7 Nov 2008

Zeno's Paradoxes of Infinite Divisibility, with Aristotle's, Spinoza's, & Leibniz' commentary, and Stoic & Deleuze's Infinitely-Divisible Present

by Corry Shores
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[The following looks first at Zeno's paradoxes as they are described in various ancient texts. I include Leibniz' arguments that seem to counter Zeno. After which, I include a brief look at Stoic temporality and its infinite divisibility. Finally, I place some of Deleuze's renditions of the notion of infinite divisibility and Zeno's paradox.]

Zeno argues that all things can be explained by the one, and that plurality produces paradox.

[Zeno's arguments pin two basic principles against each other. These are very very basic principles. The first is that there is only one fundamental substance to all of reality, and it is indivisible. According to this perspective, the many things we sense are all illusions. (Spinoza's metaphysical system provides very sophisticated and compelling theoretical grounding for this perspective: there is only one underlying substance, and the many things we see around us are modifications of that one substance). We might make many golden rings, but gold is the same element each time. We might make many rings of any metal, but still it is fundamentally a metallic substance. We might make a variety of rings of any material whatever, but still it is made of matter. So if we continue generalizing our notion of the many things around us, we see that the number of things shrinks as the underlying substrate becomes more inclusive. We can then infer that there is one underlying substance for all things whatsoever.

The second perspective is that division or multiplicity is pervasive through the world. A multiplicity would not be two things, not three, four, or millions. If multiplicity is a principle, then everything is divided infinitely into infinitely many parts. Hence the principle of multiplicity is thoroughly pervasive, and hence all things are infinitely divisible.]

I. Paradoxes of Numerical Multiplicity

Zeno's Ontological Argument

I.A: If there are many, then there is at least one, but one cannot be many.

If there are a plurality of things (by division or otherwise), each thing is a unit or a one. But, if there are many, then there are not one. Hence there can only be one One, that is, one unit.

(Simplicius 99.9 , 138.3 , 139.19

Philoponus Physics 42.9)

Destructive Division

I.B: Infinite division turns the one into infinity or nothing.

If we divide the one, either of two absurdities result:

1) we divide it infinitely into infinitely many indivisible parts, but then the one equals infinity, which is absurd, or

2) we continue dividing the one to infinity, which reduces it to parts that are reduced to nothing, but then the one would be made up of nothing, which is also absurd.

Hence there is only the indivisible one, and not a plurality.

(Simplicius 139.19 , 139.27,

Philoponus 80.23)

Extensive Existence

I.C: only finitely-extensive things can exist

Plurality can be thought to produce the infinitely great and the infinitely small in two ways.

1) By infinite division and infinite combination, we obtain a scale of sizes ranging from the infinitely small to the infinitely great.

2) Each thing must be a different size to be different, and there are infinitely many things, therefore there must be an infinitely smallest and infinitely largest thing.

If this is so, then:

a) If we add to something that is infinitely great, it will not increase in size, because its size was already greater than any given or givable magnitude.

b) If we subtract something infinitely small from some finite magnitude, we will not obtain some smaller magnitude that is any smaller. For, it is already smaller than any given or givable magnitude.

This results in two absurdities:

1) Thus, if we add an infinitely small value to something that is one unit large, for example, we will still have something that is only one unit large. For, the infinitely small value cannot be quantified in comparison to a finite value. Thus it is equal to zero. So we would have parts of the one that are nothing, but that is absurd.

d) If we subtract any value from an infinitely large magnitude, it will not become any less. This makes all finite values equal zero in comparison. But this is absurd, because finite extensive values cannot equal zero.

Zeno then claims that if something equals zero, then it does not exist. But the multiplicity that results from infinite divisibility produces infinitely large and infinitely small things which

1) must exist because they necessarily follow from the principle of infinite divisibility, but

2) cannot exist, because they have no magnitude, and thus equal nothing.

Because the argument for multiplicity is false, its contrary argument must be true: reality is made-up of one thing only.

(Aristotle, Metaphysics B 4.1001b7

Simpicius 97.13 [or 138.39], 139.5 140.34)

Leibniz' Opinion:

Leibniz explains that the beginning of a body has an unextended [and hence intensive] magnitude. So for him it is not absurd to have a magnitude that does not increase a finite value when added to it. [For his demonstration, see the entry on his "Theory of Abstract Motion."]

In-Different Plurality

I.D: multiplicity implies universality.

BoldIf there are many things, each thing is still a thing. Its magnitude may be compared with the other things, and it may be compared with itself. If we compare something's magnitude with itself, it will be the same quantity divided by itself, which equals one. So everything has a comparative magnitude of one. But if everything is one, then each thing itself equals an infinity of other things. They all equal each other. But if they all equal each other, than none is greater or lesser than any other. Hence no thing has any greatness. But something must have magnitude to exist. So we cannot presuppose that there is a multiplicity.

(Simplicius 138.30)

Finite Infinities

I. E: Multiplicity implies that all finite things have infinite magnitude.

Moreover, if there is infinite divisibility, then any given thing is really an infinity of units or self-same things equaling one. But, that means any given finite thing is infinite in magnitude, which is absurd. So there cannot be many things, there can only be one.

The Infinite Density of Finite Plurality

I.F: Multiplicity implies that finite pluralities are constituted by infinities.

We know that between any two rational numbers, there is another number, because the rational numbers are dense everywhere. Zeno uses this notion of density to claim an absurdity in the idea multiplicity. So, if there is a multiplicity of things, there cannot be more or less than the number of things that exist. So there must be a definite number of things that exist. However, between any two things there is one other, and so on (or any one thing can be divided). So if we presuppose multiplicity, we conclude both that there are a finite number of things, and also that there are an infinity of things. This is absurd, so there must only be one thing in reality.

(Simplicius 140.27)

The Infinite Density of Finite Magnitudes

I.G: Multiplicity implies that things with a finite magnitude are constituted by an infinite magnitude.

To be real, a thing must have magnitude. So each thing in a multiplicity must have a magnitude. And if something has a magnitude, it can be divided. For, if something extends so far, it is made up of smaller extensions that when totaled equal the larger one. But if we have one finite thing, then it must be made up of infinitely many smaller parts. And if each part has a finite value, and there is an infinity of them, then every finite thing is infinite in size. But this cannot be, so instead there must be an end to the division. So there must be something that is smaller than all finite things as a result of infinite division. But then we would have to say that finite things are made up of non-finite things, which is absurd. So there cannot be many things, there can only be one.

(Simplicious 138.300)

In-Grained Impossibility

I.H: Proportional portions have disproportional effects

We have ten-thousand total millet grains. When we drop them, we hear many small noises. For, each grain makes its own small noise, which together makes the total sound of the falling grains.

Now we just drop one grain and hear one sound. We do not hear the sound of the one grain's ten thousand parts falling. We just hear one sound.

But, one grain is one-ten thousandth of ten-thousand grains. So whatever holds between one grain and ten-thousand of them should hold analogically between one grain and one-ten thousandth of a grain. However, we said that one ten-thousandth of a grain does not make a sound, but one grain out of ten thousand does. This is absurd.

(Simplicius 255r)

Aristotle's Commentary:

Even the ten thousandth part can make a sound; for, if we total the air that is move when we drop a ten-thousandths part ten-thousand times, it will move the same amount of air as one whole grain dropped once.

(Aristotle Physics Book VII section 5)

II. Paradoxes of Place and Movement

No Place for Multiplicity

II.A.: Multitude requires location, but locations must have location, which implies an infinite regress.

If there are many things, they must be in different places. But any one place must itself be in some place. But then that place must be located somewhere. But if every place must have a place, then no place can have a definite location. So if there are not places, then there cannot be different things. Hence, there must only be one thing.

(Simplicius 562.1,

Aristotle Physics Book 4 sections 1-3,

Philoponus 510.2)

Aristotle's Commentary:

Mammals are healthy when they are warm. Their warmth is placed in their bodies as a physical property. But their health is placed in them as a state or status. We need not then ask where physical properties are located in the being, nor do we need to ask where statuses are located in the animal. So in other words, there need not be an infinite regress of placement. So we need not conclude that the placement of any one thing lead inevitably to absurdity [see the entry on Aristotle's explanation for a more complete account of his solution.]

(Aristotle, Physics Book IV, section 3)

Movement Has No Place

II.B: Movement cannot happen in a determinate place, so there is no place where we may find movement:

Movement either occurs in the place that it is in, or in some place that it is not in. We know that nothing can be somewhere where it is not. But, we also know that if something is set in place, it is not moving. Hence there can be no movement; for, there is no place for movement to occur.

(Diogenes, IX. 72,

Epiphanius, adv. Haer. III.II)

The Over-Pointed Arrow

II.C: Finite moving things cannot extend outside themselves at any moment, hence they cannot move over time.

An arrow has a determinate length. So even when in motion, it maintains that same length. Thus at any moment, the arrow is in a determinate place equal to its length. But, if the arrow can only be in one place at a time, it can never change place at any moment, and it hence cannot move during any given instant. So at every moment of the arrow's movement, it is at rest. In other words, while moving, the arrow is always at rest. But this is absurd, because we presupposed that the arrow is in movement.

Also we might simply say that because the arrow only takes up as much space as its length during any instant, then it is not at any instant crossing into new points of space. But if the arrow never crosses into new points of space, then it can never move.

(Aristotle Physics Book VI section 9,

Simplicius 1011.19 , 1015.19 , 1034.4 ,

Philoponus, 816.30 ,

Themistius 199.4)

Aristotle's refutation:

For Aristotle, both time and space are composed entirely of divisible parts [for more, see the entry on Book VI sections 1 and 2]. If an arrow were made-up of indivisible parts, then it would have an absolutely determinate length. And if time were made up of indivisible parts, then there would be specific moments in time without any extent, hence there would be moments when an arrow of a determinate length takes up a determinate amount of space equal to its length. But Aristotle demonstrated that time and space cannot be made up of such indivisible parts. So instead, if we were to examine any part of time during the arrow's motion, we will find that during that moment the arrow traveled a distance greater than its length, whose magnitude itself was never absolutely determinate to begin with.

(Aristotle Physics Book VI section 9)

Leibniz' opinion:

The beginning and the end of motion is its conatus, which is its tendency to change place. The conatus causes a body to extend an infinitely small amount past its own boundary. So, a point in a moving body is in many points in space at one (infinitely small) moment of time [see the entry on Leibniz' mens momentanea where he explains his theory of conatus.]

The Motion that Stops Itself from Beginning

II.D: Distance can be halved infinitely, so a body never

beginBolds moving.

Something moving to another place must first travel half that distance. But before it may reach the half-way point, it must first reach half of the halved distance (that is, a quarter of the total distance), and so on. But if space is infinitely divisible, then there will be no end to the half-divisions. As a result, there would be no beginning to the motion. For, there will always be created another half-distance standing between it and some point along its way.

(Aristotle Physics Book VI section 9,

Simplicius 1013.4)

Aristotle's commentary:

Aristotle refers us back to his explanation of the space-time continuum in Book VI sections 1 and 2. Aristotle argues that time and space are continuous in themselves, and continuously correlated to each other. So suppose there is a body in motion. If we divide its distance traveled, we proportionally divide the amount of time it takes for the body to arrive at its destination. So if the body maintains the same speed, we find that it traveled half the distance in half the time, a quarter of the distance in a quarter of the time, an eighth in an eighth, and so on. So we may divide the starting distance infinitely into infinitely many more smaller starting distances. This creates a starting distance that is so small to be practically no distance whatsoever, and given space's correlation with time, it should take practically no time at all to cross it.

(Aristotle Physics Book VI section 9)

Spinoza's Opinion:

In his Letter on the Infinite, Spinoza speaks of such a paradox in terms of dividing time. The underlying substance of all reality is eternal and indivisible, both spatially and temporally. We know this when we think about it rationally, but when we use our imaginations, we tend to divide it up infinitely, and obtain paradoxes. So for example, we might divide up an hour into an infinity of instants and think it impossible for the time to ever pass. This is because substance is eternal, and its temporality is indivisible.

it is the same thing to make up duration out of instants, as it is to make number simply by adding up noughts.
(Spinoza, Letter 12 to Meyer)

Leibniz' Opinion:

We may continue halving the beginning distance until reaching an unextended magnitude. Conatus is the infinitely small motion through this unextended space. For, conatus is the intensive beginning of motion. In fact, it is only because we can divide the beginning space infinitely that motion may begin. [For Leibniz' explanation, see the entry on his "Theory of Abstract Motion."]

Infinite Division Creates Infinite Distance

II.E: If we divide space infinitely, that creates an infinity of points to cross, which cannot be accomplished in a finite amount of time.

If space is infinitely divisible, that creates an infinite number of points for a moving body to cross. The body will have to make contact with each point. But to make contact with a point requires a finite amount of time. And an infinite number of contacts creates an infinite number of finite units of time, which means it would take infinitely long to for a body to move even the slightest distance.

(Aristotle Physics Book VI section 9, Book VIII section 8 ; De Lineis Insecabilibus 968a 18 ,

Simplicius 947.5, 1289.5 ,

Philoponus 81.7 , 802.31 ,

Themistius 186.30)

Achilles Trailing the Tortoise

II.F: Because the tortoise begins further down the track, and because he moves continuously, he perpetually creates more distance for Achilles to cross.

Achilles will race the tortoise, but he will let the tortoise begin further down the track, although both start at the same time. By the time Achilles reaches the tortoise's starting point, the tortoise is already further down the track. So now Achilles has another distance to cross before he overtakes the tortoise. So Achilles now runs to the tortoise's new leading-point. But again, by the time he gets there, the tortoise has advanced again. So long as the tortoise keeps moving, Achilles will never overtake him.

(Aristotle Physics Book VI section 9,

Simplicius, 1013.31)

Aristotle's commentary:

Because this problem also does not observe the continuous correspondence of the time and space continua, his prior solution applies here as well.

(Aristotle Physics Book VI section 9)

A Day (or Two) at the Races

II.G: Bodies whose motion is counter to each other travel more than one distance: that between one and the other, and that between either and the ground below it.

Sitting on a race track is train of four cars.

They do not move the whole time.

Another set of four cars is positioned have-way across train A.

Train B moves to the right, until reaching the finish line at the right-end of train A.

Likewise, train C is the same length as both trains A and B, and it is placed along the other half of A. Train C moves to the left, against the motion of train B.

Both train B and train C move at the same speed, and both start at the same time. Since both are equally far from their finish line, they both finish the race at the same time. So over the course of a certain duration, train B moves this far:

And train C moves this far:

So we see that train C moved four squares. But, train C moves at the same speed as train B, so it must have taken twice as long, because train C moved twice the distance of train B.

The paradox according to Aristotle is that the whole time it took C to move was twice that of B, because train C had twice the distance to cover. But both can only get equally far given their equal speeds. So somehow the half-time of B's movement equals the whole-time of C's movement. As Aristotle explains:

half a given time is equal to double that time
(Aristotle Physics Book VI section 9)

The paradox according to Simplicius is that in the same time, one train moves twice as far as the other, even though they move at the same speed. Then Zeno follows-up with the above conclusion that the time the two trains take is both double and half the other.

(Simplicius 1016.9)

Aristotle's and Simplicius' refutation:

Aristotle and Simplicius say it is wrong to conclude that the amount of time for train B to move two spaces is the same amount of time for train C to move four spaces. For, train C did not move forward the value of four spaces, but only two, as is obvious when trains C and B are compared with immobile train A, who serves as an objective measure.

(Aristotle Physics Book VI section 9

Simplicius 1016.9)

III. Time and Division According to the Stoic Chrysippus

as explained by Stobaeus:

Motion can be faster or slower. In order to make that determination, we need to know how much motion happened in a common interval.

time is the interval of motion according to which the measure of speed and slowness is sometimes spoken of; or, time is the interval which accompanies the motion of the cosmos.

Time is infinite in both directions (towards past and towards future). So both the past and the future are infinite.

Like Aristotle, Chrysippus recognizes that time is infinitely divisible, because it is continuous. But, if we can divide time infinitely, then the present has no time. For, it may be divided down to an infinitely small interval, so hence it does not extend in time. [As unextended time, this would be intense temporality.] Nonetheless, this does not mean that the present does not exist. On the contrary, it only the present exists.

If we are walking, then at that moment, walking exists for us. But when we are sitting, walking does not exist for us in that way. Rather, it exists more as something that may be predicated of us. We are beings that walk, but our walking does not exist for us when we are sitting.

In the same way, the past and the future exist as predicates of the present, but they do not exist for the present.

(Stobaeus Anthology 1.8.42)

IV. Deleuze's Zeno and Infinite Divisibility.

Deleuze speaks of two Stoic times, Aion and Chronos. Aion time is infinitely divisible. Here, the past and the future infinitely subdivide the present, which is a durationless instant standing between past and future.

it is the instant without thickness and without extension, which subdivides each present into past and future, rather than vast and thick presents which comprehend both future and past in relation to one another.

(Deleuze, Logic of Sense 188c)

Sometimes it will be said that only the present exists; that it absorbs or contracts in itself the past and the future, and that, from contraction to contraction, with ever greater depth, it reaches the limits of the entire Universe and becomes a living cosmic present.


To further articulate the infinite divisibility of this type of time, Deleuze quotes Borges as saying:

"No decision is final, all diverge into others. The ignorant suppose that an infinite number of drawings requires an infinite amount of time; in reality, it suffices that time be infinitely subdivisible , as is the case of the famous parable of the Tortoise and Hare" (72a).

Deleuze footnotes this, in which he says: "The parable of the tortoise and the hare seems to be an allusion not only to Zeno's paradox but to Carroll's as well" (77b).


1) A = B and

2) B = C, then

Z) A = C.

1 and 2 are premises, and z is the conclusion, which is arrived-at by means of an implicative inference. But between 2 and z is an inference that is also an implicit proposition. If we explicated it, we have:

3) if 1 and 2 are true, then z is true.

So in all we now have:

1) A = B and

2) B = C, and

3) If 1 and 2 are true, then z is true.

Z) A = C.

But now between 3 and z we have another implicit proposition that would be explicated as:

4) If 1, 2, and 3 are true, then z is true.

All together making:

1) A = B and

2) B = C, and

3) If 1 and 2 are true, then z is true.

4) If 1, 2, and 3 are true, then z is true.

Z) A = C.

And yet, this too creates another implicit premise, 5, which creates 6, and on and on to infinity.

[See the entry on Carroll's Achilles and Tortoise parable for a more thorough explanation.]

For Deleuze, there is an intensity between premises and conclusions that is expressed in the inference, but just implicitly. Intensities are implicated in other intensities on to infinity, like Russian dolls. So even though explicating an intensity cancels it, there is always another one inside the last that was implicitly expressed from the beginning as well. So even one simple syllogistic inference may be infinitely divided, because it is made up of intensity, which is non-extensive.

Likewise for time. We can have becoming, because there are moments between conditions that are neither one nor the other. There is an infinitely small instant when water boils to steam, when it is neither water nor steam, but is also both water and steam, because it is the pure moment of transition between. A moment of becoming is infinitely divisible down to a pure becoming that is neither what was nor what will be. It is what is, which is no more than becoming in its purity.

[Site Topic Directory]

Aristotle. Physics. Transl. R. P. Hardie and R. K. Gaye. Available online at:


Aristotle. Metaphysics. Transl. Hugh Tredennick. Available online at:


Lee, H.D.P. Zeno of Elea: A Text, with Translation and Notes. Transl. H.D.P. Lee. Amsterdam: Adolf M. Hakkert Publisher, 1967.

The Stoics Reader: Selected Writings and Testimonia. Transl. Brad Inwood and Llyod P. Gerson. Cambridge: Hackett Publishing Company, Inc., 2008.

Deleuze, Gilles. Logic of Sense. Transl. Mark Lester. London:Columbia University Press, 1990, reprinted by Continuum, 2001.

Spinoza. 12th Letter to Meyer, "The Letter on Infinity."

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