The Waltz of Reason: The Entanglement of Mathematics and Philosophy

by Karl Sigmund

"A mind-bending jaunt ... that makes clear in fascinating detail how math is more than a sum of its parts" (Publishers Weekly)

“Let no one ignorant of geometry enter here,” Plato warned would-be philosophers. Mathematician Karl Sigmund agrees.

In The Waltz of Reason, he shows how mathematics and philosophy together have shaped our understanding of space, chance, logic, cooperation, voting, and the social contract. Sigmund shows how game theory is integral to moral philosophy, how statistics shaped the meaning of reason, and how the search for a logical basis for math leads to deep questions about the nature of truth itself. But this is no dry Sigmund’s wit and humor shine as brightly as his erudition.

The Waltz of Reason is an engrossing history of ideas as vibrant as a ballroom full of dancers, one that empowers as it entertains, following the complex and occasionally dizzying steps of the thinkers who have molded our thought and founded our world.

Genres: Philosophy, Mathematics, Science, Nonfiction

417 pages, Kindle Edition

Published December 19, 2023

Reviews

[emily · Rating 4 out of 5]

'In mathematics, you do not understand things (John von Neumann). You just get used to them.'

Like part 1 and part 4 best. Brilliant. RTC later. Might even round it off to a 5 (because even if it didn't blow my mind to the degree/level that I believe will nudge this one to a 5*, Sigmund executed and accomplished everything he essentially 'promised' to do in the book (all brilliantly and clearly done)).

As to Wittgenstein, he did not even try to explain. Explaining was not his business, as he insisted. But he did his best to describe mathematical reasoning. He stressed time and again that a proof must be “übersehbar” (meaning perspicuous, or surveyable, or to be taken in with one glance—Cartesian, to use Ian Hacking’s term). The proof of Pythagoras at the start of this chapter would have given him satisfaction.

Unfortunately, only a few theorems can be reached by such proofs. Usually, the best one can hope for is a series of moves, one following the other, and each one perspicuous. The famous mathematician Paul Erdős, one of the most prolific mathematicians of the last century, claimed that God (in his words, “the Supreme Fascist”) kept these proofs locked away in a book. The highest accolade a mathematical demonstration can earn is therefore: “It is from the book.”

What happens when we experience proof? Neurobiologists have scanned mathematicians’ brains, and duly found that some bits of lobes light up, and others not. It does not explain why understanding proof provides pleasure. Can evolutionary biology provide the key? So far it has not. Mathematical hedonism remains mysterious. It is well-known that mathematical discovery precedes proof—often by ages. Insight relies on induction and analogy.

[Chris · Rating 4 out of 5]

If you love mathematics and how it underpins geometry, logic, philosophy, game theory, computation, science, politics, society, and language, you’ll enjoy this book. It assumes at least a general knowledge of math and takes interested readers on a more or less fascinating journey, depending on the reader’s particular interests.

I particularly liked the chapters covering science and calculus, computation and language. But I bogged down in game theory, utility theory, and the social contract because I have never taken to those topics. I’ll be happy if I never see another explanation of the Prisoner’s Dilemma, and all the boring noise that it generates. Having appeared in countless books I’ve read over many years, it remains an impenetrable non-intuitive menace, despite its obvious mathematical truth.

[Aaron · Rating 4 out of 5]

I received this book as a graduation gift from Professor Jeff upon completing my BA in Philosophy in May 2024. Sigmund notes early on that The Waltz of Reason is concerned with the connections between mathematics and philosophy, though not strictly the philosophy of mathematics. In practice, it reads more like a loosely connected set of essays on mathematical topics with occasional philosophical reflections.

That looseness was, to me, a weakness: the book never quite settled into a cohesive through-line. Some chapters felt more engaging than others. I especially enjoyed the discussion of game theory, which was both clear and thought-provoking. By contrast, the chapter on Euclidean geometry dragged with long, detailed proofs that felt more clunky than illuminating.

Overall, the essays were hit or miss, with moments of real insight alongside stretches that lacked focus. A worthwhile read in parts, but not consistently compelling.

[Clay Kallam · Rating 2 out of 5]

If you enjoy doing algebra problems in your head -- or if you have a pad of paper to work out formulas you haven't thought of since high school -- "The Waltz of Reason" is for you.

Otherwise, it's a book that supposedly shows how math illuminates some basic philosophical questions, but comes up short in that quest. The best example is probably the chapter that investigates the various aspects of the Prisoner's Dilemma, which is a situation in which two people must make choices that impact the outcome for both of them. The classic situation is this: If Prisoner A decides to confess, and so does Prisoner B, both get five-year terms. If one confesses and the other doesn't, the one who confesses gets 15 years and the other goes free. If both refuse to confess, they each get 10 years. But they cannot know what the other prisoner decides to do.

Karl Sigmund investigates numerous permutations of this thought experiment, using mathematical models to determine the best courses of action. But in the real world, relationships come into play in any kind of Prisoner's Dilemma, and Sigmund completely ignores them, assuming that the numbers alone will guide the decision-making process.

This blind spot extends throughout Sigmund's discussion of various issues, leaving the reader with a bunch of algebra problems and proofs that are of interest only to those who like solving them. I don't, even if I did remember much about algebra or geometry, so this book turned into a long slog that only my OCD-like compulsion to finish every book that I start made me get to the final pages.

I had certainly hoped for better from "The Waltz of Reason," but instead was extremely grateful when the music of the math finally stopped.

[Mishehu · Rating 4 out of 5]

I came to this book thinking, mistakenly, it’s a pop treatment of philosophy of math. Turns out it’s about the connection between math and philosophy. Ish. Some of the book is that. Some is general pop math. And a serious chunk is about math (game theory) in the social sciences. In short, it’s a hodgepodge. Some of the hodge I enjoyed, some of the podge not as much. In general, the author writes well. I don’t fault him for writing a book that departed from my expectations. I do fault him for explaining math unclearly in several spots. I’d rate this a decent book, not a great one.

[Bob Finch · Rating 4 out of 5]

Fascinating and intricate interweaving of mathematics and philosophy. Sigmund is an excellent writer (I loved his book about the Vienna Circle), tho he might be a bit too succinct (for non-mathematicians at least) in a few of his explanations here. I hope the author comes out with a revised edition, as there are quite a few mistakes (e.g., mislabeled figures). I also take issue with his solution to a puzzle that, if his solution is correct, must be missing some information. There are also a few passages that could benefit from added clarity. But, such quibbles aside, this is an excellent book. I highly recommend it.

[David · Rating 3 out of 5]

Some parts were interesting. It was just way to long. I am not a mathametician though.

[Thomas · Rating 3 out of 5]

Entertaining but also entirely inconsequential.

[mono]

Stopped on Randomness, had to take a tangent into Plato's numerology. This reminded of a couple of Gregory Chaitin's books - Meta Math & The Limits of Mathematics.