Unconventional Essays On The Nature Of Mathematics

This book comes from the Internet. Browsing the Web, I stumbled on philosophers, cognitive scientists, sociologists, computer scientists, even mathematicians!—saying original, provocative things about mathematics. And many of these people had probably never heard of each other! So I have collected them here. This way, they can read each other’s work. I also bring back a few provocative oldies that deserve publicity. The authors are philosophers, mathematicians, a cognitive scientist, an anthropologist, a computer scientist, and a couple of sociologists. (Among the mathematicians are two Fields Prize winners and two Steele Prize w- ners. ) None are historians, I regret to say, but there are two historically o- ented articles. These essays don’t share any common program or ideology. The standard for admission was: Nothing boring! Nothing trite, nothing tr- ial! Every essay is challenging, thought-provoking, and original. Back in the 1970s when I started writing about mathematics (instead of just doing mathematics), I had to complain about the literature. Philosophy of science was already well into its modern revival (largely stimulated by the book of Thomas Kuhn). But philosophy of mathematics still seemed to be mostly foundationist ping-pong, in the ancient style of Rudolf Carnap or Willard Van Ormond Quine. The great exception was Proofs and Refutations by Imre Lakatos. But that exciting book was still virtually unknown and unread, by either mathematicians or philosophers. (I wrote an article en- tled “Introducing Imre Lakatos” in the Mathematical Intelligencer in 1978.

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).

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