How to Read Mathematics

Look at this proof of Fermat's Little Theorem [0], can you spot the indirect bit of the proof?

By the way, when you construct a proof you almost always have some leeway in how big your indirect section are. E.g. variant A: Assume X, do bits Y, Z, then contradiction. Variant B: Do bits Y', Z', assume X, then contradiction. In the second variant Y' and Z' could be useful on their own, and might be easier to understand without the inversion. For that reason, I often try to keep the contra-factual parts of the proof as small as possible. Though as with writing any code, clear writing trumps general rules.

[0] http://primes.utm.edu/notes/proofs/FermatsLittleTheorem.html

I expect you mean this bit:

    Suppose that ra and sa are the same modulo p,
    then we have r = s (mod p), so the p-1 multiples
    of a above are distinct and nonzero ...

More completely, I expect you want them to say:

    Consider the (p-1) multiples of a given by:

        a, 2a, 3a, ... (p-1)a.  (mod p)

    These are all distinct.  To see this, consider
    otherwise, and suppose ra=sa (mod p)

... and so on.

Is that what you meant?

The point is that all writing is aimed at an audience. I wouldn't expect someone with no experience of Science Fiction to be able to read "Quantum Thief," and I wouldn't expect anyone with a reading age of 6 to be able to read "Lord of the Rings." Similarly, that proof requires some degree of familiarity with the structure of proofs. This is not an especially difficult.

I've always found that indirect proofs, or proofs by contradiction, or proofs by the contrapositive, are mostly obvious as to what they are doing, although not always.

In short, I agree with what you say, don't think the problem is as bad as you are portraying, think the example you have given is not especially good, but I'd be hard pressed to find a better one.

And finally, it's possible to come up with bad writing in every context. Some proofs are badly written, badly expressed, and badly explained. I know - I've not only read many of them, I've written some as well.

No surprise there.

Addendum: Sometimes the proofs that gave me the greatest understanding were the ones that were the most badly written, forcing me to work through the material on my own terms and understand it in my own way. Perhaps well-written, well-expressed and well-explained proofs are actually a bad thing.

My secret for reading difficult math and physics texts: write out, by hand, every equation as you encounter it. Firstly, it slows me down so I'm not tempted to skim it like a novel. More importantly, my standards for what I write are much higher than my standards for what I read; if I see something I'm not completely convinced of, I may shrug and move on, but I'm not willing to write something down unless I really understand it.

It's about modeling specific physical processes in buildings. It's computer science in the sense that the paper focuses on efficient implementation. However, the audience is the indoor environment community.

The reviewer is a skilled experimenter, whom I enlisted to check my understanding of the physical processes. (By the way, this person knew what I was saying, just didn't like the way I said it. For our internal reviews, we tend to get pretty picky.)

It's definitely not for a general audience-- there'll probably be five people in the world who can use the paper directly (once the results are embedded in software, there'll probably be several hundred who will use it).

I understand math, but I can't read mathematical symbols. Does anyone know a good website/book for applying and understanding mathematical symbols?