Zeno, Parmenides and the Continuous: Why Nothing Continuous Is Indivisible

Orientation #

He refers back to “the previous reading, the third reading,” where it was seen that time and magnitude are divided in the same way, and to earlier fragments of Heraclitus and papers he gave “on the rule of contradiction in our knowledge.” He says “we’ll see ones later on here in book six” of the apparent contradictions that caught Hegel. The previous class treated false imagination as a source of deceptive knowing; the next concludes the general treatment of the continuous and begins reading 5 on the indivisible now.

The class, in order #

1. Opening aside: Stevenson’s vocabulary and the three kinds of minds (“the wits, the the dim wits and the nitwits”) #

Not part of the reading; a word-hunt in The Black Arrow (illustrated by Wyeth, set in the War of the Roses) leads back to a favourite division.

• “pottle” for bottle: P and B change into each other because they are made alike in the mouth.
• “tosspot” (those who rush into battle): he guesses a drunkard, literally one who tosses a glass; “I don’t know.”
• “shuttle-wit”: he wonders whether it means one who always changes opinion with the times (the character watches which side is winning before joining). He does not know the phrase and doubts Shakespeare uses “shuttle” this way.
• The three kinds of people (Hesiod; Aristotle in the Nicomachean Ethics; Thomas; St Basil the Great): those who discover the great things themselves (very few); those who cannot discover but can learn from discoverers; those who can neither discover nor learn. His names: wit, dimwit, nitwit. He classes himself as an “upper-class dimwit,” one who knows where the wits are; the lower-class dimwit does not.

2. Reading 4 begins: magnitude and time unlimited in the same way (“two ways in which you could speak of the unlimited”) #

Point: the comparison of magnitude and time from reading 3 is broadened to both senses of “unlimited.”

• Unlimited in the extremities vs. unlimited in divisibility: a line with no beginning and no end takes a time with no beginning and no end to traverse; if one has no extremities, neither has the other; if one is limited, so is the other.
• Divisible forever (already seen): if magnitude is divisible forever, so is time, and vice versa.
• Conclusion: the two are limited or unlimited in the same way, in both senses.

3. Zeno’s problem and the ad hominem answer (“what they call in an argument ad hominem”) #

Point: from the equal divisibility of time and magnitude, a corollary answers Zeno.

• Who Zeno is: pupil of Parmenides, who denied motion and change; Aristotle discusses Parmenides but not as a natural philosopher, since nature is the beginning and cause of motion and rest, and to take away motion is to take away all natural science (as Thomas says denying free will takes away the whole of ethics and practical philosophy). Zeno defends his master by showing that the affirmers of motion are in a ridiculous position.
• Zeno’s argument as he reconstructs it: (1) the continuous is divisible forever, so a finite distance A to B contains an infinity of points (like the modern mathematician’s line composed of points); (2) one must go to each point before the next, and it takes some time to go from one point to a later one; (3) therefore it takes an infinity of time to cover a finite distance; so “one will never be able to leave a room,” never get home (“Gotta call home”).
• Aristotle’s reply: time and magnitude are divisible in exactly the same way, so on Zeno’s own assumption a finite time XY would be composed of an infinity of nows, one now for each point; there are enough nows to go through all the points and the time is still finite. Zeno takes “infinite” to mean a time with no end point, whereas what follows is only infinite divisibility.
• Thomas calls this ad hominem: Aristotle does not think a line or time is really composed of indivisibles (shown before), but answers “the man” on his own assumption.
• Student: isn’t ad hominem attacking the person’s character? Answer: no, it answers his objection not in reference to the truth but in reference to what he thinks. He grants it is “almost like an aside” in Aristotle.

4. Student’s question: what did Parmenides do with the senses? The two roads (“the road from the senses”) #

Point: Parmenides denied the data of sense as illusion because the road from the senses seems to lead into contradiction.

• Heraclitus’s fragments as the road from the senses: the healthy become sick, day becomes night; but “becomes” means comes to be, and if the healthy is sick it both is and is not healthy, a contradiction. Heraclitus says men admire Hesiod though he “didn’t even know day and night; they are one” (Hesiod’s Works and Days treating day for labour, night for rest as two things); likewise the waking and the sleeping. He thinks Heraclitus may not really have believed day is night, since another fragment says “we should not act and speak like those asleep,” which presupposes the difference; but the fragments point to an apparent contradiction in what we sense.
• The same happens in the history of science: Newton’s hypothesis of light as a shower of particles vs. Huygens’s wave; a 19th-century experiment seemed to settle it for the wave; Einstein’s 1905 Nobel-prize explanation of the photoelectric effect required pinpointed particles; quantum theory arose to overcome that apparent contradiction; then general relativity and quantum theory contradict each other when confronted, so one does one or the other, and string theories try to resolve them. Apparent contradictions from the senses are not peculiar to the Greeks. (Heisenberg’s Gifford Lectures cited for the history.)
• Parmenides’s second road: from the impossibility of a contradiction in things; one cannot even think that something both is and is not. Those who say day and night are the same he calls “two-headed mortals”: one head to think it is so, another to think it is not; and a monster is something opposed to nature, so it is against the very nature of our mind to think a thing both is and is not. “That’s kind of a beautiful way.”
• Plato’s Parmenides: Parmenides and Zeno come to Athens and examine the young Socrates, who contradicts himself as the men Socrates examines elsewhere do; Plato thus represents Socrates learning the method of examination from them, and that method is based on this: seeing whether the things you think fit together or contradict.
• For Parmenides the road from the senses is the false road, “the road of illusion.”

5. Aristotle’s untying: the apparent contradiction in change (“suppose the cook is a pianist”) #

Point: Aristotle keeps both, the impossibility of contradiction and the reality of change, by showing the contradiction in change is only apparent.

1. Parmenides is right that it is impossible to both be and not be; Aristotle spends much of Book Four of Wisdom refuting attempts to deny it.
2. But it is absurd to deny change, which is clear in experience.
3. So what seems a contradiction in change is an apparent contradiction in things (it could be a real one in our thinking) under which something hidden lies, as Heraclitus said the hidden harmony is better than the apparent.

• The dilemma: if the healthy cannot be sick, they cannot come to be sick and will always be healthy; if the sick cannot be healthy they are always sick (“tough luck”). So one must admit the healthy become sick, the light dark, the hard soft, or there is no change; and we all speak so, not just madmen. Aside: otherwise “we’d be Calvinists,” the good always good, the bad never becoming good.
• The cook and the pianist: if the cook is a pianist we say “a pianist cooked dinner” and “the cook played the piano,” because being a pianist happens to the cook; but the cook as such cooks and the pianist as such plays. So it is not the healthy as such that becomes sick but that to which health happens, the body, which as such becomes sick; and not the sick as such that becomes healthy but the body.
• Hence the deception is the first kind of mistake outside of words, the mistake of the accidental. It is apt to deceive because necessarily before you become healthy you are sick, before soft you are hard, before light you are dark: something that happens necessarily to what becomes, though it itself as such cannot become that.
• If one cannot untie this, one must choose between the two roads; Aristotle, seeing both the impossibility of contradiction and that change exists (clearer than anything), understands the road from the senses into reason. The appearance of contradiction is important in the development of knowing: it points to something hidden; the Greeks spoke of change between contraries without seeing clearly the third thing, the subject as such of the change.

6. Those who admit contradictions in things (“a contradiction I’ve learned to live with”) #

Point: down through history men speak as if there really were contradictions in things.

• Hegel, Marx; Lenin’s concise statement that dialectics in the Marxist sense (dialectical materialism) is the study of the contradiction within the very essence of things.
• Modern mathematicians he has heard say that some higher math cannot be done without accepting contradictions.
• Weizsäcker (with Heisenberg) wanting another logic for subatomic events, not observing the two axioms of being and non-being: it is impossible to both be and not be, and it is necessary to be or not be. Shakespeare’s “To be or not to be, that is the question” touches both: it is a question because you cannot both be and not be and must be one or the other. Heisenberg is wiser, having more of Aristotle and understanding potency (ability): there is nothing between being in act and not being in act, but in a way potency is between the two.
• A Heideggerian told him the principle of Parmenides “has taken quite a beating lately.”
• His old teacher Kasurik at a Chicago meeting caught a speaker in a contradiction and asked what he would do about it: “that’s a contradiction I’ve learned to live with.”
• Aside on current politics: the US must deal with Arafat (as representing the Palestinians and for Arab support before going after Hussein) while knowing from documents he supports terrorism, so the policy looks contradictory. Student: isn’t this contradictory? Answer: perhaps in the course of action, but not in the thinking; they are not thinking both that he is and is not a terrorist, only that other reasons make it imprudent to go after this one, as they went after the Taliban.

7. Student’s question: why the infatuation with denying non-contradiction? (“it is so because it is so”) #

Point: deniers assume the axiom in the very act of denying it, which is a sign they know it.

• Aristotle in Book Four of Wisdom: some deny it out of perverse stubbornness; more reasonable people deny it because something seems to contradict it (night turning into day). But they deny it in words because something contradicts it, that is, because they accept it; so their objections are invalid, assuming as premise what they try to deny as conclusion.
• Parallel: his brother Mark on attempts to prove Euclid’s fifth postulate (doubted since ancient times, even in Proclus): every proof assumes the thing to be proved among its premises, which makes the argument invalid; Mark defends the postulate as known by itself, not as obvious as some things but really obvious.
• Shakespeare, Two Gentlemen of Verona: the maid, asked why Proteus is best, has “no reason but a woman’s reason: it is so because it is so”; not a reason really.
• Mark’s second point: that they always assume it is a sign they really know it; the same for deniers of the axioms about being and non-being.
• So the two roads are not two roads but one, from the senses into reason: reason begins from sense, and what it first understands among statements is what Parmenides pointed to; the apparent contradictions puzzle reason, which knows they cannot be so, and their untying becomes the discovery of what was hidden. “It took some time to get there.”
• Heraclitus: “the things that can be seen, heard, and learned are what I prize the most.” Empedocles: you cannot see or touch God, and this (sight and touch) is “the broadest road leading into the mind of man.” Why sight and touch: sight is the sense of clarity and of distance, touch the sense of certitude, and they are the only senses that know shape; we recognise a cat, a chair, a man by shape, not colour. Other fragments name sight and hearing, as Heraclitus does.

8. Back to the text: nothing continuous is indivisible (“nor is it what is it indivisible”) #

Point: having shown nothing continuous is composed of indivisibles (equivalently, every continuous is divisible forever, or divisible into things always divisible, the second definition of continuous), Aristotle now makes explicit that no continuous thing is itself indivisible.

• Thomas divides the text formally against almost everything before; he says it is almost unnecessary. Why: if the continuous is that whose parts meet at a common boundary, parts are in its very definition. From logic, quantity divides into discrete and continuous; every quantity consists in a kind of multiplication of parts (Thomas elsewhere); in continuous quantity the parts meet at a common boundary, in the discrete, as seven into three and four, the parts meet nowhere. So how could the continuous be indivisible? And what is divisible forever cannot be indivisible.
• Still Aristotle makes it explicit; one must keep a kind of suspension of wonder and ask whether time, or a magnitude, or the distance or time travelled, could be indivisible.
• Student: does he have some other common notion of continuous, since he seems to argue from examples (magnitudes, time)? Answer: he is not thinking explicitly of the definitions there.
• Text read: “It is clear from the aforesaid that neither line nor surface nor body nor time nor generally any continuous thing will be indivisible, not only through what has now been said, but also because it will happen that the indivisible be divided.”

9. The argument from the faster and the slower (“the indivisible therefore will be divided”) #

Point: assuming a continuous thing is indivisible, the indivisible gets divided.

1. At all time there is the faster and the slower; the faster goes through more in equal time, and there is some ratio between them; take 3:2 (indifferent which).
2. Let the faster go one and a half the distance in the same time; divide its magnitude into three indivisibles AB, BC, CD, and the time into three indivisibles KL, LM, MN.
3. The slower, two thirds as fast, goes through EF, FG (the distance EG) in that time; since it goes an equal distance in equal time, it goes EF in half the time, and to divide three indivisibles for it you must take one and a half. So the indivisible is divided, and the body “will go through the partless not in an indivisible but in more.”

• One could turn it round and divide EG into three instead: the same argument. Simpler still: if in one indivisible unit of time the slower goes a certain distance, the faster goes that distance in less than one unit, which is less than an indivisible of time.
• Why this is not the same as what went before: something can be not composed of indivisibles yet indivisible, as a point is, or the one that is the beginning of number.
• Modern math divides the continuous as if dividing number and divides the one into halves; but the one is simpler than the point, “infinitely,” since the Greek geometers say a point is a one having position; if the point is indivisible, even more so the one.

His words #

• wit / dimwit / nitwit: his names for Hesiod’s three kinds of minds (discoverers, learners, neither).
• unlimited in its extremities vs. divisible forever: the two ways magnitude or time is unlimited.
• now / instant: the indivisible of time corresponding to a point of magnitude.
• ad hominem: answering a man’s objection on his own assumption, not in reference to the truth.
• the two roads: from the senses; from the impossibility of a contradiction (Parmenides).
• two-headed mortals: Parmenides’s name for those who think a thing both is and is not.
• the mistake of the accidental: first kind of mistake outside of words; taking what happens to a thing for the thing as such.
• the subject as such of the change: the third thing hidden under change between contraries.
• dialectics (Marxist sense): Lenin’s “study of the contradiction within the very essence of things.”
• the two axioms of being and non-being: impossible to both be and not be; necessary to be or not be.
• potency, “ability” [ed.: potency]: in a way between being in act and not being in act.
• Book Four of Wisdom [ed.: Metaphysics IV].
• continuous: that whose parts meet at a common boundary (first definition); divisible into things always divisible (second definition).
• discrete quantity: parts meet nowhere, as three and four in seven.
• a point is a one having position: the Greek geometers’ phrase.

Texts #

Read in class

• Aristotle, Physics VI, reading 4 (handout page 5), first part on time and magnitude unlimited in the same way and Zeno; last two paragraphs, “It is clear from the aforesaid that neither line nor surface nor body nor time…” and the argument from the faster and the slower (3:2, AB BC CD, EF FG, KL LM MN), with Thomas’s commentary.

Mentioned

• Aristotle, Physics VI, reading 3 (prev.: time and magnitude divided in the same way).
• Aristotle, Book Four of Wisdom (Metaphysics IV) on refuting deniers of contradiction.
• Aristotle, Nicomachean Ethics; Hesiod; St Basil the Great (three kinds of minds).
• Hesiod, Works and Days.
• Heraclitus, fragments (day and night; those asleep; things seen, heard, learned; hidden harmony).
• Parmenides, fragment (two roads; two-headed mortals).
• Empedocles, fragment (God not seen or touched; the broadest road).
• Plato, Parmenides.
• Thomas Aquinas on denying free will taking away ethics (locus not given); Thomas “elsewhere” on quantity as multiplication of parts.
• Heisenberg, Gifford Lectures; Weizsäcker on a new logic.
• Lenin on dialectics.
• Euclid, fifth postulate; Proclus.
• Shakespeare, Two Gentlemen of Verona; “To be or not to be” (Hamlet, inferred); Shakespeare’s history plays on Lancaster and York.
• Robert Louis Stevenson, The Black Arrow (Wyeth illustrations).

His questions #

• If you travel a line with no beginning and no end, how long will you be travelling it? A time with no beginning and no end, an infinite time.
• If I have to go through an infinity of points to get out of this door, how much time will it take? On Zeno’s assumption, an infinity of time; but with an infinity of nows in a finite time, only a finite time.
• Who is Zeno? A student of Parmenides, defending his master’s denial of motion.
• Can the healthy be sick? No, that would be a contradiction; it is the body, to which health happens, that becomes sick.
• Does the pianist as such make a nice dinner, and the cook as such play the piano? No; the cook as such cooks, the pianist as such plays.
• If it is impossible for the healthy to be sick, why do we all speak that way? Because of the accidental: being sick happens necessarily before becoming healthy.
• Is the Arafat policy contradictory? Perhaps in the course of action, not in the thinking.
• Why do they deny the axiom of contradiction? Because something seems to contradict it, which means they accept it; a sign they really know it.
• If the continuous is that whose parts meet at a common boundary, how could it be indivisible? It could not; parts are in its very understanding.
• Can something be not composed of indivisibles and yet indivisible? Yes, a point, and the one that is the beginning of number.