Introduction to Introduction to Mathematical Systems Theory

Many in the ACT community view Jan C. Willems as a proto-ACT theorist, even if he did not use category theory. Reading the introduction to his Introduction to Mathematical Systems Theory: A Behavioral Approach you can see some of the ideas that we are still grappling with now in full relief,

We start this book at the very beginning, by asking ourselves the question, What is a dynamical system

Disregarding for a moment the dynamical aspects—forgetting about time—we are immediately led to ponder the more basic issue, What is a mathematical model? What does it tell us? What is its mathematical nature? Mind you, we are not asking a philosophical question: we will not engage in an erudite discourse about the relation between reality and its mathematical description. Neither are we going to elucidate the methodology involved in actually deriving, setting up, postulating mathematical models. What we are asking is the simple question, When we accept a mathematical expression, a formula, as an adequate description of a phenomenon, what mathematical structure have we obtained?

We view a mathematical model as an exclusion law. A mathematical model expresses the opinion that some things can happen, are possible, while others cannot, are declared impossible. Thus Kepler claims that planetary orbits that do not satisfy his three famous laws are impossible. In particular, he judges nonelliptical orbits as unphysical. The second law ofthermodynamics limits the transformation of heat into mechanical work. Certain combinations of heat, work, and temperature histories are declared to be impossible. Economic production functions tell us that certain amounts of raw materials, capital, and labor are needed in order to manufacture a finished product: it prohibits the creation of finished products unless the required resources are available.

We formalize these ideas by stating that a mathematical model selects a certain subset from a universum of possibilities. This subset consists of the occurrences that the model allows, that it declares possible. We call the subset in question the behavior of the mathematical model.

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