Is there a mathematical way of thinking? In my experience, there is. Some naturally think like one; some are welltrained to; and, some learn it the hard (and timetested) way of trailanderror. If you belong to the last category, here is a book that would grease your path and ease your journey — I am referring to Kevin Houston’s How to think like a mathematician.
Houston, in his preface to the book, describes the aim of the book as follows:
The aim of this book is to divulge the secrets of how a mathematician actually thinks. As I went through my mathematical career, there were many instance when I thought, ‘I wish someone had told me that earlier.’ This is a collection of such advice. Well, I hope it is more than such a collection. I wish to present an attitude — a way of thinking and doing mathematics that works — not just a collection of techniques (which I will present as well!)
I am glad to report that Houston succeeds admirably in his presentation of the attitude — at least as far as a nonmathematician like me (albeit with a keen interest in reading, understanding, using and to some extent even writing mathematics) can tell.
From the preface, I also understand that most of the methods described in the book had been tried and tested in the classrooms.
The book consists of six parts, namely,

[1] Study skills for mathematicians

[2] How to think logically

[3] Definitions, theorem and proofs

[4] Techniques of proof

[5] Mathematics that all good mathematicians need

[6] Closing remarks
Each part, in turn, consists of four to eight chapters; for example, the first section on study skills consists of one chapter on sets and functions (mainly to serve as a model mathematical text — to be used later for discussions), one chapter on mathematical reading, two on writing, and one on problem solving.
The chapters are full of exercises (both solved and unsolved); each of them also begin with a nice quote and end with a nice, short summary.
There is also a very useful appendix which gives the recipes for proving different types of mathematical statements.
Running into nearly 260 pages or so, and flowing nicely without any rough patches, here is a book that I would have recommended strongly — except for one small glitch; the book costs nearly nineteen pounds (nearly Rs. 1500/), which makes it a book for libraries and not for individuals — I do not see many undergraduates who can afford such a costly book, especially when it is not a prescribed textbook, or, that they would buy even if they could afford it, considering that it is not even a hardbound edition.
I hope Cambridge brings out a lowpriced edition soon. Till then, having exhausted my book grant for the year, I am going to be satisfied with a recommendation of the book to our library and with the downloaded nearfinal version of the book (available on Houston’s homepage as a pdf) for a personal copy.
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