Glossary of Noetic & Geometric Terms

Greek, Scholastic Latin & Mathematical Lexicon of Loci Geometrici et Noetici

Systematic lexicon of thirty-six foundational Aristotelian concepts governing demonstrative science (epistēmē), syllogistic necessity, continuous vs. discrete quantity, potential infinity, imaginative presentation (phantasma), and abstraction in intelligible matter (hylē noētē).

ἐπιστήμη (epistēmē)

  • Greek: ἐπιστήμη | Scholastic Latin: scientia

  • Rendering: Demonstrative scientific knowledge

  • Primary Bekker Loci: Anal. Post. I.2, 71b9–16; Metaph. E.1, 1025b18–28

Unqualified scientific knowledge obtained when we know the necessary cause (aitia) through which a fact holds and see that it cannot be otherwise.

ἀπόδειξις (apodeixis)

  • Greek: ἀπόδειξις | Scholastic Latin: demonstratio

  • Rendering: Scientific demonstration

  • Primary Bekker Loci: Anal. Post. I.2, 71b17–25; I.4–6, 73a21–75a37

A syllogism productive of scientific knowledge (syllogismos epistēmonikos), proceeding from premisses that are true, primary, immediate, better known than, prior to, and causes of the conclusion.

νοῦς (nous)

  • Greek: νοῦς | Scholastic Latin: intellectus principiorum

  • Rendering: Intuitive intellect / intellectual intuition

  • Primary Bekker Loci: Anal. Post. II.19, 100b5–17; De Anima III.4–5, 429a10–430a25

The supreme noetic state (hexis) that apprehends immediate, indemonstrable first principles; more exact and truer than epistēmē itself, serving as the principle of principle (archē archēs).

ἐπαγωγή (epagōgē)

  • Greek: ἐπαγωγή | Scholastic Latin: inductio

  • Rendering: Noetic induction

  • Primary Bekker Loci: Anal. Post. I.1, 71a8; I.18, 81a38–b9; II.19, 100b3–5

Not probabilistic enumeration, but the genetic process by which sense-perception and memory bring the universal nature to a stand in the soul until nous intuits the necessary connection.

ἀξίωμα (axiōma)

  • Greek: ἀξίωμα / τὰ κοινά | Scholastic Latin: dignitas / communis animi conceptio

  • Rendering: Axiom / Common Notion

  • Primary Bekker Loci: Anal. Post. I.2, 72a16–17; I.10, 76a37–b2; Metaph. B.2, 996b26–33

An immediate principle that anyone who is going to learn anything must grasp of himself—prototypically the quantitative axiom ‘if equals be subtracted from equals, the remainders are equal’ (Euclid C.N. 3).

αἴτημα (aitēma)

  • Greek: αἴτημα | Scholastic Latin: petitio / postulatum

  • Rendering: Postulate

  • Primary Bekker Loci: Anal. Post. I.10, 76b23–34

A demonstrable or constructive assumption adopted without the learner’s prior conviction or contrary to the learner’s opinion; precursor to Euclid’s five geometrical Postulates.

ὑπόθεσις (hypothesis)

  • Greek: ὑπόθεσις | Scholastic Latin: suppositio

  • Rendering: Scientific hypothesis

  • Primary Bekker Loci: Anal. Post. I.2, 72a18–24; I.10, 76b3–39; Topics VIII.3, 158b30

A thesis asserting the existence (to einai) of the primitive objects of a science’s genus—such as units in arithmetic or points and lines in geometry—from which attributes are demonstrated.

ὁρισμός (horismos)

  • Greek: ὁρισμός | Scholastic Latin: definitio

  • Rendering: Essential definition

  • Primary Bekker Loci: Anal. Post. I.2, 72a21–24; I.10, 76b35–39; Topics VI.4, 141a26–142b19

A thesis stating what a thing is (ti esti) through prior and more intelligible terms without asserting its existence—e.g. defining a line as breadthless length.

καθ’ αὑτό (kath’ hauto)

  • Greek: καθ’ αὑτό | Scholastic Latin: per se

  • Rendering: Essential / intrinsic predication

  • Primary Bekker Loci: Anal. Post. I.4, 73a34–b24

An attribute belonging to a subject either inside its essential formula (as line in triangle) or containing the subject itself inside the attribute’s own formula (as straight and curved belong to line).

καθόλου (katholou)

  • Greek: καθόλου | Scholastic Latin: universale commensuratum

  • Rendering: Commensurate universal

  • Primary Bekker Loci: Anal. Post. I.4–5, 73b25–74b4; I.24, 85a13–86a30

That which belongs to every instance (kata pantos), essentially (kath’ hauto), and qua itself (hēi auto)—exemplified by the 2R angle-sum belonging commensurate-universally to triangle qua triangle rather than isosceles.

μετάβασις εἰς ἄλλο γένος (metabasis)

  • Greek: μετάβασις εἰς ἄλλο γένος | Scholastic Latin: transitus in aliud genus

  • Rendering: Prohibition of genus-crossing

  • Primary Bekker Loci: Anal. Post. I.7, 75a38–b20; De Caelo I.1, 268b1

Aristotle’s structural law that an arithmetical proof cannot establish a geometrical theorem concerning continuous magnitudes unless the genera stand in formal subalternation (as optics to geometry).

τὸ ὅτι καὶ τὸ διότι (to hoti / to dioti)

  • Greek: τὸ ὅτι / τὸ διότι | Scholastic Latin: demonstratio quia et propter quid

  • Rendering: Fact vs. reasoned causal fact

  • Primary Bekker Loci: Anal. Post. I.13, 78a22–79a16

Distinction between empirical sciences that know the fact (optics, harmonics, observational astronomy) and pure mathematical sciences that supply the formal geometrical or arithmetical reason why.

ἀκρίβεια (akribeia)

  • Greek: ἀκρίβεια | Scholastic Latin: exactitudo / certitudo

  • Rendering: Scientific exactitude

  • Primary Bekker Loci: Anal. Post. I.27, 87a31–37; Metaph. M.3, 1078a9–17

Precision proportional to conceptual simplicity and priority: arithmetic is more exact than geometry because the arithmetical monad lacks position, whereas the geometrical point adds position (thesis).

ποσόν (poson)

  • Greek: ποσόν | Scholastic Latin: quantitas

  • Rendering: Quantity

  • Primary Bekker Loci: Categoriae 6, 4b20–6a35; Metaph. Δ.13, 1020a7–32

That which is divisible into constituent parts each capable of being a single determinate ‘this’; the proper ontological domain of mathematics, bifurcated into discrete multitude and continuous magnitude.

διωρισμένον (diōrismenon)

  • Greek: διωρισμένον ποσόν | Scholastic Latin: quantitas discreta

  • Rendering: Discrete quantity

  • Primary Bekker Loci: Categoriae 6, 4b20–37

Quantity whose parts join at no common boundary (koinoi horoi), prototypically number (arithmos) and spoken logos.

συνεχές (syneches)

  • Greek: συνεχές | Scholastic Latin: continuum

  • Rendering: The continuum / continuous quantity

  • Primary Bekker Loci: Categoriae 6, 4b20, 5a1–14; Phys. V.3, 227a10–17; VI.1–2, 231a21–233b32

Quantity whose parts share a common boundary where their extremities become one, and which is divisible solely into parts that are themselves always further divisible (eis aei diaireta)—line, surface, body, time, and place.

ἀδιαίρετον (adiaireton)

  • Greek: ἀδιαίρετον / ἀμερές | Scholastic Latin: indivisibile

  • Rendering: Indivisible / partless limit

  • Primary Bekker Loci: Phys. VI.1, 231a21–b18; De Anima III.6, 430b6–20; Metaph. Δ.6, 1016b24–31

That which lacks parts in quantity or species; in geometry, the point is indivisible in every dimension yet possesses position, serving as limit (peras) of a line without ever composing it.

στιγμή (stigmē)

  • Greek: στιγμή | Scholastic Latin: punctum

  • Rendering: Geometrical point

  • Primary Bekker Loci: Anal. Post. I.27, 87a36; Topics VI.4, 141b5–22; Phys. VI.1, 231a24–b18; Metaph. Δ.6, 1016b25

Aristotle’s standard term for the geometrical point (prior to Euclid’s sēmeion): an indivisible substance having position (ousia athetos vs. thetos), functioning as boundary and division of continuous length.

μονάς (monas)

  • Greek: μονάς | Scholastic Latin: unitas

  • Rendering: Arithmetical unit / monad

  • Primary Bekker Loci: Anal. Post. I.27, 87a36; Metaph. Δ.6, 1016b24–26; M.8, 1084b26

That which is quantitatively indivisible in every respect and without spatial position (athetos), serving as the primary measure of discrete number.

μέγεθος (megethos)

  • Greek: μέγεθος | Scholastic Latin: magnitudo

  • Rendering: Continuous geometric magnitude

  • Primary Bekker Loci: De Caelo I.1, 268a6–13; Metaph. Δ.13, 1020a11–14

Measurable continuous quantity: divisible in one dimension as line (grammē), in two as plane surface (epipedon), and completely in three as solid body (sōma).

πλῆθος (plēthos)

  • Greek: πλῆθος | Scholastic Latin: multitudo

  • Rendering: Countable multitude / plurality

  • Primary Bekker Loci: Metaph. Δ.13, 1020a7–11; I.1, 1053a20–30

Quantity countable into discontinuous parts; when limited and measured by the unit, multitude constitutes number (arithmos).

ἀνταναίρεσις (antanairesis)

  • Greek: ἀνταναίρεσις | Scholastic Latin: subtractio reciproca

  • Rendering: Reciprocal subtraction (anthyphairesis)

  • Primary Bekker Loci: Topics VIII.3, 158b29–35

The pre-Eudoxian definition of equality of ratios (preserved in Euclid Elements X.2–3): two magnitudes have the same ratio as two others when they undergo the exact same sequence of reciprocal subtractions.

ἀσύμμετρον (asymmetron)

  • Greek: ἀσύμμετρον ἡ διάμετρος | Scholastic Latin: incommensurabile

  • Rendering: Incommensurability (of the diagonal)

  • Primary Bekker Loci: Anal. Prior. I.23, 41a23–30; Anal. Post. I.2, 71b26; Metaph. A.2, 983a15

Aristotle’sparadigm of apodeictic necessity by reductio ad absurdum: if the diagonal of a square were commensurable with its side, odd numbers would become equal to even numbers.

χωρισμὸς τῇ νοήσει (chōrismos)

  • Greek: χωριστὰ τῇ νοήσει | Scholastic Latin: separatio secundum intellectum

  • Rendering: Separation in thought

  • Primary Bekker Loci: Phys. II.2, 193b31–35; Metaph. M.3, 1078a17–31

The mathematician’s legitimate methodological act of considering surfaces, volumes, lengths, and points apart from physical motion and sensible matter without asserting Platonic separate existence.

ἄπειρον (apeiron)

  • Greek: δυνάμει ἄπειρον | Scholastic Latin: infinitum potentiale

  • Rendering: The potential infinite

  • Primary Bekker Loci: Phys. III.4–7, 202b30–208a4

Not an actually completed magnitude (which is impossible), but an open-ended potentiality whereby another part can always be taken—inwardly by bisection in magnitudes, outwardly by succession in numbers.

τόπος (topos)

  • Greek: τόπος | Scholastic Latin: locus

  • Rendering: Place

  • Primary Bekker Loci: Categoriae 6, 5a8–14; Phys. IV.1–5, 208a27–213a11

The first motionless boundary of the containing body (to tou periechontos peras akinēton prōton, 212a20), grounding positional order without positing an empty self-subsistent void.

φαντασία (phantasia)

  • Greek: φαντασία | Scholastic Latin: imaginatio / phantasia

  • Rendering: Sensible imagination

  • Primary Bekker Loci: De Anima III.3, 427b14–429a9

A movement generated by actual sense-perception that preserves spatial configuration and figure apart from external matter, bridging sensation (aisthēsis) and intellectual thought (noēsis).

φάντασμα (phantasma)

  • Greek: φάντασμα | Scholastic Latin: phantasma

  • Rendering: Imaginative diagram / presentation

  • Primary Bekker Loci: De Anima III.7, 431a14–17, 431b2; III.8, 432a3–14; De Mem. 1, 449b31–450a7

The concrete imaginative presentation in which the intellect beholds universal formal structure; as the geometer draws a determinate finite triangle yet proves solely from its universal triangularity.

νόησις (noēsis)

  • Greek: νόησις / ὁρᾶν τῇ νοήσει | Scholastic Latin: intellectio / intuitus mentis

  • Rendering: Intellectual vision / apprehension

  • Primary Bekker Loci: Anal. Post. I.12, 77b30–31; De Anima III.4–8, 429a10–432a14

Direct intellectual insight reading formal necessity inside mathematical constructions; highlighted by Hoenen as the Aristotelian text refuting blind symbolic formalism.

τὰ ἐξ ἀφαιρέσεως (aphairesis)

  • Greek: τὰ ἐν ἀφαιρέσει / τὰ ἐξ ἀφαιρέσεως | Scholastic Latin: abstractio mathematica

  • Rendering: Objects of mathematical abstraction

  • Primary Bekker Loci: Anal. Post. I.18, 81b3; De Caelo III.1, 299a15; De Anima III.7, 431b12–17

Aristotle’s technical designation for mathematical entities obtained by stripping away sensible contraries (weight, heat, hardness) while retaining continuous quantitative determination.

ὕλη νοητή (hylē noētē)

  • Greek: ὕλη νοητή | Scholastic Latin: materia intelligibilis

  • Rendering: Intelligible matter

  • Primary Bekker Loci: Metaph. Z.10, 1036a9–12; Z.11, 1036b32–1037a5; H.6, 1045a33–35

Pure continuous spatial extension remaining after sensible qualities are abstracted; individuates multiple congruent circles or triangles that share the exact same formal definition.

ἐνέργεια (energeia)

  • Greek: ἐνέργεια | Scholastic Latin: actus / operatio

  • Rendering: Actuality / constructive activity

  • Primary Bekker Loci: Metaph. Θ.6–9, 1048a25–1051a33

Priority of act over potency; in Metaphysics Θ.9 (1051a21–33), geometrical theorems are discovered by actualizing latent divisions through auxiliary lines, making the causal middle term immediately manifest.

δύναμις (dynamis)

  • Greek: δύναμις | Scholastic Latin: potentia

  • Rendering: Potency / latent geometric capacity (and square)

  • Primary Bekker Loci: Phys. III.6, 206a14–25; Metaph. Δ.12, 1019b33 (square in geometry); Θ.9, 1051a24

Both ontological potentiality (as latent chords and parallels exist potentially in a figure prior to construction, and infinite bisections exist potentially in a segment) and the technical Greek geometric term for a mathematical square.

ἐντελέχεια (entelecheia)

  • Greek: ἐντελέχεια | Scholastic Latin: entelechia / actus perfectus

  • Rendering: Completed actuality

  • Primary Bekker Loci: Phys. III.1, 201a10–11; De Anima II.1, 412a27; Metaph. Θ.8, 1050a21–23

Aristotle’s coinage signifying internal completeness or attainment of form (telos); used in Physics III.1 to define motion as the entelecheia of the potential qua potential.

τὸ τί ἦν εἶναι (to ti ēn einai)

  • Greek: τὸ τί ἦν εἶναι | Scholastic Latin: quod quid erat esse / quidditas

  • Rendering: Essence / formal quiddity

  • Primary Bekker Loci: Anal. Post. II.3–10; De Anima III.4, 429b10–22; Metaph. Z.4–11, 1029b1–1037b7

What it is for a thing to be what it essentially is—e.g. ‘to be straight’ (to euthei einai) vs. the extended straight thing (to euthy) composed of form in continuum.

τὸ καλόν — τάξις, συμμετρία, τὸ ὡρισμένον (to kalon)

  • Greek: τάξις, συμμετρία, τὸ ὡρισμένον | Scholastic Latin: pulchrum: ordo, commensuratio, determinatio

  • Rendering: Mathematical beauty: order, symmetry & definiteness

  • Primary Bekker Loci: Metaph. M.3, 1078a31–b6

Aristotle’s defence against Aristippus: mathematics pre-eminently exhibits the chief forms of beauty—order (taxis), symmetry/commensurability (symmetria), and determinate precision (to hōrismenon)—and acts as their causal explanation.

ἀριθμός (arithmos)

  • Greek: ἀριθμός | Scholastic Latin: numerus

  • Rendering: Discrete number (multitude measured by the unit)

  • Primary Bekker Loci: Categoriae 6, 4b23–31; Phys. III.7, 207b7; Metaph. Δ.13, 1020a13; I.6, 1057a3

Limited multitude (plēthos peperasmenon) composed of indivisible units without spatial position; contrasted throughout Aristotle with continuous geometric magnitude (megethos).

αἰσθητὰ κοινά (aisthēta koina)

  • Greek: τὰ κοινὰ αἰσθητά | Scholastic Latin: sensibilia communia

  • Rendering: Common sensibles (movement, rest, number, figure, magnitude)

  • Primary Bekker Loci: De Anima II.6, 418a17–20; III.1, 425a14–28; III.3, 428b22–25

Quantitative and kinematic determinations—movement, rest, number, figure (schēma), and magnitude (megethos)—perceived jointly across multiple senses (chiefly sight and touch) and retained in spatial phantasms.