How to Read Mathematics

I've read a couple of texts on the history of math. Most of them were dry, a few were very entertaining like Crowe's on Vector Analysis. But I haven't found anything that beats mac tutor. I used to read it nearly everyday many years back.

The section on Al-Khwarizmi, a key innovator in algebra shows how laborious the task was (site down linking cache):

http://webcache.googleusercontent.com/search?q=cache:nOT6h0c...

a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.

  > The way to really understand the idea is to re-create what the author left out.

If reading mathematics requires re-creating what the author left out, why not leave it in?

Sure, it will be longer, but if the purpose is communication, wouldn't that be better? The reasons I can think of are not beneficial for communicating knowledge: that's how the game is played (tradition); it excludes the uninitiated/untalented; it's neater to leave out the truth of discovery; it makes the author seem superhuman; there's satisfaction for the reader in understanding the puzzle.

EDIT the reader can skip explanations he can work out himself (or use them as a check); papers are already structured with details in deeper, skip-able sections. One can have a summary that excludes details altogether (like an abstract, or equivalent to a present maths paper). In the article, the parts left out are not "known", but steps that the author could work out themselves, perhaps after many dead-ends to find the right combination. To avoid repetition of known specific concepts (like vocab), one could explicitly reference them, or assume them for a given audience.

Perhaps the essential problem is that the omitted steps are not a single concept (like a well-known term), but many concepts, combined according to other concepts (like a complex expression), so they can't be easily be referenced, nor assumed. Someone, somewhere will have to work out the combination - I'm suggesting it is more efficient for it to be the one writer than the many readers.

Imagine how poorly your idea would be communicated if you included definitions or explanations for every word longer than six letters. It would be hard to cut through all the noise to get to the interesting part of your message. By assuming a particular level of background information (in this case, vocabulary), you're able to focus on the interesting and important insights.

The same is true in math. You don't leave out random steps; you leave out steps that you expect your audience to have already mastered. You leave out steps where the working-out process doesn't contain anything particularly new or useful. Anything that comes down to "apply a bunch of lower-level math in a tedious way" doesn't belong in your paper. Re-creating it shouldn't be necessary for getting the basic idea.

When this is done properly, the result is clear and concise communication.

Oh cool, it was in issue #5 (http://hackermonthly.com/issue-5.html).

I'm sure I have the text lying around somewhere. I can't remember the title and author, but in any case the text was just a compilation of relevant excerpts from primary sources to be used as reference. It didn't provide any analysis. All the analysis was provided by the professor, Karen Parshall, whose courses I can enthusiastically recommend for anyone at U.Va. interested in the history of math or science (bonus: they satisfy your 2nd writing requirement). Her website is here: http://www.math.virginia.edu/~khp3k/

I credit her for pointing out the fact that I note in my original comment, that an equation can be equivalent to a large tract of text. She made the point in the context of why more "advanced" mathematics couldn't have developed before symbolic notation had matured sufficiently to support it. Here, I'm generalizing her point to explain why reading a mathematical text dense in equations might take longer than the page count alone might suggest.