We (at Coho) talk a lot amongst ourselves and with other folks about why we love math. You likely have reasons of your own (assuming you do, indeed, love math). Some of the most common answers have to do with the beauty of the subject, the ability to rely entirely on logical argument, the objectivity of the subject, and/or the abstract thinking necessary. These are all, of course, extremely true and extremely valid reasons for loving math.
One reason that we hear less about, despite it being (in our opinion) very likely to be one of the more compelling reasons, is just how f*****g addicting it is to learn and do math. I really believe that every mathematician is hopelessly addicted to the "aha!" feeling. There's truly nothing like it.
In this little note I'd like to examine the "aha!" feeling, its various sources, and how hopelessly addicted we are to it. To be very clear: it's a great thing to be addicted to. I really wish more people could get addicted to it, and in many respects I think that one of the most important aspects of math education as a whole is getting people addicted to figuring things out.
To begin, let's clarify two different kinds of "aha!" moments — the learning "aha!" and the doing "aha!".
*Disclaimer: I'm going to be making drug references throughout, since that's a typical example of a bad thing that people get addicted to. To anyone reading this who may have had (or still have) struggles with addiction (to substances), please know that the following analogies are not intended to trivialize any such health issues. We also do encourage you to seek help (physical and mental) in regards to addiction.
The Learning "Aha!"
The "learning aha!" is the daily dose for math addicts. It's what we need to keep going. It's that little rush that you get when a long bout of confusion abruptly vanishes.
Often, this takes the form of several minutes, possibly hours, of staring at a textbook or a paper with a furrowed brow. You're reading something and it just doesn't quite make sense. Maybe the first time you read it, it's utter gibberish. Then you kind of start making sense of it, but you know you don't have a full grasp of it. You work out a couple examples, and maybe those examples even make sense, but there's a general structure, pattern, or reasoning that you know you're still missing.
You might be staring at the same page, or flipping back and forth between the same two pages, for a full hour. You might even have to get up and walk away for a while, or just read on and come back to that passage, or do some of your other work. For me, the best (and hardest) approach is to shake things up entirely — to give up, but only temporarily.
Somehow, some way, for reasons that I don't think anyone on planet Earth has a full understanding of, something eventually clicks and you just "get it". That missing piece to a puzzle whose shape you didn't even know comes into existence, and somehow you just know that you now understand the thing. That's the "learning aha!" moment, and it's awesome.
The Doing "Aha!"
Related, but separate, to the Learning Aha is the Doing Aha. This is the "aha!" that comes when you figure out a problem. This problem could be one of thirty homework problems, it could be one of 8 problem-set problems, or it could be an original problem that no one on planet Earth has ever solved before and now you get to write about it for the first time. These experiences all give out the same high, in various doses. In fact, each is the gateway drug to the next.
Getting addicted to this feeling is quite possibly the best thing you can do for yourself as a mathematician. Become addicted to it and then become as greedy for it as a Wall Street banker is for commissions. Don't let others take this rush from you, and don't take it from yourself either.
People (including yourself) can take this rush from you by giving you hints and/or giving you solutions. When you are addicted to this high, you know that an 8-problem problem set is not 8 problems to be solved, but rather 8 chances to feel that rush. If it were just 8 problems to be solved, then looking up solutions would obviously be the way to go. But the only way for you to get those 8 sensations is to do them yourself. There's no shortcut.
In fact, I believe that the only reason it is a rush at all is because of the struggle. If you didn't struggle to solve the thing, you won't get the rush. You have to live with the crappy feeling of failure, confusion, and frustration before you get the reward. But if you stick with it long enough, the reward is oh so sweet.
Everything Is Reversed
This high — this rush — is the complete opposite of all the "bad" highs out there. Drugs are a good example, though any vice would do. Drugs are expensive, give you a pleasurable high with virtually no work, and usually have terrible after-effects. Math is virtually free, gives you a pleasurable high only after lots and lots of work, and has positive after-effects — added skills and knowledge.
Drugs open you up to a vicious cycle of pay, get high, feel terrible, pay again. Math opens you up to a virtuous cycle of work, "aha!", feel great, work some more.
To be clear, this is not a "drugs are bad" seminar that I'm trying to put on here. This is a "math is good" seminar. Let yourself become a hopeless, unrecoverable math addict — an addict to the drug of figuring things out.
Engineering the "Aha!"s
Unfortunately, too many people never feel the high that math has to offer. They stop before the drug can really kick in. When things get hard, they stop, and will often say that they're just "not math people" or some such thing. But in reality, they're often just one "aha!" away from being hooked. Once you have a small "aha!", you enjoy it for maybe an hour (or a day), but then you want a bigger one. Some people just never get that first taste, which is unfortunate.
Society often views mathematicians — and some mathematicians view themselves — as some kind of masochists. Why would anyone voluntarily do this really hard thing called "math"? And this is a fine image to have of us, because math is indeed hard and requires lots of blood, sweat, and tears (well, hopefully no blood, and rarely any sweat, but certainly tears).
But this also doesn't paint a selfish enough picture. And in this case, I think the selfishness is good. We know what the rush of figuring something out feels like, and we love it. We therefore want more and more of it, and mathematics gives the best arena for such things because we don't have to wait for experimental results or other peoples' opinions — we can prove it to ourselves even if no one else is around to see it.
So I now happily identify as an addict, and will happily admit that that's a large reason why I love math — because it gives me a powerful high unlike anything else I've experienced, and this high is safe, healthy, and (mostly) free.