Ekland: Mathematics and the Unexpected

by Carson Reynolds

“Besides deterministic and stochastic systems, there seems to be room for at least one other type of dynamical system, which is closer to evolutionary processes and may therefore be called Darwinian. In such systems the past history determines various possibilities for the present state, among which the system will choose in response to external stimuli, such as changes in the environment.” Each evolutionary stage looks like the final goal toward which the species was striving; but this illusion is shattered as the next stage sets in, and the ‘final’ state is shown as one in an endless succession of states, a step in an aimless march to infinity”

This passage points towards the space in which adaptive systems lie. Not all adaptive systems are deterministic. Certainly iterative techniques like Newton’s Method are, but they aren’t fully random either. A deterministic adaptive system creates 1-to-1 maps at each stage of iteration. This new type creates 1-to-N maps at each stage, and then uses some input (random or otherwise) to choose among the N.