Review of The Great Math War: How Three Brilliant Minds Fought for the Foundations of Mathematics by Jason Socrates Bardi

When The Great Math War became available on NetGalley, you would not believe how quickly I windmill-slammed to request it. My university education is actually in mathematics (I hold an Honours Bachelor of Arts in Mathematics, a Bachelor of Education, and minors in English and philosophy). I wrote my honours thesis on ZFC set theory and the Banach–Tarski paradox. So I was already familiar with the overall ideas Jason Socrates Bardi presents here—yet I was not prepared for the wealth of biographical and historical background he also provides. What a treat of a book. I received an eARC in exchange for a review.

Put simply, this is a history book first, a philosophy book second, and a math book a distant third. There is very little actual mathematics in this book, so for anyone who is less well-versed in set theory or algebra, don’t be scared off! Indeed, I think Bardi largely succeeds at simplifying explanations of what these mathematicians are actually talking about to the point where a layperson will likely understand. Mostly, this book is a loose biography of Bertrand Russell, David Hilbert, and L.E.J. Brouwer. By tracing the course of their lives against the backdrop of the Boer War, the First World War, and the years leading up to the Second World War, Bardi examines the influences on these men that led them to develop, respectfully, logicism, formalism, and intuitionism. Honourable mention as well to Cantor, whose early work on set theory and the continuum hypothesis get a detailed treatment before Bardi turns to Russell and Whitehead’s contributions. Also, I appreciate how Bardi makes an effort to highlight the great women mathematicians—particularly Kovalevskaya and Noether—who deserve all the flowers.

I very much enjoyed this opportunity to learn more about the humans behind these ideas! Though I knew a lot about foundational theories already—and particularly Russell’s connections to set theory—I didn’t know much about Russell himself. I had no idea what a cad he was. At one point, he inveigles his longtime lover (Ottoline Morell, whose biography I now want to read) into hosting his new, hotter, younger American girlfriend in London, even though he admits he’s no longer interested in the new girl romantically but just wants to smash her. It felt like a Real Housewives episode, and I had to call up my bestie—who is decidedly not into math or philosophy in this way—to give her this tea. Likewise, I had only the vaguest idea of who Hilbert was, aside from his name on a bunch of theories and lemmas and proofs, and to be honest if I had heard of Brouwer it was literally only in connection to the name intuitionism.

Bardi excels at making this history come alive. When we learn about these mathematicians in the course of our mathematics education, we usually hear about them in isolation—names attached to theorems, as I said of Hilbert above. Bardi emphasizes the humanity of these so-called greats—and the history that shaped them. Living as they did at the beginning of the twentieth century, events like the First World War truly, deeply affected their outlook on life—and the mathematics they developed as a result. I can’t understate how Bardi helps us connect contemporary events with Russell, Hilbert, and Brouwer’s philosophies.

In the same way, Bardi treats the entire subject of the foundation of mathematics with reverence. He makes it sound as interesting as I believe it to be: here we are, only a century on from the war he chronicles within this book. We owe so much of modern mathematics to these mathematicians and those who took up their call to arms (regardless of for which school of thought). It’s so unfortunate that most people’s math education in school is so stunted and truncated that they never learn of the foundational math crisis let alone how its legacy has reverberated through the decades.

I will say that, at times, Bardi’s writing lingers or digresses to an annoying degree. He luxuriates in his own wordplay to the point of self-indulgence. This book took me nearly a month to read. Some of that is the result of other things going on in my life that gave me less time to read—but mostly, I suspect, it was Bardi’s style. The same qualities that help him bring these people to life also, paradoxically, put me to sleep! I kept shouting, “Get on with it!” Your mileage may vary here; maybe I was just being impatient. Nevertheless, The Great Math War is an example of the great need for editors: there isn’t much in this book that I would cut, yet there is much in this book that I would recast.

Style issues aside, I cannot fault this book for its informativeness, breadth, or depth. I wish Bardi had featured a few mathematicians who seem to be absent from this story—Fraenkel, who put the F in ZF set theory, and Skolem. Dedekind and Peano. I was waiting in deep anticipation for Gödel to show up near the end of the book, and indeed he does, yet with a curiously short time in the spotlight. Bardi depicts his incompleteness theorems as more a whimper than a bang to the ending of the Great Math War, whereas at the time when I was studying this, they felt to me as apocalyptic as I imagine they felt to those so heavily invested in proving their foundational theory consistent and complete. Yet maybe, as Bardi suggests, the fact that Hilbert, Brouwer, and Russell had long since passed the torch has something to do with that.

In any event, The Great Math War proved to be exactly the Kara catnip I hoped it would be. Lengthy and detailed, this is not a book for the faint of heart. However, it is far more history than math, for anyone more into the former than the latter. I quite appreciate Bardi putting this period in the history of mathematics into focus.