books, companions to books, and varying levels of detail

When I was taking my first graduate algebra class, I was desperate for any sort of supplementary material. One of our textbooks was Lang (we also used Hungerford) and so naturally I tried to find anything on the Internet that would explain what Lang was saying; Lang's textbook can be terse at points. (Incidentally, there's a note on the back of my copy of Hungerford, quoting from a review of the text, which says something like "the book is sufficiently understandable that an averagely prepared graduate student can read it and understand most of it." I found this to be true, at least at those times when I was actually willing to sit down with the book instead of desperately flipping through it to find the theorem that would magically make my homework work out, and I remember being quite amused by the fact that this was sufficient praise that Springer would actually print it on the book.

At one point during my searching I found George Bergman's A Companion to Lang's Algebra, which is basically what it sounds like -- a couple hundred pages of supplementary material, which Bergman wrote in his attempts at teaching the basic graduate algebra course at Berkeley. This material includes proofs of things that Lang doesn't bother giving a proof of, notes about other things that one might like to think about when reading a certain passage, alternate notations, some supplementary exercises, and so on. (For example, at the beginning of the fourth section there's an explanation of where matrix multiplication comes from; no graduate textbook would include this, because it's pretty much standard, but it's good to see how it fits into the new framework one is learning.)

In general mathematicians have a habit of leaving out some details when writing -- some of which are crucial, and some of which are not. This is usually explained by saying that the reader should be able to fill in the details. While ideally this is true, sometimes one just doesn't have the time to fill in those details, or -- more importantly -- sometimes one just doesn't have the ability, and seeing some sort of example of a proof or a calculation would make it easier to fill in similar details later. Now, it would be unnecessarily pedantic to fill in all the details of every calculation. But I suspect that in the case of a book like Lang which is widely used, all the details have been worked out by someone. All the interpretive notes that one might want are in someone's mind. Wouldn't it be nice if, when you came to a sentence that said "it is easy to see" and you don't see it, you could just click and see an explanation -- either written by Lang, or by someone else who has either struggled with the same point or watched their students struggle with it? (And yes, I know Lang is dead, so we can't ask him to do this.) In general, having a web site full of various explanations of such classic texts would be useful. A lot of this material is already out there on the Internet; people have written it up for the use of their classes. But having it all in one place would be useful. These various lecture notes, supplemental handouts, and so on could be threaded together into a sort of virtual textbook. Indeed, one could probably learn from the notes people have written on the Internet on any classic textbook without having the textbook itself.

And not just for books; research papers, too, could benefit from this. It's often written in a paper that "the reader can see that"; but you know that the author of the paper has actually done the calculation, and wouldn't it be nice if you could just click on the paper and have it appear? Sometimes we shoot too much for terseness, at the expense of clarity. And sometimes terseness is good; one doesn't want to be disrupted by the unnecessary details.

This is different from the idea of, say, a textbook which is in the form of a Wiki, which anyone can edit. What I'm proposing is that the original text is inviolable, not least because in many of these cases there may be a lot of printed copies out there, but that anybody who wants to -- or at least some reasonably large set of people -- could write comments on the text, instead of everyone sitting around and sweating through the details on their own. Something like having a paper which lots of people have scribbled notes in the margin with lots of different colors of pen, except you could actually read the notes, you could decide that you only wanted to read certain people's notes (the ones who seemed to struggle at the same points you did, or who were particularly good at explaining things, and so on). I'm not sure to what extent the mathematical culture would like this, because we don't place much value on mathematical exposition, and so exposition of exposition might be thought even less valuable. But it's interesting to think about.

(Although I think that this is a new idea -- I don't recall seeing anything like this before, which combines multiple people's notes on a text -- I am not under the delusion that nobody's thought of this before; I'd appreciate pointers.)