Formalisations of Evolutionary Biology 501On this conception of scientific theories, the empirical world in all its complexity
is formalised in the language of mathematics with all its richness.^24 Hence, the
formalisation of a theory rests on finding the most appropriate domain of math-
ematics and mathematical techniques. As van Fraassen^25 has pointed out, the
slogan that motivated Suppes’ conception of scientific theories was that philoso-
phy of science should use mathematics not meta-mathematics.^26
The broadest and simplest definition of a mathematical model^27 construes it as
a specification, using a domain of mathematics, of the kinds of entities involved
in a system and the dynamics of their behaviour. Things are more complicated
than this definition captures, however, because a mathematical modelof empirical
phenomenawill always involve an idealisation of the phenomenal world.
There are at least four major benefits of conceiving of theories as mathematical
models.^28 First, a mathematical model introduces precision which, in part, results
from a removal of the ambiguity that plagues ordinary language — however, part
of the cost of precision is idealization and a high level of metrication. Second,
mathematical models provide a powerful machinery for exploring the dynamics of
the phenomena. From now on, I will refer to the collection of properties of a class
of phenomena as a phenomenal system, and to the collection of properties of a
mathematical model as a system(a designation which is short for model system).
Hence, the expression, “dynamics of the system” refers to the dynamics of the
model which is purported to represent the dynamics of a phenomenal system.
(^24) If one accepts logicism (the view that all of mathematics can be reduced to deductive logic),
then, in principle, any mathematical model can be reduced to a formalisation in logic. The most
well known advocate of logicism was Bertrand Russell. Logicism is still the subject of debate.
For the most part, the major challenge to logicism was G ̈odel’s discovery that the derivation of
all mathematical truths requires a logic that it is impossible to formalise. Although an important
issue in the foundations of mathematics, whether all of mathematics can be reduced to logic is
too esoteric an issue in the context of the formalisation of scientific theories. Even if logicism is
correct, the complexity of such a reduction will be great but, more importantly, it will provide
no added value to actual scientific theorising.
(^25) van Fraassen [1980, p. 65].
(^26) Meta-mathematics, following Tarski [1935; 1936], encompasses the semantic and syntactical
examination of formal languages and systems. It is the logical examination of rules specifying
the permissible combinations of symbols and numbers in mathematics and their uses, as well as
the logical examination of mathematical methods and principles. First-order predicate logic with
identity figures prominently in the study of the foundation of mathematics and has become the
general study of the logical structure of axiomatic systems. Hence, appeal to first-order predicate
logic in the formalisation of scientific theories is using meta-mathematics, not mathematics.
(^27) It is important to be precise about this because the term “model” in science (and other
contexts) is open to numerous interpretations. A mathematical model, in the context of the
formalisation of a scientific theory, is the formulation of the dynamics of a system using the formal
calculus of a domain of mathematics that embodies a specific interpretation of the calculus. The
formulation of the dynamics in such a language consists of specifying the formulae that encode
all the relevant dynamical properties. Sometimes this may be a single equation or a small set of
equations such as the three equations used by Lorenz to formalise the dynamics of turbulence
[Lorenz, 1963].
(^28) The first benefit is also realized within the syntactic conception. The remaining three are also
realized within the semantic conception but less fully. This is not surprising since the Galilean
conception is merely an extension of the semantic conception.