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Zeno's arguments on the Size of a Body

Zeno's arguments on Motion

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.
 

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.
 
Argument 1:  Dichotomy Argument 3:  Arrow
Argument 2:  Achilles (and the tortoise) Argument 4:  Moving Rows in the Stadium

Aristotle, Physics VI 9.239b5-240a18, presents all four arguments.  To follow the arguments in the order of Aristotle's text, follow the links:
Text 1:  Arrow Text 4:  Dichotomy (again)
Text 2:  Dichotomy Text 5:  Moving Rows in the Stadium
Text 3:  Achilles

 
 
  • 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.

  • 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.

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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