reading math

[gusl]

Mathematical books can be roughly divided into two parts: the obvious, which you breeze through; and the insufficiently explained parts, in which you can get stuck for hours. For any given book, this partition will be different for every reader.

When reading fiction, you're given a single perspective: you're seeing the world moment-to-moment, and your incomplete understanding is an important component of the fun. Just keep reading, and the mysteries of the past may be uncovered later.

When reading math, the same does not happen (partly because writers have no idea what background an individual reader has)... which is why readers of math may choose to be stuck for hours, so that they can completely understand what they're reading before moving on.

Comments

[gustavolacerda.livejournal.com]

I wouldn't go so far as to say there's no such thing as partial understanding in math.

[neelk]

I often do breeze past hard-to-read definitions. That's because definitions are often complicated or weird in order to make an important theorem's proof go through, and if you know what the theorem is, you can go back to look at the definition and use the theorem to help understand why the definition is the way it is.

[gustavolacerda.livejournal.com]

yeah, so in this case, your intuitions are at a higher level than the details of the definitions...

[neelk]

I think there are two kinds of intuition. First, there's the conceptual intuition, which is usually higher level than its realization (but not always so -- see [1]). It's easy to forget this intuition once you have a formal definition, but IMO it's very important not to if you want to do good research.

Then, there's the intuition about how to formalize an idea. My first few guesses are almost always wrong; the right formalization is the one that makes proofs transparent, and to figure this out I don't know anything better than just doing the proofs and seeing what happens. When I read mathematics, I assume the author went through the same process him- or her-self, which is why I will look for the theorems before returning to the definitions. If I were a better reader of mathematics than I actually am, I'd not just look at theorems, but work through the proofs myself, so that I can see exactly how the definition works to finesse difficulties in the proof.



[1] Sometimes, when I want to find a new general concept, I 1) will work out how to formalize a special case-concept, 2) generalize the formalization, and 3) from the generalization, reverse-engineer an informal concept. This happens a lot in PL research, because you during language design you often want to go backwards from some specific examples to find general principles.

[the-locster.livejournal.com]

I'm often horrified by how concepts I find easy to understand and intuitive to think about turn into horrendously complex mathematics when formalised.

[http://users.livejournal.com/_wirehead_/]

hmm. maybe this explains why i have such a hard time reading math. whenever i get to a complicated or difficult part, i tend to glaze over and try to skip past it, assuming it will make sense when i can see where the author is going. too much fiction reading?

[stepleton.livejournal.com]

I often read math textbooks and similar works in an iterative, coarse-to-fine fashion. First I go for the "what", then return again for the "how". Moreover, this procedure is lazy or semi-lazy. Beyond the first several passes, depending on whether I'm busy or if the material is a bit afield for me, deeper understanding happens on a need- (or want-)only basis. Occasionally it has been years between the last pass and subsequent refinements.

Sometimes I have no choice about this. I get the gist but can't understand the details until I learn more, elsewhere, and then return to the material.

[gwillen.livejournal.com]

I think it varies a lot. I never quite got the hang of reading textbooks, I think...

[surrey-sucks.livejournal.com]

I'm not mathematically inclined, and when I attempted to read my calculus text, I felt like throwing up. My brain could not translate what I was seeing into something that I could understand.