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## CHAPTER II |
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Arguments about atomism and infinite divisibility were first developed in detail by Zeno of Elea (born c. 490 bc) in the form of his now famous paradoxes. Since these paradoxes have had a very important influence on subsequent disputes regarding atomism and divisionism, it is important to identify them here. I shall give the basic thrust of each paradox by a short statement, and then, in order to show the logical form of the underlying argument, I shall offer an expanded version of it. I shall defer my critical comments on the paradoxes and the relevance to my subject to the next chapter.

I can succinctly state the Achilles paradox as follows: Achilles and the tortoise race; the tortoise is given a head start. By the time Achilles reaches the spot where the tortoise was, the tortoise will have moved on. So Achilles can never catch the tortoise.^{(1)}

To appreciate the logical structure of the paradox, the relevant premisses and conclusions must be identified and set forth in explicit terms. Some of the premisses are merely implicit in the brief statement of the paradox.

A basic assumption of the argument is this:

AP0 ^{*}To catch the tortoise, Achilles must eventually occupy the same spot as the tortoise.

This premiss is present in the argument in a contrapositive form:

AP0' If it is always the case that Achilles does not occupy the same spot as the tortoise, then Achilles never catches the tortoise.

The information given in the argument ostensibly does not permit the conclusion that Achilles does catch the tortoise. What the premisses do ostensibly warrant is indicated as follows:

The initial premiss is:

AP1 The tortoise is given a head start. AI1 AP1 ==> AC1 AC1 The tortoise is ahead of Achilles. AP2 If the tortoise is ahead of Achilles then Achilles runs toward the tortoise. AI2 AC1 & AP2 ==> AC2 AC2 Achilles runs toward the tortoise. AP3 If Achilles runs toward the tortoise, then the tortoise runs on ahead to another spot. AI3 AC2 & AP3 ==> AC3 AC3 The tortoise runs on ahead to another spot. AP4 If the tortoise runs on ahead to another spot then, when Achilles reaches the spot previously occupied by the tortoise, the tortoise will occupy a spot different from the one Achilles occupies. AI4 AC3 & AP4 ==> AC4 AC4 The tortoise occupies a spot different from the one Achilles occupies. AP5 If the tortoise occupies a spot different from the one Achilles occupies, then the tortoise is ahead of Achilles. AI5 AC4 & AP5 ==> AC5 AC5 The tortoise is ahead of Achilles.

But AC5 is just AC1; the argument leads to the original premiss. No valid reasoning with these premisses leads to any other conclusion. No matter how many times the argument is followed through, the conclusion is always the same -- the tortoise is ahead of Achilles. No premiss or conclusion in the argument leads to Achilles being in the same spot as the tortoise.

We could argue that what is true of each instant is true of the whole race, but that would involve the fallacy of composition. __A stronger way to conclude__ that Achilles can never catch the tortoise is to use mathematical induction on each iteration of the argument. AC1 is true after the first iteration. Assume AC1 is true after N iterations. Then, by applying the premisses in order, AC1 is also true after N + 1 iterations. These two conditions satisfy the requirements for mathematical induction and allow us to conclude that it is always the case (after every iteration) that Achilles does not occupy the same spot as the tortoise. (If always AC4, then Achilles can never catch the tortoise.) However, since mathematical induction was not available to Zeno, we may appeal to additional premisses.

AP6 There is no end to an infinite sequence (of steps or acts). AP7 Achilles's repeated attempts to catch the tortoise constitute an infinite sequence. AI6 AP6 & AP7 ==> AC6. AC6 There is no end to Achilles repeated attempts to catch the tortoise. In other words, Achilles can never come to the end of his sequence of attempts to catch the tortoise -- that is, Achilles can't catch the tortoise.

It is interesting to note that this argument is valid without reference to the speeds of either Achilles or the tortoise. It is clear that if Achilles runs slower than the tortoise or at the same speed as the tortoise then we have no difficulty with the conclusion. But if Achilles runs faster than the tortoise the conclusion is absurd. Since Achilles is "the fleetest of all Greek warriors"^{(2)}, we may assume:

AP8 Achilles runs faster than the tortoise. AP9 If Achilles runs faster than the Tortoise, and he runs toward the tortoise, then Achilles will be closer to the Tortoise when he reaches the spot the tortoise left.

With these two additional premisses, it is possible to conclude, validly, that Achilles is always getting closer to the tortoise.

AI7 AC2 & AP8 & AP9 => AC7 AC7 Achilles is closer to the tortoise.

But nothing in these premisses allows us to conclude that Achilles actually catches the tortoise. The paradox is that an apparently valid argument with acceptable premisses yields such an unacceptable outcome.

The dichotomy has two forms.

1. For Achilles to reach any point he must get half way. Then he has to get half the rest of the way. Since there will always be a fraction to go, he can never reach any point.

This argument must also be unpacked and stated in the form of premisses and conclusions. Here are the relevant premisses:

D1P1

For Achilles to reach another point (his destination) he must first get half way to it. D1P2

If Achilles is at a point not his destination, then he moves toward his destination point. D1P3

If Achilles moves toward his destination point then he first reaches a point half way towards it. D1P4

If Achilles is at a point half way towards his destination, then he is at a point which is not his destination. D1P5

Achilles is at a point which is not his destination.

And here is the form of the argument:

D1I1

D1P2 & D1P5 => D1C1 D1C1

Achilles moves toward his destination. D1I2

D1C1 & D1P3 => D1C2 D1C2

Achilles reaches a point half way toward his destination. D1I3

D1C2 & D1P4 => D1C3 D1C3

Achilles is at a point which is not his destination.

But D1C3 is just D1P5. As in the Achilles, the argument leads to the original premiss. No valid reasoning leads to any other conclusion. No matter how many times the argument is followed through, the conclusion is always the same. Nothing in the argument leads to Achilles being at the other point. As in the Achilles, this form of the Dichotomy rests on the premisses (1), that an infinite series has no end and (2), that Achilles is traversing an infinite series of points. Since an infinite series has no final term it is concluded that Achilles cannot reach the end of the series. But this is just what he must do in order to reach the other point.

2. The second form of the paradox can be explained this way:

For Achilles to reach any point he must get half way. To get half way, he must get to half that, etc.. To get anywhere, he must have already covered an infinite number of points.^{(3)}

In this form of the Dichotomy, the infinite series of points that Achilles is to traverse is "reversed" from that of the first form. As the points are enumerated, the second is one quarter of the way to the destination in the second form while it is three quarters of the way to the destination in the first form. Between Achilles and any other point there is an infinite series of points. For Achilles to get to any one of these points, he must have already traversed, in reverse order, the infinite series of points which, according to the premiss in the first form, he can not come to the end of. This form of the Dichotomy entails a premiss which is less clearly acceptable.

D2P1

Achilles cannot traverse an infinite number of points. D2P2

Between Achilles and any point is an infinite number of points. D2P3

To traverse the distance to any point, Achilles must traverse all intervening points. D2I1

D2P2 & D2P3 => D2C1 D2C1

To traverse the distance to any point, Achilles must traverse an infinite number of points. D2I2

D2C1 & D2P1 => D2C2 (Modus Tolens) D2C2

Achilles cannot traverse the distance to any point.

This argument is more straight forward, but it is also clear that premiss D2P1 is not obviously true. However, this argument and the preceding first form of the Dichotomy exhibit symmetry. Support for premiss D2P1 can be achieved with a additional premisses -- namely:

D2P4

Achilles can traverse an infinite sequence of points if and only if he can come to the end of a infinite series.

D2P5

Achilles can not come to the end to an infinite series.

D2I3

D2P4 & D2P5 => D2P1 (Modus tolens)

D2P1 falls out as a conclusion from these two less questionable premisses. Since it is clear that Achilles can not come to the end of an infinite sequence, it must also be the case that he can not traverse an infinite sequence of points.

The first form of the Dichotomy concludes that Achilles can't reach any point, and the second form concludes that he can't even get started. Although the arguments appear to be valid, both conclusions are clearly absurd.

The arrow cannot move. To do so requires that it be in one place equal to itself during one part of an instant and another during another. Also, it would occupy a space larger than itself in order for it to have room to move.^{(4)}^{,}^{(5)}

RP1

Everything at a place equal to itself is at rest. RP2

A flying arrow is always at a place equal to itself at every instant in its flight. RI1

RP1 & RP2 => RC1 RC1

A flying arrow is at rest at every instant in its flight. RP3

That which is at rest at every instant does not move. RI2

RC1 & RP3 => RC2 RC2

A flying arrow does not move.

While the above rendition of the argument suffers from the fallacy of composition, it is possible to render the argument in a form not subject to this fallacy. This can be done as follows:

Def: |
RD1 |
An instant is an indivisible minimal element of time. |

Def: |
RD2 |
Something is at rest (instantaneously) iff it is in its place (one place equal to itself) in one instant and it is in the same place (equal to itself) in different instants (remains at rest). |

Def: |
RD3 |
Something moves iff it is not at rest. |

RI3 |
RD2 & RD3 => RC2 | |

RC2 |
Something moves iff either (A) it is not in one place (equal to itself) in one instant or (B) it is in different places (equal to itself) in different instants. |

I will present the two disjuncts as separate cases.

Case 1:

It would appear that this case could be disposed of immediately by noting that it seems to contradict RP2 directly. -- RP2 could be interpreted that everything is always in a place equal to itself. -- However, it is instructive to analyze more deeply. We can consider the 'not' as applying to "one place" or alternatively to "one instant". Let us first consider the 'not' applied to "one place". "Not one place" becomes "different places".(A) That which moves is not in one place (equal to itself) in one instant.

RI42 RC2 => RC22 RC22 Something moves iff it is at different places in the same instant. (An instant has more than one place.)

Although this interpretation is practically inconceivable to us, it is the interpretation intended by the argument. But we can think of it as like the blurred photograph of something in motion. The object is apparently at (many) different places (equal to itself) at the "instant" the photograph was taken.

RP4

If something is at different places during the same instant then it is not at one place equal to itself. RI5

RP4 => RC3 RC3

If something is at one place equal to itself then it is not at two different places during the same instant. RI6

RP2 & RC3 => RC4 RC4

An arrow in flight is not at different places during the same instant. RI7

RC22 & RC4 => RC5 RC5

An arrow in flight does not move.

The "also" clause has more the form of an "otherwise" clause.

RP5

Something cannot occupy a space smaller than itself. RP6

If something is not at a place equal to itself then it occupies a space either smaller than or larger than itself. RI8

RP5 & RP6 => RC7 RC7

If something is not at a place equal to itself then it occupies a space larger than itself. RI9

RP4 & RC6 => RC7 RC7

If something is at different places during the same instant then it occupies a space larger than itself. RI10

RC22 & RC7 => RC8 RC8

If something moves then it occupies a space larger than itself. (An arrow must occupy a space larger than itself if it is to move.)

RC22, however is also in direct conflict with a widely held premiss.

RP7 Nothing can be in two different places (equal to itself) during the same (one) instant.

This case concludes that the arrow cannot move during an instant. We are left with case 2.

Case 2:

(B) That which moves is at different places (equal to itself) in different instants.

RI41 RC2 => RC21 RC21 Something moves if it is at different places (equal to itself) in different instants. (This is our usual understanding of motion.)

We are considering whether an arrow can be in motion in an instant. By the above case, something could move only if it were in different places in __different__ instants. Therefore, for it to move in the one instant under consideration, that instant would have to have two parts which were also instants. It would be these "sub-instants" in which the arrow were at different places. But, by RD1, an instant is indivisible; so, it has no such parts which are instants. Consequently, *at each instant* it is not possible for the arrow to be at different places in *different instants*.

In either case the arrow cannot move. Consequently the logical disjunction of the two cases also yields an unmoving arrow.

Oppositely marching rows of soldiers pass each other and a standing row of soldiers in the same time. Since oppositely moving rows pass twice as many bodies as each passes stationary bodies, "Zeno concluded that 'double the time is equal to half'".^{(6)} Vlastos states that "Aristotle and all our other ancient informants understood this as a (supposed) paradox of relative motion" and attributes the interpretation which follows to Paul Tannery.^{(7)}

If extension and duration are atomic, that is, there are minimum amounts of each, then an analogy can be made between atoms of extension moving in jumps of atomic time and rows of soldiers drilling in a stadium. Consider three rows of soldiers, one standing (A B C), one marching to the right (1 2 3), and one marching to the left (4 5 6). As the row of soldiers marching to the right passes the row of standing soldiers, it takes one (minimum) unit of time for the soldiers to move one unit of distance -- from a position opposite certain standing soldiers to a position opposite the next ones.

1 2 3 => After one => 1 2 3 A B C time unit A B C

On the other hand, the other row of soldiers, marching left, also move one unit of distance in one unit of time.

A B C after one A B C 4 5 6 <= time unit 4 5 6 <=

The problem is that the relative change between the soldiers marching left to those marching right is twice the distance in the same amount of time.

1 2 3 => After one => 1 2 3 4 5 6 <= time unit 4 5 6 <=

Soldier 6 ends up opposite soldier 1 after 1 unit of time. In order to get there he had to pass soldier 2. This must have taken one unit of time, and passing from there on to soldier 1 must have taken another unit of time. Hence the expression "double the time". It is reasonable to presume that "half the time" refers to the immediately preceding antecedent (the doubled time) rather than to the fixed unit of time which got doubled. Otherwise, we would be looking for a relationship of 4 to 1 instead of 2 to 1. Vlastos confirms this interpretation in his quotation:

So it follows, he thinks, that half the time equals its double [that

t/2 =t] (Aristotle,Physics239b35).^{(8)}

It is difficult to get the sense of the conflict because we are accustomed to thinking of time as continuous. The perplexing nature of the situation may be illustrated by noting that there must be some time when soldier 6 is opposite soldier 2, while the argument says that there is not. Unpacking the argument into premisses and conclusions yields the following:

SP1

Soldier 6 passes from soldier C to soldier B in a minimum unit of time. SP2

The instant at which soldier 6 is opposite soldier C is the instant that soldier 6 is opposite soldier 3, and the instant at which soldier 6 is opposite soldier B is the instant that soldier 6 is opposite soldier 1. SI1

SP1 & SP2 => SC1 SC1

The minimum unit of time that soldier 6 takes to pass from soldier C to soldier B is the minimum unit of time that soldier 6 takes to pass from soldier 3 to soldier 1. SP3

If two instants are separated by the minimum unit of time there is no instant between them. (Time is atomic.) SI2

SC1 & SP2 => SC2 SC2

At one instant soldier 6 is opposite soldier 3 and at the next instant soldier 6 is opposite soldier 1, and there is no instant between these two instants. SP4

Any soldier passing from soldier 1 to soldier 3 must pass all those in between. SI3

SC2 & SP4 => SC3 SC3

Because soldier 2 is between soldiers 1 and 3, soldier 6 must pass soldier 2. SP5

If soldier 6 passes soldier 2 there must be an instant at which this happens. SI4

SC3 & SP5 => SC4 SC4

There is an instant at which soldier 6 passes soldier 2. SP6

If there is such an instant, it must be between the instant that soldier 6 is opposite soldier 3 and the instant that soldier 6 is opposite soldier 1. SI5

SC4 & SP6 => SC5 SC5

The instant at which soldier 6 passes soldier 2 is between the instant that soldier 6 is opposite soldier 3 and the instant that soldier 6 is opposite soldier 1. SC6

SC5 contradicts SC2

This paradox can be tersely stated as follows: Ultimate parts must have no magnitude or they would not be ultimate parts. But an extended object cannot be made up of parts with no magnitude. Parts of zero size add up to zero size. So an extended object must be "so small as to have no magnitude". The parts must have magnitude. But an infinity of extended parts must have infinite extension. So an extended object must be "so large as to be infinite".^{(9)}

Expanding this statement to show more fully the premisses, inferences, and conclusions yields:

PP1

The size of an extended object is not zero. PP2

Extended objects are made up of parts. PP3

Ultimate parts have no magnitude (zero size). PP4

The number of the parts [of] an extended object is infinite. PP5

If something is made up of parts then its size is the sum of the size of its parts. PP6

Parts of zero size add up to zero size. PP7

An infinity of parts of non-zero size adds up to infinite size.

If an extended object has parts, there are two cases to consider: the parts are ultimate or the parts are extended.

PI1

PP2 > PC1 OR PC2 PC1

An extended object is made up of ultimate parts. PC2

An extended object is made up of extended parts.

Case 1: The parts are ultimate (PC1 holds).

PC1

An extended object is made up of ultimate parts. PI2

PC1 & PP3 => PC3 PC3

An extended object is made up of parts of zero size. PI3

PC3 & PP5 => PC4 PC4

The size of an extended object is the sum of parts of zero size. PI4

PC4 & PP6 => PC5 PC5

The size of an extended object is zero size. ("It is so small as to have no magnitude").

Case 1 concludes that an extended object has __no__ size -- a clearly unacceptable result. Case 2 fairs no better.

Case 2: The parts are extended (PC2 holds).

PC2

An extended object is made up of extended parts. PI5

PC2 & PP1 => PC6 PC6

An extended object is made up of parts of non-zero size. PI6

PC6 & PP5 => PC7 PC7

The size of an extended object is the sum of parts of non-zero size. PI7

PC7 & PP4 => PC8 PC8

The size of an extended object is the sum of an infinite number of parts with non-zero size. PI8

PC8 & PP7 => PC9 PC9

The size of an extended object is infinite size. (It must be "so large as to be infinite".)

Case 2 concludes that an extended object must be infinite in size -- an equally unacceptable result.

The paradox lies in the following: it cannot be denied that things are made up of parts; but if things are made up of parts then there are two possibilities, and both possibilities lead to absurd conclusions.