Categories, Chapter 6 on Quantity: Intentions of the Mind, Number and the Unity of Piles

Orientation #

He does not say what was done last time or what comes next; he does say these Trinity texts are “not maybe appropriate in studying the categories, but makes you stop and think,” and that the fundamental divisions of quantity “we’ll need for more advanced parts of philosophy.” The previous class treated the species of continuous quantity and quantity’s closeness to substance; the next finishes chapter six (properties of quantity) and begins chapter seven on relation.

The class, in order #

1. Three ways a conception of the understanding stands to the thing (“conception of the understanding”) #

Point: reading a Thomas text (source not stated; on the handout) on how what the understanding conceives is related to the thing outside the soul, “tripliciter.”

1. Sometimes what is conceived is a likeness of a thing outside the soul, as with the name “man”; this has an immediate foundation in the thing, whose conformity to the understanding makes what is understood true, and the name is properly said of that thing.
2. Sometimes what the name signifies is not a likeness but follows from the way of understanding a thing outside the soul: the intentions the understanding introduces, e.g. “genus” (back to Porphyry’s predicables). The near foundation is in the understanding, not the thing, because outside the mind animal is not separated from dog, cat or elephant; but the remote foundation is in the thing, so the understanding introducing these intentions is not false.
3. “Similiter est de omnibus aliis quae consequuntur ex modo intelligendi,” as the abstraction of mathematical things — a statement Thomas “doesn’t develop right here” and which “has to be developed some time.”

• Remote foundation of genus spelled out: things have something in common; genus is to difference as matter to form, though genus is not matter (matter is a part of the thing); we understand things fused first, and the mind separates out what they have in common.
• Aside (De Potentia, reread “today”): God cannot make the genus not be said of the species — number not said of seven, animal not said of dog. “That’s pretty confining, to the immortal God.”

2. The four orders in the Proemium to the Ethics, and the missing mathematics (“order is compared to reason”) #

Point: Thomas’s division of the sciences by the order reason considers, and his (noted, unresolved) disagreement with the Leonine editors over where mathematics goes.

• Thomas seems to violate “the rule of two or three,” but the division is two: order reason does not make (though it can consider it) vs. order reason makes; the latter is subdivided into three.
• Order reason makes in its own acts: genus before difference in the definition (a square is first a quadrilateral, then equilateral and right-angled; a perfect number is a number before it is perfect) — logic. Order reason makes in acts of the will: virtue as the road to happiness, vice to misery — ethics, domestics, politics. Order reason makes in exterior matter: the table, house, clothing; carpentry, “the art of Christmas cookies” — mechanical, liberal, civil arts.
• Order reason does not make: natural philosophy, “and metaphysics is also about order not made by our reason” (“I didn’t make the order of the angels”).
• Thomas does not mention mathematics. Some say he must mean it, since Aristotle divides theoretical philosophy into natural, mathematical and first philosophy, and Boethius (De Trinitate) gives the fuller statement; but “he doesn’t say it.” “We point to this text” — the abstractio mathematicorum follows on the way of understanding. So mathematics apparently goes with logic, “not that they’re the same”: different sciences, one more in imagination, the other more in reason; but not with the first (unmade) order. “Subtle, very subtle point.” Left as “a question mark in your mind.”

3. Can God make a contradiction? Two straight lines between two points (“God cannot make a square circle”) #

Point: an objection from Christ’s passing through the wall (and the sealed tomb) seems to show God can do what geometry says is impossible; Thomas’s answer shows what mathematical abstraction is.

• Aside: the boyhood challenge “can God make a stone so big he can’t lift it?” — a stone too big for God to lift involves a contradiction.
• Objection: when the body of Christ and the body of the wall are together, a point in each coincides, likewise two other points; so there are two lines between two points, contradicting geometry; hence God makes contradictories (he did not roll the stone back; “I think the church won’t say that”).
• Thomas’s reply: they are not the same points; one point is in the body called the wall, the other in the body of Christ. Mathematics considers line in separation from body, and in the abstract only one line is possible between two points; between two points in Christ’s body that coincide in place with two points in the wall there can be two lines, one in each body. Still miraculous, but no contradiction with mathematics. “Something strange about mathematics abstraction”; “you’ve got to keep on reading these things” (texte oublié).
• Aside: his mother’s view that Christ sparing his mother the pain of childbirth was “a wonderful thing” (birth harder since the fall).

4. Sentences text: aggregation follows division of the continuum, so this number is not in God (“aggregatio, how would you translate”) #

Point: first Sentences passage — aggregatio (flocking together; grex, gregis; the parish’s “Flocknote”) and such passions follow the division of the continuous (Avicenna, Metaphysics III), and that kind of number is not in God.

• Three Persons are not a continuum divided into three; those would be three parts and God no longer simple. Parallel objection (second article): if God’s power is infinite it has parts.
• Reply to the fourth: in absolute number there is composition and aggregation — in three, three ones are brought together (Euclid: number is a multitude composed of ones); it is less certain than one, its beginning; so the multitude is measurable by one not only as regards the understanding but in the thing (“I’m counting”). In the number of relations or Persons there is no order of certitude (one not more certain) and no composition in the thing; number in God is not a multitude measured by one, “except perhaps according to the understanding only,” which forms affirmative propositions of things not composed.

5. Second Sentences text: is three in God an integral whole? (“ternarius, of which one part”) #

Point: objection and reply on whether number in God is an integral (composed) whole.

• Objection: as continuous quantity is made whole from its parts, so is number; ternarius has one as one part and two as another; therefore an integral whole in God. (He usually says “composed whole,” as against universal whole; but the word is “integral.”)
• Reply: in God unity or duality is not a part of three except according as our reason takes it, because number, and unity, are in God and in creatures in different ways:
  1. One is what is undivided in itself and divided from others; each created thing is distinguished from others through its essence, so its essence, as undivided and distinguishing, is its unity.
  2. Many such unities constitute a number of created persons — many essences congregated, nothing of one being numerically in another (“you and I, really two”).
  3. So number in creatures has the notion of distinction and of “a certain piling up” of things distinct in essence, and hence of an integral whole.
  4. The personal unity in God is the relative property (Father from Son, Son from Father), not the essence; the essence is one in number in the three (“The Father and I are one” — unlike his own father and himself, one in nature but not in number).
  5. Therefore no piling up, only distinction; the number is not an integral whole “except perhaps in our mind,” as reason unites the definitions of the three personal properties — an integration in reason only.
• Aside: forty years of classes were distinct “piles”; former students reintroducing themselves at St. Mary’s — “one heap after another.”

6. Is a number one simply? Dionysius and the numbering of God with creatures (“partakes of what? Unity”) #

Point: further handout texts on the unity of number and on what things can be numbered together.

• A text on creatures: either the unities are flocked together by piling up, which does not make something one simply and so does not constitute a species of being (“number is not some species of being”); or, if number is one simply and not through itself, what makes it one from many unities must be said, “which cannot be assigned.” “I have to think about that some more.”
• Dionysius (as Thomas explains): all things partake of one, proved from the case that seems least so, number, which is opposed to one as divided to undivided; every number partakes of one (“seven is one number” — “isn’t it seven ones though?”). Distinction: number taken by itself (two, three) vs. as naming a part (half, third, tenth).
• Simply and properly only things that come together univocally in something one are numbered (you and your chair “two”: two bodies, common in matter). God and creature come together in no one way, yet by the community of analogy (creatures imitate Him) God and the angel can be said to be two things, not simply and properly; hence God is not a part of something, nor are God and the angel greater than God, but simply many.
• His own question raised in passing: whether the ten categories (highest genera) are a number in the species of quantity, since they do not come together univocally in one — left open.
• Further passage: things coming together in species, genus “or in some order” can be numbered; everything one in creatures is a part of number with another. “That’s rough stuff.”

7. Metaphysics VII: many things in act do not make one thing simply (“something uniting and some way binding”) #

Point: Thomas’s argument against man being one by diverse forms, and the pile as one secundum quid.

1. Many things existing in act do not become one simply unless something unites and binds them.
2. If Socrates were animal and rational by diverse forms, a third thing would be needed to unite them; none can be assigned.
3. So man would be one only by putting together, like a pile — secundum quid unum, simpliciter multa — and, since a thing is a being as it is one, not a being simply.

• Fallacy of secundum quid and simpliciter: easily mixed up. His own cases: “white philosopher,” “black philosopher,” “philosophical grandfather,” “Christian logician” — each a putting together (baptized to become Christian; logician by reading Porphyry and Oesterle), not one thing simply.
• Asides: Oesterle’s logic, still in circulation, used by his granddaughter at Benedictine College; Oesterle’s exercise “he killed his daughter Ruth, therefore he is ruthless” — an equivocation; “your argument is like an unsharpened pencil: it’s pointless.”

8. Democritus: no one from two, because potency and act not distinguished (“impossible for one to come to be from two”) #

Point: Democritus speaks rightly that two things in act never make one, but, not distinguishing ability and act, made indivisible magnitudes the substances of things.

• He wanted no many in potency in what is one, so each magnitude indivisible; but a straight line is one in act, many in potency (geometry has a theorem to divide it).
• The thought experiment: divide bodies every way they can be divided; what is left is either sizeless things or nothing; not nothing, since nothing comes from nothing; so atoms — motes in the sunlight.
• His problem with the cone (made by rotating a right triangle): cut it parallel to the base; the two facing circles are either equal (then the cone is a cylinder) or unequal (then the side is not straight). “Very subtle though these problems” — “these Greeks.”
• Applied to number: if number is a putting together of units, either duality is not something one, or unity is not in act in it — “really three actually distinct ones?” — “tearing your mind apart.” Duality would then be not two units but something composed from two, otherwise a pile.
• A pile of rocks or the trash bag: one “in some way,” very secundum quid. The biologists’ “you have as many lives as cells.” The three Persons are “not a pile of persons.” Does arithmetic study piles? Euclid’s theorems must be about something one per se and truly.
• What the categories give: being into ten, quantity into discrete and continuous, discrete into number and speech — the fundamental divisions needed for the difficult questions.
• Aside: the Greek’s thanks for being born a man, a Greek, and for having met Socrates.

9. Metaphysics VII: three ways a composite takes its species (“gathers its species from something one”) #

Point: the composite is one simply when it takes its species from a form, a composition or an order; from a multitude of parts collected it is one only secundum quid.

• Form: the mixed body, where the mixed bodies do not survive in act. Composition: the house (“I have one house, or a pile?”). Order: the syllable and the number — back to discrete quantity’s two species.
• A line of a Shakespeare sonnet (“From fairest creatures…”) or Macbeth (“Fair is foul…”) is not a pile of letters: distinction of syllables plus an order, meter — a kind of unity. “Then it is necessary that the whole composed is one simpliciter.”
• Pile and people: species from a multitude collected, one secundum quid only.
• Asides: whether to say “composed” or “put together”; Thomas’s whole question of eight articles that God is not put together in any way, distinguishing about six ways of composition. The campaign talk of “bringing the country together”: simpliciter or secundum quid?

His words #

• tripliciter; the three ways a conception stands to the thing outside the soul
• immediate / remote foundation (in the thing vs. in the understanding)
• intentions the understanding introduces; genus as an intention
• ability [ed.: potency] and act
• abstractio mathematicorum; “abstraction of mathematical things”
• “rule of two or three” (his rule for divisions)
• aggregatio; “flocking together” (grex, gregis)
• integral whole, which he prefers to call “composed whole,” vs. universal whole
• “piling up,” heap, pile (aggregation of distinct essences)
• personal unity = relative property, not the essence
• ternarius; number of three
• secundum quid unum / simpliciter multa; fallacy of secundum quid and simpliciter
• compositum; “put together”; composition, form, order as sources of a composite’s species
• syllaba; “speech” as the second species of discrete quantity [ed.: oratio]
• texte oublié; “forgot a text”

Texts #

Read in class

• Thomas, on the threefold relation of the understanding’s conception to the thing (work not stated; handout)
• Thomas, Commentary on the Nicomachean Ethics, Proemium (the four orders)
• Thomas, Commentary on the Sentences: reply “to the fourth” on aggregation and number in God; objection and reply “to the second” on ternarius as an integral whole (loci not stated)
• Thomas explaining Dionysius on every number partaking of one (work not stated; probably Commentary on the Divine Names, inferred)
• Thomas, Commentary on the Metaphysics, Book VII: many things in act; Democritus; the three sources of a composite’s species
• Aristotle, Categories, ch. 6 (per verified sources: handout “Quantity - Chapter Six,” pp. 15-17) Mentioned
• Thomas, De Potentia (God’s power; genus said of species; infinite power; stone and square circle) — article not stated
• Thomas, the question in eight articles on God’s simplicity (locus not stated)
• Avicenna, Metaphysics, Book III
• Boethius, De Trinitate (division of theoretical philosophy)
• Euclid, definition of number; theorem dividing a straight line
• Porphyry, Isagoge (predicables) (inferred)
• Oesterle’s logic textbook
• Shakespeare, Sonnets (first sonnet) and Macbeth, single lines quoted

His questions #

• Where does Thomas put mathematics among the four orders, since he never mentions it? Apparently with logic, as following the way of understanding — different sciences, but not with the unmade order; left as a question mark.
• Between two points, can you draw two straight lines — and did Christ passing through the wall do it? No: the coincident points are two points in two bodies; mathematics abstracts from body.
• When there are three Persons, did you divide something continuous into three? No; that would make three parts and God not simple.
• How does Euclid define number? A multitude composed of ones.
• “The Father and I are one” — one what? One in number in essence; unlike his own father and himself, one in nature only.
• Isn’t seven seven ones? What makes a number one simply? Left open (“I have to think about that some more”).
• Are the ten categories a number in the category of quantity? Left open.
• Is a pile of rocks, or a trash bag, something one? In some way, very secundum quid.
• If you cut a cone parallel to the base, are the two circles equal or unequal? Either answer breaks the cone: Democritus’s problem, left as a puzzle.
• What kind of unity does a line of Shakespeare have? Order: distinction of syllables and meter, so one simpliciter.