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Contents
Chaos Theory
Gödel's Incompleteness Theorem
Chaos Theory
Chaos theory is one of the most famous branches of mathematics. Perhaps the most famous chaotist is Ian Malcolm from Jurassic Park. However, the effects are far from fictional and are very interesting.
Chaos theory was first discovered by a graduate student in the 1960s, who was running a primitive computer program to generate the results of weather in a closed system. Once he had run through the entire set of effects, he fed the starting data into the computer, letting the algorithms and other physical laws stay the same. But this time he got completely different results. Wondering how this was possible, the student went over the data. It turned out that he had shortened the decimala from the original experiment from more than 10 to 7. This seemingly minor effect changed the weather in the system completely and spawned chaos theory.
In its essence, chaos theory states that every change, no matter how minor affects the entire system. This is exemplified in the "butterfly flaps its wings in New York.." statement. This is a "thought experiment," in which a butterfly whos wings flap in a certain manner in New York, changing the weather in Japan days later. The other major portion of chaos theory is fractals. These look like art but are very well-worked mathematical equations and formulas. If you look at the fractal itself, ignoring any background, and then zoom in, you will see the exact same thing. The entire shape is composed of a certain pattern repeating itself. Try making this fractal:
1)Start with a straight line.
2)Now draw two lines that form an obtuse angle in the middle of the first line.
3)Draw another set of lines in the middle of the two from step 2.
4)Continue until bored.
This illustrates the basic principle of fractals. It also brings us to the final point of chaos theory. Suppose you want to find the length of the coast of England. Lets say you use a yardstick, walking along the beach. That would be a basic estimate. Now try using a ruler. Measure in inches, half-inches, etc. all the way to the subatomic level. The coast of England will eventually become infinitely huge, because you can always, theoretically, use something smaller. Of course this is not practically possible, but it remains an interesting point.
Gödel's Incompleteness Theorem
What Heisenberg's Uncertainity Principle is to Quantum Mechanics, Gödel's Incompleteness Theorem is to mathematics. Both describe the incompleteness of their respective systems. The idea is actually fairly basic, but the implications are tremendous.
The basic idea is that in anycomplex formal system, there are inherent incompletions. Gödel meant that systems, such as arithmetic, may never be used to prove their own completeness or consistency. (Complete means all statements will be true. Consistent means there will be no contradictions.) Gödel assigned different statements a number in his proof. Then he made a statement x say that "Statement x cannot be proved." If x is true, then it of course may not be proved. If x is false, then it cannot be proved without eliminating consistency. So x must be unprovable.
While the idea may seem fairly obvious, many mathematicians of Gödel's time thought that every axiom of mathematics, specifically arithmatic, could be proved by the system. It was startling to find this was not so. In addition, the Theorem is important in AI, where machines have to be told everything and can think it through and prove it. The limitations still cramp AI designers today.
© Robert Litzke, 2002
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