It is a hard a question to pick one book to represent my favourite book. This challenge was set by person who started to organise a group of people to read mathematics books in a book club of sorts. What follows is what I wrote for them on that subject.
I’m going to exclude standard text books as they are used to practice the ideas we are playing with.
I’m going to exclude history of mathematics books, while I read them there are parts of history where I get excited, usually by thinking about how the person being discussed got to the point where they did whatever thing it was that is documented. For instance, Ian Stewart’s very nice ‘Taming the infinite the story of mathematics’, starts with the wonderful line, ‘Mathematics did not spring into being fully formed. It grew from the cumulative efforts of many people, from many cultures, who spoke many languages.’ Knowing this means that I can’t know how or why that happened to be, I can, I suppose, guess but I’ll not be certain without evidence, and then at this remove that evidence is not viewed as it was at the time, we can’t know what went on in the Babylonians minds. Frank Sweetz wrote a book titled Capitalism and Arithmetic, the new math of the 15th Century. We also know that people approximately counted seasons and years from the Mayans to the Celts through the legacy of stones in various trigonometric designs. So I will abandon these books for their distance of the journey.
This thinking has me reflecting on three of my favorite books, in the series about people, in no particular order:
Siobhan Roberts excellent book on John Conway, Genius At Play. Levels of Infinity by Hermann Weyl a collection of his essays including one on Emmy Noether, which is reason alone enough to read the book.
Lastly in a category all on it’s own Lymm Gladwell’s excellent Mathematics and Art, which if you want to characterise it you might say it is a coffee table book, it is also a nice history of mathematics but not in the style of the general history books which tend to be very much aimed at course work. It is beautifully illustrated and just a joy to dip into now and again.
This however leaves whole subsections of mathematics alone, it causes me some grief not to mention the Princeton Companion to Mathematics, edited by Timothy Gowers, or Mathematics From The Birth of Numbers, by Jan Gullberg both of which are tomes and well worth the investment. In this short survey of my shelves I’ve left out a few brilliant things, anything by the popularising group Keith Devlin, Ian Stewart, Marcus Du Sautoy.
Now comes the reason I can can’t name a single book, there is a series that takes a bit of work at times, be that as it may, you can skip the hard parts now and again and still get significant gain from working though this little set of books. It is called The Best Writing in Mathematics and it has been an annual release since 2010 edited by Pitici they are a joy in and of themselves and cover a broad swathe of the subject.
With all that in mind I think you might see why I have problems picking a single book.