I have a big problem:
When I read any mathematical text I’m very skeptical. I feel the need to check every detail of proofs and I ask myself very dumb questions like the following: “is the map well defined?”, “is the definition independent from the choice of representatives” etc… Even if the author of the paper/book says that something is a easy to check, I have this impulse to verify by myself.
I think that this approach is philosophically a good thing, but it leads to severe drawbacks:
I waste a lot of time in reading few lines of mathematics, and at the end of the day I look at what I’ve done and I realize that I managed to go through a few theorems without learning enough. Remember that when one is a (post)graduate student (s)he has plenty of things to learn, so the time is almost never enough.
This kind of learning could be affordable for undergraduate texts, but very often is almost impossible to read a paper with a skeptics point of view. At a certain point things become very complicated and the only way out is to accept results on faith.
And finally the real object of my question:
3. Despite the big effort I’ve employed in reading very carefully something, after few weeks or months I obviously forget the details. So, for example if I try to read again a proof after a while, maybe I would remember the big picture but probably I would check again the details as though I’d never done it yet.
Therefore, even if the common rules for a mathematician say that “learning” should ideally be done skeptically, I’ve finally realized that maybe this is not very healthy. Now, could you recommend a sort of royal road for reading mathematics? It should be a middle way between accepting every result as true and going through every detail. I’d like to know what to do in practice.
I have the following advice for reading papers: read them (up to) three times.
The first time through, you do not check that the claims are correct. You are attempting to get broad structural understanding. Don’t even look at the proofs. Many details will be left dangling. This is fine. This is your first pass. If you are doing a lot of reading, leave a note to yourself that you have read this paper coarsely.
If there is something in the paper justifying further understanding, read it again. This time, read through the proofs quickly. Again, don’t check any details. Just ask yourself: “Is this the sort of argument I have seen before? Does this structure of proof match the broad structure I got from my first reading?” If you are doing a lot of reading, or might come back to this paper in the future, leave a note to yourself that you have read this paper.
If there is still something in the paper to justify detailed understanding, read it a third time. Check every line. Ask yourself “why should I believe this is true” as often as possible. If a particular claim needs an additional idea (for you to believe it), record this idea in the margin nearby. If you are doing a lot of reading, or might come back to this paper in the future, leave a note to yourself that you have read this paper in detail.
You leave notes to yourself because you trust yourself to have applied the appropriate level of skepticism to the things you have read.
There are not enough hours in the day to read everything at the detailed level. For results that are entirely predictable, you should arrive at that conclusion after one or maybe two readings (and you will have recorded that you believe those results at that level). Only the results with complexity or surprise should merit a third read.
Is this method perfect? No. Does it help allocate time better? I think so.
How does this apply to textbooks? You can certainly write in your textbooks, so leaving notes should be no problem. Clearly, the “big theorems” should be read three (or more) times. Other results may only need to be believed at the “plausible” (coarse, once read) or “likely” (medium, twice read) level. Is this ideal? Probably not, but neither of us is immortal, so some accommodation of finite time must occur.