Philosophy of Biology

Formalisations of Evolutionary Biology 505

gramming languages likeMathematicaandMaple, the dynamics of systems have
be formalised algorithmically and the algorithms “run” to produce a computer
simulation of the behaviour.^36

3 THE FORMALISATION OF SOME COMPONENT THEORIES WHICH

COMPRISE CONTEMPORARY EVOLUTIONARY THEORY

3.1 Formalisation in Selection Theory

I have in various places above indicated that Mary William’s formalisation of
selection theory is not a syntactic conception formalisation. It is an axiomatisation
and draws on the deductive machinery of mathematics:

It shows this by stating them [the seven fundamental principles she

identifies as implicit in Darwin’sOrigin of Species] in a sufficiently

rigorous form to allow the deductive techniques ofmathematicsto

be used to derive from them other principles of the theory (Williams

p. 344 italics added for emphasis)

Deduction and axiomatisation, however, are features of the syntactic, semantic
and Galilean conceptions. This is quite explicit in Patrick Suppes’ set-theoretical
predicate version of the semantic view but is also true of topological versions. What
is specific to the syntactic conception is the requirement that the axiomatisation
be in first-order predicate logic with identity and that semantics are provided by
correspondence rules. Even if one weakens the conception to permit the semantics
to be given using models, it is the requirement that the axiomatisation be in first-
order predicate logic that separates it from the semantic and Galilean conceptions.
Mary William’s axiomatisation is not in first-order predicate logic. It is in set
theory, and the deductive apparatus employed is that of set theory. To anyone
who studies her formulation of the fundamental principles and their deductive
consequences, it will be clear that she has provided a set theoretical axiomatisation.
But even a cursory examination of the paper makes this clear. Consider the
following:

Informal Theorem B 3

Every theorem that mathematicians have proved about strict partial

ordering is automatically a true theorem about the ancestor relation.

Therefore biologist’s knowledge about the ancestor relation can be en-

riched by mathematician’s knowledge about strict partial orderings.

(p. 348)

The definition of “strict partial ordering” that she employs is from set theory
and the specific definition she uses is from Patrick Suppes’Axiomatic Set Theory
[1960]:

(^36) See, for example, [Gaylord and Nishidate, 1996].