Scientia est potentia

Math is hardly a contentious subject. It’s a topic that’s unlikely to evoke any sort of response from most people, beyond a casually dismissive remark like “I was never any good at that stuff.”

But for me, math is something I get very upset about. Not math itself – I’m an engineer, not a mathematician, and can lay no claim to anything but the most cursory of knowledge about rigorous math – but the way that people perceive it.  Ask most people and they’ll tell you that they aren’t any good at math. But I would put money on the fact that almost no one who professes a lack of competency or even an active dislike of mathematics could define “math” for you, or even give you an example of a mathematical theorem or concept. And that – people deriding or dismissing something that they do not understand – is a problem for me.

Why does no one know any math?

The answer is easy. People don’t know math because no one is teaching it at the primary and secondary levels. Sure, there are lots of things called “math”. But all of them boil down to mindless computation, continual application of an algorithm which is not explained or justified. “Math” education at the K-12 level in North America utterly ignores the fact that mathematics is, at it’s heart, a conceptual exercise.

Gauss called mathematics the queen of the sciences. Whatever he meant by that, it is certain that math stands apart from the natural and applied sciences. Math has no explicit object – it is like a living being in that the only goal of mathematics is to accomplish, well, math. The first thing to understand about math is that it is, at a certain level, arbitrary. We seek to understand things, how they behave, how they interact with other things, and so forth – except these are not physical things. We are not motivated by the existence of some object in nature, or the desire to create something for an external purpose. Some mathematical concepts can be understood by analogy to the physical world, but many cannot.

Take numbers for example. You might say that the concept of numbers has obvious physical meaning. It’s just a way of quantifying how much of something you have. But that is only true for some numbers. You could easily imagine 5 of something, or maybe even one million. What about -5? How many is that?How about 2.71828183? ?? 2 + 3i?

One of my first year calculus professors would often write some new (to me) expression on the board and then say, “I want to understand this thing.” And that is really what math is.  We begin with a system of logical axioms. Under these axioms we define certain objects. We realize that these objects can be combined, that they can interact in complex ways. We begin to see that although we started arbitrarily, patterns emerge from the logically derived results of our assumptions. And herein lies the beauty of math. Aside from the initial axioms, there can be no more arbitrariness. Everything must be logically justified and there can be no contradictions. And as we unravel how our artificial constructs behave under these few restrictions, we see an amazing interconnectedness. We see that these objects, operations and relationships that we have derived in complete generality and abstraction can be adapted to model the realities of the physical world. But it is still so much more than that. From a lot of scratch marks on a piece of paper we’ve conjured up a whole new universe of the mind, the ultimate expression of the human search for truth and meaning.

Everything I’ve written in the last 3 paragraphs probably seems alien to you. From K-12 you were never permitted to take even the faintest glimpse of this world. You could not be trusted with it. Kids couldn’t possibly understand hard questions like why and so had to be told the answer to the much easier question, how. Why justify a result when instead you can just accept it? You spent years memorizing multiplication tables – a computation which, on it’s own, is utterly useless – but were never asked why the product of two numbers should exist, and what it’s definition should be. You spent a great deal of time memorizing how to compute the areas of various shapes (as if, later in life, people would routinely demand that you ascertain the area of a trapezoid or be shot), but none asking what area means, questions how to define it, and given a definition, if we could come up with a general way of computing it.

Someone probably told you about a thing called sine (and his friend cosine), and what it meant in the context of right-angled triangles. But no one ever told you what sine really is (quick way to stump a high school math teacher – ask how your calculator knows what the sine of, say, the square root of 5 degrees is). It turns out that this function has lots of clever applications (and by lots I mean pretty much everything in your house that uses electricity), and none of them are at all related to triangles.

And if all this stuff seemed hard, if it never really made sense to you, it’s totally understandable. You were never allowed to see the big picture, the great tapestry of mathematics, but were instead forced to make sense of a few arbitrarily selected crumbs of knowledge. Your teachers always wanted you to know the answer, but math isn’t about getting the answer. The answer to a math question is more questions – is there, in general, an answer to that type of question? If so, can I always find it? If I get the answer, how will I know that it is right? Can I find a way of expressing ALL answers to ALL variations of the question? I can only figure out one answer – can I show that it is the ONLY one? What features characterize the solutions to this type of problem? Can I relate them to other, similar, but harder problems?

These aren’t questions that are only relevant to mathematics. They’re the type of probing questions that people in general need to ask about a lot of things. A framework is much more useful than an answer, and type of rigorous, logical thought taught by mathematics would serve everyone well. To realize where results come from, to question every step of the process as well as the initial assumptions, is the difference between an informed, responsible citizen and a mere subject. So please, give math a chance.

If you thought this was the least bit interesting, please read A Mathematician’s Lament by Paul Lockhart and A Mathematician’s Apology by G.H. Hardy.