Physics 6.1
A. (231a21-3) Aristotle briefly recapitulates a definition of successive (ἐφεξῆς), touching
(ἁπτόμενον), and continuous (συνεχές). Note that the accounts he gives here are all of twoplace predicates, not of one-place predicates:
two things are continuous if their extremes are one;
two things are touching if their extremes are together (ἅμα);
two things are successive if there is nothing of the same kind (συγγενές) in between
them.
B. (231a23-231b18) Aristotle now formulates and defends a claim employing a one-place use of
continuous, namely:
It is impossible for there to be something continuous composed out of things that are
indivisible. (a24)
The example he gives is that it is impossible for a line (γραμμή) to be composed out of points
(στιγμαί), assuming that a line is continuous and that points are indivisible.
— “to be composed out of some things” translates ἔκ τινων εἶναι. I don’t feel satisfied
that I understand exactly what this means.
— I don’t know how to convert Aristotle’s notion of continuity into a one-place
predicate, and I find it strange that he does not remark on the shift he is making.
— In what follows, “components” means things out of which something is composed.
Argument 1 (a26-29): it can neither be the case that the extremes of two points are one, nor
that the extremes of two points are together [sc. but components of a continuum must have
extremes that are one or together].
(i) It cannot be the case that the extremes of two points are one, because in an indivisible
thing there is no distinction to be made between the extreme and some other part of the
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thing. [Hence the supposition that the extremes of points A and B are one entails that
point A and point B are one and the same point (?)]
(ii) It cannot be the case that the extremes of two points are together because, in fact, a
thing without parts has no extreme. An extreme is other than that of which it is the
extreme.
— The argument given under (ii) entails claim (i).
— (i) seems to show that points cannot be continuous with each other, while (ii)
seems to show that points cannot touch each other. On the other hand, the text will
proceed as if Aristotle has only argued that points cannot be continuous with each
other; he is going to give a further argument that points cannot touch each other.
— Why can’t an indivisible thing have extremes? Aristotle argues (231a26-7, 28-9) as if
an extreme (ἔσχατον) is a part—hence whatever has extremes has parts. But I
thought that an extreme or limit is not part of the thing whose limit it is. It does seem
somehow plausible that whatever has extremes must have parts, but I don’t see why
this must be so. I’m not satisfied with what Aristotle says about it.
Argument 2 (a29-b6): If something continuous is composed out of indivisible things, then
those indivisible things must either (i) be continuous with each other, or (ii) touch each other.
(i) They cannot be continuous with each other “for the reason given” (b1).
— Does this refer only to point (i) in Argument 1, or to the whole of Argument 1?
(ii) If two indivisible things touch, they must touch “whole to whole” (they cannot touch
part to whole or part to part, given that they are partless). But if they touch whole to
whole, there will not be anything continuous composed out of them, because a
continuous thing has parts that are separated in place.
— I take it that touching “whole to whole” entails occupying exactly the same spot: in
other words, being “together” (ἅμα).
— What does touching whole to whole, whole to part, and part to part mean? One
idea is that “whole” and “part” refers to wholes or parts of extremes. On this
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interpretation, for A and B to touch part to part would mean that part of A’s extreme
coincides (is ἅμα) with part of B’s extreme; this is (I take it) the most common way to
be in contact. A and B would touch whole to part if all of A’s extreme coincides with
part of B’s extreme: this will happen if A is completely surrounded by B (and A has
no holes: then A’s outer and only limit will coincide with B’s inner limit). A and B
would touch whole to whole if all of A’s extreme coincides with all of B’s extreme.
— On this interpretation, why is it necessary that things without parts touch
whole to whole? One would have to argue, I think, that if something has no parts
then its extreme also has no parts.
— Another interpretation would be to take “touching” in this context to mean
“overlapping”. Then A and B touch part to part if part of A is in the same place as part
of B; A and B touch whole to part if all of A is in the same place as part of B; A and B
touch whole to whole if all of A is in the same place as all of B. This interpretation is
suggested by Michael White, p. 29.
— On this interpretation, it is easier to see why things without parts can only
touch whole to whole. The interpretation also yields a nicer structure for
Aristotle’s argumentation: Argument 1 (a26-9) would establish that indivisibles
cannot have shared or coinciding extremes, while Argument 2 (a29-b6) would
establish that they cannot overlap in such a way as to constitute a continuous
whole. The term συνεχεῖς at 231a29 and b1 could be taken to cover both the case
of a shared extreme and the case of coinciding extremes—i.e., both continuity
and touching according to Aristotle’s official definitions—while the term
ἁπτομένας at 231a30 and ἅπτεται at b2 takes on the extended sense of
“overlapping”.
— On the other hand, it is unattractive to interpret Aristotle’s terminology in a
non-standard way, especially when the official definitions have just been stated.
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[In what follows, it isn’t clear which arguments are subordinate to which; so I will just list
them as “items.”]
Item 3 (b6-10) Moreover, a magnitude (μῆκος) or stretch of time cannot be composed out of
successive points or moments.
There are no successive points or moments, because there is always a line between two
points, and time between two moments, whereas two things are successive only if there
is nothing of the same kind between them.
[Implication: a line and a point are of the same kind (συγγενῆ) with each other, and a
moment and a time are of the same kind with each other. (?)]
Item 4 (b10-12): If a time or magnitude is composed out of some things, then it is [or can be]
divided into those things (ἐξ ὧν ἐστιν ἑκάτερον, εἰς ταῦτα διαιρεῖται, b10-11). No continuous
thing is divisible into partless things. [Therefore no time or magnitude is composed out of
partless or indivisible things. (“Partless” and “indivisible” seem to be treated as equivalent.)]
— Where does the premise that no continuous thing is divisible into partless things
come from? Aristotle expresses it using the imperfect tense, thus implying that it was
previously established: ἀλλ’ οὐθὲν ἦν τῶν συνεχῶν εἰς ἀμερῆ διαιρετόν, b11-12.
— Ross says that Aristotle is referring back to Arguments 1 and 2, i.e., the passage from
a26 to b6. There are two objections to this view, and I think they are decisive. First, the
suggestion is now in play that times or magnitudes might be composed out of
successive indivisibles (cf. item 3); but arguments 1 and 2 only showed that they
cannot be composed out of continuous or touching indivisibles. It was an explicit
premise of argument 2 that if there were indivisible components of a continuum,
they would have to touch or be continuous. Second, the present argument turns on a
connection between the “composed out of ” locution (ἔκ τινων εἶναι) and the
“divisible into” locution (εἴς τινα διαιρεῖται), and so it is crucial that the premise is
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about divisibility, as opposed to composition. But the arguments in the chapter up to
now have all been about composition, not about divisibility.
— We know from the opening of book 3 and elsewhere that Aristotle’s contemporaries
often defined continuity in terms of infinite divisibility (Phys 3.1, 200b18-20: διὸ καὶ
τοῖς ὁριζομένοις τὸ συνεχὲς συμβαίνει προσχρήσασθαι πολλάκις τῷ λόγῳ τῷ τοῦ
ἀπείρου, ὡς τὸ εἰς ἄπειρον διαιρετὸν συνεχὲς ὄν). However, I had hoped that Physics
6.1 offers a proof that what is continuous—under Aristotle’s definition of continuity—
is infinitely divisible. It now seems to me that no such proof is offered; instead,
infinite divisibility of the continuous is assumed, and Aristotle’s arguments only
concern composition.
Item 5 (b12-15): “There cannot be another kind in between”; if there were, it would have to be
divisible or indivisible, etc. Obscure. Developing and (again) rejecting the proposal that a
continuum could be composed out of successive indivisible components?
Item 6 (b15-18): Every continuum is infinitely divisible; otherwise there would be an
indivisible touching an indivisible.
— In items 5 and 6, Aristotle continues working with the concept of divisibility. I don’t know
why; I don’t understand the differences and connections between constitution out of some
things and divisibility into those things, between ἔκ τινων εἶναι and εἴς τινα διαιρετόν εἶναι.
C. (231b18-232a22) In the remaining part of the chapter, Aristotle discusses the divisibility of
magnitude, motion, and time, with most of his attention being devoted to motion.
He begins by saying that either magnitude, time, and motion are all infinitely divisible, or
none is infinitely divisible. His locution is somewhat strange: τοῦ δ’ αὐτοῦ λόγου μέγεθος καὶ
χρόνον καὶ κίνησιν ἐξ ἀδιαιρέτων συγκεῖσθαι καὶ διαιρεῖσθαι εἰς ἀδιαίρετα, ἢ μηθέν. Hardie
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and Gaye translate: “The same reasoning applies equally to magnitude, to time, and to
motion: either all of these are composed of indivisibles and are divisible into indivisibles, or
none.” Somehow I don’t find this quite satisfying, but I can’t say why.
— There is a parallel locution in 6.10, 241a23-6: μοναχῶς γὰρ ἂν κινοῖτο τὸ ἀμερὲς καὶ
ἀδιαίρετον, εἰ ἦν ἐν τῷ νῦν κινεῖσθαι δυνατὸν τῷ ἀτόμῳ· τοῦ γὰρ αὐτοῦ λόγου ἐν τῷ νῦν
κινεῖσθαι καὶ ἀδιαίρετόν τι κινεῖσθαι.
The structure of the remaining discussion seems to be this:
1. Brief argument that if a magnitude is composed out of indivisibles, then a motion over
that magnitude will be composed out of indivisibles (231b21-25).
2. Long argument (or cluster of arguments) for the absurdity of the supposition that a
motion is composed out of indivisibles (231b25-232a17).
3. Brief argument that the time of a motion is composed out of indivisibles if and only if
the motion and the magnitude traversed by the motion are composed out of indivisibles
(232a18-22).
Now in more detail:
1. (231b21-25) This seems to be assertion rather than argument. Aristotle says that if a
magnitude ABC is composed out of indivisibles A, B, and C, then motion over that
magnitude will be composed out of corresponding indivisibles D, E, and F.
— I would expect him to argue for this by reductio: given that Z traverses A by motionpart D, suppose that D is divisible into D1 and D2; then Z would traverse parts of A, A1
and A2, by motion-parts D1 and D2; but A was supposed to be indivisble.
2. (231b25-232a17) Aristotle begins this section by asserting the equivalence of “motion is
present” and “something is moving” (παρεῖναι κίνησιν and κινεῖσθαί τι). He thus allows
himself to move from the supposition that a motion is composed out of indivisibles to the
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supposition that something’s moving is composed out of indivisibles (καὶ τὸ κινεῖσθαι ἔσται
ἐξ ἀδιαιρέτων).
From a purely logical point of view, I see some point to this move: in the upcoming
argument, he is going to express a distinction by way of the perfective and imperfective
aspects of the verb κινεῖσθαι, and he therefore requires a tight connection between that verb
and the noun κίνησις, which he’s been using up to now.
Content-wise, though, I’m not sure what it means to say that τὸ κινεῖσθαι is composed out of
indivisibles, and whether the nominalized infinitive τὸ κινεῖσθαι means the same as the noun
ἡ κίνησις, or something different.
Aristotle then proceeds as follows.
[When I need to render κινεῖσθαι with an internal accusative, I’ll use the verb “traverse.”]
a.
It is impossible for something simultaneously to be moving to Y and to have moved to Y.
(231b28-232a1)
b.
If Z at one time is traversing the magnitude A, and at a later time has traversed A, then A
must be divisible. That is because while Z is traversing A, it must be in between where it
was before it began to traverse A, and where it will be once it has traversed A. (a2-6)
c.
Therefore, if Z is to make it across an indivisible magnitude A, we must suppose that Z
first has not traversed A, and later has traversed A, but never is traversing A. (Not
explicitly inferred, but is supposed at a7-8)
d.
Now let us suppose that Z undergoes a motion across the whole magnitude ABC, i.e.,
that for some time it is traversing ABC, with the motion it is then undergoing being DEF.
[If we look at the parts of ABC, the idea will be that Z first has not traversed any of it,
then has traversed A, then has traversed AB, then has traversed ABC, but never is
traversing A or is traversing B or is traversing C.]
Now we get some paradoxical consequences.
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First, there will be a motion (DEF) composed not out of motions but out of κινήματα,
since each part of ABC is gotten across not by moving through it but by a mere
transition from not-having-traversed to having-traversed (a8-10).
Second, it will be possible to have walked a distance without ever walking that distance
(a10-11).
Third, supposing that everything either is moving or is resting, it will be possible for
something to be moving and resting continuously throughout the same period of time.
For one the one hand,
throughout the time from not having traversed any of ABC to having traversed ABC,
Z was traversing ABC,
but on the other hand,
Z was resting throughout the time from not having traversed any of ABC to having
traversed A, and throughout the time from having traversed only A to having
traversed AB, and throughout the time from having traversed only AB to having
traversed ABC. Therefore Z was resting throughout the time from not having
traversed any of ABC to having traversed the whole of ABC.
Fourth, a dilemma:
if D, E, and F are motions, then it will be possible for something not to be moving
but rather resting although a motion is present to it;
if D, E, and F are not motions then a motion (DEF) will be composed out of things
that are not motions (D, E, and F).
3. (232a18-22) Aristotle takes it as already established that a motion is divisible if and only if
the magnitude traversed is divisible. [I think he has only argued for the “if ” direction, not the
“only if ” direction.] He now invokes the principle (P) that something moving at constant
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speed (ἰσοταχές) traverses a lesser distance in a lesser time (232a20-1), in order to argue that
the time of a motion is divisible if and only if the motion (and magnitude) is divisible. He
doesn’t spell out the argument; let me try to supply one:
[Motion and magnitude divisible time divisible:] Suppose Z traverses A at constant
speed by motion D, occupying time T. Suppose that A and D are divisible; let Z traverse
A1 by D1, occupying time T1, where A1 and D1 are proper parts of A and D, respectively.
Given that Z traverses a lesser distance by way of D1 than it traverses by way of D,
principle P tells us that time T1 is less than time T. [I want: that T1 is a proper part of T.]
Hence time T is divisible.
[Time divisible motion and magnitude divisible:] Suppose T is divisible, and take a
proper part T1. From principle P, Z traverses a distance less than A in T1; hence
magnitude A is divisible; hence motion D is divisible.
— What is the best way to interpret principle P? Aristotle writes: ἐν τῷ ἐλάττονι δὲ τὸ
ἰσοταχὲς δίεισιν ἔλαττον: something with equal velocity traverses less distance in less
time.
— I was interpreting ἰσοταχές as “moving with constant velocity”; that now strikes me as
suspect. I think the idea is that if Y moves at the same speed as Z, then Y traverses less
distance than Z if and only if Y moves for less time than Z.
— Aristotle clearly appeals to a second moving body, moving at the same speed as a
given body (ὁμοταχῶς κινούμενον, ὁμοταχές), in 6.6, 236b34-237a3.
— As long as only one moving body is in question, there’s no need to specify anything
about its velocity. Whether its velocity is constant or not, it will move a proper part of
the distance in a proper part of the time.
— But maybe in Aristotle ἰσοταχές means constant speed and ὁμοταχές means same
speed? In 6.7, 237b27 and 34 ἰσοταχῶς seems to mean “with constant speed.” On the other
hand, in 7.4, 249a13 and 19, ἰσοταχές clearly means “with equal speed.”
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— In addition to the question about ἰσοταχές, I wonder whether Aristotle did well to
formulate his principle in terms of “the lesser” (ἔλλατον) instead of appealing to proper
parts. The fact that there is a magnitude less then X does not immediately entail that X is
divisible, whereas the fact that X has a proper part does immediately entail that X is
divisible.
— For his present purposes, I think the “lesser” formulation will work. His arguments
concern merely possible motions, and he can stipulate that these begin at the same place
and/or time, in which case the lesser interval will be a proper part of the greater. For
example: Suppose Z traverses magnitude AB in time T. Suppose time is divisible. Let Y
move with equal speed and in the same direction as Z, starting from A, for time T2<T.
Then Y will traverse a magnitude AC<AB. Since AC shares an endpoint with AB, has the
same direction, and is lesser, AC is a proper part of AB. Hence AB is divisible.
— Aristotle seems to assume that a proper part of something X must be lesser than X. If
there are infinite magnitudes or times, this is not true. If we allow motions over infinite
magnitudes, we may affirm that a proper part is traversed in a proper part of the time,
but deny that a lesser distance is always traversed in a lesser time. This point is important
for evaluating Aristotle’s arguments in 6.7 that it is impossible to traverse a finite
distance in an infinite time or an infinite distance in a finite time. One of the steps in his
argument is that if something traverses a (finite) interval AB in an infinite time, then it
will traverse any proper part of AB in a finite time (238a8-11); probably his reason is that
otherwise unequal intervals would be traversed by the same thing in equal (because both
infinite) times. We should insist that the time taken to traverse a proper part may be
infinite, and that in this case although the time taken to traverse the part will not be less,
it will be a proper part of the time taken to traverse the whole interval.
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