Friday, 14 March 2014

Achilles and the Tortoise



I have many varied interests, some of them you know, Some of them I know, some I dare not admit to myself and some may well forever remain unidentified. But of all of them one is a curious liking to obscure paradoxes. I like the inconclusivity of them, the sheer reluctance of them to conclude to something, the ambiguity they persists and the infinite admiration that it invokes. Be it the time travelers paradox that effectively renders him a silent spectator of ages past or the Achilles paradox pointed out by the greek philosopher Zeno that is so obviously untrue yet mathematics require obscure concepts to prove it so. I take pride in the fact that these concepts are elusive and obscure to many it makes sense to me.


Now the paradox of Achilles I'd something that captured my imagination for sometime now, I will let my readers enjoy and revel in it's magnificence while I go and pen down something worth while.




In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. 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 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still 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, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.








PS : I was so lazy that I effectively copied the while definition of this paradox outright from Wikipedia.