Motion and Time Divisible Forever: Aristotle's Argument from Faster and Slower

Orientation #

He begins by placing the “second reading” against the first: in the first reading Aristotle showed for everything continuous that it is not composed of indivisibles, but it was clearer for magnitude; the second reading returns to the motion over the magnitude, and the third and fourth readings do the same for time. He recalls “what we saw earlier in the course” in the second book of Natural Hearing (the distinction of natural from mathematical philosophy) and an earlier discussion of the argument against Anaxagoras in the ninth reading. The previous class treated the senses of “end” in Metaphysics V; the next treats false imagination as a source of deception.

The class, in order #

1. Second reading: what is presupposed (“there is the same reason for composing magnitude, time, and motion”) #

Point: magnitude, time and motion stand or fall together; either all are composed of indivisibles or none.

• Presupposition 1: if magnitude ABC is from indivisibles A, B, C, the motion DEF over it, by which the mobile is moved, has each part indivisible (A moved by D, B by E, C by F).
• Presupposition 2 (“kind of obvious”): if motion is present, something moves; and if something is moved, motion is present.
• Presupposition 3 (the fourth paragraph, “which is going to get us into some troubles”): a thing moved from one place to another cannot at once move and have moved. Aristotle’s example: walking to Thebes; his own: walking home, building a house. When I am walking home I have not walked home.
• A student asks whether these are statements naturally understood; he: close to it, “a little more particular”.
• Aside: Aristotle returns to this in Metaphysics IX (“Wisdom”) to distinguish walking home from understanding, loving, seeing: when I am understanding a triangle I have understood it; when I am loving you I have loved you (used to amuse students: “what do you say to your wife… have I loved you yet?”); when I am seeing the painting I have seen it; but building the house, I have not built it.

2. The absurdities of a motion from indivisibles (“something will have walked without ever walking”) #

Point: with A partless, the mobile must have gone to A without going to A.

1. If it comes to A later than it is coming, A is divisible (for while coming it neither rested nor had come, but was between). So if A is indivisible, it will have gone to A while going, or without going.
2. Then motion is not from motions but from “having been moved” [ed.: Greek and Latin have one word here, the old momentum, “much different sense in modern science”].
3. Something will have moved without moving: gone through A without going through it, “walked without ever walking”.
4. If everything must move or rest, and it rests in each of A, B, C, it rests in the whole ABC while being moved: “something continuously resting will at the same time be moved”.
5. Dilemma: if the indivisibles of DEF are motions, motion is present but nothing is moved (contradicting presupposition 2); if they are not motions, motion is not from motions — “like saying the line is not from lines”.

3. Third reading: time, and the three ways of arguing (“Time, he says, must be indivisible and composed”) #

Point: time must be shown divisible in the same way as magnitude and motion, and Aristotle argues three ways: from two equally fast bodies, from a faster and a slower, from one body. His favourite is the middle one, “more illuminating”.

• Analogy: Euclid in Book II takes two lines to show something and then one line cut; Heath says it is easier to see with two first.
• Equally fast: if body A goes this distance in this time, the equally fast body in less time goes less distance; so time and distance are divided alike. But this takes the distance’s divisibility as known; if from the first reading the argument applies to time too, then time divisible forces distance divisible. “A little bit of overkill.”

4. Preliminaries for the faster/slower argument, with Euclid II.5 (“as some define the faster”) #

Point: Aristotle sets out what he will reason from.

• Every magnitude is divisible into magnitudes: he seems to proceed from its being impossible for the continuous to be from indivisibles to its being “always divisible into something divisible”, i.e. always divisible. Every magnitude is continuous.
• The faster goes an equal distance in less time (how we think at first) — and even a greater distance in less time, “as some define the faster”.
• Digression, Euclid II.5 (his favourite theorem there): a line cut into equal and unequal segments — the square on the equal segments exceeds the rectangle on the unequal ones by the square on the interval between the points of section. He turns it for students: can a rectangle (broad sense, square or oblong) have the same perimeter but more area? 5×5 = 25 against 4×6 = 24 (difference 1²), 7×3 = 21 (2² = 4), 2×8 = 16 (3² = 9), all perimeter 20. Then, “because of the infinite divisibility”, the square can have less perimeter and still more area: 2×10, perimeter 24, area 20. What this shows: as with the faster body, once it covers a greater distance in the same time it can, by infinite divisibility, also cover a greater distance in less time. A student objects it would “always” do so; he: “only goes so far”, at a certain point.

5. The argument: alternating two truths (“the faster has gone what further”) #

Point: the faster and slower body together show distance and time divisible forever.

• Text: A faster than B; in time FG, A goes from C to D; B, slower, is only at E. A reaches H (between) in less time FI; CH > CE in FI < FG: further in less time. He draws it on the board (“I don’t know if it’s on the tape”).
• Also the faster goes an equal distance in less time: if faster and slower went the same distance in the same time, neither would be faster. Student: is this the same as “the faster changes before”? He: a little different way of stating it. Consequences of my being faster than you: in the same time I go further; for the same distance I take less time.
• The alternation: faster goes this distance in this time → slower in that time goes less (distance divided) → faster goes that lesser distance in less time (time divided) → slower in that time goes less again → and so “divisible forever”. This is “the second definition of the continuous”.
• “Kind of marvelous he does that”: one knowledge of all continuous things as divisible forever; on these simple truths distance and time are shown divisible together, so they belong to the same science.

6. Parallels: Anaxagoras, Euclid IV, and why the natural philosopher is wiser (“you never come to a smallest”) #

Point: the same alternating structure appears elsewhere, and geometry only assumes what natural philosophy proves.

• Against Anaxagoras (ninth reading, recalled): if the parts (flesh, blood, bone) can fall below any magnitude, the whole composed of them could; but kinds of animals and plants do not have just any size (trees here vs. redwoods; ant vs. elephant); so the parts have limits. Contrast: natural things have limits of size due to their kind; in geometry there is no smallest or largest.
• Euclid IV: in any square a circle can be inscribed, in any circle a square; alternating, squares and circles get smaller forever, never a smallest (else the theorem would cease to be true). Circumscribing likewise gives bigger forever, no largest. Here the direction is toward the small.
• Geometry assumes divisibility forever (bisection theorem; “between any two points you can draw a straight line” implies points cannot touch — if my house wall touched the neighbour’s, as in old Quebec, no house could go between). Hence the natural philosopher is wiser than the geometer: he proves what the geometer assumes.

7. The higher knowledge distinguishes itself from the lower (“it always belongs to the higher knowledge”) #

Point: a principle “of universal importance”: the knowledge with more the character of wisdom distinguishes itself from the lower and considers the order of the two.

• Physics II distinguishes natural from mathematical philosophy. A student: doesn’t that belong to metaphysics? He: Metaphysics VI distinguishes mathematical, natural philosophy and wisdom; but natural philosophy, having something of wisdom relative to mathematics, can distinguish itself from it and determine how far mathematics is useful in it.
• Starting point: reason, not the senses or imagination, distinguishes reason from sense and imagination; wisdom is a knowledge of reason, hence homo sapiens. Political philosophy distinguishes itself from rhetoric and from domestic philosophy (Politics I).
• Theology: as Thomas shows in Summa I q.1, revealed theology has more the character of wisdom than metaphysics, reasoning from what God knows; so the distinction and order of philosophy and theology belong to theology.
• Why modern philosophy is in “such bad shape”: the famous thinkers gave up revealed theology yet lived among believers; having discarded the knowledge that sorts these things, they try to do in philosophy what should be done in theology, and cannot get out. Greeks vs. moderns: the Greeks philosophized before Christ’s revelation; their only religion was the imaginative one of Homer and the poets, which reason can judge (Xenophon [ed.: likely Xenophanes]: horses and cattle would make gods like horses and cattle; Ethiopians’ gods black, Thracians’ blonde). Modern skepticism, aimed at a theology based on the word of God, “is not of the same sort at all”; the moderns seek a substitute for theology, a certitude more than human (Thomas: theology’s certitude exceeds philosophy’s); the defect of the senses, known to the Greeks, becomes an obsession.
• Feuerbach, The Essence of Christianity, a turning point between Hegel and Marx: “God became man” read as a poetic way of saying man is God — a perversion never faced by the Greeks. His argument is a fallacy of equivocation: theologians say the infinite is God; man’s mind is infinite; so man’s mind is God. But God’s infinity is no limit to His perfection; the mind’s is being always able to learn more. Likewise Juliet: “my love is infinite… the more I give the more I can give”. Marx and Engels became enthusiastic Feuerbachians on this. Hegel’s philosophy of history is like the City of God attempted in philosophy.
• Asides: at Saint Mary’s College he got permission to use Feuerbach, “but it says here you’re responsible if anybody loses their faith”. Knowledge begins with the senses — you cannot understand “teleological” without telos; he objects to naming a thing by the science that studies it (“psychological problem”: a problem in your soul; “medical problem”; Buckley’s “life biologically defined” — rather biology defined by life, logos of bios).

8. Back to the two truths; the line of points and the difficulty of ability (“do you trust your senses enough”) #

Point: the argument rests on admitting faster and slower; the temptation to compose the continuous from indivisibles comes from the difficulty of understanding ability.

• Once you admit faster and slower (a student concedes “for the sake of argument” — “typical modern”), spelling out what they mean forces distance and time divisible forever; since motion takes time, motion too. Same knowledge for magnitude, motion, time: none put together from its own indivisibles.
• High-school mathematician: a straight line is composed of an infinity of points, because every cut yields a point. Nixon on the Democratic program: any way you slice it, the same old baloney — so what is it made of? Question: was the point there before the cut, actually or in ability? Infinite points in ability does not contradict the argument; cutting makes one actual. But the mind wants to imagine what is in something only in ability as actually there.
• Weizsäcker [ASR “Fight Socrates”]: perfected Kant’s theory of the solar system’s origin and showed how the sun puts out energy without exhaustion. Anaxagoras thought the sun a stone on fire (charged with impiety); Aristotle knew fire as we know it could not burn so long, so the sun is something else. Thomas: Aristotle’s reason for the sun’s eternity is not necessary — change may take longer than many generations. Sphere of radius 93 million miles: the earth is almost nothing of it, yet the energy fills it.
• Weizsäcker’s point: when we imagine something we make it actual. You cannot sense my ability to walk or talk, only my walking; ability is known through act, by reason, which knows one thing through another (“like Shakespeare taught us… discourse”).
• Anaxagoras: nothing from nothing, so everything gotten out of matter is in matter — imagined actually rather than in ability, so parts infinitely small. Heisenberg’s time: every elementary particle composed of all the rest (since each can be gotten out of each), so particles smaller and smaller forever — contradicting definite mass and size. Compare hydrogen and oxygen from water. Chairs in the next room vs. chairs in the trees: the second is “chair in ability”, hard to understand.
• Metaphysics II: the difficulty is in the thing itself — first matter; someone (left unnamed; “doesn’t have the advantage of Aristotle”) puzzles whether it is “something that’s nothing, or nothing that is something”.

His words #

• “reading”: the lectio-division of Aristotle’s text with Thomas’s commentary.
• “Natural Hearing” [ed.: Physics]; “Wisdom” [ed.: Metaphysics].
• “having been moved” / momentum: the one Greek/Latin word, different sense in modern science.
• “the faster changes before”: Aristotle’s first statement of what faster means.
• “divisible forever”: his rendering of the second definition of the continuous [ed.: infinitely divisible].
• “ability” [ed.: potency]; “act”; “in ability” vs. “actually”.
• “the higher knowledge distinguishes itself and the lower”: his principle for the order of sciences.
• “rectangle in the broad sense”: including square and oblong.
• “mathematical philosophy”, “natural philosophy”, “domestic philosophy”, “revealed theology”.

Texts #

Read in class

• Aristotle, Natural Hearing (Physics) Book VI (inferred), readings 2 and 3, with Thomas’s commentary (handout “The Matter and Order of Wisdom” p. 30; DHB Vol. III p. 479 — verified sources). Mentioned
• Physics VI readings 1, 4, 9 (Anaxagoras); Physics II (natural vs. mathematical philosophy).
• Metaphysics IX (walking vs. understanding, loving, seeing); Metaphysics VI (three theoretical sciences); Metaphysics II (difficulty in the thing itself).
• Euclid, Elements II (two lines then one; II.5), IV (inscribing/circumscribing), bisection theorem, “between any two points a straight line”.
• Aristotle, Politics I; Thomas, Summa I q.1; Feuerbach, The Essence of Christianity; Augustine, City of God; Shakespeare (Juliet on infinite love); Heisenberg on elementary particles; Weizsäcker; Xenophon [ed.: likely Xenophanes]; Heath on Euclid.

His questions #

• When I am walking home, have I walked home? When I am understanding the triangle, have I understood it? No to the first; yes to the second (and to loving, seeing).
• If A is indivisible, what follows when I go to A? I will have gone to A without going to A; motion would be from “having been moved”, not from motions.
• Can you have a rectangle with the same perimeter but more area? Less perimeter and more area? Yes to both, by Euclid II.5 and infinite divisibility (square 5×5 vs. 2×10).
• Can the slower body go the same distance in the same time as the faster? No; less distance in the same time, and the faster the same distance in less time.
• Is “equal distance in less time” the same as “the faster changes before”? A little different way of stating it.
• In geometry is there a smallest or largest? No — alternating inscribed squares and circles never reaches one.
• Does it belong to the senses or to reason to distinguish sense and reason? To natural or mathematical philosophy to distinguish them? To reason; to natural philosophy (student: metaphysics — he: Metaphysics VI distinguishes all three).
• Do you trust your senses enough to admit some things are faster and some slower? Student: for the sake of argument (“typical modern”).
• Was the point there before you cut the line, actually or in ability? In ability; cutting makes it actual.
• Can you sense or imagine any ability? No; only the act, and reason knows the ability through the act.