100 books like Man Must Measure

David Hilbert was the most important mathematician at the dawn of the 20th century. In 1900, he gave the mathematical community its homework for the next 100 years setting out the list of open problems that had to be solved by 2000. While to the rest of the mathematicians, he may have appeared as their professor, he was also the class clown. As colorful and funny as he was brilliant, you cannot but come away loving this great mathematical genius.

This classic rumination on the nature of mathematics is perhaps the most famous book ever written by a mathematician.

I’ve read it several times, and each time, it brings up another facet of the subject in some new light. Portions of what Hardy says ring eerily true, and seem to be part of the identity, the very DNA, that all mathematicians must surely share. Other parts seem so alien to be almost repugnant – such as his contention that most mathematics we classify as useful is ugly and inelegant.

I’d love to argue face-to-face with Hardy about this highly opinionated work of his. In the end, I suspect we’d find lots of common ground, since intrinsically, we both love mathematics so much.

The Scientific American columns of Martin Gardner ran for 24 years and were read by amateurs, semi-amateurs, professionals, and major mathematicians (Conway, Knuth, Diaconis...). It was the interaction with this audience (recorded in addenda) which gave these essays their special quality and will give the interested outsider a real feel for what interests mathematicians. The collected columns are being reissued by the AMA and CUP but my view that anything by Martin Gardner is worth reading is reflected in my personal library.

More mathematicians than you might expect have had eventful lives (and most of the others are grateful that they did not). Fourier, one of my mathematical heroes was twice in danger of the guillotine, played a substantial role in Napoleon’s Egyptian campaign, is known as one of the Fathers of Egyptology, and was the first person to write about the ‘green-house effect’. To mathematicians and physicists this is all secondary to his work on heat conduction and, of course, the Fourier series. Herevel is an excellent guide to a remarkable life. (Herevel also had a non-standard career — the ‘Herevel tip’ was instrumental in early successes against Enigma.)

We love all of David Macauley’s books. He uses hand-drawn black-and-white illustrations to describe the enormous effort and complicated processes involved in building some of the most magnificent structures in the world, from cathedrals to castles to pyramids. Cathedral was his first, and we think the best. Although intended for young readers, there are many builders, engineers, and architects that find wonder in Macauley’s work. 

I laughed out loud as I joined this family of twelve children on cramped car trips, through childhood pranks, adolescent rebellion, and through the daily joys and growing pains of a loud and loving family. My dad is one of eleven children and reading this book reminds me of many of his stories about growing up. I especially related to the clatter of dinner time conversation centered around morse code or math games. We never did morse code but we’ve played plenty of math games and word riddles.

From a strictly philosophical perspective, the emergence of category theory as a unifying paradigm rivaling set theory is probably the most important development in mathematics in the last half-century.

But for philosophers without a lot of mathematical background, learning even its rudiments can be daunting. Among many introductory texts (Lawvere and Schanuel, Awodey, Riehl, Spivak), Cheng’s book stands out as perhaps the friendliest and most accessible.

She does not forego rigor, but she isn’t afraid to put aside precise formalism when necessary for intuition and clearer understanding. Her book takes the reader from mathematical beginnings through category theory’s core constructions to glimpses of higher-order categories (one of Cheng’s areas of expertise).

A mathematically novice philosopher who wants to understand the basics of category theory couldn’t do better.

Far too many math books are written in a style so terse and ungenerous that all but the most mathematically gifted readers hardly have a fair chance of understanding.

On the other hand, the discursive style of much philosophy of mathematics gains readability at the expense of formal rigor. Button and Walsh strike the perfect balance in this exceptionally rich introduction to model theory from a distinctively philosophical perspective.

There’s no getting around the fact that the mathematics of model theory is hard going. But this book works through all the relevant proofs in clear and detailed terms (no lazy “we leave this as an exercise for the reader”), and the authors are always careful to motivate each section with well-chosen philosophical concerns right up front.

An Everest, but worth it.

Polya was a great mathematician who knew what counted (after all, he made major contributions to combinatorics, the mathematics of counting). He thought hard about what he was doing when working on problems in mathematics, developing a mental process that lead to creative breakthroughs and solutions. Polya’s problem-solving method is broadly applicable to domains other than mathematics, and this book features many nice puzzles to improve your thinking.

Algorithm design is challenging because it often requires flashes of sudden insight which seem to come out of the blue. But there is a way of thinking about problems that make such flashes more likely to happen. I try to teach this thought process in my books, but Polya got there first.

Many undergraduate mathematics books – even those aimed at new students – are dense, technical, and difficult to read at any sort of speed. This is a natural feature of books in a deductive science, but it can be very discouraging, even for dedicated students. Houston’s book covers many ideas useful at the transition to proof-based mathematics, and he has worked extensively and attentively with students at that stage. Consequently, his book maintains high mathematical integrity and has lots of useful exercises while also being an unusually friendly read.