Ask HN: Math books like SICP?

After that, anything by Serge Lang. He is a first-rate mathematician and a fine educator. A master of mathematical exposition, a rare talent.

I started with two books: "Principles of Mathematical Analysis" by Walter Rudin, and "Abstract Algebra" by Israel Herstein. Keyword is "started". Nearly every paragraph of either book sent me to a goose-chase of research and reading up on supporting materials. Safe to say that by the time I was nearly done with either, I have acquired about 50 other math books and had about zero social life ;-)

Good news is that good math books are cheap, compared to anything computing. You can find classics for $1-$5 in most yard sales and 2nd hand bookstores. The trick is to buy thin math books; the giant, multi-color text books they use in schools are highly confusing, at least to me. It's 200 pages of math stretched to 1200, with a similar jump in price.

Mathematics is both style and substance. Once you get the hang of the basic language, the succinct delivery style, the proofs and generalizations, how notation is introduced and then elided when it becomes too apparent .. all these will add up to help you navigate more "advanced" texts. You will come to recognize what is a well-posed mathematical problem, and what is not, even if you don't understand the domain itself.

Finally, if you have the option to study this formally at an institution, do it. The 2nd best option is to find a friend that shares this interest and discuss things. I didn't study it formally, but I was a barista at a coffee shop with plenty of mathphile customers. I had a lot of 1:1 instruction for highly qualified people on my 10 minute breaks :-)

replace "that" with "those" ( http://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics ).

the dover books are also a good series, and pretty much anything by Artin is good. i also looked at the Halmos book a couple of people have mentioned.

perhaps one thing to be aware of, though, is that you're not always going to learn things in the linear way they're layed out on the page. moreover, it might be helpful to have more than one book for any given subject. Lang's Algebra, for example, is a really good reference, but a tome if you read it like a text. so you might pick up something small and subject-oriented with a lot of exercises like Artin's Galois Theory or Atiyah's Commutative Algebra, and supplement it with a reference like Dummit and Foote or Lang's texts on Algebra as a whole.

oh, right: whatever book you choose, do the exercises.

It starts off with metric spaces and Euclidean n-space. The main goal is to cover open and closed sets, convergence, and compactness.

Why? Because those topics are crucial in continuity and uniform continuity.

Why want those? Because continuous on a compact set means uniformly continuous and, thus, that the Riemann integral exists.

Any questions?

The main content is just 'measure theory', and that's just freshman calculus grown up. Why? Because near 1900 it became clear that freshman calculus was clumsy for some important progress, especially cases of convergence of functions.

Measure theory? Well, first cut, 'measure' is just a grown up version of ordinary area. Simple.

Why interested in measure theory? Because want to cook up a new way to do integration, that is, what freshman calculus dues with the Riemann integral. Recall, the Riemann integral partitions the X axis and approximates the area under the curve with tall, thin rectangles. Measure theory partitions the Y axis: At first glance this seems a little clumsy, but in the usual cases get the same number for area under a curve and in bizarre cases, that can get from converging functions, get a nice answer that Riemann integration can't do.

Royden likes Littlewood's three principles, and they are cute. So, spend an evening on them. Yes, it's possible to use Littlewood's to do the subject, but there is a better way, also heavily in Royden, roughly called 'monotone class arguments' -- which are gorgeous and turn much of the whole book into something quite simple. So, prove the theorem for indicator functions. Then extend to simple functions by linearity. Then extend to non-negative measurable functions by a monotone sequence. Then extend to integrable functions by linearity. Can knock off much of the book this way, e.g., Fubini's theorem which is just interchange of order of integration grown up. The foundations of this little four step process is Fatou's lemma, the monotone convergence theorem, and the dominated convergence theorem.

About 2/3rds of the way through Royden is a single chapter that essentially compresses much of the rest of the book -- I would have to step into my library to find the chapter.

For the early exercises on upper and lower semi-continuity, they are a bit much and you likely won't see that topic again. So that exercise can be skipped.

Royden is elegant beyond belief; if you still have trouble finding the main themes, then chat for an hour with a good math proof who understand Royden well.

Then, don't miss the Radon-Nikodym theorem: It can be seen as a grown up version of the fundamental theorem of freshman calculus but, really, is much, much better. The role of the Radon-Nikodym theorem in 'modern' (i.e., Kolmogorov) probability theory, stochastic processes, Markov processes, martingales, etc. is astounding.