Physics VI: The Continuous Is Not Composed of Indivisibles, and Democritus' Cone

Orientation #

He opens with “let’s come back now to the text here at hand” and says these lowly things must be understood in order to understand higher things; this is “the first reading”, emphasizing the line, and he says Aristotle will manifest the same reason for motion in the second reading and for time in the third. He does not name the work or book. The previous class used the continuous/touching/next distinctions against a bodily account of reasoning; the next takes up the argument against motion being composed of indivisibles.

The class, in order #

1. The three definitions (“my house and the neighbor’s house”) #

The point: the continuous is that whose edges are one; touching, whose edges are together but not one and the same; next, that between which there is nothing of the same kind.

• Continuous and touching things are in a way “next” too, but the word is sometimes kept for what is neither continuous nor touching yet has nothing of its kind between.
• My house and the neighbor’s house: not continuous, not touching, but no house in between, so “next”. Two students in the room: not Siamese twins, not touching, but next, “like they say at the doctor’s office, next.”
• Thoughts are “next” in the strict sense: a next thought, a next topic, a next thing to be done, but not continuous or touching.
• Given the definitions, nothing continuous can be from indivisibles, “as a line from points”, if line means something continuous and point something indivisible.
• Thomas notes Aristotle develops the reason for everything continuous (motion down the line, the time it takes), but the first reading emphasizes magnitude; the later readings add “extra ways of seeing it”.

2. Could points come together at their edges? (“a point added to a line is no more”) #

The point: to build a line from points, the points would have to come together, but a point has no edge distinct from itself.

1. Two semicircles have a common edge, the diameter, end of one and beginning of the other; could two points have an edge in common, or two edges together? No: “they have no parts.”
2. Is there a distinction between a point and its edge? No. To imagine one is to imagine the point as a little circle whose circumference differs from what is inside, which gives the point magnitude.

• Objection he meets in class: students deny there is any point; Zeno’s argument that adding a point to a line makes it no longer, so the point is nothing. His reply: that assumes what is is the continuous.
• Aside: Thomas uses this in the De Potentia [locus garbled: “Magna…”]: the creature’s goodness is to God’s goodness not as a shorter line to a longer (then adding it would give a greater good) but as a point to a line, since the distance is infinite; so it adds nothing.
• Aside: Avicenna (quoted by Thomas) says God alone is liberal. Generosity is giving expecting nothing in return; when I give freely I at least get a good act out of it, but God gets nothing, not even a new act. Students find it hard to see “how utterly independent he is” of us.

3. Leading students to admit points exist (“does the surface of the ice cube”) #

The point: from bodies, which students grant (he mentions Berkeley as the exception), one is driven step by step to something with no magnitude at all.

1. Bodies exist and do not go on forever (this table ends); the end of a body is a surface.
2. The surface has length and width but no depth, else you have taken part of the body and not reached its end.
3. The squares that are the faces of a finite cube do not go on forever, else the body would; the end of a surface has no width, else you had not reached its end: a line.
4. The lines bounding the square end, else the square would go on forever; the end of a line has no length, else you had not reached its end: a point, with neither length, width nor depth.

• Conclusion applied: since the point has no magnitude, you cannot distinguish the point from its edge; so two points cannot have an edge in common while the rest is apart, nor have edges touching and the rest not.

4. Aside: limit, beginning, cause (“The point is a limit that has no limit”) #

The point: an analogy between the order of causes and the order of limits.

• Aristotle argues for a first cause in every kind of cause, individually in the particular parts of philosophy and universally in “the second book of wisdom”; in “the beginning of the fifth book” he says every cause is a beginning but not every beginning a cause, and that we first call one end “beginning” and the other “end”, but sometimes call both ends (the endpoints of a line). So beginning is more general than cause, and end or limit more general than beginning.
• Just as there is a cause that has a cause, there is a limit that has a limit: the surface limits the body and has limits (lines); lines limit the square and have limits (points). Is the point a limit that has a limit? No. “Just as God is a cause that has no cause, the point is a limit that has no limit.” “Kind of marvelous.”
• So Aristotle’s text: there is not an edge and some other part of the indivisible; nor are edges together, “for there is no edge of the partless”, since the edge and that of which it is the edge are other.

5. The either-or syllogism: continuous or touching (“either the whole touches the whole”) #

The point: points cannot make a line either by being continuous or by touching; Aristotle says there is the same reason for all indivisibles, and gives fuller particular reasons later for motion and time (as with the four causes: one common reason, then special reasons for form and end).

1. Not continuous, by the foregoing reason (no edge to be one).
2. In everything touching, either whole touches whole, or part touches part, or whole touches part: two circles touching (part to part), a small circle inside a larger (whole to part, and vice versa), the circle drawn twice (whole to whole). He sometimes adds a fourth, touching at the edge, already eliminated.
3. Part-to-part and whole-to-part require parts; the indivisible has none; so whole must touch whole (he warns “whole” may itself imply parts), i.e. the points coincide.
4. Coinciding points have no more length than one point, which has none; so a hundred, a million or an infinity of them give no line. You could have a series of points, but that is not continuous.
5. Aristotle’s own turn: if whole touches whole it is not continuous, for the continuous has one part other than another, parts separate in place, “part outside of part”. He notes this differs a little from his own development.

• Nor is a point next to a point or a now to a now: there is always a line between points and time between nows. On a straight line the endpoints are the farthest apart, but is there a point closest to an endpoint? For any point taken there is another closer. To imagine two points next to each other is “falsely imagining.” There is a next thought, but not a next point.

6. Digression: two meanings of “reason” (“Reason is the reason why man is more than a beast”) #

The point: “reason” names both an ability and a thought or statement of why one holds something.

• Model sentence from his mother: “I see, said the blind man, but he couldn’t see at all”, two senses of “see”. Likewise “reason is able to give a reason”, and “reason is the reason why man is more than a beast”: the first “reason” is Shakespeare’s ability “for large discourse, looking before and after” (genus ability, difference “for large discourse…”); the second is not an ability.
• Defining the second, with students: genus statement (every reason is a statement, not every statement a reason); a student proposes the difference “which gives a cause”. He tests it:
  • The best reason is the reason why something must be so (Socrates and Aristotle teach this). Vertical angles at intersecting straight lines: “they look equal”, “if you measure them”, “Euclid says so” are reasons but not the best; the best is that because the lines are straight a + x = two right angles and b + x = two right angles, and equals to the same are equal, equals from equals are equal. The intersection causes there to be angles; the straightness causes their equality.
  • But not every reason is a reason why it must be so. So he prefers: a thought of why someone thinks a statement is true or false (a statement is what is true or false; the thought may take the form of a statement, but he will not insist). The reason why someone thinks a statement true need not be why it is true: I might accept the Pythagorean theorem on Euclid’s authority. In the perfect case (the straight lines) the reason why I think it is the reason why it is. Thomas: the argument from authority is weakest in philosophy, strongest in theology.
  • Two Chinese restaurants A and B: a good meal at A last month, a lousy one at B: an argument by example, a reasonable guess and maybe the best reason available, but not a reason why I must eat better at A (cooks change, raw materials fail); to say “I must” would attribute more power to the reason than it has.
• His conclusion: he does put statement and cause into the definition, “but not exactly the way you did.” Compare defining statement itself: speech (or sentence) signifying the true or the false.
• Story: a steakhouse on the way to Quebec whose steaks smelled fishy and whose restroom sink was left high after the floor was lowered; a complaining husband, a long-suffering wife, and his announcement on returning from the bathroom.

7. Last paragraph: nothing else between points; denying the consequent (“negating the consequent”) #

The point: nothing of another kind can lie between points or nows.

1. If there were, it would be divisible or indivisible; if divisible, either into indivisibles or into what is always divisible, and the latter is the continuous.
2. Everything continuous is divisible into things always divisible; for if into indivisibles, indivisible would touch indivisible (continuous things have one edge and touch), and that has been shown impossible.

• Form of the argument: if a line divided into two points, the points would touch; they cannot; so deny the antecedent: negating the consequent, you negate the antecedent.
• Meno: Socrates reasons from affirming the antecedent (if virtue is knowledge it can be taught; it is knowledge; so it can be taught) and from denying the consequent (if it can be taught there are teachers; there are none; so it cannot), exemplifying the two valid forms without teaching them formally, and never from denying the antecedent or affirming the consequent.
• His logic classes: four students at the board with the four forms; someone always gets one wrong, once all four were wrong. “If that isn’t a clear enough example of the need for logic”: a mind that thinks what follows does not and what does not follow does will miss much truth and fall into many errors.

8. Surface from lines, body from surfaces; Democritus’s cone (“a pile of pancakes”) #

The point: the same reason (not stated explicitly by Aristotle here, he thinks) shows a surface is not made of lines nor a body of surfaces, and dissolves Democritus’s puzzle.

• Lines have no width, so they could not touch at an edge without coinciding; giving them width makes “a pencil stick”. Surfaces have no depth, so piling them like pancakes gives no depth: “getting something out of nothing.”
• Democritus’s cone (in the Loeb Greek mathematics): cut a cone parallel to the base and compare the two circles. If equal, then equal all the way down and the cone is a cylinder; if unequal, the side is jagged, not the straight line generated by rotating a triangle (as Euclid gets a sphere by rotating a circle about its diameter). Contradiction either way, “like Kant’s antinomies.”
• Student: the trouble is thinking of the cone as layered, composing it of circles without depth. Worked through: the two circles at any one cut are equal (in Euclidean geometry, unlike sawing wood where the blade’s width is lost), and the circles at another cut are equal to each other but not to the first pair; the two circles “weren’t really actually there until you cut it.” The contradiction returns only if one asks about “the next cut”: the solution is “no next”, as there is no next point on a line. Same in two dimensions with a triangle: admit a next cut and the triangle becomes a rectangle.
• You are inside the difficulty unless you understand the continuous as Aristotle does; it arises from thinking the divisible composed of indivisibles or from thinking there is a next, “like a pile of pancakes.”

His words #

• continuous: that whose edges are one
• touching: whose edges are together but not one and the same
• next: between which there is nothing of the same kind
• edge / end / limit: used interchangeably for the boundary of a continuum; “there is no edge of the partless”
• “second book of wisdom”, “fifth book of wisdom” [ed.: Metaphysics II, V]
• reason (1): ability “for large discourse, looking before and after” (Shakespeare’s)
• reason (2): a thought (or statement) of why someone thinks a statement is true or false
• the best reason: the reason why something must be so
• argument by example: the restaurant case; a reasonable guess, not a necessity
• statement: speech (or sentence) signifying the true or the false
• part outside of part: what belongs to the understanding of the continuous
• negating the consequent: from the denial of the consequent to the denial of the antecedent

Texts #

Read in class

• Aristotle, the text at hand on continuous, touching, next, and on the impossibility of a continuum from indivisibles (Physics, inferred; work not named), through the paragraph “it is impossible that any other kind of thing be between points and nows”
• Democritus, fragment on the cone (handout “On the Natural Fragments”, p. 33, verified; “the Loeb edition of Greek mathematics”)

Mentioned

• Thomas Aquinas, commentary on the same text
• Thomas Aquinas, De Potentia (goodness of the creature adds nothing to God’s) [locus unclear in recording]
• Avicenna on God alone being liberal, as quoted by Thomas
• Aristotle, “second book of wisdom” (first cause in every kind), “beginning of the fifth book” (cause, beginning, end/limit)
• Zeno, adding a point to a line
• Berkeley (doubting bodies)
• Shakespeare, definition of reason
• Euclid: vertical angles; sphere by rotating a circle; conic sections; Pythagorean theorem
• Plato, Meno (virtue as knowledge, teachers of virtue; handout “Meno as an introduction to logic”, p. 6, verified)
• Thomas on the argument from authority
• Kant’s antinomies

His questions #

• Could two points have an edge or limit in common, or two edges together? No: they have no parts, and there is no distinction between a point and its edge.
• Does the surface of the ice cube have any depth? No; giving it depth takes part of the body, so you have not reached its end.
• Is the point a limit that has a limit? No; it is a limit with no limit, as God is a cause with no cause.
• If two points touch, they must what? Coincide, and so have no more length than one point.
• Is there a next point on the line, a next now in time? No; for any point there is another closer, and there is always a line between points.
• In “reason is able to give a reason”, does reason have the same meaning? No: the first is the ability, the second a thought or statement of why.
• What separates the reason that is a statement from other statements? Student: it gives a cause; he: a thought of why someone thinks a statement true or false, the best being why it must be so.
• If you divide a line into two points, the two points would be what? Touching, which is impossible; so, negating the consequent, negate the antecedent.
• Democritus’s cone: are the circle above and below the cut equal or unequal, and what is the solution? Equal; the solution is that there is no next cut.