**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 may seem trivial today, but they were a major problem for ancient and medieval philosophers, who found no satisfactory solution until the 17th century, with the mathematical results on infinite sequences and calculus.

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2 The rock thrown towards a tree 3 The arrow paradox 4 External links 5 Update |

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 start 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.

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.

See http://philsci-archive.pitt.edu/archive/00001197/ for the original paper.