Book Review : Exploring Unanswered Questions of Mathematics
The Mathematical Tourist: Snapshots of Modern Mathematics by Ivars Peterson (W.H. Freeman: $17.95; 240 pages; illustrated)
The mathematics that people learn in school is very old. From arithmetic to calculus, the stuff has been around for thousands of years, and students might come away with the idea that all of mathematics is cut, dried, finished and known.
Nothing could be further from the truth, as Ivars Peterson makes clear in “The Mathematical Tourist,” a top-notch survey of the frontiers of contemporary mathematics. Far from being old and musty, mathematics is alive, vital and vibrant, full of interesting, unanswered questions that Peterson makes accessible to non-expert but attentive readers.
More Topics
Peterson’s topics include number theory, cryptography, topology, fractals, the fourth dimension, chaos, tiling and a host of diversions, asides and tangents (no pun intended), and he brings to all of them the sense of unfinished business and knowledge in process.
Discussing the mathematics of knots (yes, plain, ordinary knots), he writes, “Like physicists who are trying to make sense of the particles and forces that make up the physical world, knot theorists are looking for something akin to a grand unified theory that would explain all invariants and all knots. . . . Completely classifying knots using a simple set of easily computed invariants seems well beyond the reach of current methods.”
And it’s not just knots that have mathematicians tied in knots. Do you think geometry is complete? It isn’t. “One of the great classification quests in mathematics, which still isn’t over, is the search for a complete classification of geometric shapes,” Peterson writes.
A Hot Topic
Fractal geometry has been a hot topic for several years now, but Peterson makes clear that it has many more questions than answers. “Ever since the earliest applications of fractal geometry, description has outpaced explanation,” he writes. “Theories that depend on fractal properties seem to work, but no one really knows why. Further progress in the field depends on establishing a more substantial theoretical base in which geometrical form can be deduced from the mechanisms that produce it.”
That’s the good news. The not-so-good news is that for all its virtues, this is not an easy book. It presses on the boundary of what nonmathematicians can understand, and occasionally goes over it.
Not that that’s necessarily bad. Parts of “The Mathematical Tourist” will be comprehensible to any interested person, but other parts of it are tough. Peterson, who writes about mathematics for Science News, is very good at explaining these things, and he conveys the excitement of the hunt for mathematical knowledge.
Mathematics is constantly changing and constantly evolving. New links are found between seemingly unrelated ideas, which adds to their power and depth. Peterson patiently explains how that happens and what are the results.
A Dynamic Field
He breathes life into the subject, which, in his telling, is a very human undertaking. “Mathematics is an exciting, dynamic field that continues to generate provocative ideas and novel concepts,” he writes. “Some notions quickly find their way into applications; others rest on their own intrinsic beauty; still others occupy a modest place in the growing structure of mathematics itself.”
Peterson’s book, which is full of illustrations and 16 beautiful color plates, is about mathematical ideas, but more important, it is about mathematics itself. While the individual topics and explanations may sometimes be perplexing, the overall message comes through loud and clear: As in all of the sciences, in mathematics there is no end to knowledge. New things keep popping up all of the time, and they open into unexplored worlds of elegance and beauty. Sometimes, they even turn out to be useful, though mathematicians are usually embarrassed when they are.
Peterson’s book is a welcome addition to the growing field of mathematics for the rest of us. Unlike most such books, he gives more than the details. He tells us what the enterprise is all about.