Math Isn't What You Think

Math gets a bad wrap.

To the non-specialist, when one says the word math most people seem to conjure up images of formulae and that one math teacher you had in high school who was particularly horrible.

This is really unfortunate, because at its heart, math is really about curiosity, honesty and collaboration.

Those sure don’t seem like mathy terms, you say? Stick with me here.

Collaboration

In high school you might have memories of manipulating arbitrary symbols according to arbitrary rules, striving for the right answer. This is bullshit and isn’t math. Seriously.

Practicing mathematicians do use formulae in daily work, but they are by far not the focus of attention. Rather, we strive to elucidate concepts and make them communicable to our peers. The symbols and formulae that fly around are more a byproduct of this process, honed and polished over time.

Actually, pick up a copy of Euclid’s The Elements and try to read it. You’ll get a pretty good idea of how math language has changed over the years. Here’s an excerpt:

If a first magnitude has to a second the same ratio as a third to a fourth, then any equimultiples whatever of the first and third also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.

Yes, it’s pretty hard make heads or tails of. In modern language, we’d say something like the following:

Pick three fractions and call them \(a\), \(b\), and \(p\). If \(a = b\) then \(p \cdot a = p \cdot b\).

While being more compact, I find it considerably more readable. Also, it might seem really obvious to you written this way. If so, then great! That’s the entire point of developing new notation. It’s about getting the idea across in a clearer way.

This is the “collaboration” part of what I said above. Scary looking formulae and symbols, in a large part, are something we’ve honed over time to actually make things clearer and more communicable to our peers. Instead of a long, convoluted paragraph, why not just a quick one-liner?

Honesty

Making things clear like this also keeps one honest. For one it makes it considerably easier to find mistakes and oversights. Like in the example above, with the modern notation you might have thought something along the lines of, “sure, just cancel the \(p\) on both sides”. This is an excellent proof since it’s simple and straight to the point.

Notably, this also applies in your own individual thinking. With a simple and clear way to express your idea, it’s easier to think about it. And this is another main reason for the massive proliferation of technical jargon. For example, the word multi-touch would seem like gibberish to a person even a few decades ago. But now if you see a phone described as having a multi-touch screen, then you know that you can use it with all the fancy gestures like an iPhone. That word communicates all the ideas of pinch zoom, image rotation with a gesture, using multiple fingers to swipe and open a menu, et cetera with just a single word! None of those concepts are particularly difficult in and of themselves, it’s just that it’d be a pain to have to explain every time instead of just using a single word.

So when the mathematician talks about “the pullback along the connection on a differentiable manifold” it should sound like gibberish. That is, until you spend a little time familiarizing yourself with the numerous, but not so difficult underlying concepts. If I were to unpack that statement into “everyday language” it would be a short book. If you had enough patience to read it, then you’d likely have no trouble understanding, but once you are comfortable with the notions, then we can talk about them easily with a simple word.

Curiosity

Which brings me to the last point. No collaboration or development of new concepts—no math—would get done without motivation. For practicing mathematicians the overwhelming motivation seems to simply be curiosity.

This is where primary education miserably fails with math. In schools it’s all about teaching the prescribed rules to follow so you can get the prescribed answers to pass the prescribed standardized test. There do seem to be voices and teachers out there pushing to really engage students in STEM subjects, but math in particular seems to get short shift on institutional levels.

There’s a thing called the Convention on the Organisation for Economic Co-operation and Development. It started in 1960 and currently has 34 countries as members. Semi-regularly the contries in this collaboration get together and perform a Programme for International Student Assessment. Basically, a “let’s see how our students’ academic performance compare” effort. The United States ranked 27\(^{\rm th}\) in 2012.

Now, there’s a whole lot of room to quibble over these results, but one thing to takeaway is that it sure doesn’t seem like a lot of interest in math and science are stirring in students in the states these days.

Which seems odd to me, ’cause as someone who studies math, it seems like pretty much everyone does math all the time without realizing it. If you’ve ever thought about how something works and, especially, if you’ve talked to someone about it, then you’ve done math. That’s it.

The only real difference is that mathematicians in the field have put in the time to learn some specialized concepts and specialized symbols that turn out to be useful when you spend a lot of time thinking about how things work at really fundamental levels.

Takeaway

This is already a long and rambling post, written at one in the morning, but as a kickstart to my new blog, I wanted to argue that math isn’t some esoteric practice for the math shamans, but instead something inherinently human that you and I can share in through these random and rambling musings here.

There’s a lot more to be said, but for now I’ll leave you with your own thougts and your own mathings.