Five books
Topics:
books, maths
If you aren’t familiar with the website Five Books, I can highly recommend it. Twice a week, an interviewee selects five books on a theme (usually in their area of expertise) and explains their choices. It’s a great place to get new reading ideas. Their archive has 1000 interviews; it’s likely somebody you admire has had an interview there. Nobody particularly admires me (as far as I know), but I do have some knowledge of popular maths writing. I’m often heard complaining that most pop maths books aren’t very good at actually communicating mathematics. Too often they’re more about making the reader say “wow! maths is crazy!” or (worse) “wow! mathematicians are crazy / must be really smart!”. To remedy that, here are five (okay, six) accessible books that are mathematically nourishing.
Concepts of Modern Mathematics, Ian Stewart.
The book I wish somebody had given me before I went to university. Ian Stewart’s first popular book, it is more mathematically meaty than his later style, but still requires no more than a bit of Alevel maths and some willing to think hard every now and again. Each chapter describes the motivation for and basic results of some area of modern (pure) maths. It’s an underappreciated classic, and now available in a bargain price Dover edition.
Gödel’s Proof, Ernest Nagel & James Newman.
Gödel’s incompleteness theorems are arguably the most important mathematical results of the 20th century. This thin book doesn’t give a full technical proof, but aims for something like a ‘seminar understanding’ of the theorems: the background, why they are important, and the main ideas of the proof. No tecnical knowledge is assumed and the authors spend a good amount of time introducing the reader to ideas like formal languages and consistency of theories. This book was a major inspiration for Douglas Hofstadter’s celebrated Gödel, Escher, Bach^{1}
How Not to Be Wrong, Jordan Ellenberg.
The most straightforwardly ‘pop’ book on this list, How not to be Wrong developed out of the author’s regular ‘Do the Math’ columns for Slate. Ellenberg is not concerned with explaining mathematics so much as explaining how mathematical ways of thinking can help us make better judgements. Some of this is about correctly interpreting statistics and so on, but some of it is just pure mathematical insight. For example, why should America raise taxes to have a society more like Sweden’s when Sweden is trying to lower its taxes towards a society more like America’s? Answer: because, even in a simplified model, ‘quality of society’ doesn’t necessarily scale linearly with rate of tax paid, both America and Sweden may be aiming for a local maximum between their two tax rates.
The Pleasures of Counting / Calculus for the Ambitious, Tom Körner.^{2}
Two for the price of one, i wish I’d been given these as a teenager too. Tom Körner writes beautiful, eccentric books full of bad jokes and great mathematics. The Pleasures of Counting is a large book of short mathematical vignettes that try to explain what mathematicians think about (and how). The more recent Calculus for the Ambitious is a short first course in differential and integral calculus for those who want to understand more than just some rote rules to follow to pass your A level exams. It’s informal and will reward the motivate reader It would also serve as good preparation for a rigorous course in analysis.
The Shape of Space, Jeffrey Weeks.
An introduction to nonEuclidean space, topology, and differential geometry for the lay reader. If you’ve read Flatland and want to really get into some of the ideas behind it, without going to university for maths, this is the place to go. It really succeeds in communicating that maths is about ideas more than about formulas or proofs: it manages to communicate a lot of highlevel ideas through diagrams and thought experiments, almost never diving down into calculations. I once gave an informal lecture about topology to first year undergraduates where I nicked bits of it wholesale. If that’s not a recommendation, I don’t know what is.

I left GEB off of this list because it’s on everybody’s best maths books list; it’s a fine book though, and if you have any interest in maths or computation, and haven’t read it, you should. ↩︎

I should note that I work for the publisher of these two books. They both predate me though, so I’m not shilling anything here, honest! ↩︎