Edward Rothstein's Emblems of Mind describes the metaphor and abstraction which make music and mathematics similar intellectual pursuits. It is a personal journey offering the author's knowledge in music and mathematics, and a mapping between the two. The mapping is both formal and emotional; and the book educates as much as it instills a mood. As a former graduate student in topology, and a current music critic, the author is uniquely qualified to write this book, which is essentially his intellectual autobiography.
The purpose of the book is twofold. One is to educate people about what mathematics and music are about. The other is to note the deep similarities in the two subjects. Although the latter is the author's stated goal, it is only by way of the former that his theses are able to take root in the mind of the reader. It is the many examples from both music and mathematics that make the book work.
The book is organized into six chapters. Chapter 1 concentrates on the more formal connections between the two fields. For example:
- The language of each is difficult to decipher, yet rich in information, and often notation itself leads to new discovery.
- Both fields have an emphasis on mappings, proportions and relationships.
- Both have a sense of style. One can recognize a composer by his work as one can recognize a proof by its author.
Chapters 2-3 discuss math and music in detail. As the author admits, there is little in these chapters about mathematics/music that the mathematician/musician doesn't know. But the conjunction of the material makes it interesting even for experts in both areas. For the non-expert, these chapters are crucial for understanding the rest of the book. They answer the questions what is mathematics? what is music? Chapter 4 describes the emotional connections between math and music, and how the beauty in each is really the same. Chapters 5-6 discuss the nature of truth in the Platonic sense and the different ways that mathematics and music express truth.
The book can be read on many different levels based mainly on the reader's level of expertise in music and mathematics. Indeed, the only real problem I see with this book, is whether a non expert can comprehend it at all. For an extreme example, consider what a novice is supposed to get out of the following discussion: "Any loop, can be thought of as a mapping of a line segment -- say from 0 to 1 -- onto a surface, as if the line segment were stretched and wrapped like a broken rubber band so that the end points of the segment -- 0 and 1 -- are both directed to the same point on the surface, repairing the band on the surface. A homotopy -- which would identify equivalent loops -- is actually, then, a mapping between mappings, a function that transforms one function into another." If you already know what the author is talking about, then you understand it deeply and you nod agreement; otherwise you are reading words which convey only a thin veneer of the truth. There are similar examples of this kind of discussion for the music subject matter, which non-musical mathematicians would find just as bewildering. However, the author is aware that these ideas are deep and subtle, and to his credit will remind the reader of this, so that at least the reader knows what he doesn't really know.
Does the book work? Yes; for one thing, the above example is somewhat extreme. Most of the examples in the book are more accessible. For mathematics in particular, the book is a superb mini-history discussing topics such as the Babylonian number system, Euclid's infinite prime proof, irrational numbers from Aristotle through Dedekind, Galileo, Kepler, Copernicus and elliptical orbits, convergence of series from Euler to Weierstrass, Calculus of Newton versus Leibniz with Bishop Berkeley's critique, Cantor's infinite and set theory, a dash of group theory (Galois) and topology (Poincare), and even Robinson's nonstandard analysis. (It is not a history book so one can forgive the gap from 300- 1500). For this alone, I would recommend the book to non-mathematicians. It is a painless way to learn mathematics in an historical context.
Furthermore, even with the more complex examples, a reader can appreciate similarities between things that he doesn't completely understand. In fact, sometimes these connections clarify the examples rather than the examples clarifying the connections. This duality is a neat feature of the book.
Who should read the book? There are better books on both history of mathematics, and what is mathematics. However, I know of no book that connects these two topics to music in both formal and informal ways. The book is right for mathematicians who are looking to understand music; musicians looking to understand mathematics; and for history of math students looking for a reference relating math and music.
Finally, if there is any book that is ripe for multimedia it is this one. Unless you can read music, you long to push virtual buttons to hear the scores which are printed on the page.
Shai Simonson is Associate Professor of Mathematics and Computer Science at Stonehill College. There, he teaches an honors course called History of Mathematical Ingenuity. He is also the Cantor at Temple Adath Sharon in Sharon, MA.