How random is a coin toss?

Abstract

Probabilistic and deterministic Descriptions of macroscopic phenomena have coexisted for centuries. During the period 1650–1750, for example, Newton developed his calculus of determinism for dynamics while the Bernoullis simultaneously constructed their calculus of probability for games of chance and various other many‐body problems. In retrospect, it would appear strange indeed that no major confrontation ever arose between these seemingly contradictory world views were it not for the remarkable success of Laplace in elevating Newtonian determinism to the level of dogma in the scientific faith. Thereafter, probabilitistic descriptions of classical systems were regarded as no more than useful conveniences to be invoked when, for one reason or another, the deterministic equations of motion were difficult or impossible to solve exactly. Moreover, these probabilistic descriptions were presumed derivable from the underlying determinism, although no one ever indicated exactly how this feat was to be accomplished. In examining the differences between orderly and chaotic behavior in the solutions of nonlinear dynamical problems, we are led to explore algorithmic complexity theory, the computability of numbers and the measurability of the continuum.