Zeno's paradoxes

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Zeno's paradoxes are a set of problems devised by Zeno of Elea to support Parmenides' doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. It is usually assumed that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides' view. Thus Zeno can be interpreted as saying that to assume there is plurality is even more absurd than assuming there is only one. As such, if we are convinced by Zeno's paradoxes, we should take Parmenides' view more seriously.[1]

Several of Zeno's eight surviving paradoxes (preserved in Aristotle's Physics and Simplicius's commentary thereon) are essentially equivalent to one another; and most of them were regarded, even in ancient times, as very easy to refute. Three of the strongest and most famous—that of Achilles and the tortoise, the dichotomy argument, and that of an arrow in flight—are given here.

Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum, also known as proof by contradiction. They are also credited as a source of the dialectic method used by Socrates.

According to some historians of philosophy[attribution needed], Zeno's paradoxes were a major problem for ancient and medieval philosophers.

In modern times, calculus has been widely accepted by mathematicians and engineers as at least a practical solution for calculating infinitesimal distances. Other proposed solutions to Zeno's paradoxes from past and present philosophers have included the denial that space and time are themselves infinitely divisible, and the denial that the terms space and time refer to any entity with any innate properties at all.

Many philosophers still hesitate to say that all paradoxes are completely solved. Some philosophers state that these paradoxes still have modern relevance: attempts to deal with the paradoxes have resulted in intellectual discoveries, and variations on the paradoxes (see Thomson's lamp) continue to produce at least temporary puzzlement in discovering what, if anything, is wrong with the argument.

The origins of the paradoxes are somewhat unclear. Diogenes Laertius says that Zeno's teacher, Parmenides, was "the first to use the argument known as 'Achilles and the Tortoise' ", and attributes this assertion to Favorinus. In a later statement, Laertius attributed the paradoxes to Zeno.

Contents

"You can never catch up."

“ In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead ”

—Aristotle, Physics VI:9, 239b15

In the paradox of Achilles and the Tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is such a fast runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, in which said period the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.

"You cannot even start."

“ That which is in locomotion must arrive at the half-way stage before it arrives at the goal. ”

—Aristotle, Physics VI:9, 239b10

Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

H-\frac{B}{8}-\frac{B}{4}---\frac{B}{2}-------B

The resulting sequence can be represented as:

\left\{ \cdots, \frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}, 1 \right\}

This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.

This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion.

This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts. It contains some of the same elements as the Achilles and the Tortoise paradox, but with a more apparent conclusion of motionlessness. It is also known as the Race Course paradox. Some, like Aristotle, regard the Dichotomy as really just another version of Achilles and the Tortoise. However, they emphasise different points. In the Achilles and the Tortoise, the focus is that movement by multiple objects is just an illusion whereas in the Dichotomy the focus is that movement is actually impossible.

"You cannot even move."

“ If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. ”

—Aristotle, Physics VI:9, 239b5

In the arrow paradox, Zeno asks us to imagine an arrow in flight. He then asks us to divide up time into a series of indivisible nows or moments. At any given moment if we look at the arrow it has an exact location so it is not moving. Yet movement has to happen in the present; it can't be that there's no movement in the present yet movement in the past or future. So throughout all time, the arrow is at rest. Thus motion cannot happen.

This paradox is also known as the fletcher's paradox—a fletcher being a maker of arrows.

Whereas the first two paradoxes presented divide space into segments, this paradox divides time into points.

Aristotle, who recorded Zeno's arguments in his work Physics, disputes Zeno's reasoning. Aristotle denies that time is composed of "nows", as implied by Zeno's argument. If there is just a collection of "nows", then there is no such thing as temporal magnitude. Therefore, if Aristotle is correct in denying that time is composed of indivisible "nows", then Zeno is wrong in saying that the arrow was stationary throughout its flight despite saying that in each "now" the moving arrow is at rest.

According to Zeno, at any instant, the arrow must be at rest. However, this has been disputed, since being at rest is a relative term. One cannot judge, from observing any one instant, that the arrow is at rest. Rather, one requires other, adjacent instants to assert whether, compared to other instants, the arrow at one instant is at rest. Thus, compared to other instants, if the arrow is found to be at a different place than it was and will be at the times before and after, then we have reason to claim the arrow has moved.

A mathematical account would be as follows: in the limit, as the length of a moment approaches zero, the instantaneous rate of change or velocity (which is the quotient of distance over length of the moment) does not have to approach zero. This nonzero limit is the velocity of the arrow at the instant.

Both the paradoxes of Achilles and the tortoise and that of the dichotomy depend on dividing distances into a sequence of distances that become progressively smaller, and so are subject to the same counter-arguments.

Aristotle pointed out that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Such an approach to solving the paradoxes would amount to a denial that it must take an infinite amount of time to traverse an infinite sequence of distances.

Before 212 BCE, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. Theorems have been developed in more modern calculus to achieve the same result, but with a more rigorous proof of the method. These methods allow construction of solutions stating that (under suitable conditions) if the distances are always decreasing, the time is finite.

Several proposed solutions have at their core geometric series. A general geometric series can be written as

a\sum_{k=0}^{\infty} x^k,

which is convergent and equal to a/(1−x) provided that |x| < 1 (otherwise the series diverges).

Although these proposed solutions effectively involve dividing up the distance to be travelled into smaller and smaller pieces, it is easier to conceive of the solution as Aristotle did, by considering the time it takes Achilles to catch up to the tortoise.

In the case of Achilles and the tortoise, suppose that the tortoise runs at a constant speed of v metres per second (m/s) and gets a head start of distance d metres (m), and that Achilles runs at constant speed xv m/s with x > 1. It takes Achilles time d/xv seconds (s) to travel distance d and reach the point where the tortoise started, at which time the tortoise has travelled d/x m. It then takes further time d/x²v sec for Achilles to travel this new distance d/x m, at which time the tortoise has travelled another d/x², and so on.

Thus, the time taken for Achilles to catch up is

\frac{d}{v} \sum_{k=1}^\infty \left( \frac{1}{x} \right)^k = \frac{d}{v(x-1)} \, seconds.

Since this is a finite quantity, Achilles will eventually catch the tortoise.

Similarly, for the Dichotomy assume that each of Homer's steps takes a time proportional to the distance covered by that step. Suppose that it takes time h seconds for Homer to complete the last half of the distance to the bus; then it will have taken h/2 sec for him to complete the second-last step, traversing the distance between one quarter and half of the way. The third-last step, covering the distance between one eighth and one quarter of the way to the bus, will take h/4 sec, and so on. The total time taken by Homer is, summing from k = 0 for the last step,

h \sum_{k=0}^{\infty} \left( \frac{1}{2} \right)^k = 2h \,\, seconds.

Once again, this is a convergent sum: although Homer must pass through an infinite number of distance segments, most of these are infinitesimally short and the total time required is finite. So (provided it doesn't leave for 2h seconds) Homer will catch his bus.

In both cases, by moving at constant speeds (and in particular not stopping after each segment) Achilles will eventually catch the moving tortoise, and Homer the stationary bus. However, the solutions that employ geometric series have the advantage that they attempt to solve the paradoxes in their own terms, by denying the apparently paradoxical conclusions.

We now give a concrete example. Suppose the tortoise starts 10 meter in advance of Achilles, and moves at 1 meter per second, while Achilles moves at 10 meters per second. Then after 1 second Achilles will reach the tortoise's earlier location, and the tortoise will be 1 meter ahead of him. After 0.1 seconds Achilles will reach this location, but the tortoise will be 0.1 meter ahead of him. After 0.01 seconds Achilles will reach this location, but the tortoise will be 0.01 meter ahead of him. It seems as if Achilles will never reach the tortoise. However even though there are infinite numbers of "steps" Achilles will have to go through, it will take him a finite amount of time to do that: The number of seconds is 1+0.1+0.01+0.001+0.0001+... = 10/9 seconds. Also, Achilles will pass the tortoise 10/9 meter past the tortoise's starting point.

Zeno's paradoxes are addressed by calculus, in particular by the mathematical concept of limit, which gives a theoretical framework to deal with the problems brought up by Zeno.

  • d = distance between runners
  • t = time
\lim_{d \to 0}f(d) = t

A suggested problem with using calculus and mathematical series to try to solve Zeno's paradoxes is that these solutions miss the point. To be precise, while these kinds of solutions specify the limit point of infinite series, they do not explain how such a series can actually ever be completed and the limit point be reached. Thus, calculus and mathematical series can be used to predict where and when Achilles will overtake the tortoise, assuming that the infinite sequence of events as laid out in the argument ever comes to an end. However, the problem lies exactly with that assumption, as Zeno's paradox points out that in order for Achilles to catch up with the Tortoise, an infinite number of physical events need to take place, which seems to be impossible in and of itself, independent of how much time such an act would require if it could actually be done.

Indeed, the problem with the calculus and other series-based solutions is that these kinds of solutions beg the question. They assume that one can finish a limiting process, but this is exactly what Zeno questioned. To be precise, Zeno started with the assumption that a finite interval can be split into infinitely many parts, and then argued that it is impossible to move through such a landscape. For calculus and other series-based solutions to make the point that the sum of infinitely many terms can add up to a finite amount therefore merely confirms Zeno's assumption about the landscape (geometry) of space, but does nothing to answer Zeno's question of how we can actually (dynamically) move through such a space.

Put a different way, when these kinds of solutions tell us that Achilles passes the tortoise 10/9 meter after the tortoise's starting point, they assume that Achilles can actually reach that point, but Zeno questioned that Achilles can actually ever get to that point. Similarly, when we are told that Achilles passes the tortoise 10/9 seconds into the race, it is assumed that time can actually flow to that point, but once again we get the same problem: If there are an infinite number of time points between t = 0 and t = 10/9, how can t = 10/9 ever be reached? How, indeed, can time flow at all if it is assumed that between any two time points there are infinitely many other time points that, at least under our naive conception of time, have to occur one after the other? Thus, all these kinds of solutions presuppose that Zeno's difficulties have already been solved when trying to resolve the paradox. Which is to say, they beg the question and therefore don't resolve anything at all.

An unfortunate complication among these kinds of discussions is that many treatments of Zeno's paradox present Zeno's reasoning in such a way that calculus and series-based solutions really do work as objections to that reasoning. To be precise, Zeno's reasoning is often presented as arguing that because there are an infinite number of tasks to be done, it will take an infinite amount of time to complete all these tasks, and the calculus and mathematical series based solutions are now perfectly correct in objecting to that argument by pointing out that the sum of an infinite number of time intervals can add up to a finite amount of time. However, such a presentation of Zeno's argument makes the argument into a straw man: a weak (and indeed invalid) caricature of the much stronger and much simpler argument that does not at all consider any quantifications of time. This much simpler argument simply states that for Achilles to capture the tortoise an infinite series of physical events need to be completed, which is logically impossible. The calculus and mathematical series based solutions offer no insight into this much simpler, much more stinging, paradox.[2]

The following thought experiment can be used to illustrate the fact that time is irrelevant to the paradox. Imagine that Achilles notes the position occupied by the tortoise, and calls it first; after reaching that position, he once again notes the position the turtle has moved to, calling it second, and so on. If he catches up with the turtle at all, then apparently Achilles must have stopped counting, and we could ask Achilles what the greatest number he counted to was. But of course this is nonsense: there is no greatest number, and Achilles can never stop counting. So, Achilles can't catch up with the Tortoise, whether he has finite time or infinite time to do so.

The situation, then, is this. Any proper variant of Zeno's paradox, such as the above thought experiment, provides a mathematical and logical account of the physical process of movement through space (or time), and argues that it is impossible for Achilles to win (or move at all). So, in order for this paradox to be resolved, one needs to either show something wrong with the math or logic (which calculus and series-based solutions do for the improper variant, but not for the proper variant), or show why this mathematical analysis cannot be used in our physical world. As suggested below, maybe space and time are not so that between any two points one can always find another point, which would indeed prevent this analysis to go through, and possibly our naive conceptions of space and time are mistaken in other ways as well. But calculus and series-based proposals do not challenge any of our conceptions of space and time in any way, as they are purely mathematical analyses that say nothing about the nature of space and time at all. Hence, these kinds of solutions do not resolve the paradox in this second way either. In short, there is nothing in calculus or series-based solutions that prevents the infinite sequences to crop up that lead to the whole paradox. So, as such, they do not resolve the paradoxes.

If we more closely examine the thought experiment, it is clear that Achilles naming the positions "first", "second", and so forth, is a nonphysical/mathematical act rather than a physical act; as an illustration, try getting your friend to say the word "Bob" on the 1/2 second mark, then the 1/4 second mark, and so on... you just can't do it. Consequently, the "counting process" is a mathematical process, while the "catching up with the turtle" is a physical process. As with most attempts to peddle Zeno's paradoxes, the central element is the conflation of these two processes. But they are simply not to be identified. The mathematical "counting process" goes on to infinity, and this is never something one could complete. However the physical "catching up with the turtle" process is something that can be completed. This is shown by an elementary application of limiting process theory, with time as a parameter.

These considerations (one must divorce the mathematical and physical processes at hand) also apply to the paradox as given in the "much more stinging" form: "for Achilles to capture the tortoise will require him to go beyond, and hence to finish, going through a series that has no finish, which is logically impossible". Here the word finish has been confusingly used for both the physical process and the mathematical process in an effort to conflate the two.

The issue with the statement "Indeed, the problem with the calculus and other series-based solutions is that these kinds of solutions beg the question. They assume that one can finish a limiting process, but this is exactly what Zeno questioned." is similar. They (the vast majority) do not assume that one can finish the limiting (mathematical) process, and they do not need to. To finish the physical process it is not required to finish the associated mathematical (limiting) process. The two processes are completely different in nature, and divorcing the two is essential if one is to resolve the paradox.

Main article: Planck length

Another proposed solution to some of the paradoxes is to consider that space and time are not infinitely divisible. Just because our number system enables us to give a number between any two numbers, it does not necessarily follow that there is a point in space between any two different points in space, and the same goes for time.

If space-time is not infinitely divisible (and thus not perfectly continuous), it is "discrete" (composed of “lumps” and “jumps”). This means that motion, at the smallest physical level, may be a series of jumps from one quantum space-time coordinate to the next, each occurring over distance and time intervals that are not divisible into smaller measures.[2]

Thus the total number of quantum jumps made while traversing from point A to point B would be finite, and there is no paradox.

Augustine of Hippo was the first to posit that time has no precise "moments," in his 4th century C.E. text, Confessions. In Book XI, section XI, paragraph 13, Augustine says, "truly, no time is completely present," and in Book XI, section XV, paragraph 20, Augustine says "the present, however, takes up no space."

Some people, including Peter Lynds, have proposed a solution based on this ancient premise. Lynds posits that the paradoxes arise because people have wrongly assumed that an object in motion has a determined relative position at any instant in time, thus rendering the body's motion static at that instant and enabling the impossible situation of the paradoxes to be derived. Lynds asserts that the correct resolution of the paradox lies in the realisation of the absence of an instant in time underlying a body's motion, and that regardless of however small the time interval, it is still always moving and its position constantly changing, so can never be determined at a time. Consequently, a body cannot be thought of as having a determined position at a particular instant in time while in motion, nor be fractionally dissected as such, as is assumed in the paradoxes (and their historically accepted solutions).

Another approach is to deny that our conceptual account of motion as point-by-point movement through continuous space-time needs to match exactly with anything in the real world altogether. Thus, one could deny that time and space are ontological entities. That is, maybe we should acknowledge our Platonic view of reality, and say that time and space are simply conceptual constructs humans use to measure change, that the terms (space and time), though nouns, do not refer to any entities nor containers for entities, and that no thing is being divided up when one talks about "segments" of space or "points" in time.

Similarly, one can say that the number of "acts" involved in anything is merely a matter of human convention and labeling. In the constant-pace scenario, one could consider the whole sequence to be one "act," ten "acts," or an infinite number of "acts." No matter how the events are labeled, the tortoise will follow the same trajectory over time, and all of the acts will be "finished" by the time the tortoise reaches the finish line. Thus, the labeling of acts is arbitrary and has nothing to do with the underlying physical process being described and that it is possible to "finish" an infinite sequence of acts.

From the philosophical standpoint of Bergsonian space-time, the paradox is resolved as follows. The steps of the paradox as presented above can be summarised as:

There are an infinite number of positions defined by any finite movement.
Let movement from one position to the next be called an 'act'.
An infinite number of acts cannot be completed in a finite amount of time.
An infinite number of acts cannot even be started.
Thus movement cannot be started or completed.
Movement is an illusion.

Moving backwards, any claims about the nature of illusions or acts are intrinsically claims about the nature of experience. According to Bergson's conception of time, all moments of time are comprised of a mixture of both a 'snap-shot' extrinsic property and a durational intensive property, which are irreducible to one another.[3]

The arrow paradox makes an argument that considers only time as a measurable, extensive, homogeneous construct that can be modeled spatially (the above diagram of it with lines being a good example). Thus a conclusion concerning the nature of experience is not warranted by an incomplete proof of only partial properties of time. The point is that 'acts' are experiential in nature.

Some people state that the dichotomy paradox merely makes the point that the points on a continuum cannot be counted — that from any point, there is no next point to proceed to. However, it is not clear how this comment resolves the paradox. Indeed, as one variant of Zeno's paradox would state: if there is no next point, how can one even move at all? Also, it is not clear what this comment has to do with different orders of infinity: the rational numbers are countable, i.e. of the same order of infinity as the natural numbers, but on the rational number line, there is for any rational number still no next rational number either.

Mathematicians thought they had done away with Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century. Most philosophers, and certainly scientists, generally agree with the mathematical results.

Zeno's paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome. L. E. J. Brouwer, a Dutch mathematician of the 19th and 20th century, and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this, he followed Leopold Kronecker, an earlier 19th century mathematician. Some claim that a rigorous formulation of the calculus (as the epsilon-delta version of Weierstrass and Cauchy in the 19th century or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson in the 20th) has not resolved all problems involving infinities, including Zeno's.


Paradox of Place:

"… if everything that exists has a place, place too will have a place, and so on ad infinitum". (Aristotle Physics IV:1, 209a25)

Paradox of the Grain of Millet:

"… there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially." (Aristotle Physics VII:5, 250a20)

For an expanded account of Zeno's arguments as presented by Aristotle, see: Simplicius' commentary On Aristotle's Physics.


Recently, physicists studying quantum mechanics have noticed that the dynamical evolution (motion) of a quantum system can be hindered (or even inhibited) through observation of the system. This effect is usually called the quantum Zeno effect as it is strongly reminiscent of (but not fundamentally related to) Zeno's arrow paradox.

  • Terry Pratchett in the book Pyramids combines the "Achilles and the tortoise" paradox and the arrow paradox to create the paradox of the arrow chasing the tortoise. [1] The concept of "nows" in the fletcher's paradox is also explored in Thief of Time.
  • Dilbert has claimed that "No one ever wants to take more than half of what's left of the last doughnut. That's why I call it Xeno's doughnut" (2005-08-13).
  • Umberto Eco in his 2004 novel (English language version 2005), The Mysterious Flame of Queen Loana, has the narrator (trying to recover from amnesia by going through old books and possessions) look at a recursive image and remark: "... Chinese boxes or Matrioshka dolls. Infinity, as seen through the eyes of a boy who has yet to study Zeno's paradox. The race towards an unreachable goal; neither the tortoise nor Achilles would ever have reached the last..."
  • In Beyond Zork, there is a bridge named "Zeno's Bridge". It is impossible to fully cross this bridge, as you can only go a fraction of the distance to the destination.
  • Phillip K. Dick, in his 1953 short story "The Indefatigable Frog", uses Zeno's paradoxes as a basis for an experiment that places a shrinking frog in a tunnel, thus always increasing the length of the tunnel relative to the frog.
  • Jorge Luis Borges explores various implications of Zeno's paradoxes in several of his stories and essays. In Avatars of the Tortoise (1932) he discusses how the argument of "Achilles and the tortoise" has manifested itself in the writings of Plato, William James, Lewis Carroll and many others, and goes on to argue that the paradox demonstrates the unreality of the visible world.
  • Tom Stoppard in his play Jumpers references Zeno's paradox via the philosopher George; George manages to kill his tortoise Thumper while attempting to disprove Zeno's paradox.
  • In Knight Rider Season 1 - Episode 09 Trust Doesn't Rust - 0:24:00 until 0:24:38 - a reference was made by KITT to an upcoming duel between him (KITT) and KARR (his earlier prototype). Kitt says: "...however since KARR is as powerful and as nearly indestructible as myself, Zeno's paradoxes should be affected." Devon Miles then explains who Zeno is. Kitt continues: "Zeno first postulated a question which my twin would most certainly be aware of: To wit; what would happen if an irresistible force met an immovable object?" In a later head-on-collision scene with KITT vs. KARR, Michael says: "..remember Zeno and that immovable object thing? We are about to find out the answer." Scholars do not attribute the irresistible force paradox to Zeno, and its origin is uncertain.
  • In Harry Turtledove's novel Wisdom of the Fox, the character Rihwin employs the dichotomy paradox to trick a demon into carelessly halving its distance to him and thus leaving itself vulnerable.
  • In the movie I.Q. (1994), Catherine Boyd (played by Meg Ryan), poses Zeno's dichotomy paradox to her love interest, Ed Walters (Tim Robbins), as a flirtingly jocular explanation as to why it is impossible for her to approach and dance with him.
  • In Frank and Ernest (comic strip), Zeno's dichotomy paradox is stated and said to be used as an excuse for the reason one character did not get to work on time.
  • In House Of Leaves, Karen Green interviews Douglas R. Hofstadter who references Achilles and the tortoise as an analogy for the spacial anomalies of the house.

Zeno's paradox on a 1990 Croatian election poster
Zeno's paradox on a 1990 Croatian election poster
  • A political party in Croatia in 1990 used Zeno's paradox of Achilles and the tortoise to encourage voters to choose the path of slow and clever civilian initiatives (the tortoise) rather than militant nationalism (Achilles as an armored warrior). Croatian voters decided that Achilles would win the race.
  • Artist Mark Tansey created a painting titled Achilles and the Tortoise which shows the plume from a rising rocket, around which there are observers, which mimics the appearance of a nearby evergreen tree, with the rocket just short of passing the height of the tree. In the foreground are a group of people finishing the activity of planting a new tree. The image, suggesting the rocket as shown will not ever pass the height of the tree, and tantalizing the viewer with the idea that this has something to do with the processes being shown and that the new tree may or may not pass the rocket first--raises questions about how much static representations can have you understand what's being represented. Tansey is alluding to Zeno's own logic as a representation of what he's describing.

  1. ^ Plato, Parmenides, 128c; Kirk, p. 277.
  2. ^ The quantum leaps of electrons lends credence to this idea. Electrons jump from one energy state to another (and from one orbital level to another) instantaneously.
  3. ^ Borradori, G., "The Temporalization of Difference: Reflections on Deleuze's Interpretation of Bergson."

  • Chan, Wing-Tsit, (1969) A Source Book In Chinese Philosophy. Princeton University Press. ISBN 0691019649
  • Kirk, G. S., J. E. Raven, M. Schofield (1984) The Presocratic Philosophers: A Critical History with a Selection of Texts, 2nd ed. Cambridge University Press. ISBN 0521274559.
  • Plato (1926) Plato: Cratylus. Parmenides. Greater Hippias. Lesser Hippias, H. N. Fowler (Translator), Loeb Classical Library. ISBN 0674991850.
  • Sainsbury, R.M. (2003) Paradoxes, 2nd ed. Cambridge Univ. Press. ISBN 0521483476.

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This article incorporates material from Zeno's paradox on PlanetMath, which is licensed under the GFDL.

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