What's the point in being a "skeptical" learner

Question

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:

1. 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.

2. 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.

Answer 1

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.

Answer 2

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).

Answer 3

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...

Answer 4

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.

Answer 5

I think there's a time and place for both. On the one hand, if you can step through and see how the author got where they were going, you tend to learn the material at a much deeper, more nuanced level. One of my learning techniques is to explain how some new concept works to an imaginary friend, because that forces me to understand it myself.

On the other hand, if you're midway through the semester and you get hung up trying to figure out why something works, you may block out a week or two of critical lecture material explaining how to use it, which really sets you up for failure for the rest of the class. In this case, it's better to set aside your desire for detail until you've sorted out everything you need to know for upcoming homework and/or the next midterm.

One thing about the "faith-based" acceptance of the material, is that it really isn't in many cases. If you can take the end result and apply it to get practical results, it's obviously at least partially right. I don't have to algebraically solve a derivative to show the results of a given derivative are consistent with real-world data from my car, for example.

Another thing you mention is how you want to go back through every detail on subsequent readings. While you may have forgotten them since the last time, your brain probably retained a decent amount of it, so as you step back through it, you'll start to recall enough tidbits that it isn't as tedious and time-consuming. For things you plan to use a lot, this sort of refresher is a good thing.

However, if it's fairly abstract stuff you aren't going to use in practice, it might be better to take a bit of it on faith. After all, not only is the author a pretty smart person, but you confirmed their findings last time. There's no rational reason it's going to be different this time unless you've come up with a totally different set of assumptions you're trying to apply the result to.

Answer 6

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.

Answer 7

I have the same "problem" like you, I want to understand every detail. And by this my progress is slow and I always have the feeling that there is still so much to learn and still so much to do.

But one thing I learned as time passed by, that some domains have certain standard arguments and how to think about certain constructions behind merely formal checking. And until you have not internalized them checking them makes it sometimes hard to follow arguments or understand the reasoning behind it.

Maybe I give you an example from my recent studies. I had to study group theory approximately 6 months ago and entered the field with almost no knowledge about it behind the basic definitions. I started reading certain papers and felt almost (and still do...) overwhelmed. In the domain I am concerned with a special class of groups called Frobenius groups arise, they have one quite obvious definition in terms of permutations groups, and then many others. It took me a while to handle them all, see all the connections and all the implications. When I now look at them they are much less intimidating and it is much easier to follow certain arguments about them, and also to remind facts about them.

So if you always have such a though time, then either the paper you are reading is a though one, or it might indicate that you are not familiar enough with some basic arguments and ways of thinking in your domain.

Maybe adapt the following strategy, if you have a specific paper. Read it and take notes what is not clear and what is clear to you, and try to think more deeply about selected portions of it. Then put it aside and the next week just study the field as a whole (reading books about the topic, secondary literature, other papers) and then after approximately a week come back to your paper and do the same again, and hopefully it is now easier to read it; work through it again, taking notes, think about it, but then put it aside again for another week and do "general research" again. Do this until you understand the paper. I know this might be time-consuming, especially if you have some deadline, but if not than this might work out well and gives you good overall knowledge in the end.

Answer 8

@EricTowers answer is excellent, and it lines up the best with what I see asserted by top mathematicians about how they read proofs; there should be an initial "scanning" process for the big ideas and the overall structure of the proof. This should tell you whether the proof is "interesting" or novel in a way that deserves your time to dig into the gritty details.

Richard Lipton writes about this sometimes on his blog:

• Proofs and Elevator Rides (A guide to tell if a proof is a proof)

• Proving a Proof (How to convince someone your proof is really a proof)

• Is This a Proof? (Seeking the limits of mathematical proofs)

• Facts No One Really Checks (Basic theorems that rarely get proved in full detail)

I think that one of the points some of these guys make is that mathematics is still (or especially) a work of community, and to some extent you still have to trust/ communicate/ convince/ stand on the shoulders of other people as part of the work. I'm still wrestling with that idea myself, and you'll have to find your own way through that thicket, but it's food for thought.