Wednesday, March 01, 2006

The problem with finding the state of the light switch after it has been switched an infinite number of times is that the sequence of switching the light on and off is an ocillatory process.

To simplify matters, suppose that we take the simplest oscillatory curve in existence - the sine wave. y=sine(x) is constantly oscillating throughout its graph, back and forth between 1 and -1. If you ask what value y is at when x=infinity, then the answer is that its indeterminate. There is no problem with having an indeterminate answer either, since we could never have an oscillation, in physical reality that occurs infinitely fast. An oscillation that occurs infinitely fast would involve infinite amounts of energy (breaking the law of conservation of energy) and it would involve moving your hand to flip the switch faster than the speed of light (breaking special relativity.) Since the process isn't even physically possible to begin with, I find that its silly to wonder why rediculous answers come about as a result. In fact, from the point of view of a physics major, I find that this example gives a perfect thought experiment to try to figure out why oscillatory motion cannot occur infinitely fast. Silly results come about when you ask questions about things which are physically impossible.

However, the fact that the limit of an oscillating sequence is indeterminate has no bearing whatsoever on the limit of a non-oscillatory sequence. Since ordinary motion - that is, motion from one point to another - does not involve any oscillatory terms whatsoever, we do not have this problem. In fact, no law of physics is broken in ordinary motion either - so the light switch flipping thingy is a really poor argument.

If we are going to define passing by a point as being one particular example of an infinitesmal task, then I find no problem in likewise thinking that passing by an infinitesmal instant of time is also a task. To pass over even one femtosecond, we need to perform an infinite number of tasks - that is, we need to move through an infinite number of infinitesmal times. Doesn't any one have a problem with that definition of a task? Please note that, as Einstein showed, space and time are basically interchangeable (through d=sqrt(x^2+y^2+z^2-(ct)^2)), so if we assume that space is discrete, then time must be discrete as well. Likewise, any other problems that we find with space should necessarily apply to time as well.

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