Why the Philosophy of the Continuous Matters: Motion, Time, Zeno and the Limits of Imagination

Orientation #

He announces the new topic: “we’re going to go in now to the philosophy of the continuous,” a large part of “the sixth book of Natural Hearing,” and has fifteen pages of it copied for the class. He refers back to “the definition of reason before” (ability for discourse, looking before and after) and to a text of Aristotle on before and after the class already had; a text-matching system finds the class reading from the handout “The First Definition of Reason,” p. 104, and from DHB Vol. III, The School of Aristotle, p. 869. He also recalls “we just studied” that two points that touch coincide, and the definition of eternity (prev. lecture). The previous class treated Boethius’s definition of eternity; the next develops continuous, touching and next into the immateriality of reason.

The class, in order #

1. Learning from enemies (“how to benefit from your enemies”) #

Pre-class conversation: enemies point out your defects; friends sometimes will not, out of friendship.

• Anecdote: a man says things go “the worse for my friends and the better for my enemies”: friends tell him he is wonderful, enemies that he is an ass.
• Plutarch’s essay on benefiting from enemies: enemies do not love you, so they look for what is wrong with you, and you learn the truth about yourself.
• Extreme case: a man in power flattered by everyone, unable to get an honest statement of the situation.
• Brutus and Cassius, with things going against them: Brutus points out Cassius’ faults; Cassius says a friend would not; Brutus, that a flatterer would not. Shakespeare’s “sweet breath of flattery”; the dinner-guest’s “this is delicious” as a small lie; college teachers must encourage students but also tell them what their thing is.
• Rossellini presenting a mixed-up draft and being told “let’s start over again”: brutal, but sometimes the better foot; the saints at the end say they are always beginning; his own experience that rethinking from scratch goes better than completing an old attempt.
• Applied to the church scandals: people who keep hitting them in the papers are enemies of the church, yet something is to be learned from them (Plutarch again); Bennett and Buckley, solid Catholic laymen, calling for resignations; he declines to judge whether resignation is right, but says imprudent things were done.
• Student: Paul telling Peter he was wrong about the Mosaic law, as an example of a friend who tells you. He: Aristotle teaches this as part of friendship; you need not be carping all the time, and some faults friends will never change; the opposite extreme is encouraging a friend in his vices.

2. A philosophy of the continuous, distinct from geometry (“the philosophy of the continuous”) #

Point: Book VI deserves the name “philosophy of the continuous” even though geometry too is about the continuous, because Book VI is about all the continuous and is more basic.

• Recall from logic: continuous quantity (length, width, depth) vs. discrete quantity (number). Two definitions of the continuous: that whose parts meet at a common boundary (parts of a line at a point, of a surface at a line, of a body at a surface); and that which is divisible forever. Number fits neither. Euclid I–VI and the solid books are about the continuous, VII–IX about number; Euclid never defines the continuous.
• Geometry considers line, surface, body, but not motion. Motion over a line is continuous and divisible forever because the line is; the time it takes is continuous because the motion is. So Book VI is about everything continuous, geometry is not.
• The reason each is divisible forever is somewhat the same, and nothing continuous is composed of indivisibles: the line not of points, time not of nows, motion not of “moments” (Latin momentum, a word that has taken another meaning in modern physics). Seeing it in one, you see it in the others; Aristotle reasons from one to the other or about both together, e.g. time and distance divisible forever because there is a faster and a slower (the faster covers the same distance in less time; in that time the slower covers less distance; and so on).
• By the Posterior Analytics, if there is a common reason for many things, you should bring them together under that one thing.
• Geometry presupposes this: the postulate “between any two points a straight line can be drawn” assumes two points cannot touch without coinciding (if they touched, no line could be drawn between them), which is tied up with a line not being composed of points. So geometry depends on this and not the reverse: a sign that natural philosophy has more the character of wisdom than geometry.
• Why these topics here: back in Book III Aristotle gave two reasons for treating motion, place and time: (a) nature is defined as a beginning cause of motion; motion takes time; what is in motion is somewhere; (b) these seem common to all natural things, indeed to the first philosophers common to all things (“whatever is is somewhere at some time”). The continuous is common to all three.
• Aside: the grade-school question “where is heaven?” answered “where God is,” though God and angels are not in place; Sister Carolyn (“look at the person to whom you are speaking”), “leave a little room for your angel,” “Cannonball Corona.”

3. The continuous as the most basic thing in our knowledge (“we don’t think without an image”) #

Point: since we never think without an image, we never think without the continuous and time, and our most basic words are borrowed from the continuous.

• Text cited: the book on memory and reminiscence (he first says sense and sensible, then corrects himself).
• Sensibles (looking ahead to the books on the soul): accidental sensible vs. sensible as such (“I can see you’re angry”: no one has ever seen anger). Sensible as such divides into the proper or private sensible, private to one sense (color to eye, sound to ear, smell, flavor), vs. the common sensibles, sensed by more than one sense (surface, shape: the roundness of the glass known by touch and by sight). The common sensibles are tied up with the continuous; shape is the limit of the continuous.
• Private-sense words hardly extend: “red” goes to “communist” and little else. Words from the continuous extend everywhere:
  • beginning: first the beginning of the desk; then the foundation of the house, the prince of the city (principle from the Latin for beginning), the axioms and definitions as beginning of a science;
  • end: first the end of something continuous; then the end of a motion; then end as purpose; even the Greek and Latin words for definition, horos, terminus, mean limit or end;
  • road: first the road one walks; then his three “roads” (from the senses into reason; from reasonable guesses to reasoned-out knowledge; the private road of each reasoned-out knowledge);
  • in and out: first place, which is continuous.
• Hence Boethius in the De Trinitate: to think what is not continuous we must negate the continuous: God is incorporeal, without length, width or depth.

4. Why it matters for the soul: knowing the continuous non-continuously (“reason is not continuous”) #

Point: the road to the soul’s immateriality runs through the continuous.

1. If reason can be shown not continuous, we can syllogize that it is not a body (Book I on the soul already argues dialectically against thinking being continuous; we do say “a line of thinking,” but is it continuous like a line?).
2. If understanding is not in the body, the soul has a power not in the body.
3. Then its existence is not entirely immersed in the body; for if it existed only in the body it would operate only in the body.

• Aristotle’s discovery, “the way we know doesn’t have to be the way things are”: we understand a continuous thing in a non-continuous way without falsity, since we do not say the continuous is not continuous. Parallel: I know the past now, but do not say the past is now. Kindergarten memories (toy soldier tied to his drawer that had to stay at school; Sister Aquinas, his first teacher, struggling to open a coconut): the “now” is in my knowing, not in what is known.
• Two ways of knowing one thing: the shape of the glass known by the eye through color and by touch through hardness. If knowing had to match the thing exactly, two ways would be impossible. So it is no objection that the theologian and the philosopher both treat the immortality of the soul.
• The better you know the continuous, the better you know what is not: as knowing time and its now gave the definition of eternity. So too the definition of reason, “looking before and after”: the first meaning of before and after is in the continuous, the second (one before two) in the discrete; you must know the first before seeing how the word moves forward.
• Most people cannot transcend the continuous: asked what eternity is they say endless time, which, like an infinite line, is still continuous; sense and imagination are both tied to the continuous.

5. Motion, contradiction and the sixth book: Hegel and Zeno (“becoming is a contradiction”) #

Point: the continuous is needed to show change is possible without contradiction, and for the argument from motion to God.

• The Summa Theologiae gives one syllogism for “whatever is in motion is moved by another”; the Summa Contra Gentiles gives three, from Aristotle, some drawn from Book VI and the continuous character of motion.
• The paradox (he attributes it to Hegel): what is not a circle becomes a circle; there is a time it is not, a time it is. Is the last instant of not-being-a-circle the first instant of being one? If yes, it both is and is not a circle. If no, they are two nows, and as with two points there must be a time between them (nows cannot be adjacent: if two points touch they coincide); but in that time it is either a circle or not, and either way that time belongs to one of the two periods, so no time between is possible. So one seems forced to say they are the same instant: becoming is a contradiction.
• Aristotle’s solution in Book VI: there is a first instant in which it is a circle but no last instant in which it is not. This seems arbitrary but has a reason, to be seen in the book. Dying takes time; at the end of that time you are dead, so there is a first instant of being dead but no last instant of being alive; at the other end of life there was a first instant of being but no last instant of not being. “Very important for understanding becoming” without contradiction.
• Zeno, pupil of Parmenides (who denied change because it seemed contradictory: day becomes night, sick becomes healthy, so one opposite would be the other, a problem already solved earlier), defended his master by showing the difficulties in motion: you cannot reach the door, since you must first go half, then half of the half, and so on; Achilles cannot catch the turtle with a head start, since each time he reaches where it was it has gone further. Treated in Book VI and again in Book VIII.
• Student: this is why a deathbed conversion is a miracle, since there is no last moment of being alive; he agrees (“the last instant comes… the tree falls”).

6. Why words from the continuous extend and “red” does not (“extend the word in”) #

Student question: why are words from the common sensibles more extendable? His answer, tentative:

• The continuous is more understandable: he can define it several ways, but red can only be pointed at (the moderns say wavelength, but that is again something continuous, treated mathematically).
• Sensible qualities are more tied to matter: mathematics abstracts from matter, and Aristotle’s “understandable matter,” opposed to sensible matter, is not matter in the fundamental sense. So the private sensible cannot be moved as “in” can: the eight meanings of “in” (we in this room, teeth in the mouth, species in the genus, genus in the species, “I left my heart in San Francisco”) are all ordered from the first by likeness and proportion; nothing like that with “red.” Loci for the point: Metaphysics V ch. 1 on “beginning,” the Categories on before and after, the treatment of “in.”
• Corroboration: alteration, one species of motion, is always in the third species of quality (sensible qualities), never in the fourth (figure, shape). “Don’t worry too much about exactly why.”
• Student: touch words (hard, soft) extend more than red or green: a hard proposition, soft-headed. He: touch is the most basic sense; substance, nature and good are known more through touch than sight, touch being the sense of the interior, the eye superficial. Modern thought is tied to sight [the transcript’s “I sit therefore I am” is likely “I see therefore I am”].

7. Aside: “Thy will be done on earth as it is in heaven” (“His will is done in hell”) #

A digression offered as “extra today”: in heaven all wills are conformed to God’s, so the petition asks that ours be. But, as Augustine and Thomas say, even the literal sense can have more than one meaning. His old teacher Kasurik: you glorify God willy-nilly, His mercy if good, His justice if bad. So God’s will is done in heaven (merciful will) and also in hell (just will); asking “as in heaven” is asking mercy rather than what we deserve, i.e. rather than “as it is in hell.”

8. The text: continuous, touching, next, and the extended senses (“the continuous and the touching and the next”) #

Point: Aristotle begins “if the continuous and the touching and the next are as has been determined before” (Book V); the three are related by addition, each adding something to the previous, and “continuous” is then carried over by likeness to many other things.

• Strict sense: parts have a common boundary, one and the same limit being the end of one part and beginning of the other. Two semicircles and the diameter; the Canada–US border, with no no-man’s-land, is the upper limit of the US and the lower limit of Canada; the horizon is the boundary of earth and sky.
• Likeness uses:
  • the Arab philosophers: man is “on the horizon” between the material and immaterial worlds, end of one and beginning of the other; Democritus’ “man is a microcosm”;
  • natural philosophy and first philosophy: “metaphysics” is three Greek words, meta ta physika, “after the natural books”; the proof in Books VII–VIII that moved movers depend on an unmoved mover that cannot be a body is the end of natural science and the beginning of metaphysics, one border; the proof in On the Soul III that the human soul exists after death is another border between the same two;
  • continuous proportion in numbers, 4:6::6:9, where the end of one ratio (6) begins the next, as against 2:3::4:6; “doubling the dyad,” 2, 4, 8… The number seven’s parts (3 and 4, 2 and 5) meet nowhere, so number is not continuous strictly;
  • continuous syllogisms (the conclusion of one a premise of the next; a chain no stronger than its weakest link), continuous definitions (plane figure, rectilineal plane figure, quadrilateral, square, each defined by the previous), continuous divisions (rectilineal figure into triangle, quadrilateral…; quadrilateral into square, oblong, rhombus, rhomboid, trapezium), and induction continuous with the syllogism that uses its general statement. Any reasoned-out knowledge has these.
• We transfer words from geometry to arithmetic (square number, cube number) more than the reverse, because geometry is more sensible and continuous, and perhaps because number arises from dividing the continuous.
• The class ends as he turns to the strict sense in the text: “what does he mean by continuous in this strict sense?”

His words #

• Natural Hearing: his name for the Physics [ed.: Physics]; the “sixth book” = Physics VI.
• philosophy of the continuous: his title for Physics VI.
• continuous: (i) that whose parts meet at a common boundary; (ii) that which is divisible forever.
• discrete quantity: number; its parts do not meet at a common boundary.
• indivisibles: point in the line, now in time, “moment” (Latin momentum) in motion.
• private sensible: sensible proper to one sense [ed.: proper sensible]; common sensible: sensed by more than one sense; accidental sensible.
• beginning / end / road / in: his examples of words whose first meaning is in the continuous.
• horos, terminus: Greek and Latin for definition, meaning limit or end.
• understandable matter vs. sensible matter [ed.: intelligible matter].
• reasoned-out knowledge [ed.: science]; reasonable guesses [ed.: dialectic/opinion].
• the definition of reason: ability for large discourse, looking before and after.
• meta ta physika: “after the natural books.”
• moved mover / unmoved mover.
• continuous proportion; continuous syllogism, definition, division, induction: likeness uses of “continuous.”
• horizon: boundary of earth and sky; the Arabs’ “man on the horizon.”

Texts #

Read in class

• Aristotle, Physics VI, opening sentence (“if the continuous and the touching and the next are as has been determined before”).
• Handout, “The First Definition of Reason,” p. 104 (verified by text-matching).
• DHB Vol. III, The School of Aristotle, p. 869 (verified by text-matching).

Mentioned

• Aristotle, Physics V (definitions of continuous, touching, next); III (motion, place, time); VII–VIII (movers, Zeno).
• Aristotle, Posterior Analytics (common reason, bring things together).
• Aristotle, On Memory and Reminiscence (no thinking without an image).
• Aristotle, On the Soul I (dialectic against thinking as continuous); II (sensibles); III (immortality).
• Aristotle, Categories (before and after); Metaphysics V ch. 1 (beginning).
• Boethius, De Trinitate (negating the continuous).
• Euclid, Elements I–VI, VII–IX, solid books; postulate on the straight line.
• Aquinas, Summa Theologiae and Summa Contra Gentiles (motion argument).
• Plutarch, essay on benefiting from enemies; Shakespeare, Brutus and Cassius, “sweet breath of flattery” (inferred: Julius Caesar); Bennett, Book of Virtues; St. Paul and Peter on the Mosaic law; Hegel; Parmenides; Zeno; Democritus; Augustine and Thomas on the senses of scripture.

His questions #

• What other part of philosophy seems even more a philosophy of the continuous? Geometry.
• Does geometry consider everything that is continuous? No: not motion, nor time.
• If two points could touch without coinciding, could you draw a straight line between them? No, there are no points between them; so the postulate assumes points cannot touch.
• Have you ever seen anybody’s anger? No; it is an accidental sensible.
• Is the last instant in which it is not a circle the first instant in which it is? If so, contradiction; Aristotle answers there is a first instant of being a circle but no last instant of not being one.
• Is there a last instant in which you are still alive? No; only a first instant in which you are dead.
• Why do words from the continuous extend more than red or green? Tentatively: the continuous is more understandable and less tied to matter; “don’t worry too much about exactly why.”
• How are the continuous, the touching and the next related? Each adds something to the other.
• What makes 4:6::6:9 a continuous proportion? The end of one ratio (six) is the beginning of the next.
• What does he mean by continuous in the strict sense? Left open; the class ends there.