Playing with Infinity

Most mathematicians, if they have heard of Rózsa Péter at all, know her as the author of Playing with Infinity, an acknowledged classic of mathematical popularization. However, her many other mathematical contributions deserve to be better known. During the 1930s she helped establish the field of recursion/computability theory, publishing even before Turing and Kleene. (For a detailed account, see the recent book by Rod Adams.) Her work quickly achieved worldwide recognition, and in 1937 she joined the editorial board of The Journal of Symbolic Logic. Her colleagues on the board included such luminaries as Kleene, Church, Tarski, Bernays, Quine, and Mac Lane. The year 1951 marked the appearance of Péter’s Recursive Functions, the first monograph in that area. Many of her later publications investigated applications of recursion theory, particularly to linguistics and to computer programming languages.

Péter’s activity in research was matched by that in mathematics education. She taught for several years at the Budapest Teachers Training College and was most disappointed when it shut down and she had to relocate to Loránd Eötvös University, a move that many other mathematicians would have considered a step up. Indeed, for many years Péter taught in middle and secondary schools. With Tibor Gallai she wrote a couple of secondary school textbooks. Péter’s teaching exerted a huge influence on Peter Lax, whom she tutored for a few years until his family emigrated to the United States, when he was fifteen. She also devoted much effort to bringing women into mathematics. (Erdős once described her as “an immoderate feminist”!)

Besides her mathematical accomplishments, Péter made an impact in the realm of Hungarian literature as well. She wrote and translated poetry, and her translations of Rilke, in particular, were highly acclaimed. Her Nachlass contains a Hungarian version of Brecht’s lyrics for the Barbara-Song from The Threepenny Opera, rendered so seamlessly as to make one forget that the Magyar language does not even belong to the Indo-European family. She also moonlighted as a film critic.

All of these facets of Rózsa Péter contributed to Playing with Infinity. The book arose because of Marcell Benedek, a friend of Péter’s from Budapest literary circles. He regretted his lack of background in mathematics, so Péter wrote him a series of letters trying to convey the essence of some mathematical ideas. Benedek then suggested that the letters could form the basis for a book. As Péter stated in the Preface, “This book is written for intellectually minded people who are not mathematicians. … I have received a great deal from the arts and I would now like in my turn to present mathematics and let everyone see that mathematics and the arts are not so different from each other. I love mathematics not only for its technical applications, but principally because it is beautiful.” The aesthetic side of mathematics was a recurring theme for Péter, as was its unity. Often an idea or technique that Playing with Infinity introduces in one context unexpectedly (at least, to the lay reader) reappears later in a different setting, conveying effectively the cohesive whole that mathematics forms.

By such devices, by the images and examples she used to put across the concepts, and explicitly by her recounting of classroom incidents, Péter the teacher is well in evidence throughout the book. But one can also discern Péter the cutting-edge logician. The book concludes with what must have been one of the first (and, for my money, is still one of the best) presentations of Gödel incompleteness for the general public.

Playing with Infinity treats many of the same topics as another classic popularization, Kasner and Newman’s Mathematics and the Imagination, both books dating from around seventy years ago. Modern readers might find Péter’s book a bit old-fashioned. It certainly predates fractals, public-key cryptography, and internet search engines, to name a few staples (clichés?) of much current exposition. The fifty-year-old English translation, not totally idiomatic and including references to shillings and half-crown pieces, adds a further touch of quaintness. But Péter’s love for mathematics and desire to share its beauty still shine through timelessly.

Leon Harkleroad admits to some bias here, having done historical research on and translated many works of Rózsa Péter.