Away: Math pep talk

Away; math conference; too caffeine-tremulated to type. Polished repost. Some luring for ya.

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There’s one thing I, as a lowly TA, often tell despairing first-year students about mathematics.

Namely: Mathematics might not be your first choice, or your final choice, and it might seem a grind and a thing removed from all spheres of daily life, but don’t despair because even if you never need the plusses and integrals of mathematics, what you get from wading your way through them is a lot more universal — an inclination to careful thought.

Let me give you three examples.

First: In mathematics, you can’t just wave a hand and say: “And I suppose the other cases can be handled in a similar fashion.” (Okay, you can, but only if you’re a professor and thus know more than you say — as compared to a TA, who often says more than he knows.) To get anyone to believe you (well, except gullible first years), you need to show that all of the links in the chain hold, not just the first and the last with some emoting in-between. If there’s a weak link, your co-workers and colleagues will descend on you like a cloud of flying piranha on fire; which is as it should be.

Second: In mathematics, you’ll be laughed at if you try to assert that “since A follows from B, naturally from the fact of A being true we can infer that B is true also”.

And then the cold-eyed sarcastic TA will remind you that though falling down three floors is often followed by injury, this does not imply that just any injury must come from a fall — and suddenly you’ll be shying away from both her, and the window. (That’s “education thru adrenaline”.)

Third and, to my mind, most important of these: in mathematics, you have to define.

Okay?

Nothing irritates me more than some loudmouth ballooning on about “justice”, “good” and “the right thing”, without ever bothering to say just what is just, just what good means, and by which test or criteria do we see that a thing is right or not.

There are even worse examples — for one, the word “God” is so nebulous that an unscrupulous god-believer can deflect all of an atheist’s attacks just by repeatedly changing the meaning of “God” he defends.

If a mathematician says something (excluding the obvious kinda-sorta-generalizations and vague guesses), her (his?) words are well defined. Say this phrase:

The function $f(z) = 1/z^2$ has a second-degree pole at the origin.

If you then proceed to quiz the mathematician what a “function” means, you will get an answer. (Probably beginning with “Um, this will be sort of general and complicated, by the way of reverse-injective relations, and I forgot to mention that $f : \mathbb{C} \to \mathbb{C}$, where $\mathbb{C}$ denotes…”, and the explanation sort of balloons from there.)

Likewise for the origin, a pole, and a pole of second degree —

I shall resist the urge to comment that first-degree Poles are from Warsaw, second-degree Poles from Wroclaw.

What you won’t get is an indignant answer that “Why, the concept of a pole is intuitively obvious and, anyway, does not need a definition! And that’s not important here! Silly questions the asking of which makes you look like an idiot type!”

(To maintain the plausibility of this claim, I’m actively trying to avoid thinking of some professor-types right now.)

To refrain from further gasbagging: Mathematics tries to get you to the habit of knowing exactly what the words you use mean before you use them. Not a good lesson for success in politics or public life, but otherwise a thing you’d better learn unless you want to waltz down the road to folly and ruin.

If these three — noting details, being logical, and knowing what you talk about — are not good universally applicable lessons, then nothing is. And for that reason, time spent studying mathematics is never wasted.

That’s what I tell the first years, anyway.