506 Paul Thompson
The primary concept of set theory, that of a set, has already been used
extensively without any explicit definition. This was possible because
the meaning of set within set theory is very close to its meaning in its
everyday usage (and it was necessary becausesetis a primitive term of
set theory and hence cannot be defined explicitly). However, in order
to go further in this discussion of subclans, I will need to use a few
notions from set theory which are not self explanatory; therefore, this
section will be used to describe or define them. (p.353)She then describes or defines: null set, subset, intersection, union and relative
complement.
Consequently, her formalisation is more appropriately regarded as a semantic
or Galilean conception formalisation. Once it is seen as a semantic conception
of theories formalisation, it can be regarded properly as a powerful example of
the benefits of formalisation and as one of the family of models that make up
evolutionary theory. Although Williams uses the term “evolution” in the title of
her paper, she is quite explicit that she is formalising Darwin’s theory of evolution
as set out in theOrigin of Species.
Here, and throughout the paper, I use “Darwin’s theory of evolution”
to denotethe part of the present theoryfor which Darwin provided all
of the crucial explanatory principles; briefly this is the theory of ana-
genesis, or descent with adaptive modification. (p. 344 italics added
for emphasis)As the italicised phase makes clear, Williams acknowledges that this is only
part of the present theory of evolution.
What follows is a summary; a complete presentation of her formalisation would
require duplicating most of her paper. Those interested in her impressive formali-
sation would be better served by going to the paper itself. A flavour of the elements
of the formalisation, however, can be given easily since Williams provides both an
informal and a formal (mathematical) statement of each of the axioms and theo-
rems which she deduces from them. The formalisation and its value reside in the
mathematical formulation but the informal statement captures the essence of her
approach. The informal statement of the seven axioms (fundamental principles)
is as follows (although the sequence corresponds to that in the paper, the actual
numbering does not):
- No biological entity is a parent of itself.
- If b 1 is an ancestor of b 2 , then b 2 is not an ancestor of b 1.
- Every Darwinian subclan is a subclan of a clan in some biocosm;
- the clan of a setSis the set of all its descendants;
- a subclan is either a whole clan or a clan with several branches removed;