Divisions of Five, the One Convertible with Being, and the Species of Quantity

Orientation #

“So we left off here somewhere in the discussion of quantity… Page six.” He picks up the handout “Quantity - Chapter Six” (pp. 6–7, verified) at the sentence on multitudo absoluta. He announces no topic for next time. A short stretch of the transcript (language-learning aside, “at the bottom of this page… this distinction by measure is seen more in the species… Can we better stop now”) appears to belong to the end of the recording and is placed out of order; it is reported in segment 2.

The class, in order #

1. “What’s my number?” — divisions of five (“Here’s this good Pythagorean question”) #

Opening asides: Thomas in the prima pars on men being vegetarians before the fall; the Lebanese explanation of fish on Fridays (our Lord’s friends were fishermen). Then a “Pythagorean” game: he is attached to five (five letters in Duane, five fingers), and runs through the fivefold distinctions of philosophy, usually breaking into three and two:

• Predicables: genus, difference, species are essential; property and accident outside the nature (3 + 2).
• Quantity: divided into discrete and continuous by whether the parts meet at a common boundary; discrete into number and speech (Hamlet’s “I am sick at these numbers” shows the kinship of number and meter); continuous into five: line, surface, body (intrinsic measures) and place, time (extrinsic measures) — 3 + 2.
• Five senses of “before” in the postpredicaments: not 3 + 2 but four “central senses” plus cause-before-effect, which he likens to the thumb against four fingers.
• Soul, a “potestative whole” (“powerful whole”): five genera of powers — vegetative (later three: nourishing, growth, reproduction; Thomas compares baptism, confirmation, Eucharist to these), sense (five external, then inner: common sense, imagination, memory), intellect, appetitive (sense appetite and will), locomotion. Aristotle also distinguishes three souls and four grades of life (oysters resemble plants).
• Ethics: five virtues of reason — nous (“natural understanding”), episteme (“reasoned out knowledge”), wisdom in the looking reason; art and foresight in the practical (3 + 2). Understanding and wisdom are lowest species; episteme, art, foresight divide further (Thomas divides foresight into that of the individual, the father, the king).
• Thomas: five ways for God’s existence; five attributes of God’s substance, always simplicity before perfection (in stone, plant, cat, man the more perfect is more composed; God is perfect in a simple way), then infinity and unity; SCG puts God’s unchangingness first, ST fourth. SCG on God’s goodness: five chapters — that God is good, goodness itself, nothing bad in him, “good of every good” (Augustine’s phrase, absent from ST), summum bonum.
• Student adds: five books of Moses.

2. Displaced fragment (“this distinction by measure is seen more”) #

An aside on a St. Paul professor with a photographic memory (Chaucer in Middle English, Aristotle’s Rhetoric in Greek, Japanese; Gasset’s definition of a poet as “a man with his brains kicked out”). Then he reads the bottom of the page: “this distinction by measure is seen more in the species of discrete and continuous quantity than in the distinction between the discrete and the continuous,” proposes to exemplify it, and stops: “Can we better stop now.” This appears to be the end of the session, spliced early in the recording.

3. Plurality opposed to the one convertible with being (“sed multitudo absoluta”) #

Point: the multitude Thomas means is opposed to the one convertible with being, which opens on the closeness of quantity to substance.

• Aristotle in Wisdom [ed.: Metaphysics] says the subject of wisdom is being and one; the middle books are on being, book ten on one. God’s unity is the last of the five attributes, placed before the operations and the Trinity to stress one God.
• Body vs. body: body as one kind of substance against body as a species of quantity; people confuse them. Descartes made extension the substance of things — “Descartes never grew up,” since if size were your substance you would not be the same on growing (his grandchildren shoot up but are the same little boy and girl). Your size is not your substance, but the confusion shows the proximity of the two.
• Pythagoreans and Plato made numbers the substances of things; in Metaphysics VIII Aristotle compares natures to numbers: stone one, plant two, animal three, man four, adding life, sensation, reason. Thomas: if ten knew itself it would know what partakes of nine, eight, seven… — how God knows all things by knowing himself.
• Read this morning in the prima pars: are men equal? God could make us unequal without sin or defect of nature, pro pulchritudine ordinis; Thomas always quotes Romans 13, quae a Deo sunt ordinata sunt. Kasurik: “God hates equality.” The wise man orders; God is wisdom itself.
• Aside on order among teachers: Kasurik (wisest at St. Thomas, “compared to Aristotle I got the brain of an angleworm”), phoning Fr. Baumgartner at St. Paul Seminary, saving hard questions for De Koninck. Metaphysics XIII: the first species of beauty is order (taxis). No two angels equal; the angel just above enlightens you (Monsignor Dionne compared teachers to this). Thomas on states, orders, offices in the church: offices for its needs, states for its perfection, orders for its beauty (student: hence “holy orders”; a psalm/Old Testament text on God known in order, unlocated). Thomas on Adam as beginning and end of Eve — “going too far, even for a guy like me.”
• Aside: Tannhäuser on TV; a critic’s paper “Putting Mozart in His Place” puts him first; Wagner: Mozart above all masters in all arts and ages — “putting in order” uses the word place.

4. Multitude as quasi-genus; one as measure (“because the number is nothing other than a plurality”) #

Point: absolute multitude is a quasi genus of number, and the one that measures is opposed to number relatively, not contrarily.

1. Plato confused the one convertible with being with the one that is the beginning of number.
2. Number is a multitude measurable by one; the one, as simply indivisible being, is convertible with being. (Aside: truth and good are convertible; for reason the truth about the good is one part of truth, for the will truth is a particular good; so reason can know what the will is, and the will can will that reason learn.)
3. As it takes on the ratio of measure, one is determined to the genus of quantity, where measure is properly found; likewise multitude as “divided beings” is not determined to a genus, but as “something measured” it is determined to quantity, whose species is number (one might add discrete).
4. Being and one are “transcendental” — he prefers “sort of alls”: they transcend any one genus. Student’s reason why being is not a genus: genera are predicated univocally, and these are not said univocally of the ten.
5. Student: why quasi genus, not simply a genus? Because not said univocally. Application: in the Trinity three persons are not “more” than one; not a multitude measured by one, not in the genus of quantity — three grandchildren are more than two, not so in God; not 1 + 1 + 1 = 3 but 1 × 1 × 1 = 1, else three gods or three parts.
6. Aside on simplicity: ST gives eight articles; the first six eliminate six compositions (God is not a body; not matter and form — a chair is wood and shape, or seat and legs; not nature and individual — he is “a two-footed animal with reason born of a Swedish father and an Irish mother,” Mark more Swedish, Richard more Irish; not substance and being, which even the angel has; not genus and difference — “predicament” is a prison, God in no genus, hence infinite; not substance and accident — “I’m a white man, a philosophical man”). Article seven concludes universally: inductively from the six, and syllogistically (in every composition part is to part or parts to whole as ability [ed.: potency] to act; God is pure act).
7. Thomas does not multiply words: quasi because multitude has something of genus in being common, but not univocally common. “That’s how subtle it is.”
8. Conclusion read: one as beginning of number (a measure) and number (a species of quantity, multitude measured by one) are opposed not as contraries, like the one convertible with being and its plurality, but as ad aliquid, things towards something: one is said relatively because the other is referred to it.

• Aside: Fr. Bodai’s “mental indigestion” and constipation.

5. The second division: parts with position (“The second division of quantity is into those”) #

Point: Aristotle’s second division — quantities whose parts have a placing toward each other and those that do not — separates the same species as discrete/continuous, with one important difference: time.

• Time, though continuous, has no parts laid out toward each other, for past and future do not exist together; this brings in another formal consideration of parts, how they are united in the whole.
• Puzzle: time is “the number of the before and after in motion” (marking days by the sun, like Crusoe). Number is the first species of discrete quantity — so is time discrete? No: it numbers something continuous, as “three feet” is not discrete though three is.
• Room wall vs. time: the parts of the containing wall, of a line, surface, body, are laid out and exist together; parts of time, speech, number do not.
• Point vs. one: the point is nearest the one of anything, but the point has position and the one has none.

6. Odd and even measuring (“Can an odd number measure an even number?”) #

An odd number sometimes measures an even (3 measures 6, 5 measures 10). An even never measures an odd: every even is measured by two, so two would measure the odd and it would divide into two equal parts. Hence no even but two is prime; odds sometimes measure odds (3 measures 9); some odds no number measures (5, 7).

• Aside on seven: seven kinds of philosophy — three looking (natural philosophy, mathematics, wisdom), three doing (ethics, domestics, political), and the tool, logic; he prefers “kinds” to “parts” (parts makes them too much one). Seven gifts, seven sacraments, seven petitions of the Our Father.

7. Time divisible forever (“time, although a continuous quantity”) #

Time is continuous because divisible forever, being the number of motion, which is divisible forever. Aristotle proves motion and distance divisible forever together: the faster runner covers the slower’s distance in less time, in which the slower covers less distance, and so on. Hence Zeno’s paradoxes: half the way before the whole, “never going to get out.”

8. One genus divided many ways; no difference convertible (“possible to divide the same genus into species”) #

Point: number divides into odd/even, prime/composite, and perhaps perfect/imperfect; this connects with Thomas’s reason in the Posterior Analytics commentary why no difference is convertible.

• Beginners get “rational” as the example, but Porphyry defines difference as said of many things other in kind, i.e. of more than one species; his own example is rational mortal animal, where neither difference is convertible.
• Euclid’s square: equilateral right-angled quadrilateral. Equilateral is shared with the rhombus, right-angled with the oblong; only the combination is convertible — “that’s the way differences really go, for the most part.” Going from confused to distinct takes longer but fits our mind, like pinpointing by longitude and latitude, two lines for a point, finding a circle’s center.
• Asides: O’Reilly’s Killing the Rising Sun; two sets of planes over Saddam Hussein’s defences, one marking the target; MacArthur, David Lawrence and a New York Times military critic opposed the bomb. Three calendars in the sacristy for distinctness.
• Two is “the even prime number”: not the only even, not the only prime, the only both.
• Perfect number: equal to the sum of the numbers measuring it — 6 (1 + 2 + 3), then 28 (1, 2, 4, 7, 14); the third he forgets. Autistic twins who traded next primes.
• Proportion: if a:b :: c:d then a·d = b·c; from 28, 2·14 and 4·7 give 2:4 :: 7:14; any two factor pairs of 100 do the same. Students who say 2:3 :: 4:5 do not know proportion in numbers, which should be clearest; 4 is the same parts of 6 as 2 of 3.

His words #

• potestative whole / “powerful whole”: the soul divided into its powers
• looking reason vs. practical reason [ed.: speculative and practical]
• nous “natural understanding”; episteme “reasoned out knowledge”; foresight [ed.: prudence]
• Wisdom [ed.: Metaphysics]
• multitudo absoluta: plurality opposed to the one convertible with being
• quasi genus: common but not univocally; hence not simpliciter a genus
• “sort of alls”: his term for transcendentals (“too spooky”)
• predicament: category as a prison one cannot get out of
• ability [ed.: potency]
• pulchritudo ordinis: the beauty of order
• taxis: order, the first species of beauty
• ad aliquid: “towards something,” relative opposition
• number: multitude measured by one; time: number of the before and after in motion
• perfect number: equal to the sum of the numbers that measure it
• “kinds of philosophy,” not parts

Texts #

Read in class

• Handout “Quantity - Chapter Six,” pp. 6–7: multitudo absoluta passage through the second division of quantity and “possible to divide the same genus into species in more than one way” (Aristotle, Categories, ch. 6, with Thomas’s remarks as given on the handout)
• Romans 13:1, quae a Deo sunt ordinata sunt (quoted)

Mentioned

• Thomas, ST Prima Pars: vegetarians before the fall; equality of men; simplicity of God (eight articles); five ways; attributes of God
• Thomas, SCG: five chapters on God’s goodness; unchangingness first
• Aristotle, Metaphysics VII–VIII (substance, natures as numbers), X (one), XIII (order as first species of beauty)
• Aristotle, Categories, postpredicaments: five senses of “before”
• Thomas on states, orders, offices in the church
• Thomas, Commentary on Posterior Analytics: why no difference is convertible
• Porphyry on difference; Euclid’s definition of square
• Hamlet “sick at these numbers”; Zeno’s paradoxes; Wagner’s Tannhäuser

His questions #

• Where is the first distinction of five we meet in logic? The predicables, divided three (genus, difference, species) and two (property, accident).
• Does he distinguish five species of quantity? More than that; continuous quantity divides into five: line, surface, body, place, time.
• What’s the famous division of five in ethics? The five virtues of reason: understanding, science, wisdom; art, foresight.
• Why is being not one genus? Because a genus is predicated univocally, and being is not said univocally of the ten.
• Why quasi genus and not simply a genus? (student) Because multitude is common but not univocally common, like being.
• Is time a discrete quantity because it is a number? No; it numbers something continuous, as “three feet” is not discrete.
• Can an even number measure an odd number? Never, since two would then measure the odd; an odd can sometimes measure an even.
• How does Porphyry define difference? Said of many things other in kind, signifying how they are — so of more than one species.
• Is equilateral convertible with square? No (rhombus); nor right-angled (oblong); only the combination is.
• What’s the third perfect number? Left open (“I forget”).