American Scientist

THE WALTZ OF REASON: The Entanglement of Mathematics and Philosophy. Karl Sigmund. 448 pp. Basic Books, 2023. $32.50.

Are mathematics and philosophy entangled? The question would have seemed senseless to Plato. In ancient Greece, mathematics served philosophy. Since the days of ancient Athens, philosophy and mathematics have grown into separate, ever-expanding fields. Today, the question of how mathematicians and philosophers interact with one another is a reasonable one.

The Waltz of Reason gives us clues to some possible answers by introducing us to some of the dance partners in the history of mathematics. Karl Sigmund, a professor emeritus of mathematics at the University of Vienna, writes clearly and with subtle wit, making The Waltz of Reason highly readable, with plenty of illustrations.

In his Commentaries, Julius Caesar divided all of Gaul into three parts. In his book, Sigmund outlines four parts to separate the vast realm of mathematics. The first part encompasses geometry, numbers, infinity, logic, and computation—a staggering amount of material. Part two addresses limits, probability, and randomness. The author’s research interests are reflected in the third part, covering voting, game theory, and fairness. The final part includes topics of language, philosophy, and understanding. Sigmund is aware of the large volume of material that he covers in the book: In his introduction, he confesses to feeling like the aging fisherman in Ernest Hemingway’s The Old Man and the Sea, who caught more than he could manage.

Despite the overfishing, The Waltz of Reason is a relaxed and refreshingly direct account. Consider this lead-in to a discussion of chance:

Chance is notoriously hard to define. Is it the force that causes something to happen without reason for doing so? Something that happens when several causes intermingle? Something that can be, but also not be? This is just a small sample (a random sample) of attempts to explain the word chance. Mathematicians, however, do not try to define chance. They want to reckon with it.

Throughout the book, Sigmund explores how mathematicians and philosophers have influenced one another. For example, when considering Aristotle and the 19th-century English mathematician George Boole, Sigmund writes, “The most remarkable insight of Aristotle was to describe logical arguments by a formal calculus: the content of the proposition was irrelevant. Admittedly, he used no mathematical formulas, yet his rules begged to be formalized.”

Boole, the son of a cobbler and mostly self-educated, gave us such a formalization. He began by recognizing that the logical connectives and and or can be modeled by multiplication and addition. Another hundred years after that, algebraic language developed from Boole’s ideas would help to make digital computers possible. Since then, computers have transformed the world, mostly for the better. (Safer to say that. After all, ChatGPT might be reading this.)

The Waltz of Reason proposes other examples of mathematicians whose views were influenced by philosophy. Among them are two brilliant 20th-century mathematicians, Kurt Gödel and Alan Turing.

What logician can compare to Aristotle? According to Albert Einstein, it was Kurt Gödel. As a child, Gödel was called “Mr. Why” by his family because of his constant demand for reasons. During his studies at the University of Vienna in the 1920s, he was invited to join a private group of distinguished mathematicians and philosophers, known later to the public as the Vienna Circle. Sigmund writes, “For philosophy of mathematics, this was the best of times. Three great schools, headed by formidable thinkers, were contending with each other—and most remarkably, many mathematicians actually cared!”

The three schools to which Sigmund is referring represented versions of a philosophical movement called logical positivism. Their respective positions can be capsulized as follows: Mathematics is formal logic; mathematics is merely symbol manipulation; and mathematics is only a mental construction. Gödel remained mostly quiet during the group’s many debates. He was no logical positivist—far from it! Rather, Gödel shared Plato’s vision of mathematics as independent of us: something to be discovered, not invented. Such thoughts were considered primitive—even anathema—to logical positivists.

During one gathering of the Vienna Circle, Gödel announced the first of his two “incompleteness theorems.” Given any consistent set of arithmetical axioms, Gödel showed that there are true statements that cannot be proven from the set. One might prove them by adding additional axioms (provided the system remains consistent) but then other unprovable true statements will appear. Only one other attendee, Gödel’s former teacher Moritz Schlick, paid much attention. The belief that the Vienna Circle’s philosophy influenced Gödel’s discovery is widespread. It is also wrong.

In effect, Gödel was telling us that truth is not the same as provability. Eventually, this notion would pique the interest of many other philosophers, with countless books and articles published on variations of this theme.

Cambridge-educated mathematician Alan Turing, born in London in 1912, expanded on Gödel’s incompleteness theorems. Turing showed that a mechanical process that could test the truth of certain mathematical yes-or-no questions is impossible. In order to do this, Turing created an imaginary “computing machine,” one that manipulated strings of symbols on a strip of tape, following a table of rules. The significance of Turing’s thought experiment cannot be overstated. Sigmund writes that Turing’s machines “yielded the blueprint for the stored-program computer. Its arrival was as momentous for humanity as that of agriculture or writing.”

Turing was not a philosopher, although he is sometimes called that. One reason philosophers claim him is that he contemplated the question of whether a computer might someday be able to think, though he pronounced the question “too meaningless to deserve discussion.” Intelligence, Turing believed, is an emotional concept, not a mathematical one. In response to the question, he presented what he called “the imitation game.” The game, now called the Turing test, asks a human at a remote device to communicate with a human and a computer, and then try to decide which is which. Turing’s ideas about computers’ capacity for thought will likely be revisited in the future, as “digital assistants” answer our telephone calls and artificial intelligence is used to replace humans in the workforce.

The real motivation for Gödel and Turing did not come from philosophers. It came instead from challenges posed by David Hilbert, possibly the most influential mathematician of the late 19th and early 20th centuries. Specifically, it was Hilbert’s program, which proposed a formalization of all mathematics, that spurred both Gödel and Turing.

Philosophers were given much to think about by Gödel and Turing. But did philosophy contribute anything helpful to them, or to other mathematicians for that matter? The Waltz of Reason lacks a satisfactory answer. What the book does reveal is the zeitgeist, the spirit of the times, to which philosophy contributes. Mathematicians, like artists and musicians, absorb that spirit, at times enabling it to emerge through their work.

Gödel shared Plato’s vision of mathematics as independent of us: something to be discovered, not invented.

• Facebook
• Twitter

Although Turing created a purely theoretical machine that so productively manipulated symbols, it was the Hungarian American mathematician John von Neumann who introduced a physical design in the 1940s that would be a model for future digital computers, paving the way for the modern computer.

Von Neumann appears again in a later chapter, where we read about game theory. Game theory is applied mathematics, that is, mathematics applied to other fields. Game theory models situations in which “players” make interdependent decisions for gain or loss. The players might be having a harmless game of cards or they might be contemplating nuclear war. In 1943, von Neumann and the economist Oskar Morgenstern published Theory of Games and Economic Behavior, a comprehensive introduction to game theory. The book attracted widespread attention and glowing reviews, but game theory was slow to catch on in the social sciences, partly because of its mathematical demands. Today it is a standard tool in behavioral economics, and of great interest in ethics.

The Waltz of Reason contains an excellent introduction to game theory. We learn how game theorists drew motivation from other fields—in this case, ethics, psychology, and sociology. Of course, that is what applied mathematicians are supposed to do.

Pure mathematicians, on the other hand, must restrict their claims to statements that they can verify with certainty. Philosophical speculations inserted into mathematical journal submissions are unlikely to get past referees and editors. But that doesn’t mean that some mathematicians don’t discuss together occasional philosophical questions that appear to be related to their work. They do.

Certainly, mathematics and philosophy are in a relationship, but precisely what philosophy contributes to that relationship is a difficult question. Although Sigmund does not give a complete answer, the book does provide an entertaining look at how, during the past few centuries, philosophy and mathematics have at least watched each other from across the ballroom floor. If just for that, The Waltz of Reason is well worth a dance.