Simplicity of Complex Systems

30 04 2009

When I was younger, I was always amazed at computers. I wanted to understand them in greater detail and know how they worked. At the time I was sure that they were immensely complex machines which would take me years of study to understand, and that the inner workings surely involved a certain component of mysterious black magic.

Now when I look at a computer, after working my way from learning Java to C, all the way down to assembly, memory management, pipelined processor design, and logic gates, there is little to no magic left. They are, in fact, incredibly simple. It’s almost disappointing to think back and realize such a mysterious thing was really so simple, yet on the other hand quite gratifying to have the understanding now.

Computers are composed of nothing more than logic gates stretched out to the horizon in a vast numerical irrigation system. — Stan Augarten

And the logic gates are nothing more than switches. Given enough time and enough spare parts I could probably come up with a basic microprocessor. Systems of great complexity can be build just from a few simple components. Perhaps that should not come as a surprise, since we ourselves are such a system, composed of cells, running off of sugars and proteins, which can be broken down into basic rules from physics and chemistry. Almost anything, perhaps even the universe itself, can be broken down it to simple rules and components.

Now, as I progress in my learning of linear algebra, calculus, and discrete mathematics I ponder if someday I’ll look back with an understanding of math like I did with computers. Right now, so much if it seems confusing, with a hint of magic and trust in the fact formulas work. Am I one by one peeling away the layers of complexity in search for some simple and discrete unit of mathematics? Does the answer really just lay in the integers?

Will I one day reach a point in some number theory class, while writing away at a proof, where I look back at my days frustratingly calculating things and think to myself, “it’s all so simple! Once you understand the basics”?



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