I recently added two new pages to the blog: a bibliography for listing references I cite on multiple occasions, and a suggestions and requests page. The bibliography is likely to soon contain citations for at least some of the following books which have recently come into my possession:
- Introduction to the Theory of Computation, Sipser
- Lectures on Quantum Mechanics, Faddeev, Yakubovskii
- Representation Theory: a First Course, Fulton, Harris
- Conceptual Mathematics, Lawvere, Schanuel
- Concrete Mathematics: a Foundation for Computer Science, Graham, Knuth, Patashnik
I haven’t looked at 2 or 4 very closely yet, but so far I find 1, 3, and 5 to be among the best written textbooks I have ever read. Sipser’s book, in particular, strikes me as having found a perfect balance between brevity and clarity. His tone is conversational but finely polished, and I rather like his habit of summarizing the basic strategy of a proof before actually writing it down. Generally I am finding the book an absolute pleasure to read, which I can’t say for most of the math textbooks I’ve seen. You will likely see me blogging a little about languages and automata once I finish up my current series (right now I’m stuck on what should be a trivial proof).
Why don’t more mathematicians write like Sipser?
Annoying Precision 
I read Sipser as a high school student, and it’s still up there with Knuth for readability.
I find Sipser’s book ugly.
I like Papdimitriou/Lewis way better.
I read Sipser as an undergrad, and I liked it a lot too. However, I have been on-and-off trying to read Fulton-Harris for the last 2 years, and I always get frustrated with how wordy it is. Sometimes you just want them to get to the point. While I don’t like books that are too rigidly definition-proposition-proof, I guess I don’t like books that are too “conversational” either.
Perhaps Sipser’s book is a good balance. As I recall, it gives intuitive “proofs” and actual proofs of every(?) theorem.
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i especially liked the wonderful treatment of euler-maclaurin in gkp’s book. was a very pleasant surprise.
I definitely agree about Sipser–that was the first time I could find something on theoretical computer science that struck a balance with being both understandable and interesting.
The “proof ideas” are quite useful. I wish more textbooks did them.
The “easy” answer is that with basically every science besides math (it is debatable whether mathematics is a science at all, but I’m not going there), the practitioners are able to talk to others about their own work (if you want to be more acidic, you can also replace “are able to” with “want to”). Thus, they at least indirectly practice communication skills.
Luckily, we have some exceptions, even in more “pure” mathematics. Halmos is a delight, for example. Harris, even – I think the only reason I don’t appreciate him much is because I have very shaky foundations relating to the stuff he does, and I feel if I had a more robust background his intuition-based teaching would be great. If you want a hidden gem about algebraic topology, look for “Galois’s Dream” published by Birkhauser. Incidentally, you’ll find this conversational style in many texts written by Japanese and Russian mathematicians, hinting me maybe Americans just take themselves too seriously sometimes, which is ironically the opposite you’d expect out of most cultural stereotypes?
-YZ
I’m tempted to put most of the blame on Bourbaki, but I don’t know how fair an accusation that would be.
I am curious to hear how you find number 2). Also, number 4) is interesting, but I found reading Maclane’s “CFTWM” and having some conversations with professors about things, provide a better introduction. I would like to hear about your experience with number 4)
Definitively. You can read Sipser like a novel and still understand it. Awesome =)