The world once seemed wonderfully crafted: Newtonian mechanics made the entire universe graspable and, more importantly, predictable far into the future and, in the same way, far into the past. Using the celestial mechanics inherent in Newton's system, Urbain Le Verrier discovered the planet Neptune by inference based on perturbations of Uranus' orbit. What could better demonstrate the triumph of science than the discovery of planets by pure mathematics rather than by telescope? What would be left for science to discover?
But this was an illusion brought about by only taking partial views of the systems. Henri Poincaré pointed to problems of calculating orbits precisely under the gravitational effect of additional bodies. The result of perfect trajectories in orbits the "three body problem" meant that orbits were not truly periodic functions. The planet might return to a point near in space to that from which it departed a "year" earlier, but would not be truly periodic. This leads the author into a discussion of what is often called chaos theory. He is at his clearest in discussing a sensitivity to initial conditions and how this occurs. finally, he bridges this discussion to a way to see probabilistic outcomes as the result of seeing only part of the system.
The author then goes on to discuss catastrophe theory. This was a very hot topic 40 years ago. Perhaps it was the name that led to so many hints that the theory would explain stock market crashes, wars, and other catastrophes. This was reading in too much. The theory as developed was quite limited in discussing only dissipative systems. The author is frank in showing the limits of the theory.
The author writes with clarity. Some sections do require a mathematical view, not because they are given in equations, but because they are presented with the sort of logical rigor that a mathematics education gives one.
This book is recommended for readers who wish more than the popular approach to these topics.
