What Defines a Good Math Book?

[battousai]

what makes a "good" math book?

i wonder what you guys feel makes a math book "good". for the most part the topics for different subfields are already standardized, with the same list of contents, theorems, proofs, and even exercises. yet, even newer books come out every year, and people still prefer certain ones over another.

obviously everyone has different opinions about this matter. but what makes you guys pick a specific book over the other ones?

 

[mathwonk]

one you can learn from. to enhance your learning, try to move up to books by people who are considered knowledgeable in the field, no matter how long that takes you.

 

[Sankaku]

I look for writing on several different levels:

Clarity/Rigour: Does the book present the proofs clearly and consistently? Some books vary quite a bit in how well they present the core material. Having all the results clearly follow each other is important. A conversational style can be good as long as it doesn't obscure the rigour.

Low-Level Motivation/Intuition: Does the book motivate the proofs and give intelligent examples? Rigour is important, but you need to know the insight behind the proofs and some intuition as to why they were framed that way.

High-Level Motivation/Intuition: Does the book give a good overview of how everything fits together? It is all good to motivate each proof individually, but unless you know the big picture, the topic can still feel like a patchwork of random results.

Lastly, there is an intangible sense of "insight" you get when reading certain authors. As Mathwonk likes to point out, leading mathematicians sometime provide an insight into the material that other authors might not quite reach.

Some examples (these are purely my own judgements):

Munkres' Topology is great for clarity and low-level motivation but fails with the high-level.
Lee's Topological Manifolds is great for high and low-level motivation but it occasionally seems to obscure to rigour.
Hungerford's Algebra is rigorous but provides very little intuition of any kind.
Jacobson's BA1 and BA2 seem to strike just the right balance (for me).

 

[homeomorphic]

Intuitive and well-motivated. Doesn't make things look like they came out of nowhere and gives you an idea of how the subject fits together, why it's interesting, why the theorems are true.

 

[QuarkCharmer]

I'd like to add brevity to the already great list.

There need not be any further explanation that one that would suffice. We don't need a 3 page history of the integral sign before approaching the topic. I also much prefer brevity when it comes to images. Diagrams are great, when they fit the situation, and naturally, are required in some cases. However, there is never a need for a "Welcome to chapter 25" two page layout featuring children scuba diving just because chapter 25 deals with hydrostatic pressure or whatever.

 

[gakushya]

I thought this was kind of funny: in Lee's Intro to Topological Manifolds on page 2 there is a diagram of space curves. Just a couple of squiggly little R³ lines. Just seems like a retarded waste of space to me. But so far as I've gotten in this book, there are too many diagrams, many just utterly pointless. And space is precious in textbooks.

Anyways, I have a lot of math books on my shelf, despite having only read about half of them. Some people like going to the mall and shopping for clothes and the like. I like buying science textbooks off Amazon. I don't even ever finish reading them, I just like books. Or maybe the idea of being a reader? Who knows. But I have favorite publishers in certain fields. I think the publishers somehow enforce, or rather encourage, a sort of pedagogical or expositional standard when deciding on who and what to publish. At least for the larger portion of their books.

Math/Phys: Springer
Programming: Wrox/O'rielly
Linguistics: Mit Press/ Cambridge
Phil:Oxford

Half my book shelf is yellow.

 

[autodidude]

I like a math book that isn't too verbose, one shows you enough so that you can deduce certain things yourself within a reasonable amount of time, but not so little that it's easy to make wrong assumptions. Useful insights that motivate and make the material interesting (like Marsden + Tromba's vector calculus book) is always nice. Also, a range of routine and challenging problems with solutions is a (usually) must if I'm using the book for self-study.

EDIT: ^ all that without sacrificing rigour. (it seems to me that some people think of super terse books when they think of a rigorous math text :p)

 

[Sankaku]

However, there is never a need for a "Welcome to chapter 25" two page layout featuring children scuba diving just because chapter 25 deals with hydrostatic pressure or whatever.

Yes, I would perhaps go out on a limb and say there has never been a good math book published with random "scuba-children" pictures scattered through it. In fact, pretty much all the classic textbooks are black & white. I can think of only two decent books on my shelf that are printed in more than two colours.