Monday, December 21, 2009

Zeno's paradox

Zeno's is a particularly unsatisfying paradox for me. By Zeno of Elea's paradox I describe a generalized version of his most popular two (the arrow, and Achilles and the tortoise)

Firstly it makes you work to get what it is about. The recursive paradoxes are much more inviting in this regard. Who cannot be sucked in by the “this statement is false” paradox. After thinking about this recursive paradox for a while you can at least be comfortable in the knowledge that the only time you have wasted is the time actually spent thinking about the paradox itself.

Secondly there are several apparent solutions to Zeno's paradox.

Zeno's paradox is traditionally told as some sort of question concerning a really fast Greek god racing a turtle or some great Greek warrior like Achiles running to someplace famous. The details are not important. My favorite explanation of it is embedded in a joke.

  An engineer, a mathematician, and a theoretical physicist went to a dance. Shyly they positioned themselves against a wall where they had a good view of the dance.
  The mathematician sighed heavily and said “I wish I could go ask one of those people sitting at that table over there to dance with me, but it is impossible.”
  “Why is that?” asked the theoretical physicist.
  “If I go halfway over to the table, I will still have halfway to go” replied the Mathematician.
  “Yes” Said the engineer.
   “Then if I cover half the remaining distance I will still have a quarter of the way to go” Said the mathematician.
  “Yes” Replied the engineer.
  The mathematician continued “I can then cover half the remaining distance, but a 16th of the distance remains.”
  The theoretical physicist chimed in “Everytime you cover half the distance to the table a small but calculatable amount of distance remains.”
  “Right!” said the mathematician “So it impossible for me to go over there and ask for a dance”
  The physicist was about to commiserate with a “too bad for us” when the Engineer got up and walked over to the table.
  The physicist and the mathematician watched in amazement as the engineer asked a particularly attractive young lady to dance, proceeded to dance with her, gave her a lingering kiss, and then came back to their place on the wall.
  “How did you do that?” asked the physicist in awe.
  “Although you were correct I calculated that I would be able to get close enough for any purpose I could think of”.

I like the joke because it is one of the very few I can think of where the engineer gets the girl. Of course with competition like a theoretical physicist and a mathematician the odds are seriously stacked in his favor.

The crux of the paradox is that infinite division creates an infinite number of pieces. Any number can be used. Any situation where the finite measurable quantity could conceivably be infinitely divided can be used. The purpose this paradox is used for most often is to introduce the infinitesimal. A common place to hear it is in introductory calculus. The trouble is that as soon as some people hear it they are thinking in terms of solutions and not of the backdoor introduction to the elusive fluxion.

A couple of solutions are:
  1. The mapping solution. Suppose you tell Mr. X that he can find the special something against the far wall of a room, but instead you put it about three quarters of the way across the room. Mr. X dutifully covers half the distance to the far wall, knowing he will never reach it. He then covers half the remaining distance. Before he can move you run up to him telling him to reach down and pick up the special something.

    In this solution you know that Mr. X's proposed infinite path must cross through specific identifiable points (in this case I identified the three quarters point) along the way. You simply map the destination to one of these points so that Mr.X and the special something are coincidental.
  1. The multi-dimensional limit solution. This is the solution most desired by teachers of beginning calculus. In this one the engineer is pictured as traveling at a constant rate of speed. He travels half the distance across the room in one minute (it is a really big room). He then covers half the remaining distance in 30 seconds. Half the remaining distance in 15 seconds, half the remaining after that in 7.5 seconds. In this way as the chunks of distance get infinitesimal so to do the periods of time taken to traverse them. One can then show that if one adds up this infinite set of numbers it can take no longer than one minute to traverse the second half of the room.

    Here the number of parcels in one dimension are offset by the size of those parcels in another dimension. If I were teaching introductory calculus I would pause at this point and introduce several notational schemaes.
  1. The improbability solution. This solution may be my favorite because its heart it is fraught with complexity. What do we really know, and when do we know it? When the poor paradox-ed individual leaves from their starting point the questions of their existence are small with respect to the question at hand. We can identify a ratio of their size to the size of the distance traversed. For any reasonably sized distance that ratio is quickly stood on it's head. When the distance to be traversed is close to the size of the traverser then aspects of the traverser become important. How far have they really moved? If they breath in and their chest expands have they moved again? If they breath out are they moving backwards? Even if we shore up our paradox with some lame stipulation we do not keep trouble at bay for long. Due to the nature of geometric progressions we are soon at the size where surface irregularities rule. The border of a human, at the microscopic level, is not precise. Cells slough off, bacteria move around, strange growths blossom. Are the chunks of skin raining off the mover still part of them? The nature of the verbal tricks needed to maintain our paradox are week at this point, but more trouble awaits. Soon we are at the scale of the atom. Is the individual as large as the distance traveled by the furthest electron orbit associated with the atom closest to the destination? Since that orbit is well described as a probability cloud that extends around neighboring atoms (including those in air molecules) how do we draw a line? Are we to be cavalier and arbitrarily choose one? If we do this then aren't we arbitrarily creating a paradoxical structure. What does the paradox mean if we are to depend on arbitrary decisions in order to maintain it? We become subjective. To person B the traveler has passed into the sphere of the destination. Person C (who is more interesting than persons A or B) notes that there is a certain probability of calculatable physical interaction even before the traveler started across the room and could be said to have arrived even before leaving.

    Persons A and B often wonder why they ever invite person C to their parties.


Kold_Kadavr_flatliner said...

Wow. God blessa youse -Fr. Sarducci, ol SNL

Kold_Kadavr_flatliner said...

Your looooong reasons, brudda, R, like, so earthly and basic they don't go anywhere besides whorizontal - just to hit-you-in-the-@$$. Leave this world through prayer. Five minutes somewhere quiet. You don't gotta say ANYTHING except talk to God about your anger. Believe-you-me, bro, I had a whole #@!! lotta rage after our wreck and, with a severely co-dependant/transversalism family, whoa. All the better. Can't wait till I croak. I believe this world is a complete Hell: up and down roller-coaster. Just don't be on the LEFT; don't deny Jesus when you die. Then, you'll expience the REAL Hell fo'eva. Join me in Heaven, dude, where we'll have a BIG-ol, kick-ass, party-hardy for eons and eons celebrating our resurrection. How long is this (as I'm holding a weee, rice seed)? Our lives. And how long is eternity? Encircling the universe. God bless you --- Meet me beyond the clouds. See ya soon. Ya know that famous, French card player in the 1800s who doubted his faith? Look him up.

adult onset atheist said...

Perhaps you could try meditation Mr KK Flatliner. It produces many of the benefits of prayer without uncomfortable dogmas sticking to your hair.