502 Paul Thompson
Third, links among different phenomenal domains become apparent. This is a
more abstract benefit than the first two. Consider, however, the application,
alluded to above, of game theory to economics, ecosystems, and international
politics. Providing a model in terms of game theory for each of these domains
established the similarity of the dynamics in each area and a cross-fertilization
of ideas became possible. Fourth, the coherence of the knowledge of phenomena
is achieved; a coherence which, when the knowledge is expressed in a large and
complex number of ordinary language sentences, is less obvious and certainly less
demonstrable.
1.4 Formalisations and Theories
To this point, the relationship among the terms mathematical model, formalisation
and theory has not been explicitly described. It will be useful to provide such a
description, as well to be clear on what constitutes a scientific theory, prior to
looking at the nature of evolutionary theory and some examples of formalisations
of elements of it.
Although a theory can be given a formalisation, not all formalisations are for-
malisations of a scientific theory (consider Hilbert’s axiomatisation of Euclidean
geometry). Also, not all formalisations employ mathematical models; in the logical
empiricist view of theories, theories were not formalised as mathematical models
in the common sense of a mathematical model. In addition, not all mathemati-
cal models are formalisations of scientific theories. What, then, is the relationship
among these concepts? What I have argued, is that the most adequate understand-
ing of scientific theories holds that they are formalised by constructing mathemat-
ical models of dynamical systems, where the mathematical models are claimed to
be isomorphic (homeomorphic) to the empirical system of phenomena within their
intended scope.
If, however, not all mathematical models are, or are intended to be, a formalisa-
tion of a scientific theory, what distinguishes a formalisation of a scientific theory
from other mathematical models? The answer to this question appeals to function.
A scientific theory attempts to:
- integrate knowledge,
- facilitate a conceptual exploration of a dynamical system to discover un-
known (an perhaps empirically unknowable) properties (such as the Lorenz
attractor in system of turbulence), - render phenomena in a system explainable
- allow predictions
- guide future research (e.g., hypothesis formation, data interpretation)
The central feature of a scientific theory is its intended application to an em-
pirical system. A key element of scientific theorising is, therefore, the complex