Quantity by Its Parts and Measures: Two Definitions of the Continuous, Logician versus Wise Man, and Music versus Painting

Orientation #

He opens on “the text in the Metaphysics” which he says is being read “for the bonitatem doctrinae”, beyond what Aristotle does in the chapter on quantity; the handout “Quantity – Chapter Six” (pp. 1–5, English and Latin) is read from throughout. He refers back to “that problem we had about substance” (first and second substance) and to “what we talked about in the Isagoge” (species). He announces “we’ll see that when we get to quality” (habit as first species) and “when we get to the properties” (equal and unequal); the recording ends mid-sentence. The neighbouring classes treat the genera of being, quantity and number, and the transcendentals.

The class, in order #

1. The ratio of quantity: divisible into things that are in it (“what is divisible in those things which are in it”) #

Point: the Metaphysics V text lays down the ratio of quantity — quantum (a concrete word, he notes) is what is divisible into things which are in it — and each clause excludes a rival kind of division.

• “In it” excludes the mixed body. Elements are in the mixed body only in ability (potency), “in virtue” only, not in act.
  • Modern physics example: bombard a particle and get particles out; sometimes the resulting particles have more mass than the original, so they cannot have actually composed it — “you kind of have to go to the idea of ability”.
  • Aside: his friend Warren Murray asked Heisenberg whether he was Platonist or Aristotelian; Heisenberg admitted he was more Platonist (mathematical science of nature) but admired Aristotle’s notion of potency — see his Gifford lectures, Physics and Philosophy.
  • Contrast: parts of the pie “we’ll be eating later” are actually in the pie; bisecting a line or an angle in geometry — the parts “seem to have been in there already”, though not separated in act. Dividing a mixed body into elements requires not just division of quantity but alteration.
• “Each apt to be something one” — Greek tode ti, Latin hoc aliquid, “hoc est aliquid demonstratum”. Aside: the soul of man is called a hoc aliquid because it can exist by itself, though it is not a complete substance; “hard to demonstrate it, to point it out”.
• This clause excludes division into essential parts, matter and form, neither of which is apt to be one by itself. Example: Michelangelo’s Pietà is marble and shape; you cannot put the marble here and the shape there — a different kind of division from cutting a thing in two.

2. Species: multitude and magnitude; the question “how much, how many” (“measurement properly pertains to quantity”) #

Point: the two first species are multitude (number) and magnitude; both carry the notion of quantity insofar as one can be numbered and the other measured.

• Two definitions of number given: a multitude measured by one; a multitude composed of ones.
• Quantity answers “how much or how many”, which implies not merely a multiplicity of parts but measurement — a multiplication of parts is not yet “quantity”.
• Aside: Kasurik “banged into my head” that Aristotle uses the concrete words (quantum). Greek has one word for how much and how many (posos); Warren Murray says French combien serves for both; his own English candidate is “size” — the size of a family (multitude) vs. a magnitude; the size of a crowd, of the march in Washington.
• Aside (one line): Trump, his Indiana trip and the stock market.

3. Continuous vs. discrete: why two definitions of the continuous (“parts that meet a common boundary”) #

Point: the Categories defines the continuous by parts meeting at a common boundary; natural philosophy defines it as divisible forever; the difference follows from what each science considers.

• Preface: closeness of quantity to substance — what is quantitatively divisible may be divided into substances (worms, some plants; “divide you and me up” would be “a lot of liquid”).
• Discrete quantity is not divisible forever: seven into two and five, three into two and one, but one cannot be divided. The one is simpler than the point: both indivisible, but the point has position, the one has none (Platonists: a point is “one with position”). No such thing as two and a half points.
• Why the natural philosopher defines by “divisible forever”: whole is to parts as form is to matter (matter defined as that from which a thing comes to be, existing in it — wood in the table; parts likewise in the composed whole). Dividing into parts of parts of parts is going to the matter, and matter is what natural philosophy alone brings into its definitions. The logician deals with form, definition, so defines by wholeness: the parts come together at the boundary to form the whole.
• Proof that a line is divisible forever, as he gives it:
  1. Dividing a line yields either two lines or two points.
  2. Division could only halt if a line were composed of two points.
  3. Points composing a line would have to touch; part touching part — no parts; edge touching edge — then it is a little circle, not a point; so they can only coincide, giving one point, no length, not a line.
  4. Therefore division of a line always gives two smaller lines, which can be divided again. “Case rests.”
• Aristotle shows length and time divisible forever together: admit only that one body is faster than another; the faster covers a distance in a time in which the slower covers less; divide the distance, the faster covers the lesser distance in less time; alternate forever.
• “It’s very subtle, Aristotle… Kind of amazing that the natural philosopher and the logician both have a definition of the continuous, and it’s not the same definition.” Connection between them: you can always divide at the boundary.
• Aside: Euclid says two is a part of six but four is not — “part” in the sense of measure; Aristotle also mentions this use, “not so much important for us right now”.

4. Thomas’s Latin on the two definitions (“consideratio naturalis versatur circa materiam”) #

Point: handout p. 4, Latin: “et satis convenienter hic definitio ponitur, aliam autem in praedicamentis, quia consideratio naturalis versatur circa materiam” — the logician’s consideration is about the ratio of the species (species as in the Isagoge).

5. Logician vs. wise man: second substance, and time, place, motion as quantities (“the difference between the logician and the wise man”) #

Point: the same split between logic and wisdom that produced “second substance” explains why the Categories and Metaphysics V give different lists of quantities.

• Recall: for the wise man there is no real distinction of first and second substance — animal is not a different substance from the dog; that is “just in your mind, Aristotle”. Logic considers things as they are in reason, where there is not only Socrates but even more the universal (Albert the Great: the first thing to consider in logic is the universal); second substance signifies what Socrates is.
• Thomas, handout p. 4 (Metaphysics V): in the Praedicamenta time is a per se quantity; here per accidens. There species are distinguished by diverse ratios of measure — time an extrinsic measure, magnitude intrinsic; here by the very being of quantity, so what has quantity only from another (motion, time) is not a species. Place is a species there but not here: a different ratio of measure but not another quantity.
  • “What the hell does that mean?” Place is the inner surface of the containing body (natural philosophy): wine in a bottle — the bottle’s inner surface differs from the wine’s surface, yet both are surface, the same kind of quantity, though measured separately.
  • Sofa example: measure the wall, then measure the sofa in the shop — different measurements, same kind of quantity (length). Place is “an extrinsic measurement”; buying a 1.5-litre vs. 0.75-litre bottle, one does not measure the surface of the wine.
• “The wise man is a little more picky… interested in the being of the thing.” These points are “beyond what he’s doing in the chapter on quantity”, for the goodness of the doctrine.

6. Diverse ratios of measure: speech, area, volume; the definition of measure (“measured by the long and the short syllable”) #

Point: handout p. 5 — because logic distinguishes by ratio of measure, Aristotle shows speech is a quantity by its being measured by long and short syllables, a different measure from the one that measures number.

• Greek and Latin verse: long and short syllable (“Visus, tactus, gustus in te fallitur” — first syllable longer; the last foot left incomplete); English: accented and unaccented.
• Further diverse measures: a room in square inches, not inches; a container in cubic inches; a student’s paint bought by the quart but rated in square feet of coverage. “You don’t measure a line in square inches.”
• Definition of measure (first sense): that by which the quantity or size of something is made known to reason; later extended (“the virtuous man is a measure”). Some Latin dictionaries give “size” for measure. Hence logic, which considers things as in reason, treats different measures as different species, as it treats the species of a first substance as (second) substance; the wise man, who considers being, does not.
• Aside: man separate from Socrates, Plato, Aristotle exists only in the mind, “unless you’re a Platonist”; Aristotle’s handling of Plato equips you for Hegel, who tries to generate the universe from the vaguest notion, being; Hegel is better on the fine arts.

7. Digression: music vs. painting (“which is more beautiful, music or painting”) #

Point: prompted by Hegel ranking music above painting (against some of his colleagues), he argues music is the more beautiful and important art.

• Aristotle’s Politics (book VIII, he corrects a student who said Ethics) is brief on paintings, insistent on music; Plato likewise. Thomas can be used to show there will be music in heaven properly speaking, whereas for paintings “we’ll have the vision” — the saints’ bodies. A student: the magisterium puts music first among arts serving the liturgy; the psalms go with instruments.
• Student’s objection: if sight is the primary sense, why is what comes through hearing more beautiful — a reversal of order? His answer: the eye is more important for knowing, but music connects with the emotions and so with the virtues — it “moves the emotions in a way that’s in harmony with reason”. Marches represent courage better than a battle painting (his son Marcus: “I want to be a hero too”); Kasurik asked whether classical music helped him control his emotions; rock and roll’s effect (chairs flying, “make a monkey out of you”) makes one bestial, the good kind elevates (Washington Irving on the Westminster organ; Palestrina).
• Mozart’s 36th and Jupiter symphonies (C major) represent magnanimity; the 38th and 40th, the two acts of courage — approaching the threatening thing, and patience. Einstein: Beethoven made his music, Mozart found it — like Romeo and Juliet seeming a natural part of fiction. Mozart’s music “more beautiful than Shakespeare’s words” (he and Warren Murray agree).
• Student aside: before recordings people heard a piece rarely and must have retained it better; he: audiences demanded Mozart repeat a movement, and preferred his improvisations.

8. Why number comes before the continuous (“measure is found first or more perfectly in number”) #

Point: handout p. 5 middle — since the species are distinguished by ratio of measure, Aristotle puts discrete before continuous, because measure is more perfect in number: the one is simple, absolute and indivisible, whereas the measure of length is arbitrary and inexact.

• The king’s foot (grew as he aged); the metre as a metal bar in Paris that expands and contracts. “We don’t have to worry about the one in number contracting.”
• Puzzle he flags as “strange” and “curious”: the continuous has the affirmative definition, the discrete the negative, and affirmation is prior to negation (you must know what you negate) — yet discrete comes first, because of measure.
• Parenthesis: the same reason why habit or disposition is the first species of quality, though less fundamental than natural ability — by habit one is well or ill disposed towards one’s nature.
• Number is measured by the indivisible one; a line cannot be measured by the indivisible point. Prime numbers measure some numbers, not all; one measures every number, exactly.
• Is the continuous more known to us? He inclines yes (affirmative before negative); numbers are “an immaterial thing”.
  • Aside on etymology vs. meaning (against MacIntyre on “philosophy”): psyche means breath, yet that is not the meaning of soul; air was borrowed because it is invisible and seems to have least matter (you make ice from water, things from wood — what do you make from air?).
• Thoughts are more like numbers: not every part of a definition needs defining, nor every statement proof, else definitions and demonstrations would be divisible forever and nothing known. Metaphysics VIII compares forms to numbers, not to continuous things. Aside: evolution and “transgender nonsense” treat natures as infinitely divisible like continua. Class agrees: natures are more like numbers.

9. Equal and unequal require a common measure (“to be equal or unequal”) #

Point: the property peculiar to quantity, equal/unequal (to be treated later), presupposes things measured by a common measure — so a line and a surface, or time and length, cannot be called equal or unequal.

• Playing with it: Quebec is “eight hours away”, Saint Paul “twenty-four hours away” — time standing for length. Recording ends mid-sentence.

His words #

• quantum — the concrete word Aristotle uses: what is divisible into things which are in it.
• “ability” — his rendering; [ed.: potency]; what elements have in a mixed body, “in virtue” only.
• hoc aliquid / tode ti — “hoc est aliquid demonstratum”; something one that can be pointed out.
• “multitude” and “magnitude” — the two first species of quantity.
• number — “a multitude measured by one” or “a multitude composed of ones”.
• posos, combien, “size” — one word for how much and how many.
• continuous (Categories) — parts meet at a common boundary; (natural philosophy) — divisible forever.
• the one vs. the point — both indivisible; the point has position, the one has none.
• matter as cause — that from which something comes to be, existing in it.
• “part” (Euclid) — in the sense of measure.
• consideratio naturalis versatur circa materiam — Thomas on why the natural definition differs.
• “the wise man” — the metaphysician, who considers things as being; vs. the logician, things as in reason.
• “ratio of measure” — what distinguishes species of quantity in logic; extrinsic (time, place) vs. intrinsic (magnitude).
• place — the inner surface of the containing body.
• measure — that by which the quantity or size of something is made known to reason (first sense).
• “second substance” — the species or genus of a first substance, as logic calls it.
• “wisdom” (eighth book of wisdom) — [ed.: Metaphysics VIII].

Texts #

Read in class

• Aristotle, Categories, ch. 6, via handout “Quantity – Chapter Six”, pp. 1–5 (English and Latin).
• Aristotle, Metaphysics V, on quantity, with Thomas’s commentary (handout p. 4, Latin; p. 5, English) — “for the bonitatem doctrinae”.
• Thomas, Latin: “satis convenienter hic definitio ponitur… quia consideratio naturalis versatur circa materiam”. Mentioned
• Aristotle, Physics (inferred): continuous as divisible forever; line not composed of points; faster/slower body; definition of place.
• Aristotle, Metaphysics VIII: forms compared to numbers.
• Aristotle, Politics, book VIII: music in education.
• Porphyry, Isagoge (species) (inferred).
• Euclid, Elements: two a part of six, four not.
• Heisenberg, Physics and Philosophy (Gifford lectures).
• Thomas’s communion prayer/hymn (“Visus, tactus, gustus in te fallitur”).
• Albert the Great on the universal in logic; Hegel on the fine arts; MacIntyre on “philosophy”; Washington Irving on Westminster Abbey.

His questions #

• Were the particles actually in there composing that thing before you broke it up? No — with more mass than the original, they must have been there only in ability.
• Can seven be divided forever? No; you reach one, which cannot be divided.
• Why is the continuous defined by “divisible forever” in natural philosophy but by parts at a common boundary in logic? Parts are like matter, the whole like form; natural philosophy brings matter into its definitions, the logician defines by form/wholeness.
• How do you know a straight line can be divided forever? Division could only stop at two points, and points cannot compose a line, so you always get two lines.
• What is place, and why is it a species of quantity for the logician but not for the wise man? Inner surface of the container; a different ratio of measure but the same quantity, surface.
• Which is more beautiful, music or painting? Music, through its connection with the emotions and virtues (his view; a student’s objection from the primacy of sight answered thus).
• Why does Aristotle put discrete quantity before continuous, though the continuous is defined affirmatively? Because he distinguishes by ratio of measure, and the one measures more exactly than any unit of length.
• Is every part of every definition in need of being defined, every statement of proof? No, else nothing could be known; so thoughts are like numbers, not continua.
• Are the natures of things more like numbers than like lines? Numbers.
• Can a line and a surface, or time and length, be equal or unequal? No — the property supposes a common measure (treatment cut off).