term to distinguish between causal mathematical judgments (“divisible because extended”) and contingent sensory descriptions (“divided and black”).
1.8. Ens Extensum ut Extensum (Extended Being as Extended)¶
Latin: Ens extensum ut extensum.
Translation: “Extended being precisely as extended.”
Epistemological Significance: Geometry does not study empty “absolute space” (Newton’s receptacle or Kant’s a priori intuition, both of which Hoenen treats as entia rationis). Geometry’s authentic subject is real physical extension abstracted from qualitative physical accidents.
1.9. Iudicia Praevia (Preliminary Operative Judgments)¶
Latin: Iudicia praevia.
Translation: “Preliminary judgments” or “prior intuitive judgments.”
Context: Judgments that the mind elicits implicitly before formulating formal operations; for instance, the intuitive grasp that multitude can be added to multitude, or that addition necessarily increases quantity.
1.10. ταῦτα δ’ ἐστὶν οἷον ὁρᾶν τῇ νοήσει¶
Greek Citation: Aristotle, Anal. Post. I, 12 (77b30).
Translation: “These things are, as it were, to see by intellection.”
Significance: Aristotle’s definitive formulation for the intellectual intuition of necessary relations directly in the sensory phantasm.
2. Historical, Intertextual & Bibliographical Context¶
2.1. Plato’s Laws and Marsilio Ficino’s Latin Translation¶
Passage: Plato, Leges V, 737e–738b.
Mathematical Context: Plato chose the number 5040 for the territorial and civic divisions of Magnesia because \(5040 = 7! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7\). It has 59 proper aliquot divisors (excluding itself, or 60 divisors in total), and is divisible by every integer from 1 to 10 without remainder.
Ficino’s Translation: Hoenen cites Marsilio Ficino’s celebrated Renaissance Latin translation (Florence, 1484; Paris, 1518):
« Non in plures quam unum de sexaginta partes incidi potest. » (“It can be divided into no more than one short of sixty parts.”)
Intertextual Note: Hoenen notes that Ficino’s Latin rendering captured the exact computational essence: the tally is an empirical discovery, yet its outcome holds with immutable necessity.
2.2. G. W. Leibniz: Operative Definitions in the Nouveaux Essais¶
Text: G. W. Leibniz, Nouveaux Essais sur l’entendement humain (1704), Book IV, ch. 7, § 10.
Historical Dispute: Leibniz attempted to prove that arithmetic is purely analytic and demonstrable by syllogisms from definitions (\(2 = 1+1, 3 = 2+1, 4 = 3+1\)).
Hoenen’s Philological Critique: Hoenen identifies two unacknowledged intuitive operations in Leibniz’s text:
The succession of syllogisms itself requires counting the inferential steps.
Leibniz’s recursive definitions (\(n+1\)) are operative definitions that presuppose the prior intuitive grasp of the successor relation in sensible phantasms.
2.3. Cardinal Désiré Mercier and the Louvain School¶
Source: Cardinal Désiré Mercier, Critériologie générale ou Théorie générale de la certitude (8th ed., Louvain: Institut Supérieur de Philosophie, 1923, pp. 282–283).
Context: Cardinal Mercier was the leading figure of the Neo-Scholastic revival at Louvain. In defending the analyticity of mathematical propositions against Kant, Mercier argued that \(7 + 5 = 12\) is an analytic identity based on the principle that the parts composing the whole and the whole itself are identical.
Hoenen’s Rebuttal: Hoenen demonstrates that Mercier’s syllogism proves only that \(7 + 5\) equals some sum; to establish that this sum is specifically twelve, Mercier had to arrange groups of units: \((1+1+1+1+1+1+1) + (1+1+1+1+1)\) and verify a bi-unique correspondence (correspondentia bi-univoca), which is an experimental inspection in sensible matter.
2.4. Giuseppe Peano and Formalized Arithmetic¶
Source: Giuseppe Peano, Arithmetices principia, nova methodo exposita (Turin, 1889).
Context: Peano axiomatized natural numbers using five primitive axioms and the successor function \(S(n)\). Hoenen notes that Peano’s formalism does not eliminate sensory intuition; rather, it replaces the phantasm of physical multitude with the phantasm of syntactic typographic tokens on paper.
2.5. Moritz Pasch and the Axioms of Order¶
Source: Moritz Pasch, Vorlesungen über neuere Geometrie (Leipzig: Teubner, 1882).
Context: Pasch was the first mathematician to explicitly state the axioms of order (betweenness) and the axiom named after him (a line entering a triangle through a vertex must intersect the opposite base).
Hoenen’s Commentary: For two thousand years, geometers used Pasch’s axiom without ever enunciating it. Hoenen cites this as prime historical evidence of virtual judgments operating in actu exercito in the sensory phantasm.
2.6. August Ferdinand Möbius and Unilateral Surfaces¶
Source: August Ferdinand Möbius, “Theorie der elementaren Verwandtschaft” (1858; Berichte der Königlich Sächsischen Gesellschaft der Wissenschaften, 1858).
Philological Note: Hoenen refers to the Möbius strip under the Latin name folium Moebii. He uses physical experiments with twisted paper strips to illustrate that physical manipulation serves as the motivum seu determinativum for immediate intuitions of topological necessity.
2.7. Albert Einstein: Geometrie und Erfahrung (1921)¶
Source: Albert Einstein, Geometrie und Erfahrung: Erweiterte Fassung des Festvortrages gehalten an der Preußischen Akademie der Wissenschaften (Berlin: Springer, 1921).
The Famous Quotation:
“Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nicht sicher, und insofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit.” (“Insofar as the propositions of mathematics refer to reality, they are not certain; and insofar as they are certain, they do not refer to reality.”)
Hoenen’s Contextual Reply: Hoenen demonstrates that Einstein’s aphorism confuses the inexactitude of empirical measurement with an absence of mathematical necessity. When geometry refers to physical bodies, it refers to their intelligible matter (quantitative extension), within which relations are both real and apodictically certain.
3. Technical Glossary for Caput II¶
Latin / Greek Term |
English Rendering |
Philological & Technical Definition |
|---|---|---|
|
formal connection / nexus |
Predication where the predicate flows intrinsically from the formal essence of the subject (ratione formae importatae per subiectum). |
|
material connection / nexus |
Predication where subject and predicate merely coexist in the same suppositum accidentally (per accidens). |
|
intelligible matter |
Quantitative spatial extension conceived without sensible physical qualities; the proper object of geometry. |
|
formal abstraction |
The intellective act of isolating an intelligible form or quantitative nature from sensible matter. |
|
proper attribute / property |
A necessary characteristic flowing from the essence of a subject (e.g., divisibility flowing from extension). |
|
in the exercised act |
Knowledge operative implicitly in concrete practice or living mental operation. |
|
in the signified act |
Knowledge explicitly formulated as a formal proposition, definition, or theorem. |
|
direct verification / empirical finding |
The ascertainment of an immediate datum of consciousness or experiential fact. |
|
motive or determinant |
Cajetan’s description of the sensory datum: it provides the specific material content while the intellect intuits its necessity. |
|
“these things are, as it were, to see by intellection” |
Aristotle’s formula (Anal. Post. I, 12, 77b30) for intellectual intuition occurring in the sensory phantasm. |
|
extended being as extended |
The authentic proper object of geometry, as opposed to empty “absolute space.” |
|
Möbius strip |
The unilateral surface used by Hoenen to demonstrate that physical manipulation can yield an immediate intuition of topological necessity. |
|
preliminary judgments |
Intuitive, implicit judgments regarding the proper attributes of quantity that precede formal mathematical operations. |
|
figurable |
Capable of receiving geometric shape; a proper attribute of extension in Cartesian and Scholastic physics. |