One of my favourite books is
The Fascination of Groups, by F.$~$J.$~$Budden.
This is certainly no ordinary introduction to group theory. Here's a sample from the preface, to give you a taste of the writing style:
It takes 545 pages to cover what would be completed in most text-books in one to two hundred pages. But that is precisely its raison d'etre - to be expansive, to examine in detail with care and thoroughness, to pause - to savour the delights of the countryside in a leisurely country stroll with ample time to study the wild life, rather than to plunge from definition to theorem to corollary to next theorem in the feverish haste of a cross-country run.
And then later in the same paragraph, an explanation of the contents:
The objective is to provide a wealth of illustration and examples of situations in which groups may be found and to examine their properties in detail, and the development of the elementary theory in the light of these widely ranged examples.
As promised, while the book does also work as a textbook of-sorts, giving good explanations of the definitions and theorems, and including exercises for every chapter (some of which are decidedly nontrivial, and are sometimes inserted before the necessary material to answer the question is covered, just to get the reader thinking about the topic), the true value of the book lies in its extensive collections of examples of groups, and the book is positively overflowing with illustrations and Cayley tables, all neatly organized into the relevant chapters. To top it all off, towards the end are dedicated chapters on the applications of group theory to music, campanology, geometry and patterns (in the sense of wallpaper patterns).
The question “What am I doing?” haunts many creative people, researchers, and teachers. Mathematics, poetry, and philosophy can look from the outside sometimes as balleten pointe, and at other times as the flight of the bumblebee. Reuben Hersh looks at mathematics from the inside; he collects his papers written over several decades, their edited versions, and new chapters in his bookExperiencing Mathematics, which is practical, philosophical, and in some places as intensely personal as Swann's madeleine.
—Yuri Manin, Max Planck Institute, Bonn, Germany
What happens when mid-career a mathematician unexpectedly becomes philosophical? These lively and eloquent essays address the questions that arise from a crisis of reflectiveness: What is a mathematical proof and why does it come after, not before, mathematical revelation? Can mathematics be both real and a human artifact? Do mathematicians produce eternal truths, or are the judgments of the mathematical community quasi-empirical and historically framed? How can we be sure that an infinite series that seems to converge really does converge?
This collection of essays by Reuben Hersh makes an important contribution. His lively and eloquent essays bring the reality of mathematical research to the page. He argues that the search for foundations is misleading, and that philosophers should shift from focusing narrowly on the deductive structure of proof, to tracing the broader forms of quasi-empirical reasoning that star the history of mathematics, as well as examining the nature of mathematical communities and how and why their collective judgments evolve from one generation to the next. If these questions keep you up at night, then you should read this book. And if they don't, then you should read this book anyway, because afterwards, they will!
—Emily Grosholz, Department of Philosophy, Penn State, Pennsylvania, USA
Most mathematicians, when asked about the nature and meaning of mathematics, vacillate between the two unrealistic poles of Platonism and formalism. By looking carefully at what mathematicians really do when they are doing mathematics, Reuben Hersh offers an escape from this trap. This book of selected articles and essays provides an honest, coherent, and clearly understandable account of mathematicians' proof as it really is, and of the existence and reality of mathematical entities. It follows in the footsteps of Poincaré, Hadamard, and Polya. The pragmatism of John Dewey is a better fit for mathematical practice than the dominant “analytic philosophy”. Dialogue, satire, and fantasy enliven the philosophical and methodological analysis.
Reuben Hersh has written extensively on mathematics, often from the point of view of a philosopher of science. His book with Philip Davis, The Mathematical Experience, won the National Book Award in science. Hersh is emeritus professor of mathematics at the University of New Mexico.