Proposed solutions to Zeno's paradoxes

Proposed resolutions to Zeno's paradoxes can be divided into two classes: solutions that question the reasoning in Zeno's argument, and solutions that question the assumptions in Zeno's argument. Many mathematicians and engineers believe that Zeno's reasoning is mistaken, often using mathematics (calculus) to point out that an infinite number of terms can have a finite quantity. On the other hand, many philosophers believe that while there is nothing wrong with Zeno's reasoning, there are potential problems with his assumptions going into the argument. Thus, philosophers tend to try and resolve the paradox by pointing out that Zeno's description of the events taking place may not be truthful descriptions of what is actually physically occurring, and thus argue that metaphysical considerations of time and space are necessary to resolve the paradox.

Proposed solutions to Achilles and to the dichotomy

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

The supposed paradoxes arise from a confusion in the definition of a constant rate of speed. Consider the Tortoise and the Hare problem, where `steps' are ever-smaller and proportional to the remaining distance from the hare to the tortoise. The paradox implies that, if the hare is moving at a constant rate of speed, then the hare covers a constant number of `steps' per unit time, and so cannot take infinitely many `steps'. But a rate of speed is a distance per time, not a number of `steps' per time. If the hare runs at a constant rate of speed for 1 unit of time, the hare covers a constant amount of distance, but an ever-increasing number of `steps'. If the hare actually covered a constant number of `steps' per unit time, then the hare would indeed lose.

The difference between a constant rate of speed and the pace implied by the paradoxes was noticed by Aristotle. 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 -- that is, denying that a sum of infinitely many increasingly small numbers can be finite.

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.

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

: $asum\_\{k=0\}^\{infty\}\; x^k,$

which is 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^-1) and gets a head start of distance "d" metres (m), and that Achilles runs at constant speed "xv" ms^-1 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"/"x"²"v" s, Achilles has another "d"/"x" m, and so on. Thus, the time taken for Achilles to catch up is

: $frac\{d\}\{v\}\; sum\_\{k=1\}^infty\; left(\; frac\{1\}\{x\}\; ight)^k\; =\; frac\{d\}\{v(x-1)\}\; ,,\; exttt\{s\}.$

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

Similarly, for the Dichotomy assume that Homer walks at a constant speed towards the bus. Suppose that it takes time "h" seconds to reach half way; then it will take only a further time "h/2" s to reach three-quarters of the way, another "h/4" s to get to seven-eighths of the way to the bus, and so on. The total time taken by Homer is

::$h\; sum\_\{k=0\}^\{infty\}\; left(\; frac\{1\}\{2\}\; ight)^k\; =\; 2h\; ,,\; exttt\{s\}.$

Once again, this is finite, and so (provided it doesn't leave for 2"h" 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.

Proposed solutions to the arrow paradox

The arrow paradox raises questions about the nature of motion that are not answered by the mathematical attempts to solve the Achilles and Dichotomy paradoxes.

This paradox may be resolved mathematically 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.

The problem with the calculus solution is that calculus can only describe motion as the limit is approached, based on the external observation that the arrow plainly moves forward. Zeno's paradox however implies that if Zeno's method is followed to its logical extent, concepts such as velocity lose all meaning and there is no causal agent that is not similarly affected by the paradox that could enable the arrow to progress.

Another point of view is that 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.

Physical explanations

The calculus-based explanations given above outline a model of motion where one can certainly talk about a final state in the presence of continuity. Some people claim that such mathematical models sidestep Zeno's paradoxes, which they say are basically paradoxes about the nature of physical space and time. Some people, including Peter Lynds, have proposed alternative solutions to Zeno's paradoxes. 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).

Finitely divisible argument

One method of dealing with these paradoxes has been the claim 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 between any two points in time or space there always exists some other point in time or space. Due to the inability to prove infinite divisibility qua space and time, there can be no cogent argument in support of these "paradoxes." This, however, does not solve the arrow paradox.

References

*Zeno's Paradoxes at Wolfram Mathworld [http://mathworld.wolfram.com/ZenosParadoxes.html]
*An accessible calculus tutorial [http://www.karlscalculus.org/calc2.html]

ee also

Zeno's Paradox