Is it not effective to learn math top-down?

By top-down I mean finding a paper that interests you which is obviously way over your head, then at a snail's pace, looking up definitions and learning just what you need and occasionally proving basic results. Eventually you'll get there but is this a bad idea? Is learning each required math area by textbook the better way?

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Trying to learn this way is a very good way to discourage oneself. I have been in this position before and it made me doubt myself and my abilities. I'm not a geometer by any means but the research my former advisor does has a lot of noncommutative geometry. Unfortunately, I was far from being able to read any of the standard texts and so I struggled with reading papers. I really started to doubt myself and I wanted to give up on several occasions. I don't recommend it - even for bright students. It's better to build up to it while still challenging oneself. –  Cameron Williams Aug 5 '13 at 3:40
I could never be interested in the top without understanding what comes before it, but that's just me. –  Git Gud Aug 5 '13 at 3:41
I tried to understand Wiles' paper (FLT) top-down and am still on the first sentence of math. :P –  Enjoys Math Aug 5 '13 at 3:42
I respectfully counter @CameronWilliams point--I say it can be helpful, but it should have limits. As an undergraduate CS student, it would probably be a bad idea for me to start with elliptic curves in number theory and work backwards from there. But, smaller steps can be taken in a "top down" approach--for example, I could start by wanting to learn prime factorization, and proceed to elementary number theory (e.g. starting with modular arithmetic). That has a shorter "tree" of prerequisite topics, and won't leave me feeling discouraged. It's all about how much you take off in one bite. –  anorton Aug 5 '13 at 3:44
I like to keep such papers around as a reminder of what I still need to learn: I'll occasionally pick them up, see if I understand anything else, check a few references, then go back to beefing up my "classical" skills. Lather, rinse, and repeat. –  icurays1 Aug 5 '13 at 4:40

I think this sort of question, and pursuant discussion/answers/comments, can be very useful for people who're new in the math biz. Namely, in my considered opinion, neither a "strict" bottom-up approach, nor "strict" top-down approach is optimal (except, in both cases, for a few extreme personality types). This is plausible on general principles, since, after all, the smart money says we should do a good bit of both, as in "hedging". And this is true, and for more than those general principles, in my opinion.

The way that "do both" is the only sensible route would seem to be that the extremes have lent themselves to highly stylized, almost caricatured, and editorially-pressured extremes. Research papers very often are written in the first place not to inform and help beginners, but to impress experts, etc. Sometimes journals' editorial pressures push in this direction. Peoples' understandable professional insecurities push in this direction. Experts' boredom with beginners' issues helps drift. And, at the other end, publish-for-profit situations push ... for profitability, which in real "textbook markets" means that "new" textbooks will mostly resemble old ones. We are fortunate that some people (Joe Silverman, as Matt E. noted) manage to move things forward within that strangely constrained milieu.

An important further point, in my experience, is exactly that of the question's "follow the branching graph of references backward to reach the ground..." 's hidden, unknown fallacy. That is, (having tried this in all good faith, very many times in my life) peculiar conclusions are reached when/if one does not give up, but pursues things to their actual ends. Namely, some significant fraction of the time, no one ever proved the thing that gradually was back-attributed to them... though it is true, and by now people have figured out how to prove such things. Another is that the "standard reference" is nearly incomprehensible, and only if one knows that subsequent explications were life-savers can a beginner find a readable thing.

And, of course, these graphs going backward branch so rapidly that literally reading everything referred-to is ... well, physically impossible for most of us... and even if one tries to approximate it, the effort is ... let's say... "not repaid in kind". At various moments I did estimates of how many pages I'd need to read to correctly honor all the background. When it hit "more than 10 pages a second, for the next 20 years", I knew both that I couldn't do it and that either it was crazy-impossible or ... not the real issue. :)

Yet, at the same time, exaggerated reliance on "popular" textbooks (excepting some like Silverman's... hard for the novice to know who to trust, yes, ...) really doesn't lead one forward/upward.

After all the above blather, my operational advice would approximately be that one should try to figure out _what_to_do_. Thus, any source which cannot be fairly interpreted as giving help is instantly secondary. More tricky are the sources that do well-describe the Mount Fuji at a great distance... To see Mt Fuji is a great thing. What should one do?

At this date, it seems to me that on-line notes on topics that otherwise seem to be "standard", but have become cliched, are far better than most "textbooks". But, yes, there is some volatility in this, because, as above, Joe S.'s books on elliptic curves are excellent, even while being entirely within all conventions and such.

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I learn mathematics this way, to a certain extent. I began my studies just a few years before Wiles's paper appeared, so I didn't have that to read when I started, but from Silverman's book on elliptic curves I learned about Mazur's theorem on rational torsion points on elliptic curves over $\mathbb Q$, and started trying to read Mazur's paper (the famous Eisenstein ideal paper, discussed here in more detail; let me just note here that it is the paper which initiated the research direction in number theory of which Wiles's paper is one of the highlights).

However, I didn't try to read this paper in isolation. Using the bibliography as a guide, I tried to figure out what other material I had to learn to understand what he was doing, and went on to learn this material, from a mixture of text books (such as Serre's Course in arithemtic, for modular forms) and other papers (too many to list here!). In the end, it wasn't until much later in my career that I developed anything approaching a complete understanding of what Mazur does in that paper.

If your goal is to become a research mathematician, then you need to develop a knowledge of mathematics, as well as intuition and a sense of the big picture. You can't be dogmatic about how your learn this. Sometimes papers help; sometimes textbooks help. (E.g. if you want to learn elliptic curves, which is certainly necessary for understanding Wiles, it is silly not to use Silverman's book; it is simply a great text-book, which will teach you a lot much more efficiently than any other approach that I know of.) Having a mentor, or just other, perhaps slightly more experienced, students around, to give advice on what is good to read and what to avoid, helps a lot.

If your goal is something else, then I don't really have any advice. All my experience is based on learning mathematics with the ultimate goal of doing mathematics.

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It really depends what kind of person you are. Most of the people I know hate this. I love it.

The most important thing to keep in mind, though, is that you'll almost surely end up somewhere entirely different. You'll start with a complicated topic, take a few steps back, and then try to move forward. But by the time you've made any significant progress, you'll probably have diverged and started studying something entirely different.

This isn't a bad thing. Self-study is governed by student interest much more than traditional teaching. As such, you'll inevitably go off on tangents and become re-directed. I think this is a feature, more than a bug. You'll naturally gravitate towards subjects you find interesting. (Just make sure that you actually make progress. It's quite easy to skim through basic results and lose interest before getting to the meat of a topic.)

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In "Surely You're Joking Mr. Feynman", Richard Feynman describes being challenged by his sister to do just this, at an early point in his graduate studies when he was feeling overwhelmed by some of the papers. (And by the apparent ease with which his contemporaries appeared to absorb them.)

As I recall the anecdote, he describes staying up all night to work through one paper, which he finally grasps. He goes on to describe this as a seminal moment in his career, when he might have abandoned theoretical physics without his sister's goading.

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This was actually later in Feynman's career, and it led to his discovery of the law of beta decay. From p. 259 of Surely You're Joking: "'No,' she said, 'what you mean is not that you can't understand it, but that you didn't invent it. You didn't figure it out your own way, from hearing the clue. What you should do is imagine you're a student again, and take this paper upstairs, read every line of it, and check the equations. Then you'll understand it very easily.'" This was around 1957. –  littleO Aug 7 '13 at 11:36

In general I think the best way to understand math is to do a lot of problem solving and specific computations in the area. That way when you get around to reading a theorem about it you say "of course, that is obvious", instead of "huh?". A simple example comes up when you try to teach students that the real numbers (or the integers) are commutative under multiplication. They don't understand why you are bothering them with this since it's "obvious". It's obvious only because they've worked a bunch of multiplication problems.

Personally, I have never understood a subject until I've worked a lot of problems. I seem to need specifics, and then the generalities are easy to understand.

I wouldn't tell anyone not to do it top down, but it doesn't work well for most people.

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I might revise "problems" to "examples". There is always a risk that "problems" are just random filler. We can maybe have higher expectations for "examples". Further, careful attention to important worked examples, done by an expert, is much better than ugly-bad quasi-solutions to artificial problems done by semi-competents. –  paul garrett Aug 7 '13 at 0:13
Well, whether they are problems or examples, it is important to do it yourself, not read what is "worked". There is nothing like trying to your own proof or computations to clarify what you do not understand. And to the extent you can figure that out for yourself it will stick. And the most fun is to come up with a better proof than the "worked" example. –  Betty Mock Aug 7 '13 at 18:38

I would advise against doing this, as you will feel as if you know less, because everything will seem so new and overwhelming. Of course this depends on how much over your head you are.

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In my experience, this can be very difficult and also frustrating. If you aren't familiar with the basics, it can be completely non-obvious why a certain "big" result is actually important. Moreover, if you just look up what you need for the paper you are reading, you'll get a very restricted exposure to that area of mathematics. The ideas you've learned will only make sense in that one context and so your understanding will not be particularly robust. It is better to learn from the bottom up with lots of examples to build intuition. As you learn, try thinking of a given concept in as many different ways as possible so as to broaden your understanding.

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I'll say from my own experience this has NOT been very effective. Really, it seems that you won't really understand at all what's going on unless you have a very solid understanding of all the simpler math motivating the more advanced math you see. One example of this is from when I was an undergrad. There was a small group of us that always felt the need to take the most advanced classes possible. The big problem with this is I think I spent a lot of time being immersed in "abstraction without motivation". One example of this is I took a graduate course in functional analysis without fully adequate understanding of some basic stuff in undergraduate analysis (for example I didn't yet really understand why uniform convergence was so important). I was able to solve the problems in the class and do ok, but I had to spend time later on in my career filling in the gaps.

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In the specific case of understanding some research paper it makes perfect sense to follow a top-down approach. If you start bottom up it will take long time before you get ready to appreciate the paper you started with and probably you will lose interest in paper by that time. Also for people who are not in academic profession (like me), I believe this is the only approach possible.

Again if you follow the top-down approach seriously (like going through all the references in the paper and doing it recursively till you reach the level of knowledge you possess beforehand), it becomes more or less equivalent to the bottom up approach as you might find the related topics interesting enough to warrant their serious/independent study.

As an example from my experience I started to search the proof for Brent-Salamin AGM formula for $\pi$ and this opened up a whole plethora of interesting math for me including theta functions, elliptic function and finally leading to Ramanujan's modular equations. I got engrossed so much so in these topics that I created a whole blog out of it.

On the other I don't think I would have ever thought of learning elliptic functions independently (I had come across them long back in my undergraduate years and did not find them worth pursuing further). To sum it up while following the bottom-up approach you may probably miss the purpose and excitement.

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