Customer Reviews

Mathematics and Logic

5 star:		(6)
4 star:		(1)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

The book shows what the power of mathematics is, how it changes, and how it expands to new areas. Unlike books that aim to popularize math, the book does not pontificate a mystic view of the meaning of mathematics; rather, it gives a sober perspective of what has happened in mathematics and what can be expected of the field.

The book requires somewhat serious mathematical thinking.

A great strength of the book is the diverse mathematical concepts that it presents: homology groups, group theory, Turing machines, undecidability, Monte Carlo method. In a compact book, you learn a little about some important ideas in advanced mathematics.

Important!! This book is not written to popularize its subject matter, so it is different and definitely less entertaining than most popular science books.

The audience for ths book is people with background in mathematics. They teach many branches briefly with their examples in Chapter 1, but the presentation is at a fairly sophisticated level.

I found the historical and philosophical remarks very valuable, coming from authorities like these two men. Examples (slightly edited by me):

The great analysts of the 18th and 19th centuries (e.g., Newton, Leibniz, Bernoulli, Euler, Lagrange ...) had an almost unerring instinct for presenting valid results and plausible proofs without a firm basis in formal systems and without strict adherence to standards of logical rigor... Mathematical intuition in the hands of people of genius has such a clarity and unity that it anticipates special formalisms.

Mathematics is a science; it is also an art. The criteria of judgment in mathematics are always aesthetic, at least in part... One looks for `usefulness,' for `interest,' and also for `beauty.' Beauty is subjective, yet it is surprising that there is usually considerable agreement among mathematicians concerning aesthetic values.

It is a distinctive feature of mathematics that it can operate effectively and efficiently without defining its objects. Points, straight lines and planes are not defined... One need not know what things are so long as one knows what statements about them one is allowed to make [the axioms]... Other statements involving these undefined words can then be deduced by logic alone. This permits geometry to be taught to a blind man and even to a computer!

With new mathematics growing wild in the trees in the new century, one should review those things that have given modern mathematics direction and flavor. This classic little book sits along side of Sawyer's " Prelude to Mathematics" as the go to books for basic understanding. To these books I have added an unlikely candidate " Elliptical Curves" by McKean and Moll. If one has these three books, he will have a crack in the dam of mathematics...

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.

Kac and Ulam were both strong mathematicians; this encyclopedia article which they coauthored provides a nice counterweight to the answers-without-questions view of math which can result from a little indifferent coursework.

Easy and lively reading, written for everyone.

This is a wonderful book by two superstars and it's a miracle that it got written. It was commissioned by the Encyclopedia Britannica for their 200th anniversary. It explains very simply some of the most important examples of the turning of mathematical problems into algebraic or combinatorial ones, and then solving them. Groups are used to tell whether two braids are equal or not, Sperner's combinatorial lemma on triangles is used to prove Brouwer's fixed point theorem, and it gives a method of computing the fixed point too!

I love this book so much that I took the trouble to make up a list of corrections and additions. I didn't find any errors. It's just that some new results have been proved since 1968. The list is quite small and I am trying to find a home for it. Here I will just say that Hilbert's tenth problem was solved in 1970 and Fermat's last theorem was proved in 1995, but the other major open problems they mention are still open. For example, it is still not known whether Euler's constant C is irrational.

Just read the editorial review (Book News). It's all true (except for the publishing date, which is 1968, not 1986...)