International Encyclopedia of Systems and Cybernetics

W.S. IRVINE, in a review of D. GREENSPAN's "Discrete Models", writes: "Nature seems to abhor both a vacuum and a linear response ". This is a very serious matter, since most of our models are linear, being based on the famous… et ceteris paribus, which in most cases is merely a conforting delusion of the modeller.

IRVINE observes: "Calculus evolved to handle a continuous system of equations, resulting from a series of discrete observations. Calculus provides continuous solutions that must be matched to discrete observations for verification. We tend to forget that calculus is a compromise at best, not an act of faith. When the equation system generated by the natural phenomena is too complex for calculus to provide analytic solutions, we often make assumptions and linearize until one can be found " (1978, p.131).

An imprudent use of this method may lead us to ignore, or worse, disclaim some seemingly aberrant values that could appear within an apparently linear function. It also may lead us to believe, if not in some fallacies of infinity, at least implicitly in that other and more subtle fallacy of accelerated and unchecked growth. This is surely one of our most questionable contemporary postulates.

A.G. BARTO writes on this same topic: "… a discrete model… might be formalized as a set of difference equations or as an automaton" (1978, p.164). Automata, indeed, show a discrete behavior in time. In contrast, "The term continuous model will refer to a system whose time base is the uncountably infinite set of real numbers".

Still: "The domain of a discrete function… is a set of integers. The function is not defined for real numbers between the integers" (Ibid., p.169).

We face thus a methodological ambiguity: our selection of a continuous or a discrete model always implies a basic definition about the structure of space and time. Such a definition is necessarily an unverifiable postulate.