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ON CHANGE Zeno

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The Dichotomy Zeno’s arguments about motion which provide trouble for those who try to resolve them are four in number. The first maintains that nothing moves because what is travelling must first reach the half-way point before it reaches the end… The second is the so-called Achilles. This maintains that the slowest thing will never be caught when running by the fastest. For the pursuer must first reach the point from which the pursued set out, so that the slower must always be ahead of it. This is the same argument as the dichotomy, but it differs in that the additional magnitudes are not divided in half. (Aristotle Physics VI:9, 239b15)

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The Dichotomy

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1) In order for anything to move, it would have to complete an infinite number of changes (“tasks”) in a finite time. 2) It is impossible to complete an infinite number of tasks in a finite time. 3) [So] It is impossible for anything to move.

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The Arrow Paradox You cannot even move. 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)

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The Arrow Paradox 1. If something occupies the same space for a period of time, then it is at rest during that time. 2. Everything occupies the same space during each moment it exists. 3. [So] Everything is at rest at each moment it exists. 4. If (3), then nothing moves. 5. Nothing moves.

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The Arrow Paradox

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The Addition Principle: If a temporal interval T can be divided into some shorter intervals, the ts, then the distance traveled during T is equal to the sum of the distances traveled during each of the ts.

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The Stadium The fourth argument is that concerning equal bodies which move alongside equal bodies in the stadium from opposite directions—the ones from the end of the stadium, the others from the middle—at equal speeds, in which he thinks it follows that half the time is equal to its double…. (Aristotle Physics, 239b33)

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SCIENTIFIC CONCEPTS OF TIME AND SPACE. Time has played a central role in mathematics from its very beginnings, yet it remains one of the most mysterious.

SCIENTIFIC CONCEPTS OF TIME AND SPACE. Time has played a central role in mathematics from its very beginnings, yet it remains one of the most mysterious.

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