An old joke runs along these lines:
"I went to the mayor's speech today."
"What did he talk about?"
"He didn't say."
What this book of conference proceedings is about seems similarly hard to pin down. Not that the volume lacks interesting papers. As with most such collections, it contains a variety of articles — good, bad, and in-between. This review will later survey some of the high and low points. But the book, as a whole, lacks any real theme or focus.
According to the back-cover blurb, "This book stresses the strong links between mathematics and culture..." Presumably, those links were also intended to form the theme of the Mathematics and Culture conference in Venice that gave rise to this volume. So what constitutes culture? One can construe the term quite broadly or narrowly. Indeed, the phrase "mathematics and culture" itself sounds a bit peculiar, as would the phrase "topology and mathematics". The editor's definition of culture, to judge by these papers, includes natural sciences, humanities, technology, social sciences, and medicine. In short, culture seems to comprise anything and everything. (Although the blurb specifically refers to dance and cartoon, the book contains no articles on those topics.) Nor do the 27 contributions restrict themselves to links between mathematics and other areas. Some papers have a strictly mathematical thrust, while in others math appears tangentially or even not at all. There is not necessarily anything wrong with such diversity. Just be aware that the title and blurb convey a somewhat misleading impression.
As mentioned above, the selections vary in quality as much as in content. Several of the better papers, in one way or another, deal with the history of mathematics. For example, Harold Kuhn provided a glimpse at the early days of linear and nonlinear programming. Such first-hand accounts of the beginnings of now-classical fields are always welcome. Looking fifteen years further back, Jochen Brüning presented Mathematics and Fascism — The Case of Berlin. The article details the lives of several Berlin mathematicians, from Nazi collaborators to emigrés to Auschwitz victims. That individual-by-individual approach supplies both concreteness and nuance to the cautionary tale of what can happen when a government promotes bigotry, hatred, and paranoia. Marco Li Calzi and Achille Basile co-wrote two nice historical papers on mathematics and economics. The first traces the mathematization of the field of economics, while the second sketches the Nobel-Prize-winning work of Debreu, Arrow, Nash, and Kantorovich.
In a nonhistorical vein, Enrico Giusti described the Italian mathematical museum project Archimedes' Garden. Likewise, Richard Mankiewicz commented on his work with the MathsTours exhibit that visits British shopping malls. The promises and challenges of presenting mathematics in such formats have always held interest for mathematicians wishing to reach the general public. Probably many of us still remember the museum-exhibit competition from the young Mathematical Intelligencer. The articles by Giusti and Mankiewicz offer various insights into what might or might not succeed in the exhibit setting.
The Radon Transform and Its Applications to Medicine, by Enrico Casadio Tarabusi, is a gem of exposition. From a toy discretization of the basic situation to the Radon transform itself to generalizations to various implementations of the theory in medical diagnosis and therapy (CAT, PET, NMR, etc.), the article conveys much information in comparatively few pages, yet always remains eminently readable.
So far, so good. But Mathematics and Culture I unfortunately includes a fair amount of chaff among the wheat. I already alluded to papers with minimal or no mathematical relevance — they seem quite out of place here. And the two contributions on mathematics and music proved fairly disappointing. One, an acoustically-oriented paper, contains various statements that betray a knowledge gap in this area. The author stated, "Helmholtz's idea is that the human ear behaves as an analyser of 'frequency spectrum'... [W]e are not aware of any experiments that have seriously tested this hypothesis..." First of all, the hypothesis should be attributed to Georg Ohm (yes, the electrical-resistance Ohm), on whose work Helmholtz built. More significantly, many experiments have tested and confirmed Ohm's concept, most notably those of Georg von Békésy. Békésy's studies of human hearing are hardly obscure — they earned him a Nobel Prize in 1961. As for the other math/music selection, it deals more with numerology than with mathematics.
More misinformation: one of the papers commits errors like characterizing Fermat's Last Theorem as "having been recently solved by Faltings." Such ignorance or carelessness is particularly unfortunate, since that paper's author is yet another person involved in math exhibits for the general public! Such glitches raise the question of what kind of refereeing and editing process did (or did not) this collection undergo. Did anyone really vet the contents for correctness? And was there any copy-editing? Certainly, the renderings of the text in English often ranged from unidiomatic to bizarre. My favorite comes from the paper mentioned above in this paragraph. Writing about popularization, the author asked, "What is a 'right problem' for vulgarising people?"
In short, Mathematics and Culture I is a real hodgepodge. For a supposedly thematic conference, the topics that appear vary greatly. Likewise, the caliber of the selections ranges from well-written and illuminating to irrelevant to ill-informed. With conference proceedings, you can expect a mixed bag. This bag is mixed to an unexpected degree.
Leon Harkleroad has taught several MAA Minicourses on mathematics and music and is writing a book on the subject, to be published jointly by the MAA and Cambridge University Press in the Outlooks series. His other specialties include computability (née recursion) theory and history of mathematics.