336 A. W. F. Edwards
of potentials and fields of force in gravitational and electromagnetic theory. The
great 18th century developments in the calculus not only provided the necessary
mathematical tools but also encouraged thinking in terms of maxima and minima.
No-one supposed any longer that nature was economical. Rather, there was an
implicit appeal to the classical concept of parsimony, that the simplest mathe-
matical approach to any problem should be preferred, and this often involved an
extremum principle. Ockham’s razor cutaway theunnecessary mathematics but
offered no explanation of nature.
A quite different kind of extremum principle grew up more-or-less simultane-
ously in statistics in these centuries, the most famous example being the method
of least squares originating in the work of Legendre and Gauss. Far from being a
way of solving problems with mathematical economy, as is Fermat’s principle, it
is a method for selecting among probability models that one which most closely
fits some observed data (in Gauss’s case, the observations of cometary positions).
There is an element of arbitrariness in the choice of the criterion to be minimised
(here the sum of the squares of the residual errors unexplained by the model) but
subsequent research in statistical inference has produced plenty of justifications,
the most dominant of which is Fisher’s method of maximum likelihood.
Both these types of extremum principle have their analogues in evolutionary
studies. First, we shall look at attempts to model evolution as a process that
maximises ‘fitness’.
3 EVOLUTION AS FITNESS-MAXIMISATION. (1) FISHER’S
‘FUNDAMENTAL THEOREM’An English proverb reminds us that ‘Nothing succeeds like success’ but it was
not until 1930 that anyone put figures to it. InThe Genetical Theory of Natural
Selectionpublished that year R. A. Fisher introduced his ‘Fundamental Theorem
of Natural Selection’. We shall discuss it in more detail in due course, but for
the moment we note that its basis is a simple ‘growth-rate’ theorem: ‘The rate
of increase in the growth-rate of a subdivided population is proportional to the
variance in growth-rates’. It is obvious that if the subdivisions of a population are
growing at different rates then those subdivisions with the fastest rates of growth
will increasingly dominate so that the overall growth-rate will increase. (‘Growth’
could here be negative too, but we describe the simplest case.) In full:
If a population consists of independent groups each with its own growth-
rate (the factor by which it grows in a given time interval), the change
in the population growth-rate in this interval is equal to the variance in
growth-rates amongst the groups, divided by the population growth-
rate. [Edwards, 1994, 2002a]To prove this, let a population be subdivided intokparts in proportionspi
(i= 1,2,...,k)andlettheith part change in size by a factorwiin unit time