Summary — PART I — ON BOETHIUS’S DE TRINITATE (QUESTIONS V–VI)

Analytical Synopsis of Super Boetium De Trinitate (c. 1257–1259)

Executive Thesis

Composed at Paris between 1257 and 1259 and preserved in part in his own autograph, Thomas Aquinas’s Expositio super librum Boethii De Trinitate (Questions V and VI) constitutes the most rigorous epistemological treatise of the Latin Middle Ages on the architecture and methodology of the theoretical sciences. Confronting competing Aristotelian, Boethian, Platonic, Avicennian, and Victorine classifications of human knowledge, Aquinas grounds the tripartite division of speculative philosophy—natural science, mathematics, and divine science—not in arbitrary ontological taxonomies, nor merely in subjective psychological faculties, but in the per se differences of speculative objects (speculabilia) precisely insofar as they stand proportioned to an immaterial intellect seeking necessary scientific truth. Because the intellect is immaterial in its power and demonstrative science (scientia) demands immobile necessity in its object, speculative objects are diversified essentially by their grade of removal from matter and motion: physics treats realities that depend upon \ sensible matter both in being (secundum esse) and in definition (secundum intellectum); mathematics treats realities that depend upon matter in being yet are defined without sensible matter; and divine science treats realities independent of matter and motion in being itself.

The analytical summit of this architecture appears in Question V, Article 3, where Aquinas justifies the truth and objectivity of mathematical abstraction by mapping \ Aristotle‘s twofold operation of the intellect (On the Soul III.6) onto the real structure of essence and existence. Whereas the intellect’s second operation—composition and division, which regards a thing’s very act of being (ipsum esse)—cannot truly disjoin what is united in reality and therefore exercises separatio only where realities can actually subsist apart from matter (the domain of metaphysics), the first operation—the understanding of indivisibles, which regards quidditative nature—can truthfully consider apart what is conjoined in reality whenever the intelligible account (ratio) of one principle does not depend upon another. Within this first operation Aquinas distinguishes two irreducible modes of abstraction: abstraction of the whole (abstractio totius), whereby a universal specific nature such as man is abstracted from designated matter (materia signata) while retaining common sensible matter (the procedure of natural philosophy and all sciences generally); and abstraction of form (abstractio formae), whereby the accidental form of dimensive quantity and figure is abstracted from sensible matter (tactile and visual qualities and physical motion) while retaining intelligible matter (materia intelligibilis), namely substance apprehended solely by the intellect as the underlying subject of extension. Because accidents accrue to substance in an ontological order wherein dimensive quantity precedes sensible qualities and motion, and because quantity carries position (situs, the order of parts) as a constitutive difference within its own notion, the geometer abstracts without mendacity while retaining the numerical plurality of congruent figures—such as several equal straight lines or several equilateral triangles—required for geometric construction and demonstration.

Question VI completes this ontological division with a corresponding methodology of cognitive procedure and judgmental verification. In Article 1, Aquinas pairs natural philosophy with proceeding rationally (rationabiliter, moving discursively from sensible effects to extrinsic causes), mathematics with proceeding disciplinably (disciplinaliter, demonstrating with supreme certitude and pedagogical clarity through intrinsic formal causality alone), and metaphysics with proceeding intellectually (intellectualiter, gathering multiplicity into unity at the ultimate term of causal and formal resolutio). In Article 2, Aquinas establishes his decisive noetic distinction between the beginning (principium) of human cognition in apprehension and its term (terminus) in judgment. While all human thought in the present life originates in sense and relies upon the phantasm (phantasma) as a permanent foundation, judgment does not terminate uniformly across the three sciences: physical judgment terminates at external sense; metaphysical judgment uses \ phantasms via causality, transcendence, and negation without ever terminating at sense or imagination; and mathematical judgment uniquely terminates at the imagination (terminari ad imaginationem). Liberated from sensible perturbations yet anchored in imaginable dimensive extension, mathematical judgment attains exact geometric truths—such as a straight line touching a sphere at a single indivisible point—that surpass sensory perception while finding faithful structural verification in constructive spatial intuition.

Key Scholastic & Mathematical Concepts

Latin Term	English Rendering	Primary Locus	Doctrinal & Geometric Significance
Speculabile	Speculative object	q. 5, a. 1	The formal object of theoretical intellect, whose per se differences arise from combining the immateriality of the intellective power with the immobile necessity required by scientific habit (scientia), yielding the three grades of removal from matter and motion.
Materia signata	Designated matter	q. 5, a. 2	Matter considered under determinate spatial and temporal dimensions (hic et nunc, this flesh and these bones), which serves as the principle of individuation and singular mobility and is left behind by every universal science.
Materia sensibilis (communis)	Sensible matter (common)	q. 5, a. 2–3	Corporeal matter endowed with sensible qualities (hot, cold, colored, heavy, mobile), retained universally in natural definitions (flesh and bones in general) through abstraction of the whole, but excluded from mathematical definitions through abstraction of form.
Materia intelligibilis	Intelligible matter	q. 5, a. 3	Substance considered prior to sensible qualities solely as the subject underlying dimensive quantity, together with the homogeneous parts of continuous extension itself; comprehensible by the intellect and represented in pure spatial imagination, grounding both numerical multiplicity and material-part demonstrations in geometry.
Intelligentia indivisibilium vs. compositio et divisio	Understanding of indivisibles vs. composition and division	q. 5, a. 3	The twofold operation of the intellect from On the Soul III.6: simple apprehension grasps quidditative essence (quid est), permitting truthful non-assertoric abstraction of conjoined realities, whereas judgment affirms or denies ipsum esse and therefore requires real separability whenever it disjoins.
Abstractio totius	Abstraction of the whole	q. 5, a. 3	Abstraction of a universal essential whole from accidental and designated material parts (the universal from the particular), corresponding to the union of whole and part; proper to natural philosophy and shared by all sciences.
Abstractio formae	Abstraction of form	q. 5, a. 3	Abstraction of the accidental form of quantity and figure from sensible matter while retaining intelligible matter, grounded in the ontological priority of dimensive quantity over sensible quality and motion in corporeal substance; proper to mathematics.
Separatio	Separation	q. 5, a. 3–4	The negative judgment of the intellect’s second operation declaring that one reality does not depend upon matter and motion in its very act of being (secundum esse), whether positively (God and angels) or neutrally (common being, substance, potency, and act); proper to metaphysics.
Situs	Position (order of parts)	q. 5, a. 3, ad 3	A constitutive difference intrinsic to dimensive quantity consisting in the order of parts in the continuum, enabling the geometer to imagine and number multiple congruent lines or equilateral triangles of a single species even after sensible matter has been abstracted.
Scientiae mediae	Intermediate sciences	q. 5, a. 3, ad 6–7	Disciplines such as astronomy, harmonics (music), optics, mechanics, and the moving sphere that apply formal mathematical principles to sensible natural subjects, remaining formally akin to mathematics because their middle terms of demonstration are quantitative.
Rationabiliter, disciplinaliter, intellectualiter	Rationally, disciplinably (by way of learning), intellectually	q. 6, a. 1	The three cognitive modes appropriated respectively to physics (discursive reasoning from sensible effects to extrinsic causes), mathematics (demonstrative certitude via intrinsic formal causes and imaginable figures), and divine science (synthetic intellectual unity at the term of causal and formal resolution).
Terminari ad imaginationem	To terminate at the imagination	q. 6, a. 2	The verification of mathematical judgment in constructive spatial imagination rather than external sense or pure imageless intellection, anchored by the phantasm (phantasma) functioning as a permanent foundation in which exact quantitative relations are inspected.

Section-by-Section Analytical Breakdown

Question V, Article 1 — Division of Speculative Science

Read Primary Text: English (q. 5, a. 1) · Latin (q. 5, a. 1) · Commentary

Ten objections challenge the Boethian and Aristotelian division of theoretical philosophy into physics, mathematics, and theology. They argue successively from \ Aristotle‘s three intellectual virtues in Ethics VI (wisdom, science, understanding), \ Augustine‘s inclusion of logic within contemplative philosophy, the medieval curriculum of the seven liberal arts (trivium and quadrivium), the presence of theoretical divisions within operative arts like medicine, the apparent inclusion of mechanical and medical arts under physics, the impropriety of dividing metaphysics (whose universal subject is being) against particular sciences studying parts of being, alternative ontological divisions of being (potency and act, corporeal and incorporeal), and pedagogical versus foundational ordering.

In his responsio and ten replies, Aquinas first demarcates speculative from practical intellect by their respective ends (truth considered for its own sake versus truth ordered to productive or moral action) and corresponding matters (things not made by human work versus operable things). He then demonstrates that habits are differentiated not by accidental differences of things, but by per se differences of their formal objects: because the intellect is immaterial and scientific knowledge requires immobile necessity, speculabilia divide exhaustively into three grades of dependence upon matter in being and in understanding—yielding natural philosophy, mathematics, and divine science (no fourth combination being possible since understanding cannot depend on matter where being does not). The replies harmonize rival schemes by situating logic and the liberal arts as preparatory instruments (organa) possessing immediate opera rationis, distinguishing part-whole containment from subalternation (propter quid versus quia), isolating formal ratio considerandi in metaphysics, and reconciling metaphysical priority in itself with pedagogical posteriority quoad nos without circularity.

Key Theses:

• Speculative intellect differs from practical intellect by its end—truth considered for itself—requiring objects not produced by human agency.

• Habits and sciences are formally diversified only by per se differences of their objects as knowable, namely their degree of separation from matter and motion.

• Speculative objects divide exhaustively into three orders: dependent on matter in being and understanding (physics), dependent in being but not understanding (mathematics), and independent in being (divine science).

• Logic and the quadrivial liberal arts serve as propaedeutic pathways (viae) possessing immediate works of reason (opera rationis) such as syllogizing, numbering, and measuring.

• Subalternated sciences (such as music under arithmetic or medicine under physics) receive their propter quid principles from higher theoretical sciences without compromising the speculative purity of the higher science.

Question V, Article 2 — Natural Philosophy, Motion, and Designated Matter

Read Primary Text: English (q. 5, a. 2) · Latin (q. 5, a. 2) · Commentary

Seven objections question how a true demonstrative science of mobile and material things is possible at all: matter is the principle of individuation whereas science treats solely of universals; intellection inherently abstracts from matter; physics culminates in the immaterial First Mover; mobile things are contingent whereas science demands necessity; universals themselves do not move; physics investigates unmoved principles such as the soul, natural forms, and the earth at rest; and mutability extends to all creatures, including spiritual ones.

Aquinas diagnoses this aporia as the precise philosophical root of \ Plato‘s theory of separate Ideas: accepting Heraclitean flux in sensible particulars, Plato posited separate immobile substances to save scientific definition, failing to distinguish what belongs per se to the composite from what belongs per accidens to its intelligible form (ratio). Following Metaphysics VII, Aquinas shows that while the concrete composite (this house, this man) comes to be and moves per se, its essential nature (ratio) can be considered apart from the hic et nunc conditions of local motion and designated matter (materia signata, matter under determinate dimensions), while retaining undesignated or common sensible matter (flesh and bones in general). Considered in themselves in the intellect, these universal rationes are immobile foundations of necessary science; applied secondarily by reflexio through the lower sensitive powers, they yield genuine scientific knowledge of extramental mobile substances, while the First Mover enters physics only as the extrinsic causal terminus of motion—just as an indivisible point relates to a line as its endpoint without being a line.

Key Theses:

• Plato posited separate Ideas because he failed to distinguish the per se generation and flux of the singular composite from the per accidens mutability of its intelligible form.

• Natural philosophy abstracts the universal from the particular by leaving behind designated matter (materia signata under determinate dimensions) while retaining common sensible matter in its definitions.

• Universal rationes are immobile according to their intellectual mode of consideration, yet serve as the formal media for knowing mobile, material realities outside the soul.

• Scientific cognition attains singular mobile things secondarily through a reflexio aided by the lower sensitive powers.

• The immaterial First Mover enters physics not as part of its subject-genus, but as its extrinsic causal limit (terminus), analogous to the point terminating a line.

Question V, Article 3 — Mathematical Abstraction and Intelligible Matter

Read Primary Text: English (q. 5, a. 3) · Latin (q. 5, a. 3) · Commentary

Eight objections challenge the claim that mathematics considers material realities without matter and motion: considering material things without matter seems to violate truth as adaequatio rei et intellectus; every science must consider the parts of its subject, including matter; geometers number multiple straight lines of the same species to form triangles and squares, yet numerical multiplication within a species requires matter; geometry demonstrates via the material cause from quantitative parts (such as proving the angle in a semicircle right from two half-right angles); space measures motion; astronomy, mechanics, harmonics, and spherics study motion; astronomers and physicists prove shared conclusions like the earth’s sphericity; and \ Ptolemy assigns incorruptible mobile spheres to mathematics.

In the celebrated five-part responsio, Aquinas distinguishes the intellect’s first operation (intelligentia indivisibilium, apprehending \ quiddity) from its second operation (composition and division, judging ipsum esse). Because judgment asserts real existence or non-existence, it can truly disjoin only what is separable in reality (separatio, founding metaphysics). Simple apprehension, however, truthfully abstracts one conjoined reality from another whenever the essential ratio of the first does not depend upon the second, yielding two distinct abstractions: abstraction of the whole (abstractio totius, abstracting a specific nature from accidental designated parts, common to all sciences and characteristic of physics) and abstraction of form (abstractio formae, characteristic of mathematics). Because accidents supervene upon substance in an intrinsic order—dimensive quantity first, prior to sensible qualities and motion—quantity and figure depend in their intelligible essence not upon sensible matter, but solely upon intelligible matter (materia intelligibilis, substance grasped solely by the intellect as underlying extension, along with homogeneous quantitative parts). The replies vindicate abstrahentium non est mendacium, show that dimensive quantity carries situs (the order of parts) within its own notion to ground the imaginable multiplicity of congruent lines and equilateral triangles, assign part-whole geometric proofs to intelligible matter, and define scientiae mediae as formally mathematical applications of quantitative middle terms to natural subjects.

Key Theses:

• The intellect’s two operations—apprehension of essence and judgment of esse—ground a strict triad of epistemic distinctions: separation (metaphysics), abstraction of form (mathematics), and abstraction of the whole (physics).

• Because dimensive quantity inheres in corporeal substance prior to sensible qualities and motion, mathematical forms can be truthfully understood without sensible matter (abstrahentium non est mendacium).

• Mathematics does not abstract from all matter absolutely, but retains intelligible matter—substance as pure subject of extension and the homogeneous parts of continuous magnitude.

• Dimensive quantity includes situs (the order of parts) as a constitutive difference, enabling the geometer to imagine numerically distinct congruent lines and figures within one species.

• Intermediate sciences (scientiae mediae) such as astronomy and harmonics demonstrate conclusions about physical phenomena formally through abstract mathematical middle terms.

Question V, Article 4 — Divine Science and Immaterial Separation

Read Primary Text: English (q. 5, a. 4) · Latin (q. 5, a. 4) · Commentary

Eight objections argue that divine science cannot treat realities wholly separated from matter and motion: human reason reaches God only through visible material effects (Romans 1); Scripture, \ Augustine, and \ Plato speak of divine wisdom or the First Mover as self-moving; angels move by choice and local mission and seem hylomorphically composed since they are not pure act; metaphysics studies ens and substantia, which encompass material beings, matter, and motion; metaphysics demonstrates through efficient and final causes, which imply motion; and revealed theology treats creation and human moral acts.

Aquinas resolves the question by analyzing the nature of principles. Unlike incomplete intrinsic principles (such as the unit of number or the point of a line, studied solely within the science of their principiates), universal causal principles of all beings are also complete, subsistent natures maximally in act, and therefore wholly free of matter and motion. Such divine realities admit a twofold theology: philosophical theology (metaphysics), whose proper subject is ens in quantum est ens and which reaches God and the separate Intelligences from sensible effects as the extrinsic principles of its subject; and the theology of Sacred Scripture, which receives divine self-revelation through faith and treats God himself directly as its proper subject. Crucially, Aquinas distinguishes two senses of separation from matter and motion secundum esse: positive immateriality, where a nature can never exist in matter (God and angels), and negative or neutral immateriality, where a ratio does not depend on matter in being and can exist either with or without matter (ens, substantia, actus, potentia). In the replies, Aquinas refutes universal hylomorphism by distinguishing act-potency composition (esse and essentia, quo est and quod est) in incorruptible angels from matter-form composition, and explains why metaphysics demonstrates via efficient and final causality whereas mathematics—abstracting only in thought—does not.

Key Theses:

• Incomplete principles (such as unit and point) belong solely to the science of their principiates, whereas complete subsistent principles of all being admit both causal philosophical inquiry and self-revealing theological science.

• Philosophical theology (metaphysics) has being insofar as it is being as its subject and treats God and separate substances as the universal principles of its subject.

• Separation from matter secundum esse is twofold: positive exclusion of matter (God and angels) and ontological neutrality toward matter (ens, substantia, potentia, actus).

• Created spiritual substances are free of hylomorphic matter-form composition, yet remain composed of essence (quod est) and received act of being (quo est, ipsum esse).

• Unlike mathematical entities, which are separated only in consideration and therefore exclude efficient and final causes, metaphysical principles subsist in reality and exercise efficient and final causality.

Question VI, Article 1 — The Ratiocinative, Disciplinary, and Intellectual Modes

Read Primary Text: English (q. 6, a. 1) · Latin (q. 6, a. 1) · Commentary

Organized as a triple disputation with twelve objections and six sed contra authorities, Question VI, Article 1 examines \ Boethius‘s appropriation of three cognitive modes to the three theoretical sciences: proceeding rationally (rationabiliter) in physics, disciplinably (disciplinabiliter / disciplinaliter) in mathematics, and intellectually (intellectualiter) in divine science. Regarding physics, Aquinas distinguishes three senses of a rational procedure: arguing from logical second intentions (logica docens, proper to logic and metaphysics as common sciences), arguing dialectically from probable premises (logica utens, preparatory in all sciences and characteristic of contingent moral inquiry), and following the native discursive rhythm of the rational soul—advancing from sensible signs and effects better known to us toward extrinsic causes better known by nature—which belongs preeminently to natural philosophy, in contrast to mathematics, which demonstrates solely through intrinsic formal definitions.

Regarding mathematics, Aquinas explains that proceeding disciplinably—by way of learning (discere), generating unshakable scientific conviction—belongs preeminently to mathematics because it occupies the epistemic golden mean between physics and theology: freed from the flux, variability, and multifactorial complexity of \ sensible matter (with arithmetic even simpler and more certain than geometry, which exists ex additione), yet directly accessible to sense and spatial imagination through \ phantasms of line, figure, and number, mathematics surpasses both natural and divine science in human certitude, just as \ Ptolemy observed in the Almagest. Regarding divine science, Aquinas contrasts discursive ratio (circling around multiplicity like circumference around center or time around eternity) with unifying intellectus (grasping multitude within one simple truth). Because all human reasoning terminates via resolutio either extrinsically at the supreme separate substances or intrinsically at ens commune, metaphysics stands closest to pure intellectual intuition, serving simultaneously as the last science learned by analytical resolutio and the prima philosophia supplying synthetic principles along the via compositionis vel inventionis.

Key Theses:

• Natural philosophy proceeds rationally (rationabiliter) by mirroring the human soul’s native discursive movement from sensible effects to ontologically distinct extrinsic causes.

• Mathematics demonstrates purely through the formal cause; even when proving properties of a circle through a triangle, it actualizes figures potentially intrinsic to the quantitative continuum.

• Mathematics proceeds disciplinably (disciplinaliter) with supreme human certitude because its objects are both immutable—unlike physical flux—and directly imaginable—unlike separate substances.

• Logic must be learned before mathematics and the other sciences not because it is easier—being hardest as a science of second intentions—but because all sciences depend on its method.

• Divine science proceeds intellectually (intellectualiter) as the ultimate terminus of both causal (extrinsic) and formal-universal (intrinsic) resolutio, gathering multiplicity into simple unity.

Question VI, Article 2 — Phantasm and the Termination of Mathematical Judgment

Read Primary Text: English (q. 6, a. 2) · Latin (q. 6, a. 2) · Commentary

Six objections contend that even divine science must be led back to the imagination (ad imaginationes deduci): Sacred Scripture clothes divine truths in sensible metaphors; \ Aristotle teaches in On the Soul that the soul never understands without a phantasm (phantasma); Dionysius declares that the divine ray shines upon us only through sacred sensible veils; God is known from sensible effects; \ phantasms regulate our intellect as colors regulate sight; and cerebral injury impedes contemplation even of divine things. Counter-authorities from Dionysius’s Mystical Theology, \ Augustine‘s On the Trinity, and \ Boethius‘s Consolation warn that reducing spiritual realities to corporeal images is the fountainhead of metaphysical error.

Aquinas resolves the tension by introducing a foundational epistemological distinction between the beginning (principium) of cognition, which pertains to apprehension, and its term (terminus), which pertains to judgment where truth is completed. In the present life, all human cognition begins in sense and depends continuously upon the phantasm—not as a transient ladder discarded after abstraction, but as a permanent foundation (permanens ut quoddam fundamentum) in which the intellect inspects its objects either positively or negatively. Yet the terminus where judgment is verified differs radically across the three sciences: in natural philosophy, judgment must terminate at sense (terminari ad sensum), conforming to sensory manifestation; in mathematics, because definitions abstract from sensible matter while retaining imaginable dimensive quantity, judgment must terminate at the imagination (terminari ad imaginationem) and not at sense—for a mathematical straight line touches a sphere at a single indivisible point, surpassing any sensible ruler and bronze sphere; and in divine science, judgment terminates neither at sense nor at the imagination, employing phantasms solely as starting points via Dionysian causality, transcendence, and negation. Methodological ruin ensues whenever thinkers attempt to apply a uniform criterion of verification across physics, mathematics, and metaphysics.

Key Theses:

• Every human cognitive act must be analyzed according to both its principium (apprehension, originating in sense and phantasm) and its terminus (judgment, where truth is verified).

• In the present state of life, the phantasm remains a permanent foundation (permanens fundamentum) of intellectual operation across all sciences, analogous to first principles in demonstration.

• Physical judgment terminates at sense (ad sensum), because sensible qualities and motions belong intrinsically to the definitive nature of natural bodies.

• Mathematical judgment terminates at the imagination (ad imaginationem) and surpasses sense, verifying exact geometric relations—such as point-tangency between line and sphere—in pure spatial intuition.

• Metaphysical judgment terminates in the pure intellect alone, employing sense and imagination strictly as points of departure via causality, excess, and negation without ever judging divine things to be imaginable.