The past thirty years have seen a dramatic change in the way we understand dynamical systems. For centuries, mathematical analysis has been constrained to linear systems, primarily for practical reasons. Linear systems can be solved. Linear systems often provide good approximations of real-world phenomena. Most importantly, linear systems are stable and predictable to arbitrary degrees of precision.

But the (comparatively) new study of chaos mechanics has finally put an end to this idealism. We must now fully confront the reality of nonlinearity in real systems. No more is there room to ignore the noise and randomness of chaos; it cannot be dismissed as experimental error, measurement inaccuracy, or just plain bad luck. Chaotic behaviour is vitally real, and it is here to stay.

Chaos, and emergent properties of dynamical systems in general, is a critically important field of theoretical investigation. The development of genetic algorithms has proven that emergent phenomena can accomplish quite surprising results.

We may well owe our very existence to emergent phenomena. The genetic evolution process has been used to find a wide variety of very effective solutions to a host of formerly intractable problems. These solutions lack the strict, rigid "design" of solutions derived by ratiocination. They are often remarkably precise, but peculiarly fragile. Many components are mysteriously intertwined, and some defy any understanding at all.

It is not a coincidence that these same properties extend to biological systems; biology, after all, was the inspiration for applying genetic evolution to a host of problems.

This is but one example of sophisticated behaviour arising from emergent phenomena. For now, the best we can do is say that the genetic evolution method (and genetic algorithms in general) tend to produce solutions that share certain properties. They work, seem "organic", and are sophisticated but fragile.

Unfortunately, this is merely crude alchemy. To observe only that one thing plus another produces a third result is to fall woefully short of true understanding. It is of the utmost importance for theoretical analysis across all fields of study to be mindful of emergent phenomena in all their guises.

Emergence is in dire need of a Mendelev to organize the menagerie of phenomena and discover the underlying patterns and principles. We must begin to catalog emergent processes and their characteristics. We must attain a deep and true understanding of the way emergence works.

I feel, intuitively, that emergent phenomena will soon take a pivotal place in the history of mathematical development. The discovery and rigorous investigation of chaos - momentous as it has been - is not itself the revolution, but merely the vanguard of a much larger change.

In the end, once we understand the way complex systems work and change and produce non-obvious results, the study of emergent phenomena will produce no less a powerful tool of human advancement than did the mathematics of calculus before it.