The Waltz of Reason: The Entanglement of Mathematics and Philosophy

Sigmund (Games of Life), a professor emeritus of mathematics at the University of Vienna, explores in this spirited study the ways that math has been used to make sense of the world, from abstract controversies over logic to practical insights into elections and ethics. After overviewing concepts in computation, logic, chance, and causality, the author delves into “practical philosophy: morality and economics, politics and the law.” He identifies the intrinsic flaws of ranked voting systems (according to mathematician Ken Arrow’s impossibility theorem, no such system “can satisfy even the most reasonable” fair voting requirements, “as soon as there is a choice between more than two alternatives”) and the complexities of social cooperation as captured by the famous “prisoner’s dilemma” non-zero-sum game. Elsewhere, Sigmund traces evolving perceptions of math’s practical possibilities: as recently as 1940, some mathematicians were sure that general relativity and number theory refused “all possible applications,” but today “Einstein’s field equations are used in GPS, and number theory in all email platforms.” Wading into rich debates over math’s fundamental nature (Is math like a language? Does it have an existence independent of consciousness, or is it a purely mental construct?), Sigmund takes readers on a mind-bending jaunt through game theory, gambling systems, and more. This makes clear in fascinating detail how math is more than a sum of its parts. (Dec.)