Glossarium Technicum (Technical Glossary)

Lexicon of Thomistic Epistemology, Metaphysics, and Constructive Geometry

This lexicon defines 36 foundational Latin and Greek technical terms governing Aquinas’s philosophy of mathematics and its modern reception in Peter Hoenen and GEOMETOR. Every inline :term: tag links directly to its #term-<slug> anchor below.

Alphabetical Index of Terms

Lexicon Definitions

abstractio formae — abstraction of form

  • Tradition: Latin (Scholastic)

  • English Rendering: abstraction of form

The intellectual operation characteristic of mathematics (abstractio formae a materia sensibili), whereby the intellect considers accidental quantitative form (number, dimension, figure) apart from sensible qualities because quantity inheres in corporeal substance prior to sensible qualities (Super Boetium De Trinitate q. 5 a. 3).

abstractio intuitiva formalis — intuitive formal abstraction

  • Tradition: Latin (Scholastic / Hoenen)

  • English Rendering: intuitive formal abstraction

Peter Hoenen’s technical designation for the Thomistic insight in which the intellect directly beholds universal, necessary formal relationships (continuity, straightness, incidence, equality) shining within the concrete spatial phantasm prior to formulating an explicit proposition.

abstractio totius — abstraction of the whole

  • Tradition: Latin (Scholastic)

  • English Rendering: abstraction of the whole

Abstraction of the universal whole from individuating material parts (abstractio universalis a particulari), common to all sciences and pre-eminently characteristic of natural philosophy when defining ‘human’ with common sensible matter while omitting ‘this flesh and these bones’ (Super Boetium De Trinitate q. 5 a. 3).

abstrahentium non est mendacium — there is no falsehood in those who abstract

  • Tradition: Latin (Aristotelian Axiom, Phys. II.2, 193b35)

  • English Rendering: there is no falsehood in those who abstract

Fundamental realist principle expounded in Physics II, lect. 3 and ST I q. 85 a. 1 ad 1: to consider geometric figure apart from sensible matter is not to judge falsely that figures exist separately outside bodies, but to consider truly an intelligible aspect ontological prior to sensible qualities.

communes animi conceptiones — common conceptions of the mind

  • Tradition: Latin (Boethian / Euclidean)

  • English Rendering: common conceptions of the mind

Boethius’s and Aquinas’s technical name for Euclidean Common Notions (koinai ennoiai), such as ‘if equals be subtracted from equals, the remainders are equal’ (C.N. 3) and ‘every whole is greater than its part’ (C.N. 5) (In I Anal. Post. lect. 19; De Veritate q. 10 a. 6).

compositio et divisio — composition and division (judgment)

  • Tradition: Latin (Scholastic)

  • English Rendering: composition and division (judgment)

The second operation of the intellect (iudicium), looking directly to the esse of the thing and uniting or separating subject and predicate via the copula est / non est, wherein formal truth and falsity properly reside (Super Boetium De Trinitate q. 5 a. 3; In V Metaph. lect. 9).

continuum — continuum

  • Tradition: Latin (Aristotelian syneches)

  • English Rendering: continuum

Extended quantity whose parts share a single common boundary (quorum ultima sunt unum: a point for line segments, a line for surfaces, a surface for solids), infinitely divisible into ever-divisible parts and never composed of indivisible points (In VI Phys. lect. 1).

conversio ad phantasmata — turning to the phantasms

  • Tradition: Latin (Scholastic)

  • English Rendering: turning to the phantasms

Necessary dynamic orientation whereby the human intellect, both when first acquiring science and when exercising habitual geometric knowledge, turns toward concrete imaginative diagrams in order to behold the universal intelligible species in its singular continuum (ST I q. 84 a. 7; In III De Anima lect. 12).

dignitates — axioms / dignities / maximal propositions

  • Tradition: Latin (Translating Greek axiómata)

  • English Rendering: axioms / dignities / maximal propositions

Immediate, indemonstrable first principles per se notae to every learner upon understanding their terms, presupposed across all demonstrations (In I Anal. Post. lect. 5, lect. 18–19).

disciplinaliter — disciplinably / by way of demonstrative learning

  • Tradition: Latin (Boethian / Scholastic)

  • English Rendering: disciplinably / by way of demonstrative learning

Boethius’s adverb for the distinctive mode of procedure of mathematical science (Super Boetium De Trinitate q. 6 a. 1): proceeding with supreme demonstrative certitude that generates disciplina (rigorous scientific learning) more surely than mutable physics and more connaturally than metaphysics.

divisio figurarum in actu — division of figures in act (heuristic construction)

  • Tradition: Latin (Metaphysics IX, lect. 10)

  • English Rendering: division of figures in act (heuristic construction)

Aquinas’s analysis in In IX Metaph. lect. 10 (Cathala #1894) of how geometers discover (inveniunt) theorems: latent geometric relations exist in potency in an undivided figure, and become immediately manifest to intellectual vision (statim manifestum est videnti) when auxiliary lines or radii are drawn in act.

experimentum — experience

  • Tradition: Latin (Translating Greek empeiria)

  • English Rendering: experience

In Posterior Analytics II, lect. 20, the collative synthesis of many memories of singular instances by the cogitative power, preparing the phantasm in which intellectus grasps the universal principle.

genus subiectum — subject-genus

  • Tradition: Latin (Aristotelian / Scholastic)

  • English Rendering: subject-genus

One of the three essential elements of any demonstrative science (alongside dignitates and passiones): the unified intelligible domain whose per se attributes the science demonstrates—continuous magnitude for geometry, discrete multitude for arithmetic (In I Anal. Post. lect. 18).

indivisibilium intelligentia — understanding of indivisibles / simple apprehension

  • Tradition: Latin (Aristotelian / Scholastic)

  • English Rendering: understanding of indivisibles / simple apprehension

The first of the two fundamental operations of the intellect (De Anima III.6; Super Boetium De Trinitate q. 5 a. 3), by which the mind grasps quid est (essence/quiddity and pre-judgmental formal nexus) without yet affirming or denying esse.

intellectus agens — agent intellect

  • Tradition: Latin (Scholastic)

  • English Rendering: agent intellect

The active immaterial power of the human soul whose natural light (lumen) illuminates sensible and imaginative phantasms to render potentially intelligible forms actually intelligible (ST I q. 84 a. 6; De Veritate q. 10 a. 6).

intellectus principiorum — understanding / habit of first principles

  • Tradition: Latin (Scholastic)

  • English Rendering: understanding / habit of first principles

The intellectual virtue (nous) by which the mind infallibly grasps immediate first principles once their terms are abstracted from sense, memory, and experimentum (In II Anal. Post. lect. 20; De Veritate q. 10 a. 6).

ipsum esse — being itself / the act of existing

  • Tradition: Latin (Thomistic Metaphysics)

  • English Rendering: being itself / the act of existing

In Aquinas’s metaphysics, the ultimate intrinsic act (actualitas omnium actuum et perfectio omnium perfectionum) by which essence actually exists and to which every true judgment ultimately refers (ST I q. 7 a. 1; De Potentia q. 7 a. 2 ad 9).

locus immobilis — immobile place

  • Tradition: Latin (Physics IV, lect. 7)

  • English Rendering: immobile place

The innermost immobile boundary of the surrounding body, considered according to its invariant positional order (ordo et situs) relative to the whole cosmos—providing the experiential root from which homogeneous Euclidean positional space and rigid congruence are abstracted.

materia intelligibilis — intelligible matter

  • Tradition: Latin (Scholastic / Aristotelian hylē noētē)

  • English Rendering: intelligible matter

Substance precisely insofar as it underlies continuous or discrete quantity, considered by the intellect apart from all sensible qualities (heat, cold, color, weight, hardness). It provides the spatial continuum in which geometric figures exist, are individuated, and are defined (Super Boetium De Trinitate q. 5 a. 3; ST I q. 85 a. 1 ad 2; In VII Metaph. lect. 10–11).

materia intelligibilis communis — common intelligible matter

  • Tradition: Latin (Scholastic)

  • English Rendering: common intelligible matter

Continuous extension in general (plane, continuum, magnitude) included universally within the quidditative definition of a mathematical species, just as ‘flesh and bones’ in general enter the natural definition of ‘human’ (In VII Metaph. lect. 11; ST I q. 85 a. 1 ad 2).

materia intelligibilis individualis — individual intelligible matter

  • Tradition: Latin (Scholastic)

  • English Rendering: individual intelligible matter

Particular designated continuous dimensions (such as these lines or these two semicircles) that individuate multiple mathematical figures of the exact same species—allowing the geometer to intersect two distinct circles of equal radius as in Euclid I.1 (In VII Metaph. lect. 10, Cathala #1496).

materia sensibilis — sensible matter

  • Tradition: Latin (Scholastic)

  • English Rendering: sensible matter

Corporeal matter insofar as it is subject to sensible qualities apprehended by the external five senses (hot/cold, wet/dry, heavy/light, colored, resonant) and physical alteration. Natural philosophy includes materia sensibilis communis, whereas mathematics abstracts from all sensible matter (Super Boetium De Trinitate q. 5 a. 3).

materia signata — designated matter

  • Tradition: Latin (Scholastic)

  • English Rendering: designated matter

Matter considered under determinate, hic-et-nunc dimensions (hic homo, haec caro et haec ossa), which serves as the principle of numerical individuation within a material species and from which universal definitions abstract (Super Boetium De Trinitate q. 5 a. 2–3).

modi essendi — modes of being

  • Tradition: Latin (Scholastic)

  • English Rendering: modes of being

The diverse ways in which finite realities realize act of being—substantially or accidentally (in quantity, quality, relation, etc.)—to which the ten Aristotelean categories (modi praedicandi) are strictly proportional (In III Phys. lect. 5; In V Metaph. lect. 9 #890).

passiones — proper attributes / per se properties

  • Tradition: Latin (Scholastic)

  • English Rendering: proper attributes / per se properties

Necessary properties (per se accidentia) inhering in a subject by reason of its essence and demonstrated of it through its definition as middle term—such as a triangle’s interior angles equaling two right angles (In I Anal. Post. lect. 2 & lect. 18).

petitiones — postulates

  • Tradition: Latin (Translating Greek aitēmata)

  • English Rendering: postulates

Immediate constructive or relational assumptions requested (petita) from the student by the geometer for the sake of demonstration—exemplified by Euclid’s five Postulates (a puncto ad punctum rectam lineam ducere, etc.) (In I Anal. Post. lect. 19).

phantasma — phantasm

  • Tradition: Latin / Greek (phantasma)

  • English Rendering: phantasm

Singular sensible or imaginative representation preserved and formed by the interior sensitive power of imagination (phantasia), presenting spatial extension and figure to the illumination of the agent intellect (In III De Anima lect. 8 & 12; ST I q. 84 a. 6–7).

praecognitio — foreknowledge

  • Tradition: Latin (Scholastic)

  • English Rendering: foreknowledge

Antecedent cognition required before any scientific demonstration can proceed: knowing quid est (meaning/essence) and quia est (existence/truth) diversely distributed across principles, subject-genus, and demonstrated attributes (In I Anal. Post. lect. 2).

quidditas — quiddity / essence (what-it-is)

  • Tradition: Latin (Scholastic)

  • English Rendering: quiddity / essence (what-it-is)

The intelligible nature or quod quid erat esse of a thing signified by its definition; the connatural proper object of the human intellect is the quidditas rei materialis, including quantitative figures in intelligible matter (ST I q. 84 a. 7, q. 85 a. 1).

scientiae mediae — intermediate sciences (mathematical physics)

  • Tradition: Latin (Scholastic)

  • English Rendering: intermediate sciences (mathematical physics)

Sciences such as optics (perspectiva), astronomy (astrologia), and harmonics (musica) that are formally mathematical—applying geometric or arithmetic demonstrations as propter quid—yet materially natural because they apply those forms to sensible light, stars, and sounds (Super Boetium De Trinitate q. 5 a. 3 ad 6; In II Phys. lect. 3).

separatio — separation (negative judgment)

  • Tradition: Latin (Scholastic)

  • English Rendering: separation (negative judgment)

Operation of the intellect’s second act (composing and dividing) peculiar to metaphysics (scientia divina), whereby the mind judges negatively that ens, unum, actus, and potentia do not depend upon matter and motion in order to exist (secundum esse), unlike mathematical abstraction which belongs to simple apprehension (Super Boetium De Trinitate q. 5 a. 3).

simum et curvum — the snub and the curved

  • Tradition: Latin (Aristotelian to simon / to koilon)

  • English Rendering: the snub and the curved

Aristotle’s canonical contrast (Phys. II, lect. 3; De Anima III, lect. 8; Metaph. VII, lect. 10–11): ‘snub’ (simum) includes sensible matter (‘concave nose’) in its very definition and belongs to physics, whereas ‘curved’ or ‘straight’ (curvum, rectum) abstracts from flesh and includes only intelligible continuous extension.

suppositiones — hypotheses / suppositions

  • Tradition: Latin (Translating Greek hypotheseis)

  • English Rendering: hypotheses / suppositions

Immediate positional principles affirming or denying existence (esse or non esse) to which the learner readily assents even though they are proved only in a higher science (In I Anal. Post. lect. 5 & lect. 19).

terminari ad imaginationem — terminate at the imagination

  • Tradition: Latin (Scholastic)

  • English Rendering: terminate at the imagination

Aquinas’s signature methodological rule in Super Boetium De Trinitate q. 6 a. 2: whereas natural judgment terminates at external sense and metaphysical judgment terminates at pure intellect, mathematical judgment must terminate at the constructive imagination, whose spatial apprehension measures geometric truth.

unitas et punctus — unit without position and point with position

  • Tradition: Latin (Aristotelian / Euclidean)

  • English Rendering: unit without position and point with position

Aristotle’s and Aquinas’s foundational comparison between arithmetic and geometry (In I Anal. Post. lect. 42): the arithmetical unit is an indivisible principle without spatial position (substantia non posita), whereas the geometric point adds positio (situs) in the continuum.

verbum mentis — inner word / mental conception

  • Tradition: Latin (Scholastic)

  • English Rendering: inner word / mental conception

The interior intelligible expression (conceptio intellectus) formed by the act of understanding—either an incomplex definition in the first operation or a complex scientific proposition/theorem in the second operation (De Potentia q. 8 a. 1, q. 9 a. 5; De Veritate q. 4 a. 2).