What's the point in being a "skeptical" learner [closed]
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.
My suggestion would be to take notes the first time you read a paper or a book. For instance, each time you ask yourself a question, just write a short answer in the margin${}^{(*)}$. In this way, when you read again the document, you will have an instant access to your answers. This method is especially useful if you collect examples and counterexamples and improves your understanding of mathematical objects.
${}^{(*)}$ Or in a notebook, as the margin is notoriously too small for some people...
I wanted to post this as a comment but it got too long. So forgive me for posting it as an answer:
Funny you really remind me of myself when I was younger. Not to mention my favorite mathematician is Galois. First off let me say that your habit of checking details will serve you well. Many people advance quickly by being fast and loose and they intimidate others but if you hammer them down on details you'll find their skills set so full of holes. This happens worse the better university you attend, because they push the students to go so fast. So if I were you I wouldn't give up the habit of checking details. Just learn how to do both, how to read for higher level content and then read for details. Maybe take one pass plowing through the details, then another pass higher level, then rinse and repeat.
You don't learn carpentry by looking at cabinets.
If your aim is to be able to do mathematics rather than answering prepared exam questions, you are doing the right thing. There is no royal road to mathematics. The only access is through the stable and the kitchens.
I think yours is the mentality a student should have when learning new material: doubt everything, check every little detail that's left as an exercise, etc. You shouldn't accept things as acts of faith, rather you should keep questioning until one day you look back at those books for reference and say "Oh, right, I remember how this fact was verified", because you tried it yourself!
Having said that, I think that remembering every detail of some proofs is not really necessary most of the times, but the 'big picture' or an idea of a sketch of the proof is: Perhaps a proof uses some nice trick which might be useful later, then you should make sure to learn it thoroughly. Otherwise, if you just know the sketch, a good idea is to sit down and try to fill in the details yourself. Sometimes reading and re-reading does not do enough. That's one of the reason math books have so many exercises (and details of proofs left as exercises too).