Motion, the Continuous, and the Discrete: Why the Continuous Dominates Our Naming

Orientation #

He begins by asking whether he mentioned “last time” the Romeo passage, and says the class has been “knocked for a loop by the flu,” so he is recalling the first reading again. He twice defers matter to “later on” (the eight meanings of “in”, and Book Six on the continuous). The previous class worked toward Aristotle’s definition of motion in Book III; the next takes up Book IV on place, void and time.

The class, in order #

1. Romeo’s “bitter conduct” and wisdom as savory knowledge (“Come, bitter conduct, come, unsavory guide”) #

Point: a meditation on Shakespeare’s precision, used to say what wisdom and folly are.

• Life as a voyage: Shakespeare’s “life’s uncertain voyage”; Elizabethan voyage-insurance (“fifty-fifty”); reason as a pilot who should steer into the harbor of happiness but suffers shipwreck. Misery = wretchedness, which he takes to be etymologically related to “wreck.” Julius Caesar’s “tide in the affairs of men” as the same image.
• “Unsavory guide” = foolish guide, because Thomas explains sapientia as sapida scientia, savory knowledge: wisdom is something to savor, like wine (not water or soda pop), or like a kid with an all-day sucker. Boethius: the philosopher is a ruminating animal (Old Testament: only ruminants may be eaten); students must “chew these things over again.” Folly is unsavory, insipid, like reading the modern philosophers.
• “Bitter conduct”: manuductio by opposites, the opposite of bitter is sweet; Shakespeare speaks of “sweet philosophy.” Wisdom 8 (quoted by Thomas in the Summa Contra Gentiles): wisdom reaches from end to end, ordering all things “sweetly” (Latin for the Greek; could be “gracefully”); and, later in the same chapter, “there is no bitterness in wisdom” while every other knowledge may have some, since one tastes something besides the Lord (“taste and see how sweet is the Lord”).
• Thomas’s Adoro te devote (singled out by the Church after Vatican II): O memoriale mortis Domini, panis vivus, vitam praestat homini, praesta meae menti de te vivere, et te illi semper dulce sapere: “sweetly savor,” the two together, the exact opposite of Romeo’s pilot.
• Aside: Nicomachean Ethics on the sick thinking health is the best thing and the poor wealth, and the same man changing his opinion.

2. Why Book Three begins with motion (“since nature is the beginning of motion and change”) #

Point: motion is in the definition of nature, so if motion is unknown nature is unknown.

1. The full definition was: nature is a beginning and cause of motion, motion taken broadly to include change and rest.
2. “The road we are following is about nature”: philosophy is a methodos, following a road.
3. Therefore what motion is must not be hidden from the student of nature.

• The remaining six books of “Natural Hearing” [ed.: Physics] all turn on motion: III–IV things closely related to it (time as measure of motion); V the division of change (place, quality, growth); VI the quantitative division of motion; VII–VIII motion compared to movers, where the argument for the unmoved mover is worked out.
• Why motion is “up front”: in the proemium to the Ethics the subject of natural philosophy is “natural things or motion”; “nature loves to hide” (Heraclitus) and natures are revealed through what things do or undergo.

3. What follows upon motion: the continuous (“the things which follow upon motion”) #

Point: the first thing connected with motion intrinsically is that motion is something continuous; hence the discrete/continuous distinction must be recalled.

• In logic quantity is divided into discrete (main kind: number) and continuous (line in one dimension, surface in two, body in three; time and place also continuous in some way).
• Why the distinction matters: (a) geometry vs. arithmetica (the science of numbers, not the art of calculating); Euclid’s first six books on continuous quantity, the last books on solids, VII–IX on number; (b) in the books on the soul, from Book I’s critique of Plato: thoughts are like numbers, not continuous; reason understands continuous things in a non-continuous way, which cannot be due to the thing understood, so reason is not continuous, and every body is continuous, “therefore reason is not a body.” Thomas: I study the body to study the soul, the soul to study the angels, the angels to study God, “and that’s the end of my thinking.” (c) Motion, time, and the magnitude over which one moves are all continuous (Book Six).

4. The logician’s definition: parts meeting at a common boundary (“the parts meet at a common boundary”) #

Point: what is common to all quantity is a multiplication of parts; the logic definition distinguishes the continuous by how the parts meet.

• Line: two parts meet at a point; surface: at a line (circle and diameter: the diameter is the end of one half and the beginning of the other); the chalk: two parts meet at a surface.
• Continuous quantity: quantity whose parts meet at a common boundary. Discrete: by negation, whose parts meet at no boundary: seven divided into three and four, or two and five, meet in nothing. Thoughts likewise have no common boundary (not to be manifested fully now).

5. The natural philosopher’s definition: divisible forever (“that which is divisible forever”) #

Point: in the Physics (recalled here, unfolded in Book Six) the continuous is defined as what is divisible forever; the discrete is not.

• Seven to three and four, four to two and two, two to one and one; no further. The modern mathematician’s fractions confuse this because he is thinking of numbers on a continuum.
• Student question whether the one of pure number is divisible: no; the arithmetical one is simpler than the point: both indivisible, but the point has position and the one is “neither here nor there.”
• Hence (Posterior Analytics, end of Book I) arithmetic is more certain than geometry: more must be considered in geometry. Euclid’s theorems often have several cases; Euclid gives the hardest and leaves the rest, commentators like Proclus distinguish them (sometimes too many), and one is never sure all cases are covered. Three points may or may not lie on a straight line; three ones have no such variety.
• Five into two and a half: then you could divide the point, since you have five points; four legs of a chair or of a dog, take two away, two are left.

6. Thoughts are like numbers also in not being divisible forever (“there must eventually be something which is known, not by definition”) #

Point: definitions and proofs do not divide forever, so something is known first without definition or proof.

1. Euclid’s square = equilateral right-angled quadrilateral; quadrilateral divides into “rectilinear plane figure contained by four straight lines,” that into rectilinear plane figure, plane figure, figure.
2. If this went on forever there would be an infinity of definitions before any definition and an infinity of things to understand before understanding anything; nothing would be understood by definition. So something is known without definition at the beginning of all definitions.
3. Likewise a statement divides back into the statements it was reasoned from; if forever, an infinity of proofs before any proof; so some statements are known without proof. “Statements exist” cannot be denied without making a statement: “the ball game’s over.”

• What is known without definition: Descartes thought it was motion; Aristotle in Book Nine of the Wisdom [ed.: Metaphysics IX] says it is act.
• Dictionary comparison: words explained by other words cannot go on forever; the first words are known by association with something sensed.

7. Why one definition fits logic and the other natural philosophy (“to define by form and define by matter”) #

Point: both definitions are in terms of parts, but from opposite directions.

• Logic’s definition looks at parts coming together to form the whole (like the three lines of a triangle meeting at their endpoints): the whole is like form. Natural philosophy’s definition goes from whole down to parts by division: parts are like matter, “that of which something is made” (recalling the four causes).
• Logic is an immaterial science, more formal even than mathematics, lacking even the imaginative extension of math; natural philosophy defines with matter or motion. “That shows how artful Aristotle is”: he adapts the definition to the instrument.
• Student observation: “divisible forever” is less manifest and may need proof, whereas the logic definition seems evident. Reply: yes; that is why Aristotle recalls the logical one, since logic directs all the sciences; the proof comes in Book Six, e.g. from distance and time together: the faster body covers the same distance in less time; in that lesser time the slower covers a lesser distance; the faster covers that in still less time; so time and distance are divisible forever.

8. “Motion seems to be among the continuous things” (“the unlimited first appears in the continuous”) #

Point: Aristotle says “seems” because it is not yet proven (proven in Book Six); the word “unlimited” enters because there is something unlimited about the divisibility of the continuous.

9. The continuous dominates all our naming (“we tend to name the continuous before anything else”) #

Point: since reason in this life never thinks without imagining, and imagination always involves the continuous, common words name the continuous first and are carried over afterward.

• “Beginning” (fifth book of Wisdom [ed.: Metaphysics V]): first meaning is the beginning of the desk, a limit of a continuous thing; “I am the alpha and the omega” is a much later sense; Aristotle moves forward to the less known senses until God is the beginning.
• “Before” (Categories, before and after): first meaning is before in time; before in motion is led back to it; time is tied to before and after in motion and the magnitude.
• “In” (Book Four of Natural Hearing): eight meanings, unordered in the text: whole in the parts, part in the whole, genus in species, species in genus, “I got you in my power,” “I left my heart in San Francisco,” form in matter, “I’m in this room.” Thomas orders them following the method of Wisdom V, starting from the most obvious to sense (in a room), then the other seven, exactly placed. Offered for a later class.
• The continuous is a common sensible, known by more than one sense, unlike color or sound; proper sensibles are rarely carried over, but names of the continuous are, and all thinking in words goes back to them.
• Borrowed uses even for discrete things: “continuous proportion” (2:3::4:6, 4:6::6:9, the number ending one ratio begins the next); “continuous syllogisms” (a student: the conclusion of one is the premise of the next); “continuous definitions” (motion in the definition of nature; Euclid’s chain figure, plane figure, rectilineal, quadrilateral, square); square and cube numbers, Euclid’s “sides” of a number for our “factors.”
• “Road” in Empedocles, Heraclitus, Parmenides, Plato: originally continuous; in knowledge it names how one thing is linked to another (nature, motion, the continuous). “Above” (Shakespeare above Chaucer, Aristotle above Plato), “under,” “understanding” (to stand under): from place.

His words #

• sapientia as sapida scientia: savory knowledge (Thomas’s explanation of wisdom).
• unsavory / insipid: folly, the opposite of savory wisdom.
• manuductio: leading by the hand, here by opposites (bitter/sweet).
• dulce sapere: to sweetly savor (Adoro te devote).
• methodos: philosophy as following a road.
• “Natural Hearing” [ed.: Physics].
• “the Wisdom” [ed.: Metaphysics], with Book Five (beginning) and Book Nine (act).
• discrete vs. continuous quantity: number vs. line, surface, body.
• continuous (logic): quantity whose parts meet at a common boundary or limit; discrete: whose parts meet at no boundary.
• continuous (natural philosophy): that which is divisible forever.
• arithmetica: the science of numbers and their properties, not the art of calculating.
• common sensible: what more than one sense knows (the continuous), vs. proper sensibles like color and sound.
• continuous proportion, continuous syllogisms, continuous definitions: borrowed senses where one term ends one and begins the next.
• “sides” of a number: Euclid’s word for factors.

Texts #

Read in class

• Aristotle, Physics III, opening lines (“since nature is the beginning of motion and change… the road we are following is about nature… those determining about motion ought to try to go through the things which follow upon motion… motion seems to be among the continuous things… the unlimited first appears in the continuous”).
• Shakespeare, Romeo and Juliet (Romeo’s words before the poison), quoted and analyzed.
• Wisdom 8 (two verses), quoted.
• Thomas Aquinas, Adoro te devote, one quatrain, quoted in Latin.

Mentioned

• Aristotle, Categories: quantity (discrete/continuous); the chapter on before and after.
• Aristotle, Physics IV (eight meanings of “in”) and Thomas’s commentary ordering them; Physics V–VIII (division of change, quantitative division, movers).
• Aristotle, Physics VI (proofs that the continuous is divisible forever).
• Aristotle, Posterior Analytics, end of Book I (arithmetic more certain than geometry).
• Aristotle, Metaphysics V (beginning), IX (act known without definition).
• Aristotle, Nicomachean Ethics (opinions of the sick and poor; proemium naming the subject of natural philosophy).
• Aristotle, On the Soul I (thoughts like numbers) and III (reason never thinks without images).
• Euclid, Elements (Heath’s edition; books I–VI, VII–IX, solids; definition of square); Proclus.
• Shakespeare, Julius Caesar (“tide in the affairs of men”); Boethius on the ruminating animal; Heraclitus, “nature loves to hide”; Descartes on motion; Thomas, Summa Contra Gentiles opening.

His questions #

• If motion is unknown, what follows for nature? Nature is unknown, since motion is in its definition.
• What is common to any kind of quantity? A multiplication of parts.
• At what do two parts of a line, a surface, a body meet? A point, a line, a surface respectively.
• Do the three and four of seven meet in anything? No; discrete quantity’s parts meet at no boundary.
• Can you divide the one of pure number? No; it is simpler than the point, indivisible and without position.
• Does the division of definitions, or of proofs, go on forever? No; something is known without definition (act) and some statements without proof.
• Which definition defines by form and which by matter, and why does that fit logic and natural philosophy? Logic’s (parts meeting to form the whole) by form; natural philosophy’s (division toward the parts) by matter.
• Why does Aristotle say motion “seems” to be continuous? Because it is not yet proven; the proof is in Book Six.
• What is the first meaning of beginning, of before, of in? The limit of a continuous thing (the desk), before in time, being in a place.
• What do “continuous syllogisms” mean? The conclusion of one is the premise of the next.