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the Eleatic (problem of motion): *
Melissus of Samos (5th century BC) was the 3rd member of the school; like Parmenides, he argued that reality is un-generated, indestructible, indivisible, changeless, and motionless; reality is wholly unlimited, & infinitely extended in all directions; since existence is unlimited, it must also be one. Parmenides & Melissus built arguments starting from sound premises. Zeno, on the other hand, primarily employed the reductio ad absurdum, attempting to destroy the arguments of others by showing that their premises led to contradictions (Zeno's paradoxes).
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Zeno (490–430 BC) was also of the elatic school; none of whose works survived; his arguments on motion were repeated by Aristotle (his Physics) & Simplicius of Cilicia. His uses a method of proof called reductio ad absurdum. The most famous of Zeno's paradoxes (a set of philosophical problems) are the arguments against motion ; they were written to support Parmenides' doctrine that contrary the senses, plurality, change & motion are nothing but illusions. Some of Zeno's 9 surviving paradoxes are equivalent to one another: 3 of the strongest & most famous were: the Dichotomy argument, The arrow in flight and Achilles and the tortoise.
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The Dichotomy Argument.
Zeno argued that a runner will never reach the stationary goal line on a straight racetrack. The reason is that the runner must first reach half the distance to the goal, but when there he must still cross half the remaining distance to the goal, but having done that the runner must cover half of the new remainder, and so on. If the goal is one meter away, the runner must cover a distance of 1/2 meter, then 1/4 meter, then 1/8 meter, and so on ad infinitum. The runner cannot reach the final goal, says Zeno. Why not? There are few traces of Zeno’s reasoning here, but for reconstructions that give the strongest reasoning, we may say that the runner will not reach the final goal because there is too far to run, the sum is actually infinite.
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The arrow in flight..
Zeno argues that at each point along an arrow’s path, the arrow is not moving, as it occupies a single space in a moment of time. This means that the arrow, as it moves, is unmoving at each point. Because a thing cannot move and stay unmoving at once, all motion is impossible.
Achilles & the tortoise.
Zeno argues that Achilles in a footrace with the tortoise allows the tortoise a head start of 100 meters. Each racer starts running at a constant speed, one faster than the other. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, 2 meters. It will then take Achilles some further time to run that distance; by now the tortoise will have advanced farther; and then Achiles spends more time still to reach this third point, while the tortoise moves ahead. Whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can reach the tortoise. This argument is similar to the Dichotomy.
