Why Should I Learn Math?

posted Mar 04, 2005

"Why should I have to learn math, when I don't use it much?" is a very good question.

If I've ever seen a really good answer to this question, I don't remember it.

But I've finally figured out how to express something that has been gestating in my head for a while. I know why you need to learn math.

First, what do I mean by math? By math, I here refer to actual mathematics, which in general looks like this:

2. Trace the axiom to some logical conclusion that must be true.
4. Alternatively, take a statement that you wish to know if it is true and see if you can prove it, disprove it, or possibly prove it can't be proven either way.
5. At some point later, see where we are and see if we need different axioms, either because it works "better" (for some definition of better), or because there is an interesting alternative we could explore. (Set theory in particular has fascinating stories related to this.)

Two things of note: First, that's an abstract definition, and it's hard to relate it to real life. Second, most people never learn this sort of math. To them, math is all about numbers, mostly addition, subtraction, multiplication, and division. Math isn't really about numbers; that math is a consequence of certain axioms that involve numbers, but you can do real and useful math on a number of other things as well, such as "dots and lines connecting them" (graph theory) or topology, the study of shapes and other such things.

Why is this relevant in real life? This builds off of my Metaphor Rant and takes the next step, showing why metaphors are not just dangerous to use in a debate, but are, even more fundamentally, dangerous to use in thought as well.

There is danger because without this mathematical style of thinking, you are largely stuck with understanding things by metaphor. Metaphors work by drawing analogies from something in previous experience, then extending the conclusions reached based on the previous experience to the new thing.

Technically, that is a valid thing to do. Realistically, it is a highly treacherous and unreliable way of thinking; "unreliable" because it is extremely easy to get it wrong, and "treacherous" because if this is your only way of thinking about new things, you are left with no way of knowing or finding out how wrong you may be.

In order for a metaphor to be valid, all relevant attributes leading to the relevant conclusion must be shared between the original experience and the new thing of interest. Even a single relevant difference is enough to invalidate the metaphor... and while there's a bit of circularity in the definitions of "relevant" and "enough to invalidate the metaphor", in practice it is extremely difficult to fully analyse the relevance of differences.

Not only must all relevant characteristics be shared, there must be no relevant differences that render a metaphor invalid. This can be even harder, and what's worse, is often completely not understood by people who think by metaphor.

For all the reasons metaphors are unreliable in debate, they are doubly unreliable as a way of thinking about things.

What is the alternative? The alternative lies in the cognitive toolkit a good mathematical education will give you. It is immediately obvious that intuition and metaphor will not get you far in mathematics; if not obvious to you right now, it would be after some time spent even with relatively simple algebra. (You can over time develop a "mathematical intuition", but that takes a lot of work and effectively nobody is born with it; I'm not referring to that kind of "intuition" here.) Without the crutch of metaphor thinking to lean on, you are forced to start analysing things in terms of what they are, what attributes they have, how these attributes inter-relate, which attributes are relevant when, and to deal head-on with the astonishing interconnectedness of the real-world, where everything has a lot of relevant attributes and the reality is that only a very small handful of metaphors capture reality in any significant way.

Metaphor-based thinking is a form of intellectual laziness. If we lived in a world where it worked reliably, it would not be, but we don't and it is. Technically, you do not need a mathematical education, but what it does for you is provide a domain where you can't use metaphors (unless you prove they work, in which case you have an isomorphism, not a metaphor, and that's actually a very good thing). Evidence suggests that without this form of training, people find it very hard to develop it on their own. As well they should, it's a highly unnatural way of thinking, but "unnatural" in a good way.

For example, for better or for worse, take my Communication Ethics piece. In that entire thing, I think I use one metaphor, and that with severe misgivings and a link to my Rant. Instead of arguing about whether something is more like a newspaper or a radio station, I handle things based on what they are, not what they sort of resemble from a distance.

This is particularly important issue in the domain of communication ethics, because modern communications can not be appoached by analogy or metaphor to older communications; the differences are so large that no physical metaphor can capture the issues. All of them simply muddy it.

So, "Why learn math?" Even though you are quite likely to never be in a situation where you need to directly use Euclid's axioms of geometry in "real life", mathematics provides not just the best training field for learning real, non-metaphorical thinking, but perhaps the only known functional training ground, where metaphorical thinking is not even an option. And there is nobody, anywhere who can not benefit from being able to do a correct analysis of the various relevant factors of something and how to deal with them; businessperson, engineer, even dealing with relationships with other people is enhanced if you can think this way. Yes, you heard me, I just said that properly applied, mathematical thinking can get you laid.