A Companion for Mathematics

This is a review of the Princeton Companion to Mathematics. This book was forwarded to me by a friend. It is about a thousand pages long, so not a quick read, and in fact perhaps a book to be dipped into only occasionally (the book claims the original aim was to provide bedtime reading).

The writing is highly accessible. Most of the material should be clear to high school graduates, some of it might require a college education. The book is divided into eight chapters. The main features I liked are as follows:

1. Introduction: There is a lovely introduction which deals with the language, grammar, and goals of mathematics. I have not seen this kind of material discussed in too many other places. For example, here I learnt that math is broadly divided into the study of algebra (symbolic manipulation), geometry (shapes) and analysis (limiting processes). Of course, they have large overlaps also, such as the work of Descartes, which reduces geometry to algebra.

   
   

2. Origins: This part deals with the historical development of the subject, emphasizing the notion of proof, algorithms and the upheavals in its foundations (refer Godel).

   
   

3. Concepts: About a 100 specific mathematical concepts are described, such as the Fourier transform, groups of different stripes, categories, algebras, etc.

   
   

4. Branches: 26 branches of mathematics are enumerated, for example harmonic analysis, set theory, algebraic geometry. A table of the major algorithms (Gaussian elimination, simplex, Monte Carlo, etc.) and their originators is provided.

   
   

5. The stars: Short biographies of 96 great mathematicians are supplied. They start from traceable antiquity and end at around those deceased before 1987, so no contemporary mathematicians are included (e.g. I looked for and did not find Maryam Mirzakhani).

   
   

   A large number of juicy tidbits are scattered throughout this part - e.g. Cayley wrote a paper as an undergraduate which has since set the notation for determinants. Another one: Dirichlet went to study in Paris because he found the level of math instruction in German universities to be too low.

   
   

6. Implications: There is a substantial chapter on the applications of mathematics, in other disciplines like chemistry, biology, finance, art and cryptography. In fact the book acknowledges that there is no clear distinction between pure and applied math.

   
   

7. Outlook: The last chapter talks about the future of math, with a nice section that gives advice to a young mathematician (from heavyweights such as Sir Michael Atiyah).

Summary

This is a monumental work, designed to present the width (and some depth) of mathematics. I found it to be a useful introduction to areas which I am not very familiar with (almost everything). It was not so useful for fields that I am reasonably acquainted with (very few). I keep the pdf on my desktop and dip into it now and then. In that sense it is like the Arabian Nights - it's not really possible, or perhaps even desirable, to read it in a single sequence. But in the same sense, it is a treasure trove of mathematical gems. Most highly recommended.