Glossarium Technicum (Technical Glossary)

* Lexicon Terminorum Graecorum, Scholasticorum et Mathematicorum *

This technical glossary defines all 99 Greek, Scholastic Latin, German, and French philosophical and mathematical terms tagged with :term: across Father Peter Hoenen, S.J.’s De Noetica Geometriae (1954). Every inline :term: in the Latin and English editions links directly to its entry below, and each entry provides the original headword, language tradition, English translation, philosophical explanation, and direct back-links (LA · EN) to every occurrence in the text.

Alphabetical Index of Technical Terms

A

B

C

D

E

F

G

H

I

  • in actu exercito — In the exercised act (implicitly in performance) (Latin (Scholastic))

  • in actu signato — In the signified act (explicitly formulated) (Latin (Scholastic))

  • insieme — Set / Continuum of numbers (Italian)

K

L

  • limen exactitudinis — Threshold of exactitude (Schwellenwert der Genauigkeit) (Latin)

  • logica docens — Theoretical / Reflexive logic (Latin (Scholastic))

  • logica utens — Applied / Exercised logic (Latin (Scholastic))

M

N

  • νόησις — Intellectual intuition / Noetic vision (Greek)

  • νοῦς — Intellect / Habit of first principles (Greek)

O

P

R

S

T

U

V

Z

  • Zusammenhang — Connection / nexus / structural coherence (German)

Glossary Entries & Textual Concordance

abstractio formalis — Formal abstraction

  • Headword: abstractio formalis (abstractio-formalis)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Formal abstraction

  • Noetic & Mathematical Definition: Abstraction of the intelligible form or mathematical nature (quantity/extension) from sensible matter.

  • Occurrences in De Noetica Geometriae (2): Appendix: § 1. (LA · EN), § 1. (LA · EN)

abstractio intuitiva formalis — Intuitive formal abstraction

  • Headword: abstractio intuitiva formalis (abstractio-intuitiva)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Intuitive formal abstraction

  • Noetic & Mathematical Definition: Hoenen’s technical designation for the intellectual operation in which the mind directly intuits universal, necessary formal relationships (such as spatial continuity and geometric properties) within the concrete sensible phantasm.

  • Occurrences in De Noetica Geometriae (4): Appendix: § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 7. (LA · EN)

agere sequitur ad esse in actu — Action follows upon actual being

  • Headword: agere sequitur ad esse in actu (agere-sequitur-esse)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Action follows upon actual being

  • Noetic & Mathematical Definition: Classical Thomistic metaphysical axiom affirming that no causal agency or operation can proceed except from an entity that already possesses actual existence (esse in actu).

  • Occurrences in De Noetica Geometriae (4): Appendix: § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 3. (LA · EN)

αἴσθημα (pl. αἰσθήματα) — Sense-impression / perceptual datum

  • Headword: αἴσθημα (pl. αἰσθήματα) (aisthema)

  • Language Tradition: Greek

  • English Translation: Sense-impression / perceptual datum

  • Noetic & Mathematical Definition: In Aristotle’s De Anima (III, c. 8, 432a), the content of sensory perception from which the imagination forms phantasms and the intellect abstracts intelligible species.

  • Occurrences in De Noetica Geometriae (1): Caput III: A) (LA · EN)

αἴτημα (αἰτήματα) — Postulate (postulates)

  • Headword: αἴτημα (αἰτήματα) (aitema)

  • Language Tradition: Greek

  • English Translation: Postulate (postulates)

  • Noetic & Mathematical Definition: In Aristotle and Euclid, a demonstrable or constructible principle requested from the learner without proof within the science.

  • Occurrences in De Noetica Geometriae (3): Caput I: § 1. (LA · EN), § 5. (LA · EN) | Caput IV: § 7. (LA · EN)

αἰτεῖται — He postulates / requests

  • Headword: αἰτεῖται (aiteo)

  • Language Tradition: Greek

  • English Translation: He postulates / requests

  • Noetic & Mathematical Definition: Aristotle’s verb (Anal. Post. I 10, 76b28) for assuming a principle when the learner has no prior opinion or holds a contrary opinion.

  • Occurrences in De Noetica Geometriae (1): Caput IV: § 7. (LA · EN)

ἀναιρεῖ (ἀναιρεῖν) — Destroys / sublates (hypotheses)

  • Headword: ἀναιρεῖ (ἀναιρεῖν) (anairei)

  • Language Tradition: Greek

  • English Translation: Destroys / sublates (hypotheses)

  • Noetic & Mathematical Definition: Plato’s term in Republic VII (533c) for the dialectical method, which ‘destroys’ or transcends the provisional hypotheses of the mathematical sciences in order to ascend to the unhypothetical first principle.

  • Occurrences in De Noetica Geometriae (1): Caput I: § 1. (LA · EN)

andererseits — On the other hand

  • Headword: andererseits (andererseits)

  • Language Tradition: German

  • English Translation: On the other hand

  • Noetic & Mathematical Definition: Correlative discourse marker used in German mathematical exposition (paired with einerseits) when contrasting dual axiomatic or geometric aspects.

  • Occurrences in De Noetica Geometriae (1): Caput V: B) (LA · EN)

Anordnung — Order / Arrangement

  • Headword: Anordnung (anordnung)

  • Language Tradition: German

  • English Translation: Order / Arrangement

  • Noetic & Mathematical Definition: Hilbert’s second group of geometric axioms (Axioms of Order / Betweenness) in Grundlagen der Geometrie.

  • Occurrences in De Noetica Geometriae (1): Caput IV: 1. (LA · EN)

the apotheosis of stupidity

  • Headword: the apotheosis of stupidity (apotheosis-of-stupidity)

  • Language Tradition: English

  • English Translation: The apotheosis of stupidity

  • Noetic & Mathematical Definition: G. H. Hardy’s characterization (Mind, 1929) of reducing mathematical proof to purely mechanical, uncomprehending symbol manipulation.

  • Occurrences in De Noetica Geometriae (1): Caput V: B) (LA · EN)

Approximationsmathematik — Mathematics of approximation

  • Headword: Approximationsmathematik (approximationsmathematik)

  • Language Tradition: German

  • English Translation: Mathematics of approximation

  • Noetic & Mathematical Definition: Felix Klein’s term for empirical/applied mathematics bounded by a threshold of sensory accuracy (limen exactitudinis), contrasted with Präzisionsmathematik.

  • Occurrences in De Noetica Geometriae (3): Caput I: § 4. (LA · EN) | Caput V: A) (LA · EN), B) (LA · EN)

ἀρχή (acc. ἀρχήν) — First principle / starting point / origin

  • Headword: ἀρχή (acc. ἀρχήν) (arche)

  • Language Tradition: Greek

  • English Translation: First principle / starting point / origin

  • Noetic & Mathematical Definition: In Aristotle and Plato, the foundational starting point of being, demonstration, or cognition (such as the unhypothetical principle in Republic VI–VII or the immediate premises of science in Posterior Analytics).

  • Occurrences in De Noetica Geometriae (3): Appendix: § 4. (LA · EN), § 4. (LA · EN), § 4. (LA · EN)

ars epsilontica — Epsilon-delta technique

  • Headword: ars epsilontica (ars-epsilontica)

  • Language Tradition: Latin (Modern Mathematical)

  • English Translation: Epsilon-delta technique

  • Noetic & Mathematical Definition: Weierstrassian rigorous limit and approximation analysis using arbitrary small quantities ε.

  • Occurrences in De Noetica Geometriae (1): Caput V: A) (LA · EN)

ἄτομον — Indivisible

  • Headword: ἄτομον (atomon)

  • Language Tradition: Greek

  • English Translation: Indivisible

  • Noetic & Mathematical Definition: An indivisible magnitude or unit lacking further division.

  • Occurrences in De Noetica Geometriae (1): Caput VI: 1. (LA · EN)

ἀξίωμα (ἀξιώματα) — Axiom / Common dignity

  • Headword: ἀξίωμα (ἀξιώματα) (axioma)

  • Language Tradition: Greek

  • English Translation: Axiom / Common dignity

  • Noetic & Mathematical Definition: Self-evident first principle that every learner must possess to acquire scientific knowledge (Aristotle, Anal. Post. I 2 & I 10).

  • Occurrences in De Noetica Geometriae (4): Caput I: § 1. (LA · EN), § 1. (LA · EN), § 5. (LA · EN) | Caput IV: § 7. (LA · EN)

Beschaffenheit — Constitution / qualitative nature / character

  • Headword: Beschaffenheit (beschaffenheit)

  • Language Tradition: German

  • English Translation: Constitution / qualitative nature / character

  • Noetic & Mathematical Definition: Term used by German mathematicians (such as Killing and Klein) to denote the intrinsic spatial or structural character of geometric figures and manifolds.

  • Occurrences in De Noetica Geometriae (1): Caput V: B) (LA · EN)

betweenness

  • Headword: betweenness (betweenness)

  • Language Tradition: English

  • English Translation: Betweenness

  • Noetic & Mathematical Definition: Fundamental order relation among collinear points (B lies between A and C), introduced by Pasch and formalized by Hilbert and Russell.

  • Occurrences in De Noetica Geometriae (1): Caput V: B) (LA · EN)

Bezeichnungsweise — Mode of designation / system of notation

  • Headword: Bezeichnungsweise (bezeichnungsweise)

  • Language Tradition: German

  • English Translation: Mode of designation / system of notation

  • Noetic & Mathematical Definition: In modern axiomatics and mathematical formalism, a symbolic notation or convention for designating geometric objects and relations.

  • Occurrences in De Noetica Geometriae (1): Caput V: B) (LA · EN)

le cheminement de la pensée — The progression / pathway of thought

  • Headword: le cheminement de la pensée (cheminement-de-la-pensee)

  • Language Tradition: French

  • English Translation: The progression / pathway of thought

  • Noetic & Mathematical Definition: Émile Meyerson’s phrase (1931) for the actual cognitive movement of the mind in scientific discovery and deduction.

  • Occurrences in De Noetica Geometriae (2): Caput IV: 4. (LA · EN), 8. (LA · EN)

circulus vitiosus — Vicious circle

  • Headword: circulus vitiosus (circulus-vitiosus)

  • Language Tradition: Latin

  • English Translation: Vicious circle

  • Noetic & Mathematical Definition: Circular reasoning or definition where the definiendum is presupposed in the definiens.

  • Occurrences in De Noetica Geometriae (1): Caput IV: n. 13 (LA · EN)

cogito ergo sum — I think, therefore I am

  • Headword: cogito ergo sum (cogito-ergo-sum)

  • Language Tradition: Latin

  • English Translation: I think, therefore I am

  • Noetic & Mathematical Definition: Descartes’ existential judgment; analyzed by Hoenen in Appendix as an intuitive perception of a necessary connection between an experienced act (cogito) and existence (sum).

  • Occurrences in De Noetica Geometriae (18): Appendix: § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 2. (LA · EN), Recapitulatio. (LA · EN), § 5. (LA · EN), § 5. (LA · EN), § 6. (LA · EN), § 7. (LA · EN)

das Gedachte (des Gedachten) — That which is thought / the object of thought

  • Headword: das Gedachte (des Gedachten) (das-gedachte)

  • Language Tradition: German

  • English Translation: That which is thought / the object of thought

  • Noetic & Mathematical Definition: In Gerhard Stammler’s epistemology of logic, the objective conceptual content grasped by the mind in judgment and inference, as opposed to mere psychological acts.

  • Occurrences in De Noetica Geometriae (1): Caput V: B) (LA · EN)

das System beider Punkte — The system of both points

  • Headword: das System beider Punkte (das-system-beider-punkte)

  • Language Tradition: German

  • English Translation: The system of both points

  • Noetic & Mathematical Definition: David Hilbert’s phrase in Grundlagen der Geometrie (§ 1) designating a pair of points as determining a straight line in his axiomatic incidence relations.

  • Occurrences in De Noetica Geometriae (1): Caput IV: 1. (LA · EN)

Denkgesetze — Laws of thought

  • Headword: Denkgesetze (denkgesetze)

  • Language Tradition: German

  • English Translation: Laws of thought

  • Noetic & Mathematical Definition: In 19th- and early 20th-century German logic and epistemology, the fundamental normative principles governing valid conceptual thought and inference.

  • Occurrences in De Noetica Geometriae (1): Caput V: B) (LA · EN)

dispositio rei — Disposition / objective arrangement of the thing

  • Headword: dispositio rei (dispositio-rei)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Disposition / objective arrangement of the thing

  • Noetic & Mathematical Definition: In St. Thomas Aquinas (Metaphysics IX, lect. 11), the real ontological arrangement of a state of affairs to which the intellect’s composition or division must correspond for truth.

  • Occurrences in De Noetica Geometriae (1): Caput V: n. 4 (LA · EN)

distinguo — I distinguish

  • Headword: distinguo (distinguo)

  • Language Tradition: Latin (Scholastic)

  • English Translation: I distinguish

  • Noetic & Mathematical Definition: Classic scholastic disputation operator separating the true and false senses of an ambiguous thesis.

  • Occurrences in De Noetica Geometriae (2): Caput V: C) (LA · EN), C) (LA · EN)

la doctrine préalable — The preliminary / prior doctrine

  • Headword: la doctrine préalable (doctrine-prealable)

  • Language Tradition: French

  • English Translation: The preliminary / prior doctrine

  • Noetic & Mathematical Definition: Ferdinand Gonseth’s epistemological term for the pre-axiomatic intuitive body of knowledge that necessarily precedes and guides formal axiomatization.

  • Occurrences in De Noetica Geometriae (1): Caput V: 3. (LA · EN)

δύναμις (δυνάμει) — Potency / In potentiality

  • Headword: δύναμις (δυνάμει) (dynamis)

  • Language Tradition: Greek

  • English Translation: Potency / In potentiality

  • Noetic & Mathematical Definition: Aristotelian potentiality, contrasted with actuality (energeia / entelecheia).

  • Occurrences in De Noetica Geometriae (2): Appendix: § 1. (LA · EN), § 3. (LA · EN)

einerseits — On the one hand

  • Headword: einerseits (einerseits)

  • Language Tradition: German

  • English Translation: On the one hand

  • Noetic & Mathematical Definition: Correlative German expression used in mathematical analysis to distinguish complementary aspects of a geometric definition or postulate.

  • Occurrences in De Noetica Geometriae (1): Caput V: B) (LA · EN)

ἤλεκτρον — Amber (and by extension, the electron)

  • Headword: ἤλεκτρον (elektron)

  • Language Tradition: Greek

  • English Translation: Amber (and by extension, the electron)

  • Noetic & Mathematical Definition: Classical Greek term for amber, cited by Aristotle in Meteorology IV (388b) as a homogeneous natural substance and contrasted with modern microphysical particles.

  • Occurrences in De Noetica Geometriae (1): Caput VI: A. De extensione corporum. (LA · EN)

ἐνέργεια — Act / Activity / Actuality

  • Headword: ἐνέργεια (energeia)

  • Language Tradition: Greek

  • English Translation: Act / Activity / Actuality

  • Noetic & Mathematical Definition: Aristotle’s core metaphysical term for actuality or operation, originally derived from local motion and extended to all existential and cognitive acts.

  • Occurrences in De Noetica Geometriae (14): Appendix: § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 3. (LA · EN), § 3. (LA · EN), § 3. (LA · EN), § 3. (LA · EN), § 3. (LA · EN), § 3. (LA · EN), n. 5 (LA · EN), n. 5 (LA · EN)

ἐντελέχεια — Complete actuality / Perfection (entelechy)

  • Headword: ἐντελέχεια (entelecheia)

  • Language Tradition: Greek

  • English Translation: Complete actuality / Perfection (entelechy)

  • Noetic & Mathematical Definition: Aristotle’s term for realized form or act as holding its end/completeness within itself.

  • Occurrences in De Noetica Geometriae (4): Appendix: § 1. (LA · EN), § 1. (LA · EN), n. 5 (LA · EN), n. 5 (LA · EN)

ἐφαρμοζομένου τοῦ τριγώνου — The triangle being applied (superposed)

  • Headword: ἐφαρμοζομένου τοῦ τριγώνου (epharmozomenou-tou-trigonou)

  • Language Tradition: Greek

  • English Translation: The triangle being applied (superposed)

  • Noetic & Mathematical Definition: Euclid’s phrase in Elements I.4 for the superposition of one triangle upon another.

  • Occurrences in De Noetica Geometriae (1): Caput IV: § 3. (LA · EN)

ἐπίπεδος ἐπιφάνειά ἐστιν, ἥτις ἐξ ἴσου ταῖς ἐφ’ ἑαυτῆς εὐθείαις κεῖται — A plane surface is that which lies evenly with the straight lines on itself

  • Headword: ἐπίπεδος ἐπιφάνειά ἐστιν, ἥτις ἐξ ἴσου ταῖς ἐφ’ ἑαυτῆς εὐθείαις κεῖται (epipedos-epiphaneia-estin)

  • Language Tradition: Greek

  • English Translation: A plane surface is that which lies evenly with the straight lines on itself

  • Noetic & Mathematical Definition: Euclid’s Definition 7 of Book I (ed. Heiberg I, pp. 2–3).

  • Occurrences in De Noetica Geometriae (1): Caput IV: 5. (LA · EN)

ἐπιστήμη — Scientific knowledge (apodictic science)

  • Headword: ἐπιστήμη (episteme)

  • Language Tradition: Greek

  • English Translation: Scientific knowledge (apodictic science)

  • Noetic & Mathematical Definition: Demonstrative knowledge through necessary causes and first principles.

  • Occurrences in De Noetica Geometriae (2): Caput I: § 1. (LA · EN) | Caput V: 2. (LA · EN)

l’ère de la rigueur de Weierstrass — The era of Weierstrassian rigor

  • Headword: l’ère de la rigueur de Weierstrass (ere-de-la-rigueur-de-weierstrass)

  • Language Tradition: French

  • English Translation: The era of Weierstrassian rigor

  • Noetic & Mathematical Definition: Henri Poincaré’s characterization of late 19th-century mathematical analysis, in which Karl Weierstrass replaced naive spatial intuition of continuous curves with rigorous epsilon-delta arithmetic definitions.

  • Occurrences in De Noetica Geometriae (1): Caput V: A) (LA · EN)

Erklärungen — Explanations / preliminary definitions

  • Headword: Erklärungen (erklaerungen)

  • Language Tradition: German

  • English Translation: Explanations / preliminary definitions

  • Noetic & Mathematical Definition: David Hilbert’s heading at the opening of Grundlagen der Geometrie introducing the primitive undefined domains of points, straight lines, and planes.

  • Occurrences in De Noetica Geometriae (1): Caput IV: n. 1 (LA · EN)

εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ’ ἑαυτῆς σημείοις κεῖται — A straight line is a line which lies evenly with the points on itself

  • Headword: εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφ’ ἑαυτῆς σημείοις κεῖται (eutheia-gramme-estin)

  • Language Tradition: Greek

  • English Translation: A straight line is a line which lies evenly with the points on itself

  • Noetic & Mathematical Definition: Euclid’s Definition 4 of Book I (ed. Heiberg I, pp. 2–3).

  • Occurrences in De Noetica Geometriae (1): Caput IV: 4. (LA · EN)

facio ergo materia rei factibilis adest — I make/act upon it, therefore the matter of the makeable thing is present

  • Headword: facio ergo materia rei factibilis adest (facio-ergo-materia-rei-factibilis-adest)

  • Language Tradition: Latin

  • English Translation: I make/act upon it, therefore the matter of the makeable thing is present

  • Noetic & Mathematical Definition: Hoenen’s formulation of the existential judgment of external material reality grounded in conscious bodily action upon matter.

  • Occurrences in De Noetica Geometriae (1): Appendix: § 5. (LA · EN)

folium Moebii — Möbius strip (one-sided surface)

  • Headword: folium Moebii (folium-moebii)

  • Language Tradition: Latin

  • English Translation: Möbius strip (one-sided surface)

  • Noetic & Mathematical Definition: The non-orientable one-sided surface discovered by A. F. Möbius (1858), analyzed by Hoenen in Caput III § 2.4 to refute objections against defining surfaces as boundaries of bodies.

  • Occurrences in De Noetica Geometriae (12): Caput III: 4. (LA · EN), 4. (LA · EN), 4. (LA · EN), 4. (LA · EN), 4. (LA · EN), 4. (LA · EN), 4. (LA · EN), 4. (LA · EN), 4. (LA · EN), 4. (LA · EN), 4. (LA · EN), 4. (LA · EN)

Grenzprozess — Limiting process

  • Headword: Grenzprozess (grenzprozess)

  • Language Tradition: German

  • English Translation: Limiting process

  • Noetic & Mathematical Definition: Passage to the ideal mathematical limit from empirical approximations.

  • Occurrences in De Noetica Geometriae (1): Caput III: 1. (LA · EN)

ᾧ τὸ ποσὸν γιγνώσκεται — That by which quantity is known

  • Headword: ᾧ τὸ ποσὸν γιγνώσκεται (ho-to-poson-gignosketai)

  • Language Tradition: Greek

  • English Translation: That by which quantity is known

  • Noetic & Mathematical Definition: Aristotle’s definition of the unit or measure in Metaphysics X (c. 1, 1052b20): the primary measure by which quantitative magnitude or multitude is cognized.

  • Occurrences in De Noetica Geometriae (1): Caput VI: 1. (LA · EN)

hoc movetur ergo aliud, movens, existit — This moves, therefore another (a mover) exists

  • Headword: hoc movetur ergo aliud, movens, existit (hoc-movetur-ergo-aliud-movens-existit)

  • Language Tradition: Latin

  • English Translation: This moves, therefore another (a mover) exists

  • Noetic & Mathematical Definition: Existential causal inference from perceived motion to an existing efficient cause.

  • Edition Pages: pp. 293

hoc movetur ergo locus est — This moves locally, therefore place exists

  • Headword: hoc movetur ergo locus est (hoc-movetur-ergo-locus-est)

  • Language Tradition: Latin

  • English Translation: This moves locally, therefore place exists

  • Noetic & Mathematical Definition: Existential inference from local motion (translation) to the reality of spatial location.

  • Occurrences in De Noetica Geometriae (2): Appendix: § 2. (LA · EN), § 2. (LA · EN)

ὁμολογία — Agreement / Admission

  • Headword: ὁμολογία (homologia)

  • Language Tradition: Greek

  • English Translation: Agreement / Admission

  • Noetic & Mathematical Definition: Plato’s term (Republic VII 533c) for unexamined hypotheses accepted merely by convention.

  • Occurrences in De Noetica Geometriae (1): Caput I: § 1. (LA · EN)

ὕλη νοητή — Intelligible matter (materia intelligibilis)

  • Headword: ὕλη νοητή (hyle-noete)

  • Language Tradition: Greek

  • English Translation: Intelligible matter (materia intelligibilis)

  • Noetic & Mathematical Definition: Aristotle’s term (Metaphysics VII 10, 1036a9; VIII 6, 1045a34) for pure spatial extension considered in the imagination apart from sensible qualities.

  • Occurrences in De Noetica Geometriae (1): Caput II: 3. (LA · EN)

ὑπάρχειν — To belong to / To exist / To inhere

  • Headword: ὑπάρχειν (hyparchein)

  • Language Tradition: Greek

  • English Translation: To belong to / To exist / To inhere

  • Noetic & Mathematical Definition: Aristotelian logical and ontological verb for predicate inherence or actual existence.

  • Occurrences in De Noetica Geometriae (2): Appendix: § 6. (LA · EN), § 6. (LA · EN)

ὑπόθεσις (ὑποθέσεις) — Hypothesis / Supposition

  • Headword: ὑπόθεσις (ὑποθέσεις) (hypothesis)

  • Language Tradition: Greek

  • English Translation: Hypothesis / Supposition

  • Noetic & Mathematical Definition: In Aristotle (Anal. Post. I 2 & I 10), an assumed principle asserting that the subject exists or does not exist.

  • Occurrences in De Noetica Geometriae (4): Caput I: § 1. (LA · EN), § 1. (LA · EN), § 5. (LA · EN) | Caput IV: § 7. (LA · EN)

ὑποτίθεται — He supposes / assumes as a hypothesis

  • Headword: ὑποτίθεται (hypotithemai)

  • Language Tradition: Greek

  • English Translation: He supposes / assumes as a hypothesis

  • Noetic & Mathematical Definition: Aristotle’s verb (Anal. Post. I 10, 76b28) for assuming a demonstrable proposition that seems probable to the learner.

  • Occurrences in De Noetica Geometriae (1): Caput IV: § 7. (LA · EN)

in actu exercito — In the exercised act (implicitly in performance)

  • Headword: in actu exercito (in-actu-exercito)

  • Language Tradition: Latin (Scholastic)

  • English Translation: In the exercised act (implicitly in performance)

  • Noetic & Mathematical Definition: Knowledge or principle grasped and used directly within the living act of thinking or perceiving, prior to explicit reflexive formulation (in actu signato).

  • Occurrences in De Noetica Geometriae (12): Praefatio: PRAEFATIO (LA · EN) | Caput VI: 1. (LA · EN) | Caput VII: 2. (LA · EN), 2. (LA · EN), 2. (LA · EN), 2. (LA · EN) | Appendix: § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), Recapitulatio. (LA · EN), § 5. (LA · EN)

in actu signato — In the signified act (explicitly formulated)

  • Headword: in actu signato (in-actu-signato)

  • Language Tradition: Latin (Scholastic)

  • English Translation: In the signified act (explicitly formulated)

  • Noetic & Mathematical Definition: Explicit, reflexively thematicized statement of a concept or principle previously grasped in actu exercito.

  • Occurrences in De Noetica Geometriae (12): Praefatio: PRAEFATIO (LA · EN) | Caput VII: 2. (LA · EN), 2. (LA · EN), 2. (LA · EN), 2. (LA · EN), 3. (LA · EN) | Appendix: § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), Recapitulatio. (LA · EN), § 5. (LA · EN), § 6. (LA · EN)

insieme — Set / Continuum of numbers

  • Headword: insieme (insieme)

  • Language Tradition: Italian

  • English Translation: Set / Continuum of numbers

  • Noetic & Mathematical Definition: Mathematical set (used for the set/continuum of real numbers).

  • Occurrences in De Noetica Geometriae (2): Caput I: § 4. (LA · EN), § 4. (LA · EN)

κίνησις — Motion / Change

  • Headword: κίνησις (kinesis)

  • Language Tradition: Greek

  • English Translation: Motion / Change

  • Noetic & Mathematical Definition: In Aristotle (Physics III 1), the actuality of what is in potentiality insofar as it is in potentiality.

  • Occurrences in De Noetica Geometriae (1): Appendix: § 6. (LA · EN)

kritzelige Kurven — Scribbly / crinkly curves

  • Headword: kritzelige Kurven (kritzelige-kurven)

  • Language Tradition: German

  • English Translation: Scribbly / crinkly curves

  • Noetic & Mathematical Definition: Felix Klein’s vivid description (in Elementarmathematik vom höheren Standpunkte) of rough, imprecise empirical curves drawn by hand, contrasted with idealized analytic curves.

  • Occurrences in De Noetica Geometriae (1): Caput III: C) (LA · EN)

limen exactitudinis — Threshold of exactitude (Schwellenwert der Genauigkeit)

  • Headword: limen exactitudinis (limen-exactitudinis)

  • Language Tradition: Latin

  • English Translation: Threshold of exactitude (Schwellenwert der Genauigkeit)

  • Noetic & Mathematical Definition: The finite limit of precision inherent in all external sensory perception and measurement (Felix Klein).

  • Occurrences in De Noetica Geometriae (2): Caput I: § 3. (LA · EN), § 4. (LA · EN)

logica docens — Theoretical / Reflexive logic

  • Headword: logica docens (logica-docens)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Theoretical / Reflexive logic

  • Noetic & Mathematical Definition: The explicit scientific theory of logical rules and syllogistic forms.

  • Occurrences in De Noetica Geometriae (1): Caput V: B) (LA · EN)

logica utens — Applied / Exercised logic

  • Headword: logica utens (logica-utens)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Applied / Exercised logic

  • Noetic & Mathematical Definition: Logic actively practiced by the mind in reasoning about objects (in actu exercito).

  • Occurrences in De Noetica Geometriae (1): Caput V: B) (LA · EN)

materia intelligibilis — Intelligible matter (ὕλη νοητή)

  • Headword: materia intelligibilis (materia-intelligibilis)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Intelligible matter (ὕλη νοητή)

  • Noetic & Mathematical Definition: Continuous spatial extension grasped in the phantasm apart from sensible qualities, serving as the principle of individuation and constructibility for geometric figures.

  • Occurrences in De Noetica Geometriae (5): Caput III: 7. (LA · EN) | Caput IV: § 3. (LA · EN), § 3. (LA · EN), 2. (LA · EN), n. 2 (LA · EN)

materia sensibilis — Sensible matter (ὕλη αἰσθητή)

  • Headword: materia sensibilis (materia-sensibilis)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Sensible matter (ὕλη αἰσθητή)

  • Noetic & Mathematical Definition: Physical matter endowed with sensible qualities (color, heat, weight, hardness), abstracted from in mathematics.

  • Occurrences in De Noetica Geometriae (1): Caput IV: n. 2 (LA · EN)

minimum imaginabile — Smallest imaginable magnitude

  • Headword: minimum imaginabile (minimum-imaginabile)

  • Language Tradition: Latin

  • English Translation: Smallest imaginable magnitude

  • Noetic & Mathematical Definition: Empiricist thesis (Berkeley, Hume) claiming imagination consists of finite indivisible minima.

  • Occurrences in De Noetica Geometriae (2): Caput III: A) (LA · EN), A) (LA · EN)

minimum visibile — Smallest visible spot

  • Headword: minimum visibile (minimum-visibile)

  • Language Tradition: Latin

  • English Translation: Smallest visible spot

  • Noetic & Mathematical Definition: The smallest angular or spatial extent discernible by the eye.

  • Occurrences in De Noetica Geometriae (3): Caput III: A) (LA · EN), D) (LA · EN), n. 3 (LA · EN)

moveri sequitur esse — Being moved follows upon existing

  • Headword: moveri sequitur esse (moveri-sequitur-esse)

  • Language Tradition: Latin

  • English Translation: Being moved follows upon existing

  • Noetic & Mathematical Definition: Scholastic principle expressing the necessary existential connection between motion and actual existence.

  • Occurrences in De Noetica Geometriae (2): Appendix: § 1. (LA · EN), § 1. (LA · EN)

movetur ergo est — It moves, therefore it exists

  • Headword: movetur ergo est (movetur-ergo-est)

  • Language Tradition: Latin

  • English Translation: It moves, therefore it exists

  • Noetic & Mathematical Definition: Hoenen’s primary paradigm (Appendix § 1) of an intuitive existential judgment connecting perceived motion with actual existence.

  • Occurrences in De Noetica Geometriae (7): Appendix: § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 2. (LA · EN)

νόησις — Intellectual intuition / Noetic vision

  • Headword: νόησις (noesis)

  • Language Tradition: Greek

  • English Translation: Intellectual intuition / Noetic vision

  • Noetic & Mathematical Definition: Direct intellectual apprehension of intelligible natures and first principles in the phantasm.

  • Occurrences in De Noetica Geometriae (2): Caput I: § 2. (LA · EN) | Caput VII: 1. (LA · EN)

νοῦς — Intellect / Habit of first principles

  • Headword: νοῦς (nous)

  • Language Tradition: Greek

  • English Translation: Intellect / Habit of first principles

  • Noetic & Mathematical Definition: In Aristotle (Anal. Post. II 19), the intuitive intellectual faculty that grasps immediate first principles.

  • Occurrences in De Noetica Geometriae (3): Caput I: § 1. (LA · EN) | Caput II: 5. (LA · EN) | Caput V: 2. (LA · EN)

Omne agens agit propter finem — Every agent acts for an end

  • Headword: Omne agens agit propter finem (omne-agens-agit-propter-finem)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Every agent acts for an end

  • Noetic & Mathematical Definition: Foundational Aristotelian-Thomistic principle of finality, cited by Hoenen as an existential synthetic judgment connecting causal action with teleological determination.

  • Occurrences in De Noetica Geometriae (1): Appendix: § 4. (LA · EN)

οὗ μέρος οὐθέν — That of which there is no part

  • Headword: οὗ μέρος οὐθέν (ou-meros-outhen)

  • Language Tradition: Greek

  • English Translation: That of which there is no part

  • Noetic & Mathematical Definition: Euclid’s Definition 1 of a point (Σημεῖόν ἐστιν, οὗ μέρος οὐθέν).

  • Occurrences in De Noetica Geometriae (1): Caput III: B) (LA · EN)

πέρας — Boundary / Limit / Extremity

  • Headword: πέρας (peras)

  • Language Tradition: Greek

  • English Translation: Boundary / Limit / Extremity

  • Noetic & Mathematical Definition: Aristotle’s and Euclid’s ontological category for surfaces, lines, and points as indivisible boundaries of continua.

  • Occurrences in De Noetica Geometriae (1): Appendix: § 4. (LA · EN)

φάντασμα (φαντάσματα) — Phantasm (imaginative representation)

  • Headword: φάντασμα (φαντάσματα) (phantasma)

  • Language Tradition: Greek

  • English Translation: Phantasm (imaginative representation)

  • Noetic & Mathematical Definition: Sensory/imaginative representation in which the intellect beholds universal mathematical and philosophical natures.

  • Occurrences in De Noetica Geometriae (1): Caput III: A) (LA · EN)

φύσις (dat. φύσει) — Nature / by nature

  • Headword: φύσις (dat. φύσει) (physis)

  • Language Tradition: Greek

  • English Translation: Nature / by nature

  • Noetic & Mathematical Definition: In Aristotle’s Posterior Analytics and Metaphysics, that which is prior or more intelligible ‘by nature’ (φύσει / secundum naturam) as opposed to what is prior ‘to us’ (πρὸς ἡμᾶς / quoad nos).

  • Occurrences in De Noetica Geometriae (1): Caput V: B) (LA · EN)

ποιοῦντες γιγνώσκουσιν — By constructing/making, they know (facientes cognoscunt)

  • Headword: ποιοῦντες γιγνώσκουσιν (poiountes-gignoskousin)

  • Language Tradition: Greek

  • English Translation: By constructing/making, they know (facientes cognoscunt)

  • Noetic & Mathematical Definition: Aristotle’s dictum (Metaphysics IX 9, 1051a31) that geometric truths become manifest by actively drawing and dividing figures.

  • Occurrences in De Noetica Geometriae (1): Caput VII: § 1. (LA · EN)

πῶς ἔστιν — How it is / the mode of its being

  • Headword: πῶς ἔστιν (pos-estin)

  • Language Tradition: Greek

  • English Translation: How it is / the mode of its being

  • Noetic & Mathematical Definition: Aristotle’s question in Physics IV regarding the ontological mode of existence of entities such as place, time, and the mathematical continuum.

  • Occurrences in De Noetica Geometriae (1): Appendix: § 6. (LA · EN)

Präzisionsmathematik — Mathematics of ideal precision

  • Headword: Präzisionsmathematik (praezisionsmathematik)

  • Language Tradition: German

  • English Translation: Mathematics of ideal precision

  • Noetic & Mathematical Definition: Felix Klein’s term for pure, exact mathematics operating with idealized limits, contrasted with Approximationsmathematik.

  • Occurrences in De Noetica Geometriae (2): Caput I: § 4. (LA · EN) | Caput V: A) (LA · EN)

πρᾶγμα (pl. πράγματα) — Thing / objective reality / state of affairs

  • Headword: πρᾶγμα (pl. πράγματα) (pragma)

  • Language Tradition: Greek

  • English Translation: Thing / objective reality / state of affairs

  • Noetic & Mathematical Definition: In Aristotle’s logic and metaphysics, the extramental reality or objective state of affairs that grounds the truth of spoken and mental propositions.

  • Occurrences in De Noetica Geometriae (2): Appendix: § 6. (LA · EN), § 6. (LA · EN)

préalable — Prior / preliminary / antecedent

  • Headword: préalable (prealable)

  • Language Tradition: French

  • English Translation: Prior / preliminary / antecedent

  • Noetic & Mathematical Definition: In French philosophy of mathematics (Gonseth, Poincaré), designating the intuitive cognition that must precede formal axiomatic construction.

  • Occurrences in De Noetica Geometriae (1): Caput IV: § 7. (LA · EN)

principium individuationis — Principle of individuation

  • Headword: principium individuationis (principium-individuationis)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Principle of individuation

  • Noetic & Mathematical Definition: That whereby multiple individuals of the same species are numerically distinct (intelligible matter in geometry).

  • Occurrences in De Noetica Geometriae (2): Caput III: CAPUT III (LA · EN), CAPUT III (LA · EN)

propositio per accidens — Accidental / Contingent proposition

  • Headword: propositio per accidens (propositio-per-accidens)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Accidental / Contingent proposition

  • Noetic & Mathematical Definition: A judgment joining subject and predicate through contingent coexistence rather than essential necessity.

  • Occurrences in De Noetica Geometriae (6): Appendix: § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN)

propositio per se — Essential / Self-grounded proposition (per se nota)

  • Headword: propositio per se (propositio-per-se)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Essential / Self-grounded proposition (per se nota)

  • Noetic & Mathematical Definition: A necessary proposition whose predicate belongs to the intelligible nature of the subject.

  • Occurrences in De Noetica Geometriae (3): Appendix: § 1. (LA · EN), § 1. (LA · EN), § 1. (LA · EN)

προϋπάρχειν — To pre-exist / exist beforehand

  • Headword: προϋπάρχειν (prouparchein)

  • Language Tradition: Greek

  • English Translation: To pre-exist / exist beforehand

  • Noetic & Mathematical Definition: Technical peripatetic verb (used by Aristotle, Themistius, and Simplicius) signifying that the moving cause, mobile subject, or first principles must actually exist prior to motion or demonstration.

  • Occurrences in De Noetica Geometriae (1): Appendix: n. 21 (LA · EN)

ψυχή (dat. ψυχῇ) — Soul / mind

  • Headword: ψυχή (dat. ψυχῇ) (psyche)

  • Language Tradition: Greek

  • English Translation: Soul / mind

  • Noetic & Mathematical Definition: In Plato and Aristotle, the vital and cognitive principle whose intellectual faculty (νοῦς) beholds mathematical and metaphysical truths.

  • Occurrences in De Noetica Geometriae (1): Caput III: A) (LA · EN)

ῥύσις τοῦ σημείου (acc. ῥύσιν τοῦ σημείου) — Flow / fluxion of the point

  • Headword: ῥύσις τοῦ σημείου (acc. ῥύσιν τοῦ σημείου) (rhysin-tou-semeiou)

  • Language Tradition: Greek

  • English Translation: Flow / fluxion of the point

  • Noetic & Mathematical Definition: Ancient Greek kinematic conception (reported by Aristotle in De Anima I, 4 and Proclus) according to which the motion or flowing of a point generates a line.

  • Occurrences in De Noetica Geometriae (1): Caput IV: n. 8 (LA · EN)

rigueur — Mathematical rigor

  • Headword: rigueur (rigueur)

  • Language Tradition: French

  • English Translation: Mathematical rigor

  • Noetic & Mathematical Definition: Logical and deductive exactness in mathematical demonstration, central to the debate between Poincaré, Klein, and the formalist/logicist schools.

  • Occurrences in De Noetica Geometriae (1): Caput III: B) (LA · EN)

Sachverhalt — State of affairs / objective complex

  • Headword: Sachverhalt (sachverhalt)

  • Language Tradition: German

  • English Translation: State of affairs / objective complex

  • Noetic & Mathematical Definition: In German phenomenology and epistemology (Brentano, Geyser, Stammler), the objective relational structure in reality corresponding to a true judgment.

  • Occurrences in De Noetica Geometriae (1): Caput V: n. 4 (LA · EN)

σημεῖον — Point / Sign

  • Headword: σημεῖον (semeion)

  • Language Tradition: Greek

  • English Translation: Point / Sign

  • Noetic & Mathematical Definition: Geometric point (Euclid, Elements I Def. 1).

  • Occurrences in De Noetica Geometriae (2): Caput III: B) (LA · EN), n. 6 (LA · EN)

Strecke — Line segment

  • Headword: Strecke (strecke)

  • Language Tradition: German

  • English Translation: Line segment

  • Noetic & Mathematical Definition: Hilbert’s term in Grundlagen der Geometrie for a bounded straight segment AB defined as the system of two points.

  • Occurrences in De Noetica Geometriae (1): Caput IV: 1. (LA · EN)

συγγενές (pl. συγγενῆ) — Kindred / connatural / of the same genus

  • Headword: συγγενές (pl. συγγενῆ) (syngenes)

  • Language Tradition: Greek

  • English Translation: Kindred / connatural / of the same genus

  • Noetic & Mathematical Definition: In Plato’s Republic VII and Aristotle’s Posterior Analytics, the affinity and structural kinship of the mathematical sciences and their principles.

  • Occurrences in De Noetica Geometriae (1): Caput VI: 1. (LA · EN)

τὰ ἐξ ἀφαιρέσεως — Things derived from abstraction (mathematical objects)

  • Headword: τὰ ἐξ ἀφαιρέσεως (ta-ex-aphaireseos)

  • Language Tradition: Greek

  • English Translation: Things derived from abstraction (mathematical objects)

  • Noetic & Mathematical Definition: Aristotle’s standard technical designation for mathematical entities abstracted from sensible qualities.

  • Occurrences in De Noetica Geometriae (4): Caput I: § 2. (LA · EN) | Caput II: 3. (LA · EN) | Caput IV: § 7. (LA · EN) | Caput VII: 2. (LA · EN)

τέλος — End / Final cause / Completion

  • Headword: τέλος (telos)

  • Language Tradition: Greek

  • English Translation: End / Final cause / Completion

  • Noetic & Mathematical Definition: The goal or terminal perfection toward which an action or motion tends.

  • Occurrences in De Noetica Geometriae (4): Appendix: § 4. (LA · EN), § 4. (LA · EN), § 4. (LA · EN), § 4. (LA · EN)

termini significativi — Significant / semantically meaningful terms

  • Headword: termini significativi (termini-significativi)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Significant / semantically meaningful terms

  • Noetic & Mathematical Definition: Terms possessing real intensional meaning and reference to intelligible objects, contrasted by Hoenen with the empty uninterpreted symbols of pure formalism.

  • Occurrences in De Noetica Geometriae (1): Caput IV: n. 29 (LA · EN)

termini transcendentes — Transcendental terms

  • Headword: termini transcendentes (termini-transcendentes)

  • Language Tradition: Latin (Scholastic)

  • English Translation: Transcendental terms

  • Noetic & Mathematical Definition: Terms such as ens (being), unum (one), res (thing), and aliquid (something) that transcend the ten Aristotelian categories.

  • Occurrences in De Noetica Geometriae (3): Caput IV: § 7. (LA · EN), n. 29 (LA · EN) | Caput V: 1. (LA · EN)

θέσει — By position / having position

  • Headword: θέσει (thesei)

  • Language Tradition: Greek

  • English Translation: By position / having position

  • Noetic & Mathematical Definition: In Aristotle’s distinction between arithmetic and geometry: whereas the arithmetic unit is without position (ἄθετος), the geometric point is an indivisible having position (θέσιν ἔχον / θέσει).

  • Occurrences in De Noetica Geometriae (1): Caput IV: § 2. (LA · EN)

θέσις (acc. θέσιν) — Position / positing

  • Headword: θέσις (acc. θέσιν) (thesis)

  • Language Tradition: Greek

  • English Translation: Position / positing

  • Noetic & Mathematical Definition: In Aristotle’s Posterior Analytics (I, c. 2, 72a14), an immediate principle of demonstration that cannot be proved but is not indispensable for learning everything (divided into hypotheses and definitions); in geometry, spatial position.

  • Occurrences in De Noetica Geometriae (1): Caput IV: 2. (LA · EN)

τί ἐστιν — What it is (quiddity / essence)

  • Headword: τί ἐστιν (ti-estin)

  • Language Tradition: Greek

  • English Translation: What it is (quiddity / essence)

  • Noetic & Mathematical Definition: Aristotle’s question of essential definition, distinguished from whether a thing exists (εἰ ἔστιν).

  • Occurrences in De Noetica Geometriae (1): Appendix: § 6. (LA · EN)

τιθεμένου τοῦ σημείου — The point being posited / placed

  • Headword: τιθεμένου τοῦ σημείου (tithemenou-tou-semeiou)

  • Language Tradition: Greek

  • English Translation: The point being posited / placed

  • Noetic & Mathematical Definition: Greek genitive absolute describing the constructive positing of a geometric point in spatial intuition.

  • Occurrences in De Noetica Geometriae (1): Caput IV: § 3. (LA · EN)

τόδε (τόδε τι) — This something (concrete individual)

  • Headword: τόδε (τόδε τι) (tode)

  • Language Tradition: Greek

  • English Translation: This something (concrete individual)

  • Noetic & Mathematical Definition: Aristotle’s expression for a determinate, demonstratively indicated individual entity.

  • Occurrences in De Noetica Geometriae (2): Appendix: § 1. (LA · EN), § 1. (LA · EN)

umgekehrt (die umgekehrte) — Conversely / the converse

  • Headword: umgekehrt (die umgekehrte) (umgekehrt)

  • Language Tradition: German

  • English Translation: Conversely / the converse

  • Noetic & Mathematical Definition: Mathematical and logical term designating the converse of a proposition or geometric relation.

  • Occurrences in De Noetica Geometriae (2): Caput V: B) (LA · EN), B) (LA · EN)

eine Utopie — A utopia / an unattainable ideal

  • Headword: eine Utopie (utopie)

  • Language Tradition: German

  • English Translation: A utopia / an unattainable ideal

  • Noetic & Mathematical Definition: Felix Klein’s characterization of the claim that pure formal logic can generate mathematical science without any appeal to spatial intuition.

  • Occurrences in De Noetica Geometriae (1): Caput I: § 4. (LA · EN)

Verknüpfung — Connection / Incidence

  • Headword: Verknüpfung (verknuepfung)

  • Language Tradition: German

  • English Translation: Connection / Incidence

  • Noetic & Mathematical Definition: Hilbert’s first group of geometric axioms (Axiome der Verknüpfung / Axioms of Incidence) in Grundlagen der Geometrie.

  • Occurrences in De Noetica Geometriae (1): Caput IV: 1. (LA · EN)

Zusammenhang — Connection / nexus / structural coherence

  • Headword: Zusammenhang (zusammenhang)

  • Language Tradition: German

  • English Translation: Connection / nexus / structural coherence

  • Noetic & Mathematical Definition: In German philosophy of mathematics (Hilbert, Klein, Stammler), the necessary structural interconnection linking axioms, concepts, and geometric configurations.

  • Occurrences in De Noetica Geometriae (2): Caput III: 4. (LA · EN), 4. (LA · EN)