What’s New in Wolfram’s New Kind of Science?

Abstract

Wolframfs monumental best seller entitled "A new kind of science" was based almost entirely on brute-force computer simulations. In sharp contrast, this lecture presents a rigorous analytical theory based on attractors from a nonlinear dynamics perspective. New results and concepts to be presented include the partitioning (via Felix Kleinfs Vier- ergruppe) of all 256 local Boolean rules studied empirically by Wolfram into 88 global equivalence classes, one of which contains 4 topologically- conjugate rules capable of universal computation, and endowed with a 1/f spectrum. Another major result is the rigorous characterization of the time-asymptotic dynamics (attractors) of 112 local rules via an explicit generalized Bernoulli shift formula. Even more surprising, we have dis- covered the attractors of 170 local rules are blessed with the remarkable property of time-reversality. For such rules, the past evolution in time can be recovered from the future evolutions of a corresponding gtwinh rule. Only 86 local rules exhibit an garrow of timeh. One of our most fascinat- ing discoveries is a new phenomenon, dubbed an gIsle of Eden,h having no counter part in hyperbolic differential equations, which has neither a past, nor a future! In addition to providing a mathematical founda- tion for brainlike dynamics, the discoveries cited above provide simple dynamical mechanism for mimicking many exotic phenomena from brain science, quantum physics, cosmology, etc.