Formalisations of Evolutionary Biology 487claim, correctly I think, that providing semantics via models makes the syntactic
formalisation otiose.
The syntactic conception was forged within the framework of logical posi-
tivism/empiricism and was deeply informed by the rise of symbolic logic and the
conviction that it was the appropriate formal calculus for the formalisation of sci-
entific theories. The semantic conception was fashioned against this background,
denying that first-order predicate logic with identity was the most appropriate
formal calculus and denying that correspondence rules provided the required se-
mantics. The label “semantic conception” captures the rejection of the logical
empiricist method of providing a semantics but it does not reflect fully the rejec-
tion of first-order predicate logic as the appropriate formal calculus. Once this
constraint on the appropriate formal system is relaxed, the formal calculus of any
domain of mathematics can be used to provide a formalisation of a theory. This
is entirely consistent with the semantic conception but is not fully reflected in
the name and the emphasis of its early advocates on set theory and state-spaces
obscures the richer mathematical resources available. Hence, I suggest dubbing a
mature conception of the formalisation of scientific theories “the Galilean concep-
tion,” echoing his emphasis on mathematics as the revealer of genuine scientific
truth. As the opening quotation establishes, Bradwardine takes temporal prece-
dence in advocating that mathematics is the key to understanding nature but
Galileo, in his famous quotation, is more explicit about the role of mathematics as
the language of science. The Galilean conception explicitly expands the scope of
the mathematical domains that are employed to formalise a theory to encompass
all of mathematics. This is not in conflict with the semantic conception; it is an
extension of it. The Galilean conception contends that the tacit limitation on a
formalisation to set theory or state spaces, as contained within central exposi-
tions of the semantic conception, is an unintended consequence of the framework
(logical empiricism) with which the conception was being contrasted during its
development.^5
This explicit expansion of the resources available is especially important in light
of recent biological theorising with respect to (non-linear dynamical systems (e.g.,
neurobiology, population dynamics).^6 However, as we shall see below, the use
of the rich array of mathematical resources has a prior history even in evolu-
tionary biology. Population genetics from Fisher, Haldane and Wright onward
has provided a formalisation of population dynamics using probability theory and
statistics.
Each of these conceptions can be viewed as a temporal stage in the develop-
ment of a rich conception of the formalisation of scientific theories. Each stage
is a refinement of the previous stage; the different conceptions are less in conflict
than they are points in an intellectual journal. In what follows, I will describe
(^5) A careful reading of F. Suppe, P. Suppes and B. van Fraassen, the key developers and
promoters of the semantic conception, makes clear that they understood “models of theory” in
a rich way that was not at all restricted to set-theoretical or state space presentations.
(^6) See, for example, [Segal, 1989; Amit, 1989; Maurer, 1999; Murray, 1993; Renshaw, 1991].