Don\u2019t 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?<\/blockquote>\n\nThat's<\/em> how you work through a math text\u2014with 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\u2014exploring each function with a range of inputs, tracing stepwise through the algorithms, exposing data structures with print<\/code> statements or a debugger, etc., until he becomes so well-versed in the code's architecture and purpose that he could rewrite it himself in a different way.<\/p>\n\nI have a hunch that this approach generalizes beyond math, that e<\/em>very<\/em> thing you read\u2014be it a blog post, or a paper, or even a novel\u2014presents 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.)<\/p>\n\nI'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<\/em> 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\u2014I 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<\/em> trying to understand them, like if I were preparing\u00a0to answer hard questions on the subject.<\/p>\n\nSo 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,\u00a0kenjitsu<\/em>, from ken<\/em> = \"one's range of knowledge\" and jitsu<\/em> = \"fighting art\":<\/p>\n\n\n - 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\u2014don'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.<\/li>\n
- 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<\/em> you found it important. Rather than writing \"yes\" or \"interesting,\" think about what led you to\u00a0agree or how<\/em> it's interesting. Be contentful, specific, and concrete\u2014all the time.<\/li>\n
- Think like Feynman:\n
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\"<\/p>\n\n
So he\u2019d stop always, and say, \"Now let\u2019s 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\u2026 But not quite, because the head is a little bit too wide \u2014 it would break the window as it came by.\"<\/p>\n\n
Everything we\u2019d read would be translated<\/em> (as best we could) into some reality\u2026 And I learned to do that \u2014 everything I read I try to figure out what it really means, what it\u2019s really saying.<\/p><\/blockquote>\nImagine 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.<\/li>\n - 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\u2014i.e., which parameters to change\u2014so 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.<\/li>\n
- Be adversarial. For every position you run into\u2014and nearly every blog post, article, paper, or magazine feature takes a side\u2014put 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?<\/li>\n
- 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\u2014the 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.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"
You can sort of let a novel run through you: the language is loose enough that you don't need to chew on sentences\u2014you 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 […]<\/p>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-290","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/jsomers.net\/blog\/wp-json\/wp\/v2\/posts\/290","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jsomers.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jsomers.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jsomers.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jsomers.net\/blog\/wp-json\/wp\/v2\/comments?post=290"}],"version-history":[{"count":37,"href":"https:\/\/jsomers.net\/blog\/wp-json\/wp\/v2\/posts\/290\/revisions"}],"predecessor-version":[{"id":324,"href":"https:\/\/jsomers.net\/blog\/wp-json\/wp\/v2\/posts\/290\/revisions\/324"}],"wp:attachment":[{"href":"https:\/\/jsomers.net\/blog\/wp-json\/wp\/v2\/media?parent=290"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jsomers.net\/blog\/wp-json\/wp\/v2\/categories?post=290"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jsomers.net\/blog\/wp-json\/wp\/v2\/tags?post=290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}