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 16^th 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.
   

2. 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.
   

3. 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.
   
   

Period three implies chaos

Today I went into AOD's school and did a short presentation on Chaos. I've always liked chaos as a concept.

Chaos passes through the times in life where the unstable mixture of frustration and confusion and rage have been ignited by a spark of love. The devastation left by this conflagration leaves tiny fragments of life, each of which bears a similarity to the whole. The deeper you look the greater the number of similar patterns emerge. As you zoom out to try and take in the whole the patterns fit together in a jumbled superpatern with familiar resemblance to each piece. Because the fragments fit together it is not as if the world has broken; more like a veil has been removed and detail beyond the powers of human resolution are revealed.

Of course in going in to talk to AOD's advanced Jr. high math-class I take a slightly less maudlin approach to the subject. I also wanted to avoid the use of complex numbers. So I found a really cool little program and rewrote it cludgily (word?) to run on Linux and printed up a giant multi-page Mandelbrot poster. I am too embarrassed of my own code to post it, but here is a link to the original coder. Perhaps I'll post a picture of the poster if I go back to school before it gets destroyed. Something about being in a Jr. High at all that makes my skin crawl. All I really know about Jr. High is how to engage in trouble so profound the very mental state of the delinquents is eroded. Often we must not teach from our own experiences, but instead from the things we read in books.

The Mandelbrot set is such an easy concept considering it is infinity complex. The way I presented it was by 1^st going over the Koch curve, Menger sponge, a few nature fractals and then the logistic equation. I then presented the following equations:

Which I described as iterative functions. I presented the output generated by iterating from a couple of closely spaced points (carefully chosen so one of them was in the Mandelbrot set and the other was outside). The picture looked like this:

I explained that I would put a black dot where the starting point of the spiraling-in iterative path began. “This black dot” I said “is IN the Mandelbrot set”.

I had this projected on the wall, so instead of going into the Pythagorean distance formula to determine the deviation from the starting point I picked up a meter stick and began to measure the distances right on the wall. This created a dramatic pause, filled only with mad gesticulations by a middle aged man. Then I exhaled the statement “when these measurements exceed two we count the number of iterations it took to get here”. I turned to the audience and changed the slide to a spectral rainbow with a scale of integers beside it. “I then look up the number of iterations on this arbitrary color scale and color in the dot.”

“The”

I paused

“result”

I changed the slide.

“of filling in all the dots outside the Mandelbrot set on the Cartesian coordinate system is this”

“...and if we zoom in here”

I pointed to a small box on the slide

“we get this”

and I put up this slide.

“And if we zoom again we get this.”

“Again and this”

“and this”

“again”

“again...”

I thought it went over OK. Perhaps I get a little to theatrical for a third period Jr. High math-class. Of course “period three implies chaos”.

Chaos from life

I really like the Logistic equation. It comes in a couple useful forms, the simplest is the iterative form that can be used to model a time discrete biological population.For instance if one has a population that produces new members at a rate “r” of the current population we can generate the new population by iterating the equation:

Where an+1 is the next years population, spawned from this years. “r” is obviously the fertility rate (or better the “fecundity” rate) . This equation just demonstrates an ever expanding population. Lets assume that there is an upper limit to the population given by some constant like “K”. Then one model of the population is given by the logistics equation:

In order to make it simpler I can define A=a/K or simply set K to 1 (which may appear weird, setting a population maximum to 1, but I can use units like “metric tones of wheat” or “boxcarloads of bunnies” so the maximum population of 1 represents more that one individual). This gives me the super simple equation:

This is the equation of a parabola. To show it is nothing special here is a plot of it. I choose a few different values for r to give a feel for what it does. It is no less elegant than any parabola.

We cannot get this equation to do interesting things until we begin using it iteratively.

What I mean by this is that we use the output of running the equation as the input of running the equation again. We do this over and over and see what happens. If we choose r-2.9 the iterations look like this:

If I clean off the first 500 iterations you can see that the oscillations shown in the fist graph have settled down to a single value. If I choose a value less than 2.d the iterations also settle down to a single number. With decreasing values for r this number decreases to 0 at r=0.

If I choose a value for r slightly higher than 3 this is what happens.

That’s right we get a stable oscillation between two values.

If I choose r=3.5 I get four values like in this graph:

If I choose an r=3.7 a bizarre thing happens here is the graph for the first 100 iterations:

It almost looks like there could be oscillations settling down to a pattern. If I just plot the iterations after say 100 you can see that they apparently do not.

This is cool. In fact there is an infinite level of complexity here. If I vary r from 0 to 4 and plot the values obtained for 100 iterations after iterating the function 1000 times I would get a picture like this.

I can stare at this picture for hours. Look at the areas where order appears to peek out of chaos. Is this a metaphor for life from the plot of an equation that rustically describes life? What about those apparent lines in chaos? I love to blow up the picture and look at the bits of it. The more it is enlarged the more complexity it reveals.

Part of the cool thing about this is that if people used the generally available computation methods available when I was born (slide rules!) then a person calculating this picture would only recently have finished. This is a pattern almost unknown to previous generations. Now one can buy a t-shirt with it printed on it.

More will be revealed!