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The Great Math War: How Three Brilliant Minds Fought for the Foundations of Mathematics
A stirring account of the mathematicians who went looking for the bedrock philosophical foundations of their field and witnessed a house of cards collapse instead
“A fascinating romp through one of the most consequential conflicts of the early 20th century.”—Johnjoe McFadden, author of Life on the Edge
As the nineteenth century ended, mathematicians were celebrating a century of triumphs that—surprisingly—made clear how little they What is the nature of infinity? Is math free from self-contradiction? And what does math have to do with reality? This was the Foundational Crisis in mathematics.
In The Great Math War, Jason Socrates Bardi tells the story of three competing efforts by mathematicians to resolve it—and the firefight that ensued. Bertrand Russell thought we could achieve certainty if we treated math as an extension of logic. David Hilbert believed redemption lay in accepting mathematics as a formal game of arbitrary rules, no different from the moves and pieces in chess. And L. E. J. Brouwer argued math is entirely rooted in human intuition—and that math is not based on logic but rather logic is based on math. It was a bitter struggle, intellectually and personally, as the three vied to set the course for mathematics in the twentieth century.
Set against the backdrop of international warfare unfolding alongside it, The Great Math War brings the Foundational Crisis to radiant life—and shows how it indelibly shaped twentieth-century intellectual life.
“A fascinating romp through one of the most consequential conflicts of the early 20th century.”—Johnjoe McFadden, author of Life on the Edge
As the nineteenth century ended, mathematicians were celebrating a century of triumphs that—surprisingly—made clear how little they What is the nature of infinity? Is math free from self-contradiction? And what does math have to do with reality? This was the Foundational Crisis in mathematics.
In The Great Math War, Jason Socrates Bardi tells the story of three competing efforts by mathematicians to resolve it—and the firefight that ensued. Bertrand Russell thought we could achieve certainty if we treated math as an extension of logic. David Hilbert believed redemption lay in accepting mathematics as a formal game of arbitrary rules, no different from the moves and pieces in chess. And L. E. J. Brouwer argued math is entirely rooted in human intuition—and that math is not based on logic but rather logic is based on math. It was a bitter struggle, intellectually and personally, as the three vied to set the course for mathematics in the twentieth century.
Set against the backdrop of international warfare unfolding alongside it, The Great Math War brings the Foundational Crisis to radiant life—and shows how it indelibly shaped twentieth-century intellectual life.
416 pages, Hardcover
Published November 4, 2025
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Displaying 1 - 7 of 7 reviews
December 18, 2025
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.
Originally posted on Kara.Reviews.
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.
Originally posted on Kara.Reviews.
July 28, 2025
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).
November 26, 2025
weird but fun to read book.....I like his judgmental and sweeping tone and the general irreverance....overall it is light on mathematical detail though when he chooses to explain concepts and ideas he does it pithily well....good read if you are breezing thru the subject but definitely not a book for serious math students.
November 5, 2025
Quite frustrating.
October 19, 2025
The Great Math War: How Three Briliant Minds Fought for the Foundations of Mathematics is the story of how Bertrand Russell, David Hilbert, and L.E.J. Bruwer fought to solve The Foundational Crisis in mathematics in the early 1900’s.
This is the first non-fiction book that I have ever read that was in the intellectual realm, and I’m sort of torn on how much I enjoyed it. As a middle school math teacher, I found the actual math discussed in the book fascinating and enjoyable. I have already had discussions in my class based on some of the things that I read in this book. I did feel like there were many chapters in the book that could have been left out because they didn’t really pertain to the mathematical discussion. Chapters about the wars going on in the world at the time seemed to come out of nowhere to me, but perhaps this is completely normal and I’m just not used to it yet!
Overall, I enjoyed the mathematical discussions and this will not be my final non-fiction book that I will read. Thanks to Basic Books and NetGalley for access to the ARC in exchange for an honest review.
This is the first non-fiction book that I have ever read that was in the intellectual realm, and I’m sort of torn on how much I enjoyed it. As a middle school math teacher, I found the actual math discussed in the book fascinating and enjoyable. I have already had discussions in my class based on some of the things that I read in this book. I did feel like there were many chapters in the book that could have been left out because they didn’t really pertain to the mathematical discussion. Chapters about the wars going on in the world at the time seemed to come out of nowhere to me, but perhaps this is completely normal and I’m just not used to it yet!
Overall, I enjoyed the mathematical discussions and this will not be my final non-fiction book that I will read. Thanks to Basic Books and NetGalley for access to the ARC in exchange for an honest review.
December 18, 2025
Mathwars
Well, this was the most thoroughly enjoyable biography on math and figures in the mathematical field that I've come across. I only knew of Hilbert from the space that shares his namesake and had not heard of Brouwer (yes, quite a few historical figures come up when you take 2 years of calculus and diff equations in engineering undergrad, but I realize that this is child's play for anyone who has seriously studied math). Bardi not only completely fleshes out their 'beef' but does so in a fun, humorous and positively irreverent manner. I found the staccato language he used throughout as innovative and complementary to the story, but YMMV. It did feel like 2 separate works - one about Russell and then one about Hilbert and Brouwer; neither one was per se better than the other as I enjoyed them both but by the time it's ending I had already forgotten much of the Russell overview as there was little interplay between the 2 stories. But overall really enjoyed this one and look forward from future endeavours from the author.
Well, this was the most thoroughly enjoyable biography on math and figures in the mathematical field that I've come across. I only knew of Hilbert from the space that shares his namesake and had not heard of Brouwer (yes, quite a few historical figures come up when you take 2 years of calculus and diff equations in engineering undergrad, but I realize that this is child's play for anyone who has seriously studied math). Bardi not only completely fleshes out their 'beef' but does so in a fun, humorous and positively irreverent manner. I found the staccato language he used throughout as innovative and complementary to the story, but YMMV. It did feel like 2 separate works - one about Russell and then one about Hilbert and Brouwer; neither one was per se better than the other as I enjoyed them both but by the time it's ending I had already forgotten much of the Russell overview as there was little interplay between the 2 stories. But overall really enjoyed this one and look forward from future endeavours from the author.
November 12, 2025
Math And History Nerds Will Love This. Solid History For Everyone Else Too. Ok, so I'm a dude that got a Computer Science degree 20 yrs ago and came within literally half a dozen classes of getting degrees in Mathematics and Secondary Mathematics Education at the same time... who also took HIST classes as electives just because they sounded interesting. In other words, this book may as well have been titled "Jeff Sexton Will Beg To Read This Book", because I damn near did. But clearly, I *am* the very math and history nerd that my title of this review above explicitly said would love this book.
Clocking in at 19% bibliography, it has a healthy enough backing so that it doesn't drop a star on that account, and really the only even quibble I have here is that in choosing to end in 1938, Bardi doesn't even attempt to show how the war he describes here played out in the Post Second World War world where the rise of computers and computer science changed pretty well everything in mathematics.
Still, for what it is and what it actually covers, this is an utterly fascinating book that gives enough of the backstories of everyone involved to show how they arrived at the moments they did and why they mattered when they got there without going too deep into unnecessary detail too often. Yes, even when it looks like Bardi is going off on some wild tangent that can't possibly be related to The Great Math War (such as the Boer War), he ultimately manages to bring it right back around and show how that tangent actually played a role in shaping the thinking of one of his principle targets or at minimum someone closely linked to them who also shaped their thinking.
For me, this was an utterly fascinating look at a period of math history I hadn't previously really considered, featuring several names I had known of from other exploits and adventures, but had never really dived into from this angle. And yes, at least some of the most famous of their era make appearances here, up to and including one Albert Einstein.
I do think that one of the more fascinating tidbits that I had not known, particularly as someone who learned the basics of Set Theory in 2004-2005, was just how new that particular branch of mathematics was - and how controversial it had been even a century before I learned it, at which time it was (clearly) being routinely taught to Senior level college students. (And I still hold that we can actually teach it even in elementary school, as one key feature of it ultimately revolves around remainder division.)
Truly an excellent book written without the dryness that can accompany such texts in the hands of less skilled storytellers, this is absolutely one of those kinds of books that many will enjoy and a few will absolutely love.
Very much recommended.
Clocking in at 19% bibliography, it has a healthy enough backing so that it doesn't drop a star on that account, and really the only even quibble I have here is that in choosing to end in 1938, Bardi doesn't even attempt to show how the war he describes here played out in the Post Second World War world where the rise of computers and computer science changed pretty well everything in mathematics.
Still, for what it is and what it actually covers, this is an utterly fascinating book that gives enough of the backstories of everyone involved to show how they arrived at the moments they did and why they mattered when they got there without going too deep into unnecessary detail too often. Yes, even when it looks like Bardi is going off on some wild tangent that can't possibly be related to The Great Math War (such as the Boer War), he ultimately manages to bring it right back around and show how that tangent actually played a role in shaping the thinking of one of his principle targets or at minimum someone closely linked to them who also shaped their thinking.
For me, this was an utterly fascinating look at a period of math history I hadn't previously really considered, featuring several names I had known of from other exploits and adventures, but had never really dived into from this angle. And yes, at least some of the most famous of their era make appearances here, up to and including one Albert Einstein.
I do think that one of the more fascinating tidbits that I had not known, particularly as someone who learned the basics of Set Theory in 2004-2005, was just how new that particular branch of mathematics was - and how controversial it had been even a century before I learned it, at which time it was (clearly) being routinely taught to Senior level college students. (And I still hold that we can actually teach it even in elementary school, as one key feature of it ultimately revolves around remainder division.)
Truly an excellent book written without the dryness that can accompany such texts in the hands of less skilled storytellers, this is absolutely one of those kinds of books that many will enjoy and a few will absolutely love.
Very much recommended.
Displaying 1 - 7 of 7 reviews