Quantity, Time and Reason: Individuation, the Properties of Quantity, and Number as Abstraction

Orientation #

He opens by recalling the two divisions by contradictories in chapter two of the Categories (“we have seen”) and refers to what “we talked about before” on how a name becomes equivocal by reason and on the definition of reason as looking before and after. Next time: “the unity of number is difficult to understand. We’ll come back to this next weekend.” The previous class worked through quantity, number and the transcendentals; the next takes up Thomas’s threefold account of how conceptions relate to things and the unity of number.

The class, in order #

1. Two divisions by contradictories need not give four parts (“we do not always get four parts”) #

Point: Aristotle’s two divisions of quantity crisscross into three species-groups, not four.

• Categories ch. 2: universal/particular × substance/accident gives four. But parts of a plot divided by before/not-before and after/not-after do not give four: no part is neither before nor after anything.
• Quantity: parts either have a common limit or not; and either have position toward each other or not.
  1. Line, surface, body, place: common limit and position.
  2. Number and speech: neither.
  3. Time: common limit but no position.
  4. Position without common limit: “does not seem to be a real possibility.”
• Natural vs. logical difference: natural philosophy says continuous quantity is “divisible forever” (matter; parts are like matter); the logician says its parts meet at a common boundary (whole is like form; the logician talks of forms — species means form). Aside: he “used to love to talk about the forms of fiction.”

2. Why quantity is divided twice and quality once: individuation (“individualized by themselves”) #

Point: quantities having position are individualized by themselves, not by their subject, which shows quantity’s closeness to substance.

• Eucharist: the dimensions of the bread are individual, yet not individuated by a substance; the colour and taste of the bread are individuated by being in the quantity of the bread. “Kind of amazing.”
• Errors that show the affinity: Descartes confusing body (species of quantity) with body (species of substance); Plato and the Pythagoreans making numbers the substances of things. Aristotle puts quantity right after substance.
• Cause of individuation of material substances: matter as subject to quantity. Five men in the room because there is enough flesh, blood and bones to go around; many chairs of one kind because enough wood; grandma’s dozens of Christmas cookies because enough dough.
• Angels: no matter, so the difference between angels is wholly formal; hence Thomas says no two angels are the same in species.

3. Summa Contra Gentiles on quantitas dimensiva (“habet autem hoc proprium”) #

Point: dimensive quantity alone among accidents is individuated secundum se, because positio — the order of parts in a whole — is in its definition.

• “Quantitas dimensiva” = continuous quantity, but perhaps not all of it: it does not fit time.
• Distinguish positio as order of parts in a whole from positio the category (sitting, lying down).
• This “perhaps explains why the order of the species is reversed in the second division”: quantities with position have the special likeness to substance.

4. Thomas on the modes of quantity (“quantum per se”) #

Point: a second Thomas passage (three steps) confirms quantity’s nearness to substance.

1. Quantum per se (a line) vs. quantum per accidens (ut musicum).
2. Quantum per se is twofold: signified per modum substantiae et subjecti — line, surface, number (he talks about numbers “as if they’re something existing by themselves”), each defined as a quantity (a line: quantity continuous according to length, divisible, finite); or signified by way of habitus or passion of such.
3. Quantity is propinquius to substance among accidents, whence some (Descartes, Pythagoreans) thought quantities substances — “terrible mistake, but understandable.” Reason: only quantity has division in its own parts after substance; whiteness cannot be divided, so is individuated only through its subject. Hence only in quantity are some things signified as subject, others as passions.

5. The special place of time; digression on noun and verb (“a special place for time”) #

Point: the two divisions include the same species except time — time is continuous yet very much the number of something (“How old are you?”), its parts having order but not position.

• Anticipating the study of statement: statement divides into noun and verb (onoma, rhema); both are names (vocal sounds signifying by agreement, no part signifying by itself); the verb signifies with time, the noun without.
• Greek and Latin have one word for name and noun; English two. The common name becomes “equivocal by reason”: one of the two keeps it as its own, the other gets a new name — the one that stands out.
• Parallel: doing vs. making (doing said of both; behind the two virtues of reason, art and foresight; art is right reason about making — grandma’s cookies, “I aim to please”). Making gets the new name because “what do you got to show for it?” — a cookie, a child, a chair (his line to the kids on sex without children). Not that making is better.
• So the verb gets the new name: time is the measure of motion, and things in motion “sooner catch the eye than what not stirs.” This confirms that the first sense of before and after is in time; one cannot even make a statement without a verb. Stretching language for God: “Before Abraham was, I am.”
• Shakespeare, “market of his time”: time there means his life, because our life is so measured and known by time (dividing up the day; the bell rings). Aside on teaching as an art and watching the clock.

6. Numbered number, order in time, time and reason (“numbering number and numbered number”) #

• The two divisions “conspire,” like dividing quantity into divisible-forever and not, which gives discrete/continuous; the oddball is time. Some want to say time is a number: “eight days” is a numbered number (like ten grandchildren), the abstract ten a numbering number; time is numbered number. Richard II in prison: “numbering clock.”
• That the parts of time have order explains, following Thomas, why the last six genera have two for place (where, and position = order of parts in place) but only one for time: before and after are already in the idea of time. “That’ll be on your final exam.”
• Before and after have their first meaning in time; the second sense is in being, Aristotle’s example one before two (one can be without two, not the reverse; each of the five of us could be without the five).
• Time would not be fully without reason: I can count my folders because they are all here, but yesterday is gone and tomorrow not yet; I remember and anticipate. Aristotle asks after treating time whether time would be without the mind — not fully.
• Reason (Shakespeare) is the ability to look before and after; first sense in time. Christ’s “Before Abraham was, I am” — not “I was” — uses the later sense, expressed as well as language allows. John the Baptist, “he who comes after me is before me”: before in excellence, after in time; one must know the senses of before to expound it.
• Categories is ordered to Peri Hermeneias (Thomas explains): the truth of statements — speech signifying the true or false — has the verb, which signifies with time, as its formal part.
• Asides: God never changes, “sounds awfully boring” vs. “variety is the spice of life”; his retirement reading (Summa Contra Gentiles, prima pars, then De Potentia, “most powerful disputed questions ever written,” important for the Trinity).

7. Quantities per accidens (“quantities through happening”) #

Point: the species are quantities through themselves; other things are quantified through them.

• Much white when the surface which is white is large; a trip long when the time it takes is long. The white wall is big not as whiteness but as being in the wall.
• Why per se / per accidens for quantity but not substance or quality: quantity is the subject of certain qualities — colour is in the surface, so colour is spread out because of the surface. This again brings out the order of the two highest genera.

8. The three properties of quantity (“does not have a contrary”) #

• Second property, not said more or less: one three is not more three than another, one three-foot line not more three feet; whereas one sweet thing is sweeter. Shared with substance; shows quantity closer to substance than quality; follows upon the first.
• Third, equal or unequal: a property in the stricter sense, since the first two are common to many. Aside: “all men are created equal” — but not in quantity (basketball players; men and taller women; Roy Lepak pursued by Margot), nor equally healthy, strong, beautiful, virtuous.
• First property, no contrary, shared with substance; Aristotle spends most time on it, refuting those who confuse relations following on quantity with quantity. Large and small seem contrary quantities but belong to relation (“towards something,” as Thomas speaks): the same quantity is large and small toward different things; if contraries, a thing would be contrary to itself. Any line divided forever yields parts small compared to the whole, larger than others.
• Order puzzle: Aristotle treats relation after quantity and before quality, though he enumerates quality before relation; because Plato’s people defined relation as what is said of another, Aristotle must first separate what is said relatively in a way (knowledge is of something) from what is essentially a relation.
• Aristotle seems to exclude place from this property; the definition of contrary takes its words from place: species furthest apart in a genus — black/white in colour; virtue/vice in habit, continence and incontinence being nearer (the incontinent man is ashamed; the drunkard is not). Aside: Seamus, “no shame in Seamus.”

9. Appendix: number and the continuous (“division of the continuous”) #

Point: his gathered texts on whether number in the categories is numbering or numbered number.

• Avicenna, Metaphysics: the unity and number of the arithmetician are not the unity and multitude found in all beings, but only as in material things, plurality caused by division of the continuous. Hence numbers increase forever (a line divisible forever into lines). Contrast multitudes not from division of the continuous: the two (or three) species of quantity; the ten categories (else infinite highest genera); three divine persons — Thomas goes back to only two intrinsic operations, understanding and loving. Passions of numbers, infinity in numbers (Physics III), unity in potency every number.
• Argument: if numbering number = numerus absolutus, which is not separate from numbered things except in the understanding, and the categories divide being outside the soul, then the number in the categories is numbered number (time too is numbered number).
• Summa text: objection — where there is number there is whole and parts, so number of persons is repugnant to divine simplicity. Reply: number is twofold, absolute (two, three, four) or in things numbered (two men, two horses); taken absolutely, nothing prevents whole and part in God, but only in the way our mind takes it; taken in numbered things, one is part of two in creatures, but not in God, since as much as is the Father, so much is the Trinity.
• Counter-worry: two men or two horses seems to involve more than one highest genus, like “white man” (substance + quality); and logic considers things as in reason, while abstract number is in reason only. Left open.

10. Thomas, Sentences: conceptions of the understanding in three ways (“a most subtle text”) #

1. A likeness of a thing outside the soul (“man”): immediate foundation in the thing, which by conformity makes what is understood true.
2. Not a likeness but something following on the way of understanding (“genus,” met in the proem, from Metaphysics V): proximate foundation in the understanding grasping animal as common to many (unless you are a Platonist), remote foundation in the thing; “simile est de omnibus aliis quae consequuntur ex modo intelligendi, sicut abstractio mathematicorum et huiusmodi.” “Very subtle.”
   • Hence the controversy over the Ethics proem: order compared to reason in four ways — not made by reason (natural philosophy, and wisdom; mathematics not mentioned), and made by reason in its own acts (logic), in acts of the will (practical philosophy), in exterior matter (mechanical arts). Warren Murray and he wonder whether mathematics has some peculiar abstraction with a likeness to logic: remote foundation in things, proximate in the mind. Two abstractions: mathematical from sensible matter (a geometrical sphere neither shatters, bounces, nor sticks; the interior angles of a triangle), and the more abstract two and three — “does that one really exist outside my mind?” Continuous quantity easier: eat the whole pie and he’ll complain; “go get me a three.”
3. No foundation, proximate or remote: a chimera, a unicorn; neither likeness nor following on true understanding; the conception is falsa.

• Ratio is said to be in the thing when what the name signifies is in the thing, i.e. when the conception is a likeness of it. Stop at 4:30.

His words #

• “divisible forever”: the natural philosopher’s account of continuous quantity, vs. the logician’s “parts meet at a common boundary”.
• “position” / positio: order of parts in a whole (in quantity’s definition), distinct from the category positio [ed.: posture].
• quantitas dimensiva: continuous quantity, but not time.
• “individualized by themselves”: quantities with position, unlike other accidents.
• quantum per se / quantum per accidens (ut musicum); per modum substantiae et subjecti vs. by way of habitus or passion.
• propinquius: quantity nearer to substance.
• onoma, rhema: noun and verb; both “names”; verb “signifies with time”.
• “equivocal by reason”: a common name kept by one thing, the other getting a new name.
• “art and foresight”: the two virtues of reason about making and doing [ed.: prudence for “foresight”].
• “numbering number” / “numbered number”: abstract ten vs. ten grandchildren, eight days; time is numbered number; numerus absolutus.
• “towards something”: relation, as Thomas speaks.
• “mechanical arts”: Thomas’s term for carpentry, cooking.
• “conception of the understanding … three ways”: immediate foundation, remote foundation, no foundation (falsa).
• “chimera”: an imaginative thing, e.g. a unicorn.

Texts #

Read in class

• Aristotle, Categories, ch. 6 (from his handout “Quantity – Chapter Six”): the two divisions, per accidens quantities, the three properties.
• Thomas Aquinas, Summa Contra Gentiles, habet autem hoc proprium quantitas dimensiva (locus not stated).
• Thomas Aquinas, passage on the modes of quantity, quantum per se … ut musicum (locus not stated).
• Avicenna, Metaphysics, on unity and number of the arithmetician (locus not stated).
• Thomas Aquinas, Summa Theologiae, objection and reply on number in the divine persons (locus not stated).
• Thomas Aquinas, Commentary on the Sentences, conception of the understanding in three ways (locus not stated).

Mentioned

• Aristotle, Categories, ch. 2 (four parts from two divisions); Aristotle on the parts of a plot (inferred: Poetics).
• Aristotle, Physics III (infinity in numbers); Aristotle on whether time would be without the mind (inferred: Physics IV).
• Aristotle, Metaphysics V (“fifth book of wisdom”) on genus.
• Aristotle, Peri Hermeneias; Thomas on the Categories being ordered to it.
• Thomas Aquinas, Proem to the Commentary on the Nicomachean Ethics (order compared to reason in four ways).
• Thomas Aquinas, De Potentia; prima pars (his reading plan).
• Shakespeare: “market of his time”; Richard II, “numbering clock”; reason as looking before and after (inferred: Hamlet).
• John 8, “Before Abraham was, I am”; John the Baptist, “he who comes after me is before me” (inferred: John 1).

His questions #

• Why is it more appropriate for the natural philosopher to say “divisible forever” and the logician “common boundary”? Parts are like matter (natural philosophy); the whole is like form, and the logician deals with forms/species.
• Why does Aristotle divide quantity twice and quality only once? To emphasize that quantities with position are individualized by themselves, being close to substance.
• What is the cause of the individuation of material substances — how can there be five men in this room? Matter as subject to quantity: enough flesh, enough wood, enough dough.
• Why does the verb get the new name and the noun keep the common one? The verb signifies with time, the measure of motion, and things in motion catch the eye.
• Why are there two of the last six genera for place but only one for time? Order of parts is already in the idea of time (before and after), but not in “where.”
• Is the number that is a species of quantity numbering number or numbered number? Numbered number, since absolute number is only in the understanding — but “two men” seems to span two genera; left open.
• Does two or three really exist outside my mind? Two or three men do; abstract number is in reason only — left open, to be resumed next weekend.
• What kind of multitude is the ten categories, or the three persons? Not one arising from division of the continuous; the persons go back to two intrinsic operations.