These are just a few examples of what are sometimes referred to as complex adaptive systems. They have many interacting parts that change in response to local inputs and as a result change the global behavior of the complete system. The relatively smooth operation of biological systems – and even our human-constructed Internet – is in some ways mysterious. Individual parts clearly do not have an understanding of how other individual parts are going to change their behavior. Nevertheless, the ensemble ends up working.
We need a new mathematics to help us explain and predict the behavior of these sorts of systems. In my own field, we want to understand the brain so we can build more intelligent robots. We have primitive models of what individual neurons do, but we get stuck using the tools of information theory in trying to understand the “information content” that is passed between neurons in the timing of voltage spikes. We try to impose a computer metaphor on a system that was not intelligently designed in that way but evolved from simpler systems.
My guess is that a new mathematics for complex adaptive systems will emerge, one that is perhaps no more difficult to understand than topology or group theory or differential calculus and that will let us answer essential questions about living cells, brains, and computer networks.
We haven’t had any new household names in mathematics for a while, but whoever figures out the structure of this new mathematics will become an intellectual darling – and may actually succeed in designing a computer that comes close to mimicking the brain.
Rodney Brooks directs MIT’s Computer Science and Artificial Intelligence Laboratory.