Sunday, April 4, 2010

Zeno's paradox

I've always been intrigued by paradoxes in Physics, my favorite being the Zeno's paradox.  While there are various paradoxes in different fields, most I've come across are too mathematical in nature and do not directly relate to real world examples.  Zeno's on the other hand is a pretty interesting one and I've never had to try too hard to explain it to someone.   I was recently having a conversation about the Zeno's paradox with one of my colleagues and he had an interesting explanation and reasoning about the paradox.

The Zeno's paradox involves a segmented view of space or time. The tortoise and the rabbit (or Achilles) paradox says that in a race between a tortoise and a rabbit where the rabbit begins the race after a non-zero time delta from the time the tortoise began the race, it is impossible for the rabbit to ever win the race. This is because, say the starting position of the race is A, and while the rabbit began the race, the tortoise had moved to a position A+in the time interval d mentioned above.  Now, by the time the rabbit reaches the position A+x, the tortoise would have reached a different position A+x+y. This can be generalized to any position of the rabbit during the race, where, when it reaches a previous position T of the tortoise, the tortoise would have reached a different position T+farther from the rabbit, as the tortoise is in continuous motion.  Therefore, the rabbit always stays behind the tortoise and can never win the race, irrespective of how much faster the rabbit moves compared to the tortoise. It's also pretty easy to write a proof for this using mathematical induction.

Here is the reasoning that convinced me: the paradox is caused due to a distorted understanding of finiteinfinite, and infinitesimal distances. The laws associated with finite distances cannot hold true when the space is divided into infinitesimally short segments, as in the above case of the tortoise and the rabbit.  My colleague had a clearer explanation for this: basically the distance between any two points A and B in space can be divided into an infinite number of segments. Mathematically, the sum of an infinite number of distance segments is infinity.  However, when the individual distances themselves are infinitesimally short, their sum need not necessarily be infinity, and can be finite.  Therefore in the race between the tortoise and the rabbit, although there were infinite number of points, their sum is still finite (as the race is for a finite distance), and the above induction logic wouldn't hold true. 

Another of Zeno's paradoxes, the "Arrow of Flight" paradox is interesting too, and the above explanation seems to apply to that as well.

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