Zeno's Paradoxes, the Socratic Method, and the Four Kinds of Conversation

Orientation #

He says the arrow argument is placed “in the beginning” because it is connected with what was just seen before, that time is not composed of nows (prev. lecture). He refers back to Book One for the first Parmenidean difficulty, and forward to Book Eight for the fuller solution. Next time: the first two arguments for God’s existence from motion in the Summa Contra Gentiles, to see the connection between the definition and continuity of motion and the unmoved mover. The previous class treated motion and rest in the “now” and the divisibility of the continuous; the next turns to Romeo and Juliet on wisdom’s slowness.

The class, in order #

1. Who Zeno is; Parmenides versus change (“only a two-headed mortal could think”) #

Zeno is Parmenides’ pupil; in Plato’s Parmenides the two come to Athens and examine the young Socrates into contradictions (the Sophist also refers to this meeting), and Socrates speaks reverently of Parmenides.

• Parmenides insists it is impossible to be and not be; Heraclitus spoke as if change involved day being night, the young being the old; so Parmenides concludes change is an illusion.
• The famous dichotomy: if you must choose between the impossibility of being-and-not-being and the existence of change, which do you keep? Student: the impossibility. He: yet more people in history have held to change, since knowledge starts in the senses and “things in motion soon catch the eye” (Shakespeare). Parmenides: only a two-headed mortal thinks something both is and is not.
• Argument that the choice itself concedes Parmenides:
  1. You are forced to choose only because change contradicts the principle;
  2. those who reject the principle because something contradicts it say “it isn’t so because it is so”, i.e. they refuse a contradiction, which is the principle. Conclusion: the attempt to reject the principle rests on it. Parallel: the man who says statements do not exist cannot say so without making a statement.
• So Parmenides is “a man to be feared” (Plato), yet change obviously exists; hence there must be a way of understanding change without contradiction. Not until Aristotle is that way seen: one in Book One, another here in Book Six. “That’s kind of the key to the human mind going forward”: never accept a contradiction, but treat an apparent one as pointing to something hidden to unravel.
• Student: why not leave it as a problem? He: Zeno came to his teacher’s defence with further absurdities, which is why Zeno fits Book Six rather than Book One.
• Aside: people threw things at Parmenides to see if he would duck.

2. Preview of Zeno’s six arguments; the arrow pulled out first (“time is not put together from indivisible nows”) #

Aristotle will touch six arguments: four from local motion, one from becoming, one against circular motion. The arrow is briefly mentioned first because it is connected with what preceded: if everything rests when in a place equal to itself, and what is carried is always in the now, the arrow is immovable. Aristotle’s reply: this assumes time is made of indivisible nows, which was rejected for anything continuous, “just as no other magnitude is”.

3. First argument: the half before the end (“before you can walk out the door”) #

The dichotomy restated: before reaching the door you must reach half, then half the remainder, forever; so you never get out.

• Thomas calls Aristotle’s earlier reply “a kind of ad hominem”: time is divisible as much as distance, so there are as many nows as places to reach, one-to-one; a finite distance in a finite time.
• But that does not solve it: if each half is taken as actually distinct, it is like counting to infinity. Three illustrations of “an infinity you never get through”: a child reciting an alphabet with infinitely many letters; Euclid using A to prove B, B to prove C (a “large discourse”, Shakespeare) — an infinity of conclusions and no theorem; Euclid defining square by quadrilateral, quadrilateral by rectilineal plane figure, that by plane figure, that by figure — if it went on forever nothing would be defined.
• Solution (student, confirmed): the distance is divisible forever in ability, but you do not actually divide it. “He’s making something that’s there in ability actual.”
• Weizsäcker (quoted, as before): sense and imagination do not know ability; what is imagined is made actual in imagination, hence a problem you cannot solve.
• Locke on the triangle in general: are its lines equal, unequal, right-angled? He says none of these, an apparent contradiction. He: all of these in ability, none in act; adding “equilateral” or “scalene” makes actual one ability, never both in one.
• The fuller solution comes in Book Eight.

4. Second argument: the Achilles (“said with a certain tragedy”) #

The swiftest never overtakes the slowest given a head start, because the pursuer must first reach where the fleer started, by which time the fleer has moved on.

• Why “Achilles”: swift runner in the Iliad, fleeing the river god angered by the bodies polluting the river (his gloss on Thomas’s explanation, which he finds not the best).
• Greek says it is stated “speaking tragically” (Latin quadam tragoedia): tragedy strikes the imagination with wonder; Thomas: magnifying the words to move wonder.
• Board example: F twice as fast as S; S has one unit head start. In t1 F reaches S’s start, S has gone half a unit; in t2 F reaches that, S another quarter; and so on. They should meet at two miles, but F “always takes half of what remains”: 1, ½, ¼, ⅛… never two. Same argument as the door, in substance, but with a tragedy.
• Digression, restating the same truth to arouse wonder: Euclid II.5 (line cut into equal and unequal segments; the rectangle of the unequal segments plus the square on the line between the sections equals the square on the half). Restated: two rectangles with the same perimeter, one with more area (same fence, more property); even more area with less perimeter, which surprises people. The “crooked geometers” who sold land by perimeter (an L-shape) gave more perimeter for less area. His proportion: as perimeter is to area, so words are to what you say; you can say more with fewer words (a page of Aristotle vs. a book elsewhere). Asked “greater, less, or equal?” students do not know; then “by how much?” — always the square on that line, the same difference on either side. “They’re substantially the same truth, but it rouses more wonder.”
• Text: “it will be overtaken if one grants that the limited can be gone through” (he corrects the reading to “limited”): unlimited in potency, limited in distance and time. Both arguments solved later by not counting the divisions as actual. The danger: our tendency to make the potential more actual than it is.
• Aside: Msgr Dion on the mind’s tendency to make words univocal; John of St Thomas makes “sign” univocal for sensible and mental sign, whereas in Thomas it is always an objection whether angels have signs; Augustine’s definition, a sign strikes the senses and brings to mind something other than itself, so the extended use is equivocal by reason — like this: making an ability actual.

5. Third argument: the arrow (“there will not be a syllogism”) #

Text: the arrow carried along stands still; this results from taking time as composed of nows; that not given, there is no syllogism. If each now has no motion and time is made of nows, the body rests the whole time. A student asks whether this is the one from the beginning; yes, Thomas says it was mentioned first for its connection with what preceded but stands third in Zeno’s order.

• Aside: Louis de Broglie, Physics and Microphysics (preface by Einstein, who singled this chapter out), connects quantum theory’s problems with Zeno: instantaneous velocity is motion in the indivisible; Schrödinger wondered whether it was fictional; Newtonian physics puts straight-line and circular motion together where the tangent touches at a point — “but there isn’t such a thing”. The English edition omits the chapter; he has the French.

6. Fourth argument: equal masses on the race course (“the half time equal to the whole”) #

Text: the fallacy is thinking the equal magnitude, equally fast, passes the moving and the resting in equal time. Board (his simplified set-up, “may not be exactly” Aristotle’s): rows A (at rest), B and C moving contrary. When B has passed one A it has passed two C’s; so equally fast bodies would be twice as fast. Student: “in relation to what?” — “That’s the point.” Measured by the resting A, one; by the contrary-moving C, two. Student: like relativity. He: Aristotle’s point is precisely the fallacy of comparing to something at rest and to something in motion.

7. Change between contradictories (“no last now in which I am alive”) #

These four concerned change of place, the most known motion. Next paragraph (bottom of p. 14): nothing impossible in change between contradictories. Zeno’s argument: changing from man to not-man, you are not changing while still a man nor when no longer one, so you must both be and not be a man.

• Aristotle’s reply here: what changes is a body, extended; between fully white and in no way white there is something in between, so a divisible thing can be white and not white while changing. Book Eight deals with the other part.
• Student: does the change from man to not-man take time? No, except for the preceding changes in something else disposing for death. Freezing in the snow: the temperature falls over time; the end of that time is a now in which I am dead; there is no last now in which I am alive.
• Aside: the Epicurean consolation, you never die during your life and you are not there to experience your death.

8. Last argument: the revolving sphere (“one part succeeding to the part of the other”) #

A sphere rotating stays in the same place, so how is there motion? Student: one part takes the place of the other. Thomas says the same of a rotating column. Different from a whole going from place to place, “a strange kind of change of place”, but still one part succeeding where another was. Zeno picked up such objections one after another to defend his master.

9. The Socratic method and the principle of Parmenides (“three admissions is all Socrates needs”) #

Plato’s showing Socrates learning his method from Parmenides and Zeno is illuminating: the method sees whether what you think fits with the other things you think.

• Slave boy: admits (1) doubling the side doubles the square, (2) four is double of two, (3) sixteen is not double of four. From two you syllogize to the contradictory of the third. Simpler: width 2, length 6, area 10; any two contradict the third, and “this doesn’t tell me which is wrong”; you reject the least known. Einstein, The Evolution of Physics: relativity arose from a contradiction; the statement lacking experimental evidence was rejected.
• So Socrates’ method rests on the impossibility of being and not being (double and not double) at the same time in the same way.

10. The four argumentative conversations, all in the Meno (“What difference does it make whether you are refuted”) #

Aristotle distinguishes examination from sophistical conversation (On Sophistic Refutations): refutation involves contradiction; a sophistical refutation only appears to; examination finds whether you really contradict yourself and can lead to true refutation. The other two: teaching (demonstration) and dialectical.

• Meno: (i) Meno claims to know virtue, Socrates examines, Meno does not know; (ii) Socrates proposes a dialectical inquiry, neither knowing; (iii) Meno’s sophistical objection: how direct yourself to what you don’t know (filling station: “where do you want to go?”), which would destroy logic; Socrates’ reply falls into the same mistake; (iv) first slave-boy conversation is examination, purging the should-be student who thinks he knows; second is teaching (doubling by the diagonal), since Socrates knows and knows he knows; (v) third part, whether virtue can be taught, reasons with probability to it and to its contrary: the dialectical conversation as Aristotle describes it, from probable opinions even to contradictory conclusions.
• Proper order: teaching, dialectical, examination, sophistical, from more perfect down; the sophistical is bad, ordered to victory or the appearance of knowing.
• Grounding: two people either both know (no conversation needed), one knows (teaching), neither knows (dialectical); since it is unclear who knows, examination may be needed first.
• Examination and sophistical resemble each other, so Socrates is accused of being a sophist; his reply: what difference whether you or I am refuted, we get closer to the truth.

11. Hunting down a definition (“the cat’s a lazy hunter”) #

Greek thēreuein, from the word for wild beast, used in the Posterior Analytics and Sophist for arriving at a definition by division: like natives beating drums to corner a tiger, division narrows the room to escape. Locke’s Essay compares the philosophical life to a hunt; Plato earlier. Vestigium, footprint: to investigate is to track. Why he loves cats: lazy hunters, like philosophers.

12. With the truth everything harmonizes (“what a tangled web we weave”) #

Nicomachean Ethics Book I: with truth everything fits together. From Metaphysics Book IX (finishing it in another class): the end of knowledge is the first cause (Proemium: the mind wants the cause of the cause), and knowledge of the better is better (De Anima), so the end is the best thing. Two ends unless they coincide: act is before ability (what goes to act does so by something in act), so the first cause is pure act, and act is better than ability, so pure act is best; they fit. Those who make matter the first cause (Lenin: mind the highest product of matter) have “schizophrenia”. Falsehood does not fit: “tangled web”, the liar under Perry Mason’s cross-examination.

His words #

• change vs. the impossibility of being and not being: the dichotomy Parmenides forces
• ability: what is there potentially, divisible “in ability” [ed.: potency]
• act / actual: what is made actual in imagination or by a difference
• speaking tragically (quadam tragoedia): stating an argument to strike wonder
• ad hominem reply: Thomas’s name for the earlier time-divisible-as-distance answer
• equivocal by reason: extended meaning of “sign” to the non-sensible
• examination conversation; sophistical (contentious); teaching (demonstration); dialectical: the four argumentative conversations
• thēreuein: to hunt down (a definition)
• vestigium: footprint; investigate
• schizophrenia: two ends of knowledge that do not coincide

Texts #

Read in class

• Aristotle, Physics, Book VI, Zeno’s arguments: the arrow, the four reasons, change between contradictories (p. 14 of his text), circular motion (ch. 9, inferred)
• handout, “Reasoning and the Four Kinds of Argument”, p. 5 (four conversations; verified)
• Euclid, Elements II.5 (restated)

Mentioned

• Plato, Parmenides; Sophist; Meno (slave boy, Meno’s paradox)
• Thomas Aquinas, commentary on Physics VI (ad hominem reply, tragedy, column)
• Aristotle, On Sophistic Refutations; Posterior Analytics; Nicomachean Ethics I; Metaphysics IX and Proemium; De Anima
• Louis de Broglie, Physics and Microphysics (Einstein preface); Schrödinger
• Einstein, The Evolution of Physics
• Locke, Essay Concerning Human Understanding
• Augustine’s definition of sign; John of St Thomas on sacraments
• Homer, Iliad (Achilles and the river god)
• Thomas Aquinas, Summa Contra Gentiles, first two arguments from motion (next time)

His questions #

• If you had to choose between the impossibility of being-and-not-being and the existence of change, which would you hold? The impossibility; but most in history held change, since it strikes the senses.
• Which is more known to us and more certain? Change seems so, from experience; yet rejecting the principle presupposes it.
• If there were an infinity of letters, could the child recite the alphabet? An infinity of conclusions, ever reach the theorem? No; so actual infinite divisions would keep you from the door.
• So the solution to the first argument is what? The distance is divisible forever in ability, not actually divided.
• How is the Achilles the same as the first argument? He always takes half of what remains and never reaches the meeting point.
• Equally fast bodies twice as fast — in relation to what? To a body at rest vs. a body in contrary motion; the fallacy.
• Does the change from man to not-man take time? No, only the disposing changes do; no last now alive.
• How can there be motion when the sphere stays in the same place? One part succeeds to the place of another.
• What are the four conversations and why in this order? Teaching, dialectical, examination, sophistical: from more perfect down.
• Is the end of our knowledge the first cause or the best thing? Both, since pure act is first and best; otherwise “schizophrenia”.