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Sunday, February 23, 2014

Zeno's Paradox

 This is the first if the typed rough drafts, I must get it down in some form to refine it and welcome any part of the discussion.


"His essential question was how can you do an infinite amount of tasks in a finite amount of time?"
How does this idea of addition/subtraction  solve this statement.
I often introduce Zeno's paradox in my math classes as a way to discuss asymptotic behavior and limits.  I am not the only one to do this for many teachers find it a nice way to consider these things.  The general arguments against Zeno involve limits but I was wondering if there was another way around it for even philosophers don't feel that such  mathematical disprove ends the philosophic argument.  In my musings and considerations, I stumbled upon both a simpler mathematical approach and perhaps a different way to answer it in terms of metaphysics.  At the center of Zeno's argument is that you can not divide something and have nothing. This is true. Even if the limit definition says the same the thing for the limit is not about the point of dividing by infinity but what that point would look like if we could do so.  Calculus added an extra definition for continuity at that point in case we care about its existence.  If one wants to argue against the paradox using the mathematics of limits one must also include that the lim f(x)= f(a) and that f(a) exists.
Zeno, like all good Greek thinkers, loved proportions.  He argued that the arrow would never make its target because it would always be 1/2 the distance closer. This simple statement is where he skipped a stepped of mathematics, logic, and philosophy.  A proportion is a form of division.  Division is the inverse of multiplication and multiplication is repeated addition.  We do perceive, measure, and understand the passage of the arrow or Achilles by addition.  (Recall that subtraction is just a different form of addition.)  While we can never divide something to be nothing or multiply nothing to be something, we can easily add something to nothing or take away to be nothing.  This element is true in the scenario but it was not expressed in the paradox.  In all logical or mathematical arguments, we need to express these things in the beginning for the argument may lose sight of simple definitions.  We turn the passage into a problem of division, which does not allow for the zero space but our original conditions do so.
The illusion of motion is turning the addition of intervals into a zero sized one.  It is viewing flight as photograph and stating that because it does not travel in zero time so it never travels.  Consider this photograph if we were in it, for that is how Zeno presented it.  If time is frozen, we will not perceive any motion or much of anything for we will always be puppets of the infinitesimal.  This moment is one of those semi-pointless exercises in philosophy for we know the arrow to travel and we know time to pass.  Theoretical physics can change the speed of time, and even pause it for light, but not for us. 
As for that original question, of which, I need to refind the source. Consider the whole path the arrow travels as one.  It goes 1/2 the distance in 1/2 the time, and therefore makes 2 intervals.  We can keep dividing the distance and increase the number of intervals.  It will go 100 intervals of 1/100th the time.  In this manner, we can find an infinite amount of tasks in a finite amount of time.  The challenge really highlights our analytic need to make things discrete and countable and the simple continuity of a line. 
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