by James Somers, February 8, 2010
You can sort of let a novel run through you: the language is loose enough that you don’t need to chew on sentences—you can swallow them whole, steadily one by one, and still have a perfectly clear picture of who everyone is and what’s going on.
Mathematics textbooks are different. If you churned through even an introductory text at anything close to a novel-y clip, you probably wouldn’t be able to solve the most basic exercises. If given an exam on the subject, you’d fail.
I think that’s roughly what Paul Halmos had in mind when he penned this excellent advice:
Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
That’s how you work through a math text—with lots of chewing, and brooding, and musing. You have to play with the stuff in the same way that a programmer might play with another person’s code: not by reading it straight through, but rather, by running it on his own machine—exploring each function with a range of inputs, tracing stepwise through the algorithms, exposing data structures with
I have a hunch that this approach generalizes beyond math, that every thing you read—be it a blog post, or a paper, or even a novel—presents you with the option to “fight it,” to “run it on your own machine” instead of merely reading. The trouble is in breaking the habit to be passive and, more critically, figuring out what sorts of questions to ask. (Because obviously you won’t get very far with “How does the proof use the hypothesis?” in non-mathematical contexts.)
I’ve thought a lot about this recently. I read a fair amount, but I’m afraid that too little of it sticks. Even if you asked me to describe an article just after I’ve read it, there are too many times where I’d hand-wave and stammer my way through a patchy explanation. And part of the problem, I’ve surmised, is that I take too much on face—I don’t engage, or wrestle with, 90% of the sentences that I encounter. Occasionally I’ll look up words or Wikipedia entries, sure, but I don’t attack most texts in the way I would if I were actually trying to understand them, like if I were preparing to answer hard questions on the subject.
So I’ve tried to develop a modest set of techniques to overcome my own readerly inadequacy. Think of them as the basic tenets of what I’ll call “the art of knowledge-fighting” or, more succinctly, kenjitsu, from ken = “one’s range of knowledge” and jitsu = “fighting art”:
- Try to become like the kind of pestering student who slows down classes. Incessantly ask questions and restate what the “teacher” says in your own words. Read at the speed of understanding—don’t disengage from the hard stuff just to finish an article. When you start to glaze, or skim, or you feel like you’re just sort of scanning over the forms of words, reboot.
- Read with a pen. I’ve perused the books and notebooks of my smart friends, and one thing these people have in common is that (a) they pack their reading with margin-notes and (b) these notes seem to harass the author. They’re highly critical, in that they go past just trying to figure out what the author means and ask, “What would that imply? What other theories fit these facts? Isn’t this a kind of wishful thinking?…” So every time you highlight a passage or circle a word, think about why you found it important. Rather than writing “yes” or “interesting,” think about what led you to agree or how it’s interesting. Be contentful, specific, and concrete—all the time.
- Think like Feynman:
Imagine actively. Use the phrase “that would mean…” to force yourself to think on your own terms with your own vivid images. It’s easier said than done, but I’m convinced that this little trick is what made Feynman such a great explainer. Because everything he ended up teaching to someone else he had already taught himself, that first time he encountered it and tried to translate it into his own words and pictures.
We had the Encyclopedia Britannica at home, and even when I was a small boy, he used to sit me on his lap and read to me from the Encyclopedia Britannica. And we would read, say, about dinosaurs. And maybe it would be talking about the brontosaurus or something [. . .] or the tyrannosaurus rex. And he would say something like “this thing is twenty-five feet high, and the head is six feet across”
So he’d stop always, and say, “Now let’s see what that means. That would mean that if he stood in our front yard, he would be high enough to put his head through the window… But not quite, because the head is a little bit too wide — it would break the window as it came by.”
Everything we’d read would be translated (as best we could) into some reality… And I learned to do that — everything I read I try to figure out what it really means, what it’s really saying.
- One thing that good philosophers and lawyers are good at is generating counterexamples. For each of your assertions, they seem to be able to conceive of a simple scenario where your thesis doesn’t hold. Or if you present a thought experiment, they somehow know which knobs to turn—i.e., which parameters to change—so that it no longer serves your point. I’m not sure how exactly they do this, or what sort of practice one needs to develop the skill, but it can’t hurt to constantly throw caveats at the general claims you might encounter in a day’s reading.
- Be adversarial. For every position you run into—and nearly every blog post, article, paper, or magazine feature takes a side—put yourself in the shoes of someone arguing the opposite. What would their objections be? Would they feel that their position is being represented fairly by the other guy? What in the argument would they be forced to concede, and what would they be inclined to push back on?
- Explain stuff. There is no easier way to expose the holes in your own understanding than to try teaching someone else. Or if you really want to go nuts, try writing up the ideas that make you uncomfortable—the process, while painful, will clarify your thinking. The point is to never let ideas cross your mind without being engaged, or debated, or somehow extruded through discourse. When in doubt, hash it out.