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I have a long-term goal of acquiring graduate-level knowledge in Analysis, Algebra and Geometry/Topology. Once that is achieved, I am interested in applying this knowledge to both pure and applied mathematics. In particular, I am interested in various aspects of smooth manifolds, co/homology and mathematical physics. I have acquired a smattering of knowledge in all of these areas but feel that I need to become more focused to make make coherent progress. I have a very bad habit of picking up a book, reading a bit, working out a few details, and then moving on to other random topics in other random books. In doing this, I don't really feel like I accomplish much.

To rectify this admittedly undisciplined approach, I have decided to select core source material from each of the three major areas listed above and focus on it until I have assimilated all the information in that material. For analysis, I have selected Amann and Eschers' Analysis, volumes I, II, and III. I made this choice because out of the analysis texts I have surveyed, theirs seems to be the most comprehensive and treats elementary and advanced analysis as a unified discipline.

My basic strategy is to treat each theorem, example, etc. as a problem and give a fair amount of effort to proving before consulting the text. I think this is probably the best way to approach the material for maximum understanding but it requires a considerable amount of time. There are probably thousands of these sorts of "problems" among the three volumes. Ulitimately, I would like to end up with a notebook (which would probably number in the thousands of pages) that contains all of the details to all of the theorems completely worked out, as much as possible, with my own thoughts. Again, this seems like it will take forever and my time on this earth is unfortunately finite. I'm reasonably confident though that the production of such a set of notes would lead to at least a fair level of mastery of the material in question.

Can anyone suggest an alternate strategy that might be more effective in terms of time but that would lead to a comparable level of mastery?

It is also a problem that I might actually prove a fact completely on my own but then, a month later, might not be able to recall it in a time of need. What strategies are helpful for best ingraining this material (other than the obvious "Work lots of problems" approach)?

Would appreciate any tips or pointers.

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I'm always interested in these sorts of questions, but I'm also compelled to point out my uneasiness from a "don't reinvent the wheel" perspective. I have to think there's some kind of balance between how much time and effort you should spend working through the details and how much this will help you to actually understand and know how to best apply the tools you're building, which is presumably the end goal. –  matt May 23 '11 at 22:14
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@Matt: Yes, I struggle with this too but how is one possibly supposed to figure out which details to work through and which ones to skip? Invariably if I skip something, say, Theorem 297 then that result will be used to verify Theorem 431. I will then feel compelled to prove Theorem 297. During the course of which I discover that Theorem 98, which I also happened to skip, is of importance there. Hence, one must either accept many results on faith or must commit to proving everything. Not really sure how to handle this. –  ItsNotObvious May 23 '11 at 22:52
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The best way to learn something is to explain it to someone else. –  Mike Jones Sep 30 '11 at 21:11

6 Answers 6

up vote 31 down vote accepted

This was supposed to be a comment, but it is too long and there may be lessons in it from what I've experienced.


This is a great question (I would add a bounty from my own rep if this wasn't community wiki), and it is also a very personal issue that I struggle with myself. I have a similar style of study to the one you described when it comes to things I am really interested in, rather than things that happen to be part of the syllabus on an undergraduate/graduate course I am taking (those subjects falling into the latter category tend to all get the same treatment from me -- go to the lecture, absorb the main ideas, briefly look at my notes later to see they make sense, then ignore it until I need it for some problem set or an examination).

However, I find that your style of study (call it the hard grift method) means that I am often a little behind classes or lectures, despite the fact that I am trying to pursue a deeper understanding of the material I enjoy. Unfortunately, the style of exam questions means that this level of understanding rarely helps. One can ask oneself whether it is worth the trouble, and ultimately the answer to that question depends on what you want to get out of your learning.*

I am keenly aware of the approach of what Stefan Walter above calls the 'superficial' mathematician, which I do not see as at all disparaging (and for the record, I do not think he does either). I do not really believe in innate talent, but there are many mathematicians smarter than I who seem to pick up just as much knowledge as I might from the hard grift method, by instead coasting from an article to a textbook, to a set of exercises, to a pre-print, while making minimal notes and seemingly picking up the salient points naturally, and then having a fruitful discussion with others about their new findings almost immediately (call this the flowing method). From what I read of Terence Tao's blog this is the natural progression from an undergraduate mathematician to a so-called 'post-rigorous' mathematician.

The flowing method seems to reap more benefits, but it also doesn't seem to be a ticket you can buy. I have a few friends already on their PhDs (I am about to finish my humble MMath) and, without wanting to make this sound like a cop-out, their brains seem to work in a different way to mine. It may very well be the case that I am yet to make the transition because I have not yet put in the hours, but I believe that 'putting in the hours' boils down to passion. If you aren't passionate about what you are studying, you won't put in effective hours, and you won't make the transition to post-rigour.

(Aside: I would like to think that one day I might make that transition, but as it stands, I am not sure the life of a professional mathematician is for me!)


*To answer your question succinctly: you need to find out what it is you want out of your learning. If it is pure mastery, an effective method for you to try might be: stick to hard grift for a little while, but if you find you've reached a level where your intuition is guiding you more than rigour is, then stop and evaluate, and consider taking your learning to a higher level where the details of proofs are not the most important thing any more. In particular, re-read Terry Tao's post in the above link.

If, however, you enjoy learning for its own sake and want to pursue personal understanding (which I think the hard grift method is best suited for) then you should always keep this goal in mind. Personal understanding is more gratifying than pure mastery; it should be the goal of any true autodidact (see the last section of this great article by William Thurston).

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All of the answers have been great so far. During the process, the arduous process of memorizing definitions and applying them to formulate theorems, remember they are for a greater thing.

This greater thing that will be implicitly supported by the work of many, the work of those that worked hard to the solution of a problem, or those that luckily stumbled upon an interesting solution to the problem, by looking inside their coffee, thinking about how well it tastes...and EUREKA!

Whether you be in the former or latter, remember the more mathematics you know implies that you will be more prepared. But according to Seneca, the Roman philosopher, (preparation) $*$ (opportunity) = luck! So the "lucky" people in our latter paragraph really were prepared, or had a lot of chances to make it.

Question: Would you rather belong in the group that decided to prepare themselves by learning a lot of mathematics and not even realizing it in the process $-$

OR $-$

belong in the group that is given a lot of opportunity, such that no matter how many times you manage to fail at something, $P(Success) = 1$? (That's actually mathematically impossible).

(Hint: Suppose two people $A, B$ have the same amount of luck. If $A$ is more relatively prepared, what does that imply about $B$'s preparedness? Use a proportion. And why is so much doggone opportunity bad?)

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I believe it is unlikely that you're able to prove a theorem pretty much on your own but unable to recall it well a month later: you will certainly be better able to recall it proceeding as you outline than if you had only read through a proof and applied it to, say, one example. We learn best by "doing", by "owning" the material we're studying, so to speak: making it "our own" by interacting with it, reconstructing it, proving conjectures, or finding counter-examples, etc. It will also generate questions and motivate you to pursue your studies at a deeper level.

Such an approach can seem overwhelming at times and involves a lot of solitary time without some sort of support/mentor available when you really feel stuck. That's where a site like this comes in. It gives you a community within which you can participate: both asking and answering questions. I know, for me at least, I often get a much deeper appreciation of a topic when attempting to answer a question than if I were to have just asked. So it's give and take, ask and answer. It all boils down to learning!

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I can't quite parse the beginning of your answer. Do you recommend proving the theorem or skimming through the proof and applying the theorem first? –  Stefan Walter May 23 '11 at 20:47
    
Sorry...let me correct! There now, I hope it makes more sense! Thanks for the comment. Essentially, I'm with you all the way. –  amWhy May 23 '11 at 20:57
    
Yes, now I understand. –  Stefan Walter May 23 '11 at 22:08
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+1 for “owning” – I've always felt that this is very central to understanding. –  k.stm Jan 26 '13 at 15:43
    
Thanks, @K.Stm. I'm with you on that: I'm sort of a cognitive constructivist in terms of learning. –  amWhy Jan 26 '13 at 16:55

“He who seeks for methods without having a definite problem in mind seeks in the most part in vain.” -David Hilbert

Most of us don't have the power to learn everything in the world in sufficient depth to make meaningful progress. A way around this is to use a "lens"...learn what is needed to solve a specific relevant question in a desired field. Then keep doing this until you have your own personal toolbox and viewpoint on some specific area.

The pleasant surprise is: If you go deeply enough into any rich area of mathematics, you will end up learning the other areas in a very connected way. At least that is what the `unity of mathematics' tells us.

Best of luck in your studies!

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I like your approach in principle, but:

Not all theorems (or rather their proofs) are of the same difficulty level. So when you try to do some harder proof, you will often be forced to "peek" at the proof in the textbook. It could require a lot of discipline not to peek to often.

Also, in most cases knowing the proof of a theorem is not as important as knowing its statement and having a feeling how it is usually applied.

For these reasons I would prove only easy propositions and just read the proofs of the harder theorems, yet read them carefully. I would put the time you save by that into doing exercises. To this end, choose a textbook which has many exercises separated by small bits of text. This way you get a good balance between passive reading and working things out on your own. There are great differences between textbooks concerning the style of the exercises. Some authors just list things they want to mention and don't think too much about the solutions. What you want is books in which great care is taken to ensure a uniform difficulty level among the exercises.

Another aspect: Make sure you are really interested in the subject you are learning about. If you approach it with the mindset "It's boring, but I need it as a prerequisite for Cool Theory XY", then you are probably better off just having a go at Cool Theory XY and returning to the boring stuff later when you have gathered some motivation.

Finally: Although I like your strategy and do something similar myself, I have the impression that most mathematicians are more superficial, i.e. just reading through books and papers, yet arrive at an equally deep understanding in a significantly smaller amount of time. I would really like to hear the opinion of a research mathematician on this.

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I think you bring up a good point. I believe, in the earlier stages of intensive study, students need to plan on spending a great amount of time and effort going through proofs, etc...(perhaps not) soon enough, that will get easier, and at the expert level, it becomes "second nature" so to speak...not only due to familiarity with the content and methods of mathematics, but also having come to know oneself as a learner, and how one learns best. –  amWhy May 23 '11 at 22:17
    
You make some good points. Indeed, I know it would be very difficult (hopeless?) for me to attempt proofs of theorems such as the Fundamental Theorem of Algebra, for instance, without guidance and instead work on the details that the author invariably leaves out. Many results though are approachable but it is sometimes hard to figure out which ones are of the "hopeless" variety. Another of the reason I chose the A&E text is there are many small results "Remarks" throughout the text that do serve as good exercises. I too would be very interested to hear commments on your last paragraph –  ItsNotObvious May 23 '11 at 22:44
    
"Also, in most cases knowing the proof of a theorem is not as important as knowing its statement and having a feeling how it is usually applied." -- Really? I always felt like only the proof of a theorem can give me real insight into what's going on. –  Sam May 24 '11 at 8:53
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@Sam: I've always felt that the proof of a theorem gives me more insight about the definitions and theorems used in the proof than about the theorem itself. –  Stefan Walter May 24 '11 at 9:45
    
@Sam, I think what you said does apply to some proofs, and then in other cases what Stefan said might be an apter description. There's also proofs that are just so technical and mechanic that they don't give much insight into anything at all. Perhaps those can be subsumed into Stefan's category, though, as they might show you how to use certain tricks or tools from other areas of mathematics in a context not originally foreseen. So I guess in a sense no proof is completely devoid of value and insight, but it seems to me it's hard to generalize what that value is. –  Ryker Jul 16 '13 at 17:10

I think that your approach is just fine. Just stay disciplined about it.

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