****** - Verified Buyer
4.5
when learning mathematics or any subject in general, it is important to draw some connections between what you're currently studying with what's out there. these bridges between different areas give you a sense of the greater beauty at work as you start to see the whole picture. kac and ulam have done this masterfully with their gem of a book, "mathematics and logic." in so doing, the authors convey to the reader some idea of why mathematicians think the way they do.imagine yourself in a room with two seasoned mathematicians. a conversation is struck and the two mathematicians start talking about some topic that they find interesting. this topic has some connection to another topic, so they begin talking about that. now, a third topic shows up on the scene, so the mathematicians start expounding a bit on this newly arrived animal, only to get to yet another animal on the horizon. and so forth. that's how things go for the first hundred pages or so as the zoo grows! in general, such a style could quickly become an incoherent rambling mess, but this is not the case here. the transitions are not too abrupt and the reader does get a sense of why things pop up when they do. the book even closes with some chapters explicitly laying out the common threads that have woven these selected topics together. very nice.the topics covered include the usual suspects that often show up in popular expository math books, subjects like elementary number theory, combinatorics, basic group theory, probability, gödel's incompleteness theorems, turing machines, special relativity, and so on. however, the authors also throw in some topics from left field such as braid theory, information theory, and homology. i was pleasantly surprised that homology was covered since whenever algebraic topology shows up outside of bona fide textbooks, it's usually homotopy that makes an appearance, not homology. kac and ulam make the effort to treat the harder to understand and arguably more useful of the two. that's noteworthy.unless you've already attained a good amount of so called "mathematical maturity," and/or have seen some of these topics before, then there's a very good chance that you're not going to understand everything in this book. the topics are vast and could easily take up an entire lifetime to really study. don't worry if you don't get something! read that part lightly or even skip it! read this book to have some idea of what's out there, read it for culture, and read it to understand the process of synthesis. the curious reader, regardless of background, should also be able to pick up some new pieces of math from this book and that should give the reader some avenues to explore later in greater depth.compared to the usual popular math books, this book is significantly harder to read. an upper level undergrad or a graduate student will get the most out of this book, although everyone will get something out of it. if you're a graduate student looking for some casual reading material for your subway/bus commute, but don't want the math dumbed down too much, then this book is perfect for you. note that struggle is a natural part of the mathematical learning process so anyone who's even remotely interested in this book should give it a shot. it's an affordable dover book so there's very little to lose!lastly, for those who enjoy this sort of coverage of a wide number of mathematical topics without compromising too much on the meat of the matter, i highly recommend "the princeton companion to mathematics," edited by timothy gowers. it deals with almost all areas of higher math at a sophisticated level, while sweeping away enough of the technical details to still be considered "casual" reading. "the princeton companion to mathematics" is "mathematics and logic" on steroids.
