Zeno's Arguments for the Irrationality of Plurality and the Rationalityof Monism

From Marbaniang, Domenic. Epistemics of Divine Reality © 2007.

To the attacks of the pluralists, Zeno of Elea, disciple of Parmenides offered several arguments in form of paradoxes that demonstrated the utter absurdity of commonsense realism. Since absurdity is a sign of falsity, it is false that reality is many. Hence, Zeno argues that reality must be one. It may be noted that the paradox may also mean, contrary to Zeno’s contention, that reason is false and experience is true. However, since it is difficult to label reason as false without the use of reason itself, the certainty of rational reality looms over that of experience. Few of Zeno’s most famous proofs are as follows:

The Paradoxes of Plurality

The Argument from Denseness

If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited.[1]

The paradox is that things appear to be as many as they are, that is as limited, whereas rationally speaking they must be unlimited; a pair of two is separated by a third, which pairing with its next is separated by a fourth, and so on ad infinitum. Thus, the view that reality is many, or numbered plurality, involves a rational impossibility.

The assumption is that it takes something to separate an other. That means that if the ‘separator’ theory is abandoned the paradox doesn’t exist. Why can’t it be said that the things are separated by the void? In that sense, the void (meaning nothing) could rationally not separate anything; for to be separated by nothing is not to be separated at all. However, if empirically understood, the void (space) separates things in the sense that in between things there is the void. Thus, the rational-empirical paradoxical situation is not resolved but heightened by the different meanings of void by reason and experience. The paradox, consequently exists because the rational (immaterial) is applied to the empirical (material) and the fusion creates an either/or situation in which experience is ultimately dismissed as illusion.

The Argument from Finite Size

… if it should be added to something else that exists, it would not make it any bigger. For if it were of no size and was added, it cannot increase in size. And so it follows immediately that what is added is nothing. But if when it is subtracted, the other thing is no smaller, nor is it increased when it is added, clearly the thing being added or subtracted is nothing.

But if it exists, each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever. For no such part of it will be last, nor will there be one part not related to another. Therefore, if there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited.[2]

The first part of the argument which purports to show that if there are many things they cannot possess size is missing. The second part shows that if they do not possess size they are nothing. The third part shows that if reality is plural and, thus, composed of different parts, the following paradox results: Each part is divided into a front and a rear part. Each front and the rear part have a front and a rear part of their own respectively, and so on ad infinitum. Thus, the size would be zero and unlimited, which is paradoxical.

The Argument from Complete Divisibility

1. If a line segment is composed of a multiplicity of points, then the line segment is infinitely divisible; that is to say an infinite number of bisections can be made in it. One cannot come to a point where further bisection of the line segment is not mathematically possible. No singular point can thus be found. Therefore, a line segment is not composed of a multiplicity of points.
2. The line, which is made up of points, has a particular measurement (just as many points as it is and nothing more) and so is limited. It is a definite number, and a definite number is a finite or limited number. However, since the line is infinitely divisible, it is also unlimited. Therefore, it's contradictory to suppose a line is composed of a multiplicity of points.[3]

Speaking thus, then, the existence of plurality is rationally impossible. For, according to each of the above the paradox of the limited and unlimited can be seen. Rationally speaking, things, if not one but many, involve infinity by divisibility. However, they must of necessity be limited in order to be numbered as many. Thus, the phenomenal experience is proved to be rationally untenable.

The Paradoxes of Motion

The Dichotomy

The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal.[4]

Suppose a runner is standing at point A and must reach point B in order to finish the race. The only way he can reach point B is by reaching the halfway point, say A_1, between A and B, before reaching B. But then the only way he can reach halfway point A₁ is by reaching the halfway point, say A₂, between A and A₁, and so on ad infinitum in order to finish the course. Thus in order for the runner to reach point B, he will have to traverse an infinite number of points in a finite time, which is impossible. Therefore, motion is absurd.

Achilles and the Tortoise

Suppose Achilles and a tortoise begin a race. Achilles allows the tortoise to have the head start since he is confident that the slow tortoise will never win the race. But now in order for Achilles to get past by the tortoise, he will first have to reach the point left behind by tortoise; but by that time the tortoise would have already gone by farther from the point, and so on ad infinitum. In other words, if A₁ is the point where the tortoise is presently and Achilles has to reach this point before he can overtake the tortoise, by the time Achilles would have got to point A₁ the tortoise would have gone a bit away and be at point A₂ which would then become the next point which Achilles would have to reach in order to overtake the tortoise, but by the time he gets to A₂ the tortoise would have gone a bit more farther, and so on ad infinitum. In this way, logically Achilles can never overtake the tortoise. But empirically Achilles is seen to overtake the tortoise, and therein lies the paradox. Empirically Achilles overtakes the tortoise but logically he cannot. And since overtaking the tortoise is seen as logically absurd, it cannot be true.

The Arrow

Consider an apparently flying arrow, in any instant. At any given moment, the arrow occupies a particular position in space equal to its length. But for an arrow to occupy a position in space equal to its length means that it is at rest. However, since the arrow must always occupy such a position in space equal to its length, the arrow must be at rest at all moments. Moreover, since space as quantity is infinitely divisible, the flying arrow occupies an infinite number of these positions of rest. But the sum of an infinite number of these positions of rest is not a motion. Therefore, the arrow is never in motion. The absurd conclusion would then be that the flying arrow is ever at rest, which is impossible. Therefore, motion is false.

The Stadium

The fourth argument is that concerning equal bodies [AA] which move alongside equal bodies in the stadium from opposite directions – the ones from the end of the stadium [CC], the others from the middle [BB] – at equal speeds, in which he thinks it follows that half the time is equal to its double…. And it follows that the C has passed all the As and the B half; so that the time is half … . And at the same time it follows that the first B has passed all the Cs.[5]

The stadium is an argument from the relativity of motion to the absurdity of motion. Stumpf [6] has a good illustration of passenger cars  for this argument. Imagine three passenger cars of equal length on tracks parallel to each other, each car having eight windows on a side (see Figures 1 & 2).

One of the cars is at rest; the others, moving in opposite directions at the same speed. In order for the two cars (B & C) moving in opposite direction of car A, to come to the position in Fig. 2, car B’s front has to cross one more window of car A, while car C has to cross two windows of car B. Each window represents a unit of distance, and each such unit is passed in an equal unit of time. Since car B went past only one of car A’s windows, while car C went past two of car B’s windows, and since each window represents the same unit of time, it would have to follow that one unit of time is equal to two units of time or that one unit of distance equals two units of distance, which is absurd. The mathematical solution to this third paradox is as follows:

Speed of car B towards A	=	S m/s
Speed of car C towards A	=	S m/s
Speed of car C towards B	=	2S m/s
Distance to complete motion	=	2D (2 windows or units)
Time needed to complete motion	=	2D/2S
	=	D/S = 1unit of time

Therefore, one unit of time was needed for car C to cross the two windows of car B. The paradox is, thus, resolved; nevertheless, at the expense of absolute motion. The only way this paradox is solved is by accepting that no absolute motion exists. Motion is relative. The speed of car C, thus is seen to be twice greater in relation to car B, than car A. But saying that no absolute motion exists is similar to saying that motion does not exist. What may seem to be motion to one may not seem to be motion to another, and so on. Thus, no absolute statement regarding motion can be made. Thereby, then, Zeno wins.

Thus, the phenomenal world of empirical plurality is shown to be false. The main parts of the arguments of Parmenides and Zeno are summarized as follows:

1. Being cannot arise out of non-being, for then it would have to be even before it arises out of non-being; therefore, being is eternal and ungenerated.[7]
2. Being is indivisible, for it cannot divide itself from itself.
3. Being is one and not many, for if it were many it would have to be diversely differentiated by something other than being, namely non-being, which means to be differentiated by nothing.
4. Being cannot be falsified; for if spoken of, it must be; if not spoken of, then nothing is spoken of. If being is not, then nothing is.
5. Being is indestructible, for change cannot be predicated of it, it being absolute.
6. The phenomenon of plurality is absurd, for it involves the paradox of the limited and the unlimited in the one divisible unit.
7. The phenomenon of change is absurd, for it involves completion of an infinite series in a finite time, as Zeno’s paradoxes show.

Thus, reality is one, eternal, indestructible, immutable, and thus, absolute.

Implications for Divine Existence

Either of the following implications results from the supposition that being is eternal and singular:

1. God is being and the only one reality; all plurality of selves is an illusion.
2. God as an ontological distinct does not exist, for reality is one.
3. God is not, only being is; if the individual definitions of ‘God’ and ‘being’ are to be retained and not confused.

However, though Parmenides and Zeno have attempted to solve the ontological problem of the nature of reality, they have left the cosmological problem of the same unanswered. If reality is one, what accounts for the plurality that is manifest; or why does or how did reality come to appear as many? To this Parmenides and Zeno remain silent, and since a theory that doesn’t take into consideration the whole avenue of the subject in question cannot be considered to be complete and unified, attention must be turned to the Indian philosophers to see whether they have a rational answer to this cosmological question. Nevertheless, this far the contradictions between reason and experience have been aptly demonstrated by the Grecians. And the culmination of their rational search in the Eleatics was anticipated; for if reason alone is trustworthy, then experience must be dispensed with, as Zeno clearly showed

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[1] Simplicius as cited in “Zeno’s Paradoxes,” http://plato.stanford.edu/entries/paradox-zeno/

[2] Ibid.
[3] “Zeno of Elea,” http://www.utm.edu/research/iep/z/zenoelea.htm
[4] Aristotle as cited in “Zeno’s Paradoxes,” http://plato.stanford.edu/entries/paradox-zeno/
[5] Aristotle as cited in “Zeno’s Paradoxes,” http://plato.stanford.edu/entries/paradox-zeno/
[6] Samuel Enoch Stumpf, Socrates to Sartre, p. 20
[7] Samuel Enoch Stumpf, Socrates to Sartre, pp. 16, 17

© Domenic Marbaniang, 2007