Entries for August 2006
Inequality
I remember watching someone getting arrested out front of a convenience store one night several months ago. Six police cruisors were crowded into the tiny parking lot with all of their headlights spotlighting the arrest. One man was face down on the ground with two officers handcuffing him while another six or seven officers stood to the side and watched. "Yeah, take that!" I thought to myself. "That's what you get for making my neighborhood into a piece of crap!"
A couple of minutes later, I pulled into my driveway and shut off the car. What was that song they were playing as I was leaving the party? As I fumbled for my house keys, I tried to think of a few lyrics that I could google to grab a title. Once inside my house, I walked to the computer and opened DC++ to begin searching for the mystery song that was stuck inside my head.
A light passed down the street in front of my house. The police right drove past a robbery in progress.
Math and Philosophy
Okay, I thought of this several months ago while writing a math paper. Here goes the ramble:
There are 3 operations in Euclidean construction:
1. Drawing a line between two points
2. Given the points A and B, drawing a circle centered at A with a radius extending to B.
3. Drawing the points of intersection between lines and circles.
Although these operations seem general enough to allow you to build any polygonal figure, one can prove mathematically that there exist certain figures that cannot be constructed. An example of this is a 7-sided polygon.
If one extends the rules, then certain constructions become trivial. For example, there exists no method to trisect arbitrary angles; however, as soon as one allows for distances to be measured, the construction becomes possible.
Here's where the philosophy part enters in. Suppose there is an omnipotent being named Bob with the ability to do anything. You tell this being to construct a 7-sided figure using only the rules of Euclidean construction. After a cloud of magic, Bob hands you the figure and poofs off into a wisp of smoke.
But wait a minute. You have already proven that it is impossible to construct a 7-sided figure in a finite number of steps. This leaves 3 possibilities:
1. Bob used some method that you didn't think of
2. Bob cheated
3. Bob works really really (infinitely) fast
Well, case 1 is already out of the question. The proof of nonconstructibility is an existance proof that demonstrates that no possible method exists. Case 3 is a possibility, although it somehow seems that performing an infinite number of steps in a finite amount of time is cheating. Thus, it only leaves case 2.
Uh oh. Something just went wrong. It appears that a set of 3 simple rules has suddenly become more powerful than an omnipotent being. At least, that was my first thought. After a while, though, I was able to reconcile what I observed with what I wanted to believe. I mean, an omnipotent being would be able to perform an infinite number of moves in a finite number of time, so case 3 really wasn't a contradiction. Besides, I've already shown that the construction becomes possible as soon as you add additional rules. Maybe these rules were just a meaningless curiousity, much like the statement "This statement is false".
Unfortunately, I soon thought of another example that I could not repair. Imagine that you have a rod with 5 flexible henges in it:
------*------*------*------*------*------
You want to bend this rod at the hinges to shape it into a tetrahedron (ie, the shape where every side is a triangle). After fiddling with it for a little while, you start to start to look around online for the solution.
In fact, the problem is impossible, and there are several ways to prove it. The most convincing argument, however, is probably the fact that you can iterate through every possible arrangements of the rod until you have exhausted all possibilities. In every case, you will be left with at least 1 segment of the rod that cannot reach its necessary location.
Now suppose you ask Bob to perform this task. He hands you a tetrahedron and again vanishes. This time, you're not convinced though. Any idiot could break the rod apart and glue it into a tetrahedron; you want to see how he actually did it.
But in this case, it is impossible for Bob to show you. Every possible solution terminates after a finite set of moves, so he cannot pull the whole "infinite-move" trick.
So which is more powerful? God, or the rules that God imposes?
So it begins...
Well, I've had this blog for a good year now without writing a single entry. I guess it's about time I change that :-P