The Great Math War is a book about the ‘Foundation Crisis’ (Grundlagenkrise) which occurred in Mathematics from (roughly) 1883 to 1938. During that time, the philosophies of Logicism, Intuitionism and Formalism all vied to provide an account for what it is that grounds Mathematics, and thus accounts for the certainty which is characteristic of mathematical proofs.

Bertrand Russell (and Alfred North Whitehead) promoted Logicism, claiming that Mathematics can be reduced to logic. To justify that approach Russell developed his ‘theory of types’ to solve the paradoxes confronting set theory.

But his solution came at the cost of importing potentially ‘ad hoc’ principles into the logic which mathematics was supposed to be reduced to.

David Hilbert’s Formalism argued that Mathematics was just a set of rules, like those for playing a game. Games do not need justifying outside of their rules. If people enjoy the game and find it useful, then that is the foundational justification for playing the game.

But the problem which that approach raises is that it completely ignores the fact that physical nature seems to play the same (mathematics) game as mathematicians choose to play. So how can Mathematics be nothing more than a formal set of rules?

L. E. J. Brouwer’s Intuitionism took an entirely different approach. Putting logic and game rules aside, he (intuitively) applied a ‘constructivist’ methodology which insisted that mathematicians could only use concepts and proofs which they could construct from specific procedures (or from previously constructed concepts or procedures).

That approach does indeed provide a complete foundational justification for mathematics, but it only achieves that goal at the heavy cost of discarding significant portions of what Mathematicians previously counted as mathematics.

The ‘great math war’ was arguably brought to a close when Godel presented his argument for incompleteness. If there are truths which cannot be proven (or disproven) then that potentially undermines the very possibility of justifying a specific set of foundations for the entirety of mathematics.

This story of the war between Logicism, Formalism and Intuitionism is well-researched and skillfully-narrated, with some elegant turns of phrase. The book does particularly well to bring to life an abstract set of issues and to show why the issues mattered so much to those arguing about them.

But I think the book was also a little over-written and veering towards irrelevance in places (eg Boer war in chapter 4, Nazis in chapter 17). With a firmer editorial hand the book could have been significantly shorter, while still providing a good summary of the issues between Logicism, Formalism and Intuitionism.

Textually the book read well, with an engaging prose style. Around 20% of the book was committed to notes and follow-up materials.

Overall, this is a book which anyone interested in mathematics, science or philosophy could enjoy, and it is accessible to readers of any background. I found this a hard book to rate because I would like to give 5 to the parts of the book which were relevant and focused on logicism, formalism and intuitionism. But I found the irrelevant sections increasingly distracting to the point of wanting to give a 1 for those bits. So, I think that results in a 3.

(This is an honest review of a pre-publication, free, digital ARC version of the text. Please note that the final published version of a text can differ from the ARC version).

I enjoyed this historical read. I came to this only knowing the barest of terms but not knowing the history or the outcome. What I learned? The purest facts of the “war” could be described in a bare few pages. And I think that was one of the things about which that I felt most disappointed. The history was fascinating on its own, but I kept being drawn on by the question “How will this resolves? Who ‘wins’?” And then there’s a paragraph or two about Gödel . . . and then we’re done. It felt a big letdown. I felt that, interesting as it was, the history was put in because the actual “plot” is so simple and abrupt, and we needed more to tell. I realize the history is there to explain the developments, but why then so many words on Russell’s womanizing? I also felt the tone was uneven. Now we are going to be scholarly, now we’re going to be hip and current, “look, here’s a funny quip.”. There were multiple instances of really unnecessary repetitions, using several synonyms to describe the same thing. It broke the reading attention. Overall the text needs tightening, clearing out unnecessary repetitions, spending less time on irrelevant biographies.