Zeno s paradoxes

Zeno's paradoxes are a set of paradoxes conceived by Zeno of Elea to support Parmenides's doctrine that all evidence of the senses is misleading, and particularly that there is no motion.

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, that of a rock thrown at a tree, and that of an arrow in flight—are given here.

Zeno's paradoxes were a major problem for ancient and medieval philosophers, who found no satisfactory solution to them. Mathematicians thought they had done 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 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.

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Achilles and the tortoise

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.

In the modern analysis, the paradox is resolved with the fundamental insight of calculus that a sum of infinitely many terms can yield a finite result. Adding the (infinitely many) times together that Achilles needs to reach the previous positions of the tortoise results in a finite total time, and that is indeed the time when Achilles overtakes the tortoise.

The rock thrown towards a tree

The next paradox, that of the rock thrown towards a tree, is a variant of the previous one. Now Zeno stands eight feet from a tree, holding a rock. He throws his rock at the tree. Before the rock can reach the tree, it must traverse half the eight feet. It will take some finite time for the rock to fly four feet. After that time, it will still have four feet to go, and to traverse that distance must first cover half of it: two feet, and more time. After it travels two feet, it must travel one foot, then half a foot, then a quarter foot, and so on ad infinitum. Therefore, Zeno concludes, the rock can never hit the tree.

The arrow paradox

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.

This paradox is resolved by calculus 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).

The Quantum Zeno Effect

In recent time, physicists studying quantum mechanics have noticed that a quantum system's dynamical evolution (motion) can be hindered (up to inhibited) by observing the system. As this effect strongly reminds of Zeno's paradox of the arrow that cannot move because whenever it is observed it is found at a definitive position it is usually called the "quantum Zeno effect".

External links

References

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


da:Zenos paradoks fi:Zenonin paradoksit fr:paradoxes de Zénon ja:ゼノンのパラドックス sv:Zenons paradoxer zh-cn:芝诺悖论

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