So true. I learn more by writing about things than by reading about them (of course, the former includes a lot of the latter, but it's so much more productive when you're reading towards a specific goal...and if the goal is explaining the subject clearly and entertainingly you really have to have a firm grasp of the subject). It just seems to activate my brain in ways that merely reading never quite does. Doing the exercises overcomes the passivity of reading to some degree, but writing down what you learned takes it up another notch.
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Anyone know of an abstract algebra text that is well-suited for an autodidactic mathematician with a basic understanding of the subject?
: There's also Michael Artin's "Algebra" which is somewhat dense (certainly without solutions), but a very fascinating read due to its emphasis on symmetry groups. Particularly useful and enlightening in the context of modern particle physics.
1. Learning math is like topiary for your neurons. Your goal is to train your brain to model the behavior of objects that respect axioms. In programmer talk, you can think of your brain as the machine you're trying to program, the book as your source code, and that "Huh?" feeling as a compiler warning. That sheds a little light on why math is harder. SICP has an extra indirection to it. It's a book about writing code, not a book full of code.
2. 50% of the the things that mathematicians do are obvious to anybody. Another 40% are obvious in context (which is the point of "motivation" in textbooks). You're appreciation of these facts is probably what's set you on this project. The other 10% of stuff is hard, though, and I think I know why. Mathematical notation wants to be done with little animated graphics, but it's done in dry prose instead for the sake of tradition stretching back from Gauss and Euler to Euclid. I've wanted to do higher-math teaching software for a long time, and could use a collaborator.
3. Since a lot of the difficulty of modern math is in the expression of ideas that are simple to "do" in your head but hard to write down, you can get some really good catalytic effects by using multiple sources. Be careful of mismatches in "theory level" -- sometimes it's hard to tell when a highly abstract definition specializes into a definition that you already know. Even so, on the internet there are just so many resources available these days that you can almost always find the quick-and-dirty in-your-own-head explanation that practicing mathematicians give in lectures. For Algebra, I recommend you use Hungerford's text together with Dummit and Foote, with The Spellbook (from Serge Lang) as a backup. You can find good notes at: http://www.jmilne.org/math/index.html
4. Math departments often put out study materials for PhD students preparing to take their qualifiers. It's a great source of exercises and sometimes you can get solutions too.
5. Even though Steven Wolfram is kind of a pustule, a few hours spent looking at group tables in you-know-which software package can save you a week of head-scratching.
6. There's a cool paper on the classification of groups of order 16 that I absolutely love. Google it, because I'm lazy, or email me if you can't find it.
7. The people at PlanetMath are usually pretty awesome.
8. Skip things judiciously. This goes back to #3 a bit. This stuff was mostly discovered by guys writing letters to each other, so it's structure is only rarely more complicated than a tree of dependencies. So skip around if you have problems with one section. Chances are good that when you come back you'll have found something that makes the process easier. Most of math is getting used to things, as they say, and that takes both time and context.
Again, let me know if any of this is helpful to you. I'm not just saying that, either. I'm interested in how the self-teaching process works. Also, now that I've written all this dreck, you owe me.
I can't locate The Spellbook by Serge Lang. Can you give me a link?
Michael Artin's book is also a decent text aimed mostly at undergraduates:
My first-year class in field & Galois theory started with the Artin text, and by mid-semester I started daydreaming about ripping out random pages and mailing them to the author.
But I don't like colloquial exposition, so YMMV.
Lang is very good, IMO, but more difficult.
I'd hesitate to suggest to any of Lang's writings as the first go on a technical topic. If you are already comfortable with the material, his writings will give new insight and intuitions, but thats different from a first go at understanding a topic
(note to audience, we're talking about different math textbooks)
Speaking of which, does anyone have some good recommendations for a discrete mathematics textbook, specifically one that includes both theory (proofs) and application?
http://www.mhhe.com/math/advmath/rosen/ ..is what we used in college; I found it hard to read in spots, but haven't really found anything overall better.
The recommended text is Discrete Mathematics and its Applications, by Rosen, which is a mixed bag according to reviews but the consensus is generally good. AdUni provides a full series of video lectures and quite a bit of course material. Nearly all of the AdUni.org stuff is awesome (though the ancient Abelson and Sussman lectures are far better than the Yanco lectures at AdUni.org) and a great supplement to self-study. It's mostly taught by MIT grad students (which is what you'd get if you were taking the courses at MIT, so not a bad deal).