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Zeno's paradoxes are a set of paradoxes 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.

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.

Zeno's paradoxes were a major problem for ancient and medieval philosophers, who found most proposed solutions somewhat unsatisfactory. More modern solutions using calculus have generally satisfied mathematicians and engineers. Many philosophers still hesitate to say that all paradoxes are completely solved, while pointing out also that attempts to deal with the paradoxes have resulted in many intellectual discoveries. Variations on the paradoxes (see Thomson's lamp) continue to produce at least temporary puzzlement in elucidating what, if anything, is wrong with the argument.

The Paradoxes of motion

Achilles and the tortoise

"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 so fast a 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, during which 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.

The dichotomy 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 fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

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

The resulting sequence can be represented as:

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

This description requires one to travel an infinite number of finite distances, which Zeno argues would take an infinite time -- which is to say, it can never be completed. This sequence also presents a second problem in that it contains no first distance to run, for any possible first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even be begun. 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.

The arrow paradox

"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)

Finally, in the arrow paradox, we imagine an arrow in flight. At every moment in time, the arrow is located at a specific position. If the moment is just a single instant, then the arrow does not have time to move and is at rest during that instant. Now, during the following instances, it then must also be at rest for the same reason. The arrow is always at rest and cannot move: motion is impossible.

Whereas the first two paradoxes presented divide space, this paradox starts by dividing time - and not into segments, but into points.

Proposed solutions

Proposed solutions to the arrow paradox

One objection to the arrow paradox is that the arrow paradox seems to be a play on words more than anything else. In particular, the premises state that at any instant, the arrow is at rest. However, 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, the arrow would be at a different place than it was and will be at the times before and after. Therefore, the arrow moves. 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.

Another solution is that the instantaneous physical state of the arrow cannot be fully specified by its position alone: one must specify both its position and its momentum.

Proposed solutions both to Achilles & the tortoise, and to the Dichotomy

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 BC, 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.

Solution using mathematical series notation

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

${\displaystyle a\sum _{k=0}^{\infty }x^{k},}$

which is convergent and equal to a/( 1 − x) provided that |x| < 1 (otherwise the series diverges). The paradoxes may be solved by casting them in terms of geometric series. Although the 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, and for Homer to catch the bus.

In the case of Achilles and the tortoise, suppose that the tortoise runs at a constant speed of v metres per second (ms) and gets a head start of distance d metres (m), and that Achilles runs at constant speed xv ms with x > 1. It takes Achilles time d/xv seconds (s) to reach the point where the tortoise started, at which time the tortoise has travelled d/x m. After further time d/xv s, Achilles has another d/x m, and so on. Thus, the time taken for Achilles to catch up is

${\displaystyle {\frac {d}{v}}\sum _{k=0}^{\infty }\left({\frac {1}{x}}\right)^{k}={\frac {d}{v(x-1)}}\,}$ 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 s for him to complete the second-last step, traversing the distance between one quarter and half of the way 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 s, and so on. The total time taken by Homer is, summing from k=0 for the last step,

${\displaystyle h\sum _{k=0}^{\infty }\left({\frac {1}{2}}\right)^{k}=2h\,\,}$ seconds.

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

Note that it is also easy enough to see, in both cases, that 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, because they will eventually have moved far enough. 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.

Solution using calculus notation

• d = distance between runners
• t = time

${\displaystyle \lim _{d\to 0}f(d)=t}$

Problem with the calculus-based solution

A problem with using calculus to try and solve Zeno's paradoxes is that this only addresses the geometry of the situation, and not its dynamics. What is at the core of Zeno's paradoxes is the idea that one cannot finish the act of sequentially going through an infinite sequence, and while calculus shows that the sum of an infinite number of terms can be finite, calculus does not explain how one is able to finish going through an infinite number of points, if one has to go through these points one by one. Indeed, saying that there are an infinite number of points or intervals within some finite interval is of course the very assumption in the Achilles and Dichotomy Paradoxes, and it is this assumption regarding the geometry of the situation that leads to a paradox regarding its dynamics.

It should also be noted that calculus-based solutions that are offered often object to the claim that "it must take an infinite amount of time to traverse an infinite sequence of distances". However, Zeno's paradox doesn't contemplate the time it would take for Achilles to catch the Tortoise; it simply points out that in order for Achilles to catch up with the Tortoise, Achilles must first perform an infinite number of acts, which seems to be impossible in and of itself: time has nothing to do with it. Thus, calculus-based solutions to Zeno's paradoxes often make the paradox into a straw man.

Some people, using the notion of a supertask, argue that an infinite number of things can be completed in sequence as follows: do the first thing in 1 second, the second in half a second, the third in a quarter, etc. Again from a calculus point of view, this would seem to mean that after 2 seconds an infinite number of things would have been done. However, there would seem to have to be a problem with this, as getting to the end of an endless series would seem to be a contradiction in terms. Indeed, the problem with this argument is exactly the same as before. Since this line of reasoning assumes that there is a one-to-one mapping between the acts and the points (or intervals) in time during which that act takes place, one would need to go through (and finish) an infinite number of points in time in order to complete the infinite number of acts to be performed. But, once again, such an infinite sequence can, by its very infinite nature, not be finished. Indeed, following Zeno's logic as we just did now, we have created yet another version of his paradox: not only does motion through space seem to be impossible, but the flow of time itself seems to be impossible as well.

In short, trying to use calculus to resolve the paradox simply reaffirms the idea that space and time are infinitely divisible, and thus still suffers from the basic question as to how one can possibly reach the end of an endless series.

Are space and time infinitely divisible?

Given the above analyses, a rather straightforward solution to some of the paradoxes is to deny that space and time are 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. Indeed, physicists talk about Planck length and Planck time as the smallest meaningful, measurable units of space and of time, thus making measurements of both time and space discrete rather than continuous. Of course, whether or not space and time are measurable with infinite precision is ultimately irrelevant to the paradoxes and their resolution: what we as humans can know about the world is a different matter from what is or is not true or possible in the world. That is, quantum mechanics may prevent us from making infinitely precise measurements, but if time and space are continuous, the paradoxes still apply. However, if time and space are discrete, one avoids obtaining the infinite series that underlies most of the paradoxes.

Does motion involve a sequence of points?

Some people, including Peter Lynds, have proposed an alternative solution. 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 how 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).

Conceptual approaches

A final approach is to deny that time and space are ontological entities. That is, maybe we should give up on 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. Thus, compatible with the last two approaches, one denies 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.

The notion of different orders of infinity

The dichotomy paradox merely makes the point that the points on a continuum cannot be counted - from any point, there is no next point to proceed to. This is uncontroversial.

Status of the paradoxes today

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. Many philosophers, and certainly engineers, generally went along with the mathematical results.

Nevertheless, 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. It would be incorrect to say 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 resolved forever all problems involving infinities, including Zeno's.

As a practical matter, however, no engineer has been concerned about them since knowledge of the calculus became common at engineering schools. In ordinary life, very few people have ever been much concerned.

Two other paradoxes as given by Aristotle

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.

The quantum Zeno effect

In recent time, 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 Zeno's arrow paradox.

See also

• Supertask
• Zeno machine

References

• R.M. Sainsbury, Paradoxes, Second Ed (Cambridge UP, 2003)

External links

• http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp
• Zeno's Paradoxes: A Timely Solution by Peter Lynds
• Zeno's Paradoxes of Motion: a section from Kevin Brown's book Reflections on Relativity
• BBC article on shortest time measured as of 2004: 10 seconds.
• Stanford Encyclopedia of Philosophy entry
• Zeno meets modern science

Zeno's paradox at PlanetMath.

• Paradoxes
• Supertasks
• Infinity