On Abstraction
Nevertheless, I prepared the following remarks: As an undergraduate I studied mechanical engineering, not electrical engineering. The reason was that mechanical engineering was recommended by my high school math teacher, Mr Bedrosian. My father was a chemical engineer, but I didn't like chemistry -- too much memorization. In hindsight I think mechanical was better for me than electrical because mechanical mostly involves things you can see and touch and electrical does not. Looking back over decades, I see more clearly what the problem is with electrical engineering as a subject: It is too abstract. Electrical engineering begins with electromagnetics, the beautiful theory embodied in Maxwell's equations. And it is beautiful. But I don't like it because it goes against my nature. To a first-year engineering student, you can't even understand the concepts, really. The course ECE110 Electrical Fundamentals uses the text Fundamentals of Physics, by Halladay. At one point Halladay’s task is to define electric field. Electric field is too difficult to approach head-on, so Halladay tries to tip-toe up to it via temperature, which is supposed to be easier because it's a scalar field. He begins thus: “The temperature at every point in a room has a definite value.” I guess everyone knows that. But in a physics course there should be understandable definitions. What is temperature? Temperature is a concept defined in the subject of thermodynamics. Coincidently, in 1871 Maxwell himself wrote a book called Theory of Heat. His definition is this: “The temperature of a body is its thermal state considered with reference to its power of communicating heat to other bodies.” So temperature is thermal state, which is -- what? Seems like we’re being given the runaround. Let’s try a different tack. Let’s say that temperature is what a thermometer reads. From that viewpoint temperature is a measurement of the average kinetic energy of the molecules in an object or system. But what is average kinetic energy? I believe it’s something like this: Temperature at a point is a limit as the radius goes to zero of the average kinetic energy of molecules in a spherical volume centred at that point. But how can there be a limit: As the radius of the sphere gets smaller, eventually it’s smaller than the size of one molecule. So that’s what a first-year engineering student is fed: Electric field is approached via the simpler concept of a scalar temperature field, but temperature is an abstract, mathematical concept based on a dubious limit. That is, physics, which is supposed to help us understand the real world, isn't real at all -- its method is to invent a model: molecules, kinetic energy, the limit of an average, etc. How is a young person supposed to understand this? And so far I've been discussing only electric field. Maxwell's equations represent a much more complex fantasy of a model. Please understand that I am not knocking physics or its contributions. Without physics we would still be living in the dark ages. I am only saying that I don't like physics as a subject because I don't understand it, and the reason I don't understand it is because it is abstract. I liked school and wanted to stay in it, but I certainly couldn't hack electromagnetics. On the other hand I liked mathematics. Mathematics is not abstract, by which I mean that it isn't a fictional model of something real. It's like music -- a pure invention of human intellect. You can't say that Bach's Brandenburg Concertos are a model of, say, a tree. Likewise, linear algebra isn't a model of something. It exists in itself, and in that sense it is real, not abstract, in the same way that music is real. In linear algebra, if you want to see if something is true, you prove it or work out a counterexample. So, liking mathematics and being in mechanical engineering, I was led to control theory. Control theory is beautiful too, like Maxwell's equations. Control theory is one of the “sciences of the artificial,” as the Nobel Laureate Herbert Simon described. Like information theory. Information theory is based on the notion of a channel. A channel, as defined by Shannon, consists of a random variable input X, a random variable output Y, and the conditional density of X given Y. Control theory starts with similar mathematical objects, poses questions, and looks for answers. If answers are found, they are rigorously proved. Two years ago I attended a seminar in the Physics Department. The speaker was introduced as a “theoretical physicist,” meaning he didn't do experiments. He studied an infinite chain of cars moving along the highway, and derived a formula for the speed of a wave propagating because someone braked suddenly. There were no proofs of anything, just derivations. This topic became my most recent journal paper. That's typical of how I get ideas for research. I hear something and for some reason get the urge to study it. Imagine an infinite number of cars moving at constant speed along an infinite, straight road. Each driver is trying to stay a certain distance behind the car in front. Suddenly some of the drivers brake slightly or accelerate slightly and then resume driving. What will happen to the infinite string of cars? Will it go back to steady state? That’s the problem I formulated and worked on with a professor at another university. Why? Because I didn’t know the answer. If you’d like a copy of this paper, please email me. It uses differential equations in Banach space. Finally, some lessons learned about doing my kind of research:
Bruce Francis Friday, May 20, 2011 |