Formalisations of Evolutionary Biology 507- a biocosm is a set of subclans in which change through time is directed
by the laws of natural selection.
- There is an upper limit to the number of organisms in any generation of a
Darwinian subclan. - For each organism there is a positive real number which describes its fitness
in its particular environment. - Consider a subclandD 1 ofD.IfD 1 is superior in fitness to the rest ofD
for sufficiently many generations (where how many is “sufficiently many” is
determined by how superiorD 1 is and how largeD 1 is), then the proportion
ofD 1 inDwill increase;
- a subcland is a subclan which is contained within a subclan (i.e., a
sub-subclan)
- In every generationmof a Darwinian subclanDwhich is not on the verge
of extinction, there is a subclandD 1 such that: D 1 is superior to the rest
ofDfor long enough to ensure thatD 1 will increase relative toD;andas
long asD 1 is not fixed inDit retains sufficient superiority to ensure further
increases relative toD.
In their mathematical formulation, these seven axioms allow numerous theorems
to be deduced. For example (again expressed informally):
- No biological entity is its own ancestor.
- IfC 1 (m) is a subset ofC 2 (m), then for all subsequent generationsk,C 1 (k)
must be a subset ofC 2 (k). - The fitness of an organism relative to a subclan to which it belongs is always
positive. - For every Darwinian subclanDwhich does not die out, there is an infinite
sequence of subclands,D 1 ,D 2 , ..., such that each subcland is contained
in it predecessor and each is fitter than its predecessor for long enough for
natural selection to ensure that it becomes fixed inD.
In all Williams derives 20 theorems from the axioms. The utility of formalisation
is demonstrated by her ability to provide a rigorous deductive proof for each of the
theorems once a mathematical (set theoretical) formulation of each of the axioms
is given. As she correctly points out, many more theorems could be proved but
the 20 she provides more than justify the claim she and, in similar words, many
others, including me, have made:
The value of such a deductive system (or axiomatisation) is that it
provides a technique for discovering new phenomena which are too
remote from observation or too unintuitive to ever be discovered with
the unaided intuition.... (p.343)