18. Conceptions, Abstractions, and the Nature of Number

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Lecture Notes

Main Topics #

Three Types of Conceptions #

Thomas Aquinas distinguishes three ways the understanding conceives of things:

1. Likeness of External Thing: The conception is a true likeness of something existing outside the soul (e.g., “man”). This has an immediate foundation in the thing itself; the thing’s conformity to the understanding makes what is understood true.

2. Following from the Way of Understanding: The conception follows from how the understanding grasps an external thing, but is not itself a likeness of anything outside the soul (e.g., “genus”). These are called intentiones. Although the proximate foundation is in the understanding, the remote foundation remains in the thing itself. The understanding is not false in introducing these intentions.

3. False Conceptions: The conception has no foundation in the thing, neither proximate nor remote (e.g., “chimera”, “unicorn”). These are false conceptions because they are neither likenesses of external things nor follow from the true way of understanding something.

Mathematical Abstraction and Physical Reality #

Mathematical things are abstract in a particular way. Unlike simple conceptions, mathematics genuinely abstracts from sensible matter while retaining a remote foundation in things. For example:

• A sphere (whether glass, rubber, or steel) behaves differently when thrown against a wall, but a geometric sphere is abstract—it is not subject to the same physical laws.
• Mathematical lines are considered in separation (separatio) from matter; they are abstract yet have a foundation in the divisibility and extension of physical bodies.
• God cannot make mathematical truths false (e.g., He cannot make it so that two lines pass between two points, since this is a contradiction), yet physical events need not violate mathematical principles when properly understood.

Example: Christ Passing Through the Wall #

Berquist uses the Resurrection appearance of Christ passing through a locked door to illustrate the relationship between mathematical abstraction and physical reality:

• A mathematical objection claims: between two points, only one straight line can be drawn; therefore, if Christ’s body and the wall’s body occupy the same place, their points coincide, yielding two lines between two points—a contradiction.
• Thomas’s solution: The points are not the same because one point is in the body called the wall, and the other point is in the body of Christ. Thus there are two distinct points, not a violation of geometry.
• The mathematical considers the line in separation from the bodies in which those lines exist; geometry abstracts from the question of whether lines belong to different bodies.

Number and Unity #

The distinction between number and unity raises subtle philosophical problems:

• Euclid defines number as “multitude composed of ones.”
• Every number can be measured by one, both “as regards the understanding” and “as regards the thing itself.”
• A fundamental question: Is the number three truly one thing, or merely a pile of three ones?
• In created things, many distinct unities come together to constitute a number, giving number the character of composition and aggregation (aggregatio).
• Unlike created numbers, the Trinity involves a plurality of persons (three) in a unity of essence (one), without composition, aggregation, or the notion of an integral whole.

The Trinity and Divine Simplicity #

Thomas distinguishes how number exists in God versus in creatures:

• In created things: number has the notion of distinction and “piling up” of things distinct in their essence. The number three is composed of three essences, making it an integral whole.
• In God: The three persons are distinguished not by essence but by relative properties (fatherhood, sonship, procession). There is one essence in the three persons.
• Therefore, the number in God is not a multitude that is measured; it is a distinction only.
• This number in God exists only “according to the understanding,” not in the thing itself, because understanding unites the definitions of the personal properties in reason alone, not in an actual composition of the divine being.

The Rule of Two and Three (Ordo) #

Berquist notes an interpretive issue regarding Thomas’s Commentary on the Preamio (Preamble) to the Nicomachean Ethics:

• Thomas appears to validate a “rule of two or three” when dividing the orders that reason knows.
• The text distinguishes: (1) order which reason does not make but can consider (natural order, metaphysical order), and (2) order which reason itself makes (divided into three: order in the mind, order in the will, order in external matter).
• This division suggests a pattern of 2 + 3, though Berquist notes Thomas does not explicitly place mathematics in either category, creating an ambiguity worth further investigation.

God’s Omnipotence and Contradictions #

Thomas teaches that God cannot make something that involves a contradiction:

• God cannot make a square circle because it is self-contradictory.
• God cannot make a stone so large that He cannot lift it, for “a stone too big for God to lift” is itself contradictory.
• This does not limit God’s omnipotence; rather, contradiction falls outside the domain of possible being.
• Berquist notes that some theologians challenge this, claiming God could make contradictory things, but the Thomistic tradition holds firm on this principle.

Key Arguments #

The Argument from Abstraction #

• Abstract conceptions (like mathematical ones) follow from the way of understanding, yet retain a remote foundation in things.
• A geometric sphere is abstract, yet it follows from the way understanding grasps the shape of actual spheres.
• Therefore, abstraction does not sever all connection to the thing; it merely shifts the foundation from proximate (in the thing) to remote (in the way the thing can be understood).

The Argument from Trinity and Composition #

• In creatures, if many things are to constitute one thing per se and truly, there must be something uniting them (otherwise, they remain many things, not one).
• The three persons of the Trinity are not united by any additional form or essence; they share one essence.
• Therefore, the Trinity cannot be understood as an integral whole or as a composition of parts.
• The number “three” in God exists only in the understanding, not in the thing itself.

The Argument from Measure #

• In absolute number, there is composition and aggregation of ones.
• Every number is measured by one, which is simpler and more certain than any number.
• But in the Trinity, there is no such order of measure or composition.
• Therefore, the number of persons in God is not a true number in the way created numbers are.

Important Definitions #

• Intentio (intention): An intention is something introduced by the understanding that follows from the way of understanding a thing, but which is not itself a likeness of something outside the soul. Example: the intention of “genus” applied to “animal.”
• Abstractio (abstraction): The mind’s act of considering something while prescinding from other aspects. Mathematical abstraction prescind from sensible matter while retaining a remote foundation in divisible things.
• Aggregatio (aggregation): The piling together of distinct things. In created things, number involves aggregation; in God, there is distinction without aggregation.
• Separatio (separation): The way mathematics considers things as separated from sensible matter; e.g., lines are considered apart from the bodies they bound.
• Fundatio remota (remote foundation): A foundation not in the thing itself but in some aspect of the thing that justifies the understanding’s conception. Example: “genus” has its remote foundation in the material aspect of living things that they share.
• Fundatio proxima (proximate foundation): An immediate foundation in the thing itself that makes the understanding’s conception true. Example: “man” has its proximate foundation in human nature.

Examples & Illustrations #

Physical vs. Geometric Spheres #

Berquist throws three different types of balls against a wall to illustrate abstraction:

• A glass ball shatters.
• A rubber ball bounces back.
• A steel ball gets stuck or goes through.
• A geometric sphere (as a purely abstract concept) is not subject to these physical laws; it represents an abstraction from the properties of actual spheres.

The Chimera and Unicorn #

These are false conceptions because they have no foundation in external reality nor do they follow from the true way of understanding anything that exists. Unlike “genus,” which follows from understanding animal as existing in many species, chimera and unicorn are pure imaginative constructs without any grounding in the nature of things.

Christ Passing Through the Wall #

When Christ appeared to the apostles after the Resurrection, passing through a locked door, this does not violate geometric principles. The apparent contradiction—that two lines pass between two points—dissolves when one recognizes that the points belong to different bodies. Mathematics abstracts from questions about which body the geometric properties belong to.

The Division of a Cone #

If a cone is cut parallel to the base, the surfaces at the top of the lower part and the bottom of the upper part are circles. But are they equal circles or unequal? If equal, the cone would be made of equal circles, making it a cylinder, not a cone. If unequal, the cut is not perpendicular to the axis, and it is not a proper cone. This paradox troubled Democritus and illustrates the subtlety of thinking about continuous magnitude and divisibility.

The Pile of Rocks #

Berquist asks whether a pile of rocks is truly “one thing” or merely “many.” A pile is “one” only secundum quid (in a qualified sense), not simpliciter (simply). This contrasts with a house, which is one per se and truly because it is organized for a unified purpose. The unity of a number is similarly problematic: is the number three truly one, or just a pile of three ones?

Flocknotes and Parish Communication #

Berquist mentions a parish communication app called “flocknotes” (a system for flocking together—aggregatio) used to share updates about parish trips, illustrating how even modern examples embody the ancient philosophical concept of aggregation.

Notable Quotes #

“Sometimes what the understanding conceives is a likeness of a thing existing outside the soul. Just as what is conceived by this name, man… And such a conception of the understanding has a foundation in the thing, immediately, insofar as the thing itself, from its own conformity to the understanding, makes what is understood be true.” — Thomas Aquinas, quoted by Berquist

“Just as the abstraction of mathematical things and things of this sort.” — Thomas Aquinas on the remote foundations of abstract conceptions

“God cannot make the genus not be said of a species. He can’t make the genus not said of a species. He can’t make number not said of seven. He can’t make it, animal not said of dog.” — Thomas Aquinas, De Potentia, quoted by Berquist on God’s omnipotence and logical necessity

“The mathematical considers what? Line in separation from, as he says here, right? Abstractio Mathematicorum, right? You see? So it’s impossible in the abstract, you know, to have two lines between what? Two points.” — Berquist’s explanation of mathematical abstraction

“Between two points in the body of Christ that coincide in place, right, with two points in the part of the wall, there could be, what, two different lines, right? Because one is in this body and one is in that body.” — Berquist’s resolution of the paradox of Christ passing through the wall

Questions Addressed #

How does abstraction relate to truth and falsehood? #

Abstract conceptions (like mathematical ones) can be true even though they do not represent likenesses of things outside the soul, provided they follow from the true way of understanding something and have a remote foundation in the thing. Chimeras and unicorns, by contrast, are false because they have neither immediate nor remote foundation in things, nor do they follow from understanding anything that truly exists.

Can physical events violate mathematical truths? #

No, but apparent violations dissolve upon proper analysis. When Christ passed through a locked door, the apparent contradiction (two lines between two points) is resolved by recognizing that the points belong to different bodies. Mathematics abstracts from sensible matter and thus from the question of which material substance geometric properties inhere in. Physical events remain consistent with mathematical principles when both are properly understood.

Is the number three truly one thing or a pile? #

In created things, a number like three is composed of three distinct unities and has the character of aggregation. Whether it is truly “one” per se or merely “one” in a qualified way (secundum quid) is a subtle question. Unlike a house, which is organized by a single form, a pile of rocks or the number three lacks this unifying principle. In God, by contrast, the “number” three (the three persons) does not constitute a composition or aggregation at all, because the three persons share one essence and are distinguished only by relative properties.

Why does Thomas omit mathematics from his division of orders? #

Berquist notes an interpretive puzzle: Thomas divides the orders reason considers into natural order (not made by reason) and orders made by reason (subdivided into three: order in the mind, in the will, and in external matter). He does not explicitly mention mathematics. Some scholars point to his statement “Just as the abstraction of mathematical things and things of this sort” as implying mathematics falls under the same category as abstraction, but the precise placement remains ambiguous and worthy of further investigation.

How does the Trinity avoid composition while being three persons in one God? #

Thomas distinguishes how number and unity function in God versus creatures. In creatures, number involves composition and aggregation of distinct essences; in God, the three persons are distinguished only by relative properties, not by distinct essences. There is one essence in the three persons. Therefore, the “number” in God exists only in the understanding (as reason unites the definitions of the personal properties), not in the thing itself. The Trinity is not an integral whole or composition in any way.

What cannot God do? #

God cannot make something that involves a contradiction. He cannot make a square circle, a stone too large for Him to lift (for this is contradictory), or a number that is not measured by unity. Contradictions fall outside the domain of being itself; they represent non-being, not a defect in God’s power. This does not limit omnipotence but rather clarifies what “power” means.

Theological Significance #

This lecture demonstrates how understanding the nature of abstraction, number, and composition is essential for theology:

• Trinity: The Trinity is neither a composition nor an aggregation, yet involves real distinction. Understanding how number functions in creatures versus God clarifies how three can be one.
• Eucharist: The dimensions (quantity) of the consecrated bread remain individualized by themselves without a subject, a fact that presupposes understanding how quantity relates to substance and individuation.
• Divine Simplicity and Omnipotence: God’s inability to do the contradictory does not compromise omnipotence; it reflects the logical structure of being itself.
• Incarnation: Christ’s passage through material obstacles (like the locked door) appears to violate physical laws until one recognizes the distinction between mathematical abstraction and physical reality.