Points Cannot Compose a Line: Aristotle's Arguments in the First Reading

Orientation #

He refers back to “the fragment there of Anaxagoras” and to his usual way of handling it “when I’m teaching the first book here, or teaching the fragments”, to “what you showed later on about cause” (a cause that has a cause vs. one that has none), and to Physics I on hardness itself and health itself (prev. lecture, the board with the two definitions of the continuous). He does not say what comes next beyond noting the second, third and fourth readings will show motion and time continuous. The previous class introduced the two definitions of the continuous and began this same opening argument; the next continues into faster/slower bodies and the closing chapter of Book VI.

The class, in order #

1. A point has no edge (“for there is no edge of the partless”) #

The point: the definition of the continuous used in the text is the logical one (that whose edges are one), and a point cannot satisfy it, nor touch by its edges, because it has no edge.

1. A line has an end; a point does not. If a point had an end, the end would be something different from the point — “something outside of something” — so the point would not be indivisible.
2. Hence points can neither be continuous (edges one) nor touching (edges together): “for the edge and that of which it is an edge are other.” He calls this last sentence, in effect, an axiom — something known once you understand the terms, without proof.

2. Axioms: whole and part, being, before and after, beginning, end (“the whole is more than one of its parts”) #

The point: the text’s principle belongs to a family of axioms, each involving a distinction, an “otherness”.

• Thomas’s standard example of an axiom (De Koninck and his own teachers followed him): the whole is more than the part. Why that example? Whole and part in their first sense belong to the continuous; people see it there first, only obscurely in other wholes. No one, to his knowledge, has listed all the axioms.
• Metaphysics IV (“the fourth book of Wisdom”): a long defence of the two axioms about being and non-being — impossible to be and not be at the same time in the same way; necessary to be or not be. Hamlet’s “to be or not to be, that is the question”: it would not be a question if you could both be and not be, or need not be either.
• Before and after (words of Metaphysics V): nothing is before or after itself. Today is before tomorrow and after yesterday, the morning before the afternoon, but not the morning before the morning — the distinction is always of one thing to another.
• Beginning: in Physics I Aristotle says Melissus and Parmenides, denying multiplicity, do away with beginning, since a beginning is always the beginning of something other than itself. So: nothing is the beginning of itself. Beginning and before/after are tied: Metaphysics V distinguishes the senses of before and after starting from the common notion of beginning (first in being, becoming or knowing), while “first” is defined by “before”.
• End/limit (third part of Metaphysics V): sometimes we distinguish the beginning of the day from its end; sometimes we call both “the ends” or “limits” of the day. Axiom: nothing is the end of itself — a surface is the end of a body, a line of a surface, a point of a line, and none is the thing it ends.

3. Causa sui, self-made men, and statements known through themselves (“you got to be very careful that phrase”) #

The point: because every cause is a beginning, nothing can strictly be causa sui; the phrase, when Aristotle or Thomas use it, is negative in meaning.

• God is not causa sui affirmatively; you know that God has no cause. A grammatically affirmative phrase can state a negation.
• Parallel: statements known “through themselves” means known but not through other statements. Read affirmatively it would be a statement proving itself — circular reasoning. “No odd number is even”: known once you know odd and even; the whole is more than the part: known (perhaps “through its parts”, i.e. its terms) but not through another statement.
• Sartre, end of Being and Nothingness: man is the attempt to be causa sui, to be God; a contradiction, hence “man is a useless passion.” Before him Marx, wanting man to make himself; those working on clones want man to make himself so as to be his own God. “Self-made man” has a legitimate sense (worked his way up from nothing), but strictly your father and God made you.
• Objection he raises: the foundation is the beginning of the house, yet a part of it — is a beginning then the beginning of itself? Reply: we say the foundation is the beginning of the house, not of the foundation; part and whole are distinct. There is always some otherness between beginning and begun, limit and limited.

4. Back to the text: no common boundary, and the two nows (“this is also the reason why two nows”) #

The point: since a point has no end or limit, it cannot have a common boundary with anything, nor can its limits be “together”.

• Thomas, beginning of the second reading: for the same reason two nows cannot be continuous — though with a somewhat different meaning, since “touching” belongs more properly to the continuous with position (line, surface), and one speaks of one time “touching” another loosely.
• Hence the second reading manifests that motion is continuous, the third and fourth that time is; “the reason here is the same for all three, proportionally speaking.”

5. A limit that has no limit; lack vs. negation (“could you speak of a point as limitless”) #

The point: limits are like causes and beginnings — there are limits with limits and one with none — and “unlimited” said of the point is a negation, not a lack.

• A cause that has a cause and one with no cause; a beginning with a beginning and one with none (God the Father); so a limit with a limit: the four sides of a square limit the surface but are themselves limited by points; the surface limits the body but has its own limits; the point is a limit with no limit. Strictly the point is neither limited nor unlimited (unlike a line, which may have ends or go on forever).
• Order of discovery: body, surface, line, point — you are not proving their existence; you see the surface is a limit before the line, the line before the point, so the limits first seen are limits having limits.
• Could the square be limited in size if its containing lines went on forever? No — “be careful about that, Leslie.”
• Student (Don): can you say the point lacks a limit, is limitless? Student: it cannot have one, so it cannot lack one; also it has no magnitude. He: this is important. Lack (privation, the Latin word) in the strict sense is the non-being of something a thing is able to have and should have, when it should. The chair does not hear or see, but is not deaf or blind. Aristotle (Categories, Metaphysics V) distinguishes four kinds of opposites, including being/non-being against having/lacking. A straight line with no endpoints is lacking them (it can have them); the point cannot have a limit, so does not lack one. So “unlimited” is equivocal.

6. God unlimited, God’s joy, and love without hate (“no prayer shorter or grander”) #

The point: God is called infinite/unlimited in the negative, not the privative sense — no limit to his perfection — and this opens a digression on whether God is affected by our thanks.

• Thomas is always careful: God’s infinity is no lack; God is pure act, no ability to receive; so strictly you cannot give God anything. Thanksgiving: Augustine’s “Deo gratias” — no prayer shorter or grander.
• Does God rejoice in our thanking him? Student: in his human nature. He: also in his divine nature. God loves us; who loves rejoices in the beloved’s good; our thanking, loving, honouring him is our good, not an addition to his. It gives him no more joy: love and joy follow knowledge, and his knowledge is eternal, so he is not learning something new. “There is more joy in heaven over one sinner who repents” — but that joy cannot be new; my repentance is always present to his eternal knowledge, and (Jeremiah) his love is perpetual, eternal. Distinction: God’s pleasure is in himself; his joy is in himself and in [unclear in recording].
• Student: does it belong to love to sorrow over the beloved’s evil? Reply: joy arises from love, sadness from hate (when what I hate is forced on me or on those I love); no hate in God, so no sadness.
• The six in us: love/liking (liking the weaker word) gives rise to wanting/desire when we lack the thing, to joy/pleasure when we have it; hate/dislike gives rise to turning away when threatened, to pain/sadness when forced on us. Tricky exams; a disliked dish served at dinner. God lacks nothing, so no desire in him except metaphorically; always love, always joy (the Psalms).
• Why no hate in God: God is love itself; as in Physics I, hardness itself cannot be soft, health itself sick, so love itself cannot hate. God is simple — “God is whatever he has.”
• Conclusion: the point is “unlimited” a little as God is: not a lack, simply a negation.

7. How things touch: whole to whole means coincide (“either the whole touches the whole”) #

The point: the third paragraph adds that points cannot make a line by touching in any way.

• Text: the points making up the continuous must be continuous or touching (same reason for all indivisibles); not continuous, by the foregoing; in all touching, whole touches whole, part part, or whole part; the indivisible has no parts, so whole must touch whole; but then it is not continuous, since the continuous has part outside part, divided into parts separated in place. (A common limit means the parts lie on either side of it.)
• His own version (usually run together when he treats the Anaxagoras fragment): possibly a fourth way, touching at the edge, distinguishing edge from part (the four lines of a square: edge or part?); he eliminates all four. Points, having no parts, can touch only as whole touches whole, i.e. coincide; coinciding, they have no more length than one point, which is none; so ten, a hundred, a million, an infinity of points (“like the mathematician says”) make no line.
• Student: is Aristotle here proving the second definition from the first, since not-composed-of-indivisibles means divisible forever? He: “we’ll come down” — Aristotle is going to do “something kind of strange.”

8. Points cannot be “next” either (“between whom there is nothing of the same kind”) #

The point: the continuous is not made of points next to one another.

• Lining up in the sister school playground, the next student; a row of houses, the next house. Two points already have a distance between them (they cannot touch), and potentially an infinity of points in between.
• Text: neither is a point next to a point nor a now to a now, for “next” means nothing of the same kind between. My next-door neighbour: air, bushes, grass between us, but no house. Between points there is always a line, between nows time.

9. The second argument, from the second definition (“divided into those things of which it is”) #

The point: Aristotle’s second principal reason argues from “divisible forever” to “not composed of indivisibles” — and then, at the end, seems to reverse.

• Text: if a thing is divided into what it is made of, it would be divided into indivisibles; but nothing continuous is divided into the partless.
• The Marietti text is confused here; it does not give all the divisions Thomas follows (a division seems left out, perhaps lost). Thomas: first reason from the first definition, then from the second.
• Contrast: number is not divisible forever because composed of indivisibles — Euclid: a multitude composed of units; units are simpler even than a point, which the Greeks call “a one having position.”
• Last paragraphs, where Thomas says Aristotle manifests what he presupposed: nothing of another kind can be between points and nows, for it would be divisible forever and so continuous; and everything continuous is divisible into always-divisibles, “for if into indivisibles, indivisible would touch indivisible.” So the reason why the continuous is divisible forever is that it cannot be made of indivisibles — “kind of paradox there, which came first?”

10. His own way, with the Anaxagoras fragment (“there’s no smallest of the small”) #

The point: his classroom proof goes from “not composed of points” to “divisible forever”, the reverse of the second argument’s direction.

1. Draw a line, cut it in half, halve the half; students are inclined to accept it goes on forever — but that is only “probable.”
2. To make it certain: if you did not cut into something further divisible, you cut into nothing or into indivisibles.
3. Not nothing: Anaxagoras’s other fragment, “what is cannot cease to be by being cut”; you cut a thing into what it is made of (Nixon: however you slice the Democratic program it’s the same baloney), so it would be made of nothing — absurd.
4. Not two points: the touching argument, with circles on the board (part-part, part-whole, whole-whole — “I go around twice”; a fourth way at the edge, but a point distinct from its edge would be a tiny circle with something inside, hence extended). Points touch only like whole to whole (they have no whole either), i.e. coincide, no length.
5. So you always get two shorter lines, which can be cut again: divisible forever.

11. The apparent circularity, and how Thomas handles one elsewhere (“Thomas doesn’t say anything about apparent circularity”) #

The point: Aristotle seems to reason from each definition to the other; Thomas is silent; he raises the question and tries a solution.

• Circular reasoning is bad (Posterior Analytics): we reason from more known to less known; A proving B and B proving A makes A more known than itself, “before and after itself.” There must be statements known not through others, or one goes on forever.
• Famous place in the Posterior Analytics where Thomas notes Aristotle reasoning both ways between demonstrations being of the necessary and of the “as such” (kath’ hauto, per se). Thomas: the demonstration proper is from necessary to as such; reasoning from as such to necessary takes “episteme is of the kath’ hauto” as a probable opinion, common in the academy — showing a real connection without circularity.
• Digression on why Thomas notes it there: like Nicomachean Ethics I, where Aristotle, about to attack Plato, says it is hard because held by friends, but philosophers love truth: Plato is a friend, truth a greater friend; impious to put Plato before truth (Socrates in the Phaedo: little about Socrates, much about the truth; Thomas: God is truth itself). Elsewhere (Metaphysics, Physics I) he disagrees with Plato without remark — but two of the Ethics’ ten books are on friendship, so he must show disagreement is not against true friendship. Likewise it fits that Thomas raises circularity in the very book that forbids it.
• Applied here — Student: the proper demonstration would be from the definition, but the other direction is more known. He: we prove “divisible forever” more certainly from “not composed of indivisibles” (as with the fragment); halving seems only probable to students until one shows you cannot cut into nothings or points. So one solution, Thomas-style: reasoning from the second definition simply is from something probable. Student: is the second then a property of the first? Left unresolved.
• Open question: can “divisible forever” be shown independently, so as to reason from it to the other? Student: cut a line in half, zoom in on the half, it looks the same length, repeat — never changes. Class ends there, left open.

His words #

• edge / end / limit / boundary: the same word in the text; the edge and that of which it is an edge are other.
• axiom: a statement known once you understand its terms, not through other statements; example: the whole is more than the part.
• Wisdom [ed.: Metaphysics]; Natural Hearing [ed.: Physics].
• causa sui: strictly impossible; of God, means negatively “has no cause.”
• known to themselves: known, but not through other statements; not “self-proving.”
• lack [ed.: privation]: non-being of what a thing is able to have and should have, when it should; against mere negation.
• unlimited / infinite: equivocal — lack (endless straight line) vs. negation (point; God).
• love/liking, wanting/desire, joy/pleasure; hate/dislike, turning away, pain/sadness: the six as ordered in us.
• next: nothing of the same kind between.
• one having position: the Greek mathematicians’ point, adding to the one.
• as such [ed.: per se, kath' hauto].
• probable opinion: a starting point Aristotle can reason from without circularity.

Texts #

Read in class

• Aristotle, Physics VI, opening chapter, paragraphs 1-3 and the final paragraphs, with Aquinas’s first reading (inferred); Marietti Latin text consulted
• Anaxagoras fragments, “no smallest of the small” and “what is cannot cease to be by being cut” (handout On the Natural Fragments p. 33 — verified source)

Mentioned

• Aquinas on Physics VI, beginning of the second reading (nows not continuous) (inferred locus)
• Aristotle, Metaphysics IV (axioms of being); Metaphysics V (beginning, before/after, end/limit, four opposites) — Aquinas lectio 2 per verified source; handout Note for Word Beginning p. 2
• Aristotle, Physics I (Melissus and Parmenides; hardness itself, health itself)
• Aristotle, Categories (four opposites)
• Aristotle, Posterior Analytics (no circular reasoning; necessary and as such), with Aquinas’s commentary
• Aristotle, Nicomachean Ethics I (Plato a friend, truth a greater); Plato, Phaedo
• Euclid, definition of number
• Sartre, Being and Nothingness, last sentences; Marx
• Shakespeare, Hamlet (“to be or not to be”); Augustine, “Deo gratias”; Jeremiah on perpetual love; Psalms; “more joy in heaven over one sinner”

His questions #

• Does a point have an end? No: the end would be something different, and the point would have parts.
• Can anything, in a strict sense, be causa sui? No — every cause is a beginning, and nothing is the beginning of itself; of God it means he has no cause.
• Can a point lack a limit, be called limitless? Not as a lack: it is unable to have one; “unlimited” is a mere negation, as with God.
• Is the chair blind? No, it merely does not see; blindness is a privation.
• Does God rejoice in our thanking him, and does it give him more joy? He rejoices in our good, in his divine nature too, but not newly: his knowledge and love are eternal.
• Can there be sadness in God? No: sadness arises from hate, and love itself cannot hate.
• If two points coincide, how much length do they have? As much as one point — none; so no number of points makes a line.
• Can you have a next point? No: between any two points there is a line, and potentially an infinity of points.
• Which is more basic: divisible forever, or not composed of indivisibles? Left open; his suggestion: the reverse direction is only from a probable opinion.
• Can the continuous be shown divisible forever independently? Left open (student’s zooming-in suggestion ends the recording).