Mathematics: Frontiers and Perspectives [Hardcover]

This book, with which the International Mathematical Union tried to emulate the famous list of problems proposed by David Hilbert in 1900, together with his description of the state of mathematics those days, totally fails to survey the state of this science in the year 2000 and does even less to present a comprehensive list of important problems in all branches of mathematics that should be solved in the 21st century.

The editors - all of them highly respected and renowned mathematicians - didn't make a strong effort to collect the opinion of several other high-ranked colleagues; instead they asked their buddies and pupils for contribution. I really would like to read something about the topic from Elias Stein, Steven Krantz, or Serge Lang, to name just a few. Also, the editors could have done something to reach some kind of agreement among the contributors; while some pose some problems of their own interest others claim that even trying to imitate Hilbert is nonsense. Some dissert on the interaction between mathematics and physics (there's even one who claims that mathematics is a part of physics) but almost none pays attention to the newer ineractions between mathematics and other disciplines. It seems that they haven't realized that nowadays most of the motivation for difficult and interesting problems comes from such areas as economics, communications, military and computer science. Its surprising that almost all of the contributors still think that theoretical physics is the main supplier of mathematical problems.

And the worst: they didn't cover all the branches of mathematics (true, it's a difficult task, but the AMS and the IMU should be capable of doing that). There's nothing said about operator theory, hypercomplex analysis, coding theory, commutative algebra, wavelet analysis, and many other disciplines. Their major lack is applied mathematics, I mean, probability, statistics, reliability, simulation, operations research, etc. Most of the contributors blame Bourbaki for having gone away from the actual sources of mathematical knowledge, but they remain in the Bourbaki setting themselves!

Conclsion: better spend your money in fine and focused reflections about mathematics and mathematicians, like Hardy's, Polya's, Wiener's, von Neumann's, and even Bourbaki's.