I recently violated my Kindle-only rule and purchased a dead-tree book. And not just any book, but a huge book: The Princeton Companion to Mathematics, which clocks in at just over 1000 pages and 6.5 pounds. I’m not a mathematician, so I can only remark on the book from the perspective of a self-learner, but I’m really enjoying it so far. The book is a very comprehensive overview of the different areas of study in the field of mathematics, the history, theory, and mathematicians behind each area of study, and so forth. There’s no problems in the book, so it’s not a good tool for learning how to do mathematics, but if you’re interested in the theoretical side of pure mathematics and you want to get a very broad and comprehensive overview of the field, this is probably the book for you.
While accessible in the sense that it’s not aimed at the professional mathematician, I’ve found that the book is very challenging in some areas. I’m going through at a slow pace, perhaps 10-15 pages per day, and really trying to grok the concepts, but it’s tough. I took calculus, linear algebra, and statistics in college, but there’s still a lot of deep concepts here that I haven’t been exposed to. More than that, the concepts are presented from a much more theoretical basis than they were in college, which tended more towards applied math than pure math.
At any rate, if you enjoy math and you’re considering diving into some advanced study but you’re not sure where to start, I recommend The Princeton Companion to Mathematics. It’s challenging, but rewarding, and if nothing else, you’ll probably get a sense for those areas of math that you don’t enjoy.
While I’m on the subject, I would absolutely love to find books like this for other fields, like engineering, physics, biology, etc. If you know of any really comprehensive overview books for other fields of science and engineering, please share in the comments.