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of Ancient Mathematics
Zeno's arguments on the Size of a Body
Zeno's arguments on Motion
- Arguments on the Size
of a Body (assuming continuity and divisibility)
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 Step 1 (Zeno, frag.
DK B2): For if it were added to another thing,"
he says, "it would not make it larger. For given it
has no size, when it is added it could not contribute anything
to its magnitude. And so what is added in this way would
be nothing. If when it is removed, the other would be not
be any smaller, and when added the other is not increased, it
is clear that what is added isn't anything and what is removed
isn't either.
Step 2 (Zeno, frag. DK B1):
Having first proved that "if the existent does not have
magnitude it would not exist," he infers that "if it
is, each thing must have some size and bulk, and that one part
of it can be apart from another. And with regard to what
there is at this stage*, the same argument
follows. For that will have size and some part of it will
be there at this stage*. To say this
once is like saying it constantly. For there will not be
some sort of limit to this, nor will one thing not be related
to another [as a part]. In this way, if there are many,
they must be both small and large, small so as not to have size
and large so as to be infinite.
The idea behind the argument seems to be that if the pieces have
no size, the totality will be infinite (of an infinite number
of bits); however, if they don't have size, the totality will
be the totality of nothings and so will be nothing. You
never get what you started with.
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Here is the context of Simplicius' discussion:
Simplicius, Commentary on Aristotle's Physics, 138.29-139.23
But Alexander seems to have taken his opinion about how Zeno established
the one from the accounts of Eudemus. For Eudemus says in
the Physics, "And so is it that this does not exist, but
is some one? For he raised this puzzle, and they say that
Zeno says that if someone should propose to him whatever the one
is, he will have to speak of real things. But he raised
puzzles, as it seems, from the fact that each of the perceptibles
is said to be both categorically many and many by division, but
supposes the point to be nothing--namely he did not consider to
be a real thing that which neither increases something by being
added nor diminishes it by being taken away." And it
is likely that Zeno did exercises in taking each side (whence
this is called "double-tongued") and produced some such
puzzling arguments about the one. In his treatise which
contains many dialectical proofs on each side, however, he shows
that it follows for anyone who says that there are many that he
must say opposite things of which one dialectical proofs is where
he shows that if there are manny, they will be both big and small,
big so as to be infinite in size and small so as to have no size.
In this argument, he shows that where there is neither size nor
bulk nor mass there is nothing, nor would this exist. (Zeno, frag. DK B2) "For
if it were added to another thing," he says, "it would
not make it larger. For given it has not size, when it is
added it could not contribute anything to its magnitude.
And so what is added in this way would be nothing. If when
it is removed, the other would be not be any smaller, and when
added the other is not increased, it is clear that what is added
is isn't anything and what is removed isn't either."
And Zeno doesn't say this to destroy the one, but to destroy
the view that each of the many and infinite things has size, since
for each thing taken there is always another because of infinite
division. He shows this after showing that nothing has size
from the fact that each of the many is the same as itself and
one. And Themistius says that the argument of Zeno establishes
that what-is is onefrom the fact that it is continuous and indivisible,
"since if it were divided," he says, "there will
not be anything which is precisely one due to the infinite divisibility
of bodies." But Zeno seems rather to say that there
is not be many things.
Simplicius, Commentary on Aristotle's Physics, 140.27-141.8
And what should we say of the many, considering what is comes
up in the book of Zeno. For once more he shows that if there
are many the same things are finite and infinite. Zeno writes
these things, (verbatim), (Zeno, frag.
DK B3) "If there are many, it is necessary
that they be as many as they are and neither more than they are
nor less. If they are many, the existents will be infinite.
For there will always be other things between each of the existents,
and again other things between them. And way the existents
are in this way infinite." And in this way he showed
the infinite in multitude from dichotomy. But as to the
infinite in magnitude, he earlier proved with the same dialectical
reasoning. (Zeno, frag. DK
B1) Having first proved that "if the existent does
not have magnitude it would not exist," he infers that "if
it is, each thing must have some size and bulk, and that one part
of it can be apart from another. And with regard to what
there is at this stage*, the same argument
follows. For that will have size and some part of it will
be there at this stage*. To say this
once is like saying it constantly. For there will not be
some sort of limit to this, nor will one thing not be related
to another [as a part]. In this way if there are many, they
must be both small and large, small so as not to have size and
large so as to be infinite."
141.8
*The Greek word, 'proekhein' can mean
'protrude' or 'be outstanding'. However, in the context,
it is contrasted with 'apekhein' (be apart), and so, I think,
the sense is 'be there before some part of it departs from
it'' or later 'be there as a part before it departs'.
There are four argument of Zeno on montion which present difficulties
for those solving them.
Aristotle, Physics VI 9.239b5-240a18,
presents all four arguments. To follow the arguments in
the order of Aristotle's text, follow the links:
- Arguments on Motion
1: the Dichotomy
 Aristotle,
Physics VI 9.239b11-14:
First is the one concerning something not moving because the
mover must earlier arrive at the half before it reaches the end,
which we discussed earlier. (next
text)
(the standard modern interpretation--intitial
diagram) The point is that to leave, he must get halfway, and halfway
to the end, and so forth, so that he will never get started. Let D
be the distance to be traveled, and let D1 = D/2 and Dn+1
= (D + Dn)/2. There is no end to the infinite series, D1,
..., Dn, ....
(the popular interpretation,
and probably the correct one) The point is that to leave, he must get
halfway, and halfway to there, and so forth, so that he will never get started.
Let D be the distance to be traveled, and let Dn+1 = Dn
- 1/2 Dn. There is no beginning to the infinite series,
..., Dn, ..., D1, D = ..., D/2n..., D/4,
D/2, D. |
- Arguments on Motion
2: Achilles
 Aristotle,
Physics VI 9.239b14-240a18:
Second is the so-called Achilles. This is that the slower
runner will never be overtaken by the fastest. For the
pursuer must first come to where the one running away set out,
so that the slower must always be somewhat ahead. This
argument is the same as the dichotomy, but differs the extra
magnitude is not divided in two. And so it follows from
this argument that the slower is not overtaken, but this occurs
in about the same way as the dichotomy (for in both it happens
that it does not arrive at the limit from the magnitude being
getting divided in some way, except that additionally not even
the most celebrated fastest one will in its pursuit of the slowest.
Thus the solution must be the same. And the claim that
what is ahead won't be overtaken is false. For while it
is ahead it is not overtaken; nevertheless, it is overtaken if
, in fact, it is given that it traverses the finite distance.
(next text)
It is unimporant for the argument how fast Achilles and the
tortoise are running. Suppose that the tortoise has reached in
time tn a distance Dn and that Achilles
has travelled less than Dn in tn.
Then the next time one takes tn+1 = the time it takes
for Achilles to reach Dn.
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- Arguments on Motion
3: the Arrow
 Aristotle,
Physics VI 9.239b5-9
Zeno argues fallaciously. For if, he says, everything is
always at rest when it is at an equal to itself, while that which
moves is always in the now, the moving arrow lack motion.
But this is false, since time is not composed of indivisible
nows, just as no other magnitude is. (next
text)
Aristotle, Physics
VI 9.239b29-33
And so these are the two arguments , while the third was just
stated, that the arrow, which is moving, stays put
And it follows according to the assumption that time is composed
of nows. For if this is not granted, there will be no deduction.
(next text)
This puzzle is somewhat difficult to work out.
It seems that Zeno argues that the arrow is in a place equal
to itself means that it is there for a stretch of time (where
'is' implies duration, as for Plato). Hence the arrow is
there for a moment and is at rest.
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- Arguments on
Motion 4: the Moving Rows in the Stadium
Alexander's diagram according to Simplicius
A Standing blocks
B blocks moving from D to E
G blocks moving from E to D
D start of the stadium
E end of the stadium
Aristotle, Physics
VI 9.239b33-240a18
Fourth is the argument about equal masses moving oppositely along
equals masses, some from the end of the stadium, and others from
the middle, with equal speed, where he thinks it follows that
the half time will equal the double. The fallacy is that
the mass moving along one in motion is assumed to move an equal
magnitude in an equal time with equal speed as one moving along
one at rest. But this is false. For example, let the
stationary masses be AA, those starting from the middle be BB,
which are equal to the others in size and number, and let GG be
those moving from the end, with these too being equal in size
and number with those, and let them be equally fast as the B's.
It happens that the first B and the first G will be at the end
at the same time, when they are moving alongside one another.
It follows that G will traverse all the B's,* while B traverses
half (the A's). Thus, the time will be half. For each
is alongside each for an equal time. At the same time the
first B will have moved along all the G's, since the first G and
the first B will be at opposite ends [becoming in an equal time
alongside each of the B's as alongside each of the A's, as he
says], since both come to be alongside the A's in an equal time.
And so this is the argument, and it follows according to the mentioned
falsehood.
*Ross, Aristotle's Physics (Oxford, 1949) and Hardie
and Gaye (see revised Oxford translation, Princeton, 1984) unnecessarily
delete 'the B', and in the latter case add 'the A's'. Coherence
may be achieved by understanding 'the A's' in the next phrase.
Most scholars agree that the assumption is that the distances
travelled are not continuous but are conceived as minima, whereas
the dichotomy and the Achilles assume continuity. Let the
blocks be minima. Then the rows BBBB and GGGG each move one unit
at a time. But relative to each other they move two units.
To take an example, since G4 is opposite B4
and A4 and then opposite B2 and A3,
there must have been a time when G4 was opposite B3
in between A3 and A4.
Beginning of Arguments on Motion
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