Proof and the Art of Mathematics

I was really blown away by this book. Besides the quality of the diagrams and illustrations, which go far beyond any popular mathematics book I've ever purchased, the variety and depth of examples of proof techniques is really impressive. All my old favorites (like tiling a checkerboard and the mxn chocolate bar problem), and many that I've never seen before (a proof that the number of different infinities is not any of the infinities!) are beautifully illustrated and described in accessible but rigorous terms.My 7 year old daughter looks over my shoulder when I'm reading this book and asks lots of curious questions. This book has been a great introduction to the beauty and wonder of mathematics for this budding mathematician!

The author demonstrates his ability to write proofs and his inability to explain how to write them. I cannot believe that anyone would find this helpful.

This well-written book by Professor Hamkins has a variety of interesting elementary mathematics. And has many nuggets of wisdom (e.g. the number 1 and prime factorization, the use/mention distinction with fractions and rational numbers, and vacuous truth). It's an excellent choice for aspiring mathematicians.

Excellent purchase, especially like the "mathematical habits" sections, which will surely cause a moment of reflection and will serve as conversation starters.

Even if you already have substantial training in mathematics, this book is worth reading just for the elegance of the proofs.

The author of the book is a world-class set theorist who, probably more than anyone else, has promoted a multiverse realism in set theory. This argument, somewhat crudely, advances the point that the independence results obtained from the method of forcing, suggest a multiplicity of set theory universes, similar to that of non-Euclidian geometries. In addition he has also advanced the idea that the phenomenon of independence can even be found at the level of arithmetic. I think he is wrong and my particular philosophical inch is better satisfied by the work of Hugh Woodin. However the point remains that Joel David Hamkins is an exceptional, philosophically astute, mathematician whose work I have enjoyed immensely. So I was disappointed when I discovered severe errors early in the book. I even e-mailed the author and have received no reply."Dear SirOn page 3 you define the natural numbers as 0, 1, 2, 3,…. In doing so by distinguishing ‘natural number’ from 1, 2, 3,… a definition, which is common in many mathematical texts, especially undergraduate textbooks in elementary number theory. In addition, you also explain that subtraction between two natural numbers does not always result in another natural number, and use this as a justification for the integers. Of course, we could define subtraction in the natural numbers as ‘cut-off subtraction’ which would allow subtraction between any two natural numbers; however, you make no mention of this mathematical operation in the text. On page 10 you give Theorem 6. For any natural number n, the number n**2 – n is even. Clearly this is true and defined in the natural numbers as n**2 is always equal or greater than n.The problem is in proof #2 (by High-school algebra) in which you prove this theorem by rewriting n**2 – n as n(n-1). (n-1) is not always a natural number, take n=0, as your previous comments on page 3 point out. Quite naturally some may consider this a minor quibble; however, as the text has made the definitions of 'natural number' and 'subtraction', although informal, at least conceptually, clear. The proof as it stands fails in the natural numbers."This does not seem to be a misprint and this is my point as he makes a similar mistake on page 12. In short very disappointed and so cannot recommend the book, I will not continue with the book and will be returning my copy.

Proof and the Art of Mathematics is a wonderful book for multiple reasons: first of all, it really does what it says on the cover, namely to introduce aspiring mathematicians to the art of writing proofs.This is achieved through a combination of: being walked through various elegant proofs selected by Prof. Hamkins, being asked to critique mistaken proofs which contain some subtle errors, but most importantly by being provided with many carefully chosen exercises so that you can practise what you learn--any craft requires some personal effort. (At the very end of the book, the author provides answers to some selected exercises where the you can compare your own proof attempts with model proofs.)One big plus for me is the variety of mathematical areas covered: you'll find sections on topics such as the mathematics of the infinite, number theory, combinatorics, real analysis and many more (a total of 15 topics). Each section is sprinkled with bits of mathematical wisdom which are bound to improve the mathematical skills of anyone willing to attempt the exercises with them in mind. (The book also contains some very helpful coloured illustrations!)--Also, I've had a positive experience with the delivery: I received this book on my doorstep only one day after its release, despite not living in the UK!

The author is both a philosopher and a practising mathematician. He not only gives the classic proofs of some of the most beautiful theorems but also considers several proofs of the same theorem to gain extra insights.

Great book! I recommend it for anyone who is interested in math. Whether you are a beginner or advanced, you will surely find something in the book that catches your attention.