Formalisations of Evolutionary Biology 499mathematical entities (numbers, vectors, functions) used to represent states of the
system within a state space (a topological structure), the behaviour of which is
specified by three functions. These functions are commonly called laws but are not
the same as laws in the received view. In the received view, laws are statements
describing the behaviour of entities in the world. In the semantic view, laws are
mathematical descriptions of the behaviour of mathematical systems of mathemat-
ical entities. The three laws are: laws of coexistence (which specify the physically
possible set of states of the system); Laws of succession (which specify the possible
histories of the system); and laws of interaction (which specify the behaviour of the
system under conditions of inputs from interaction with other systems). A theory
also requires the specification of a set of measurable magnitudes (represented by a
function defined on the state space). Statements which formulate propositions to
the effect that a particular magnitude has a particular value at a particular time
are elementary statements. A satisfaction function determines the sets of states
which satisfy the assignment of a value to a physical magnitude.
For population genetic theory, the state space will be a Cartesian n-space where
‘n’ is a function of the number of possible pairs of alleles in the population. A
law of coexistence to the effect that only alleles at the same locus can form pairs
will select the class of physically possible pairs of alleles. States of the system
(genotype frequencies of populations) are n-tuples of real numbers from zero to
one and are represented in the state space as points. These are the measurable
magnitudes. An example of a satisfaction function for the elementary statement
“genotypeAa occurs with a frequency of 0.5” would specify the set of states in
the state space that satisfy the statement. In this case the set of states would be
a Cartesian (n−I)-space, which is a subset of the state space. For population
genetic theory, a central law of succession is the Hardy–Weinberg law.
State spaces (phase spaces) provide a powerful tool for characterising the dy-
namics of a system topologically. A simple and clear example of this power is
given by Kellert [1993]; Kellert, explicating the theoretical account of turbulence
given by Lev Landau [1944], provides a state space representation of turbulence
at increasing levels of turbulence in a river.
By way of concluding this exposition of the semantic account, let me summarise
its central features:
- Scientific theories are not linguistic but rather mathematical entities
- Theories are not appropriately formalised in first-order predicate logic with
identity. - The specification of the model(s) whose intended scope is an empirical system
is a complete formalisation of the theory. - The process for confirming a theory is complex and extra-theoretical involv-
ing a justification of an assertion that the model is isomorphic with the em-
pirical system that is its intended scope. This requires recourse to numerous
other theories (mathematical and empirical).