Atomism and Infinite Divisibility

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CHAPTER II

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ZENO'S PARADOXES

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.

The Achilles

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

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

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:

(A) That which moves is not in one place (equal to itself) in one instant.

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".
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.

The Stadium

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, Physics 239b35).(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

The Paradox of Plurality

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.

(Chapter 3)