Musings of the Masters: An Anthology of Mathematical Reflections, ed. Raymond G. Ayoub, 2004. 288+xii pp. $47.95 hardcover. ISBN 0-88385-549-6. The Mathematical Association of
In Musings of the Masters, Raymond G. Ayoub has collected 17 essays, excerpts, and speeches by renowned mathematicians who offer their thoughts on issues as diverse as mathematical pedagogy ("Thoughts on the Heuristic Method", by Hadamard), aesthetics ("Mathematics and the Arts", by Morse), ontology ("Mathematical Invention", by Poincaré), the role of the scholar ("The Community of Scholars", by Lichnerowicz) and the role of the history of mathematics ("History of Mathematics: Why and How", by Weil).
We might begin by asking why read a book about mathematics? Ayoub notes in the introduction that the collection stems, in part, from "a curiosity concerning the creative process in mathematics together with a curiosity concerning the essence of this remarkable subject" (p. vii). He ends the introduction by expressing hope that the collection will encourage mathematicians "to share with others their perceptions on this beautiful subject, or on other matters dealing more directly with the humanistic side of our nature." Ayoub's work lives up to its title and to the first goal; as to the second, only time will tell.
There is much to recommend this book. Several of the essays ought to be required reading of all who intend to become mathematicians or teachers of mathematics. Poincaré's selection gives a personal view of the role "unconscious" (what we would call subconscious) thought plays in mathematical discovery, an encouraging lesson to anyone who has ever meditated on a problem for weeks or months without approaching a solution. Meanwhile the Presidential address by Sylvester emphasizes the importance of observation in mathematical creativity: "Most, if not all, of the great ideas of modern mathematics have had their origin in observation" (p. 158). Weil explains the value of the history of mathematics: its role is to place before us examples of first-rate mathematical work (p. 204), presumably for emulation or inspiration; this is a sentiment that teachers of ordinary, non-mathematical history ought to keep in mind. Hadamard's essay on the heuristic process offers the viewpoint of someone who has clearly had much classroom experience; it is interesting to see the great similarities between the debate over mathematical pedagogy in Hadamard's time and in our own.
Unfortunately in the service of its second goal (encouraging mathematicians to share their own perceptions), many of the "Musings" are mathematicians discussing topics only distantly connected with mathematics. Lévy's "Does God Exist?" ventures into theology and notes that belief in "God" is a reflection of the tendency to accept a word as an explanation (p. 223); Morse's "Mathematics and the Arts" espouses the viewpoint that "the basic affinity between mathematics and the arts is psychological and spiritual and not metrical or geometrical" (p. 88), and Lichnerowicz's essay argues that scientists must involve themselves in political activity as scientists (p. 197).
There is no question that Ayoub's book lives up to its title and its stated intent. Many of the selections deal directly with mathematics. On the other hand, many of the selections have little connection to mathematics proper, and their presence is justified by being written by prominent mathematicians. If you are looking for an anthology of writings on mathematics or by mathematicians, this is worthwhile reading.
Jeff Suzuki, Associate Professor, Brooklyn College