Zeno's paradoxes

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

"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)

The dichotomy paradox

"that which is in locomotion must arrive at the half-way stage before it arrives at the goal." (Aristotle Physics VI:9, 239b10)

The resulting sequence can be represented as:
:
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 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 paradoxical 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

"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)

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 Achilles & to the Dichotomy

Both the paradoxes of Achilles and the Dichotomy depend on an 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.

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

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/x²v s, Achilles has another d/x m, and so on. Thus, the time taken for Achilles to catch up is

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

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 matter is not infinitely divisible; that there exist particles of matter so small that further division is not possible. (Atomism did develop as a response to these paradoxes.) However, Zeno's paradoxes are about space and time, not directly about matter. It is, however, still open to question whether 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, just because we use numbers for space and time, we can usefully endlessly measure (now or ever) moments of space and time between any two other moments of space and time. It is ludicrous, using a metre stick, to give a measurement of 0.194612065 metres because the measuring apparatus used cannot be that precise. So too, we may need to recognize that there are fundamental units of space and of time that we cannot (now or ever) measure any smaller. Physicists talk about Planck length and Planck time as the smallest meaningful, measurable units of space and of time, thus making measurements both of time and space also discrete rather than continuous. This all raises the issue of how much any mathematics is an apt description of "the world".

This argument, however, presumes that time is an ontological entity in the same way that matter is - a somewhat Platonic view of reality - instead of time being simply a construct humans use to measure change. A less Platonic approach would be to say that matter, itself discrete, can only move through discrete distances. Notice also that the paradox depends in part on stopping at each infinitesimal point, instead of simply continuing the movement through all the many points.

Status of the paradoxes today

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 Weierstrauss 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)

"… 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)

The quantum Zeno effect

See also