Sunday, August 1, 2010

The Public vs. the Language of Mathematics:

I’ve got another excellent passage from Oliver Heaviside’s “Electromagnetic Theory”. In this one he talks about the merits of mathematics and some of the things the public (in my observations) get out of the subject.

I read a rather insightful book about how the education system treats children. It’s written by a PhD Mathematician, so it carries some weight. However, I am not sure how widely read it is outside of the university community. The book is called “A Mathematician’s Lament” by Paul Lockhart and I definitely recommend reading it if you get the chance. It’s not a high level math book by any means, it is just a good read to help get insight into the world. I have read reviews on Amazon insisting that education theory has been applied better than when this book was written. However, I should mention my mother is a high school math teacher and her curriculum has only been changed by standardized tests. So, whatever new “theory” these people are professing… people are still losing out on math in literal droves. As Paul Lockhart put it “In school you learn that math is not something you do, it’s something that is done TO you.” Lockhart’s essay is post year 2000, however, I find it very interesting to read similar commentary from Heaviside in the 1890’s, over a hundred years earlier.

“‘Mathematics is gibberish.’ Little need be said about this statement. It is only worthy of the utterly illiterate.
‘What is the use of it? It is all a waste of time. Better be doing something useful. Why, you might be inventing a new dynamo in the time you waste over all that stuff.’ Now, similar remarks to these I have often heard from fairly intelligent and educated people. They don’t see the use of it, that is plain. That is nothing; what is to the point is that they conclude that it is of no use. For it may be easily observed that the parrot-cry ‘What’s the use of it?’ does not emanate in a humble spirit of inquiry, but on the contrary, quite the reverse. You can see the nose turn up.
“But what is the use of it, then? Well, it is quite certain that if a person has no mathematical talent whatever he had really better be doing something ‘useful,’ that is to say, something else than mathematics, (inventing a dynamo, for instance,) and not be wasting his time in (so to speak) trying to force a crop of wheat on the sands of the sea-shore. This is quite a personal question. Every mind should receive fair development (in good directions) for what it is capable of doing fairly well. People who do not cultivate their minds have no conception of what they lose. They become mere eating and drinking and money-grabbing machines. And yet they seem happy! There is some merciful dispensation at work, no doubt.
“‘Mathematics is a mere machine. You can’t get anything out of it that you don’t put in first. You put it in, and then just grind it out again. You can’t discover anything by mathematics, or invent anything. You can’t get more than a pint out of a pint pot.’ And so forth.
“It is scarcely credible to the initiated that such statements could be made by any person who could be said to have an intellect. But I have heard similar remarks from really talented men, who might have fair mathematical aptitude themselves, though quite undeveloped. The fact is, the statements contain at once a profound truth, and a mischievous fallacy. That the fallacy is not self-evident affords an excuse for its not being perceived even by those who may (perhaps imperfectly) recognize the element of truth. But as regards the truth mentioned, I doubt whether the caviler has generally any distinct idea of it either, or he would not express it so contemptuously along with the fallacy.” (Section 9, Electromagnetic Theory, 1893)

I think the first part of this is great and really illustrates the divide some people have. Mathematics is, literally, its own language. The array of symbols used in mathematics indeed looks like gibberish to the uninitiated. But take someone that only speaks English and send them to Russia and you have the same problem. People are incessantly lazy, and learning new languages later in life is actually quite difficult (well for most people). Not to mention a language that is based on pure logic where certain “slang” is not allowed, but it is not different than any language. Such as in English “I can’t not do that”, grammatically makes no sense. Just as 2 = 1, mathematically makes no sense. If you want to start incorporating “slang” in math, you need to prove it makes sense or define it in such a specific way. Take the concept of a gradient for example? I won’t bore you (yet), but it’s sort of similar.

Like any language it has evolved from many rudimentary levels, but, like any language, it has great descriptive power for what is happening. If you ever attempt to read old school math proofs from something like Euclid’s “Thirteen Elements” you won’t see anything like you see in modern math. Instead you see long droning paragraphs setting up a situation that would essentially say something like “sin (x + y) = sin(x)cos(y) + sin(y)cos(x)”. Isn’t that way easier to look at than a long paragraph explaining the same thing? When you look into math, you have to remember it really is short hand for a giant paragraph discussing a specific situation. While it may seem intimidating with all the crazy symbols, trust me its better. Read Newton’s “The Principia” to find out…

In this treatment I am not sure Heaviside is completely clear towards the end. However, he is attempting to illustrate the power of Mathematical Theory versus Application. This reminds me of an interesting thing concerning Tesla and Edison. In America we tend to venerate Edison as the “great American inventor”. However, very little appears to be known as to how inefficient his methods were (never mind how horrible of a person he actually was, but that’s another topic). Anyway, the Edison method can probably best be described as the “brute force” method, as one of my math teachers would have said. Basically, if you want to make a light bulb then you keep trying every kind of material for a filament until something finally starts to glow. It’s pretty messy and not a very nice way to go about doing things. Tesla on the other hand would do the math; he’d study the physics, the heat transfer of materials and so forth. He’d come to solutions much faster than his contemporaries and on top of that they are elegant solutions. This is why most people now work in this fashion and why the concept of Applied Mathematics is so powerful. I think people sorely miss out on that kind of power.

I don’t want to lead anyone astray; this stuff can get real hard. In fact if you are doing mathematical research sometimes you might delve into a particular aspect of the field that has five other people that MAY know what you’re talking about. But this is not a problem exclusive to mathematics; you can find it in about anything, even something like Economics. I even encountered Economics text books that are using Quantum Mechanics to help explain what is happening for Risk Pricing in the stock market. If that’s not a mind bowing application of Quantum Mechanics… then I’m sure there are many others, but I thought that was an interesting one.