Formalisations of Evolutionary Biology 497confirmation of a theory consists in establishing that the required isomorphism
holds. Explanation and prediction consists in demonstrating that the element(s) in
the model that bijectively maps to the empirical event being explained or predicted
is a consequence of the structure and behaviour of the model, where the model is
deemed to have an acceptable level of confirmation.^19
The actual task of confirming a theory (model) is complex involving, among
others, theories of measurement, experiment, experimental design, goodness of fit,
data modelling, etc. A theory of measurement provides an agreed upon standard
in terms of which observed phenomena are compared as well as a set of principles
governing the conditions under which measurements are made. These principles
ensure that the measurements are in accord with the theory of the experiment.
A theory of the experiment specifies a broad conceptual framework within which
experiments can take place. It specifies such things as what assumptions based on
other scientific theories can be employed (for example electromagnetic theory and
quantum theory when using an electron microscope in a biological experiment),
the possibility and role of simplifying assumptions, correct patterns of inference,
etc. A theory of experimental design specifies the exact nature of the technique of
experimentation. The appropriate methods for controlling extraneous variables are
an important component specified by a theory of experimental design. A theory of
goodness of fit (often a statistical measure such as aX^2 -test^20 (chi-squared test),
an example of which is given in the population genetics section below^21 ) specifies
when normalised data (by application of a model of data, often a probabilistic
or statistical theory, especially regression analysis^22 employing, for example, a
(^19) A theory (mathematical model) explains, or predicts, a phenomenon, or number of phenom-
ena, if:
- the system defined by the theory is isomorphic to the phenomenal system in which the
set of phenomena to be explained occur, and - the set of elements of the mathematical model which are mapped onto the set of relevant
phenomenal objects within the phenomenal system can be shown, within the mathematical
model, to be a consequence of the structure or behaviour of the model.
From an abstract mathematical point of view, explanation is a description of the structure and
behaviour of phenomenal systems in terms of the structure and behaviour of a mathematical
model. As such, the validity of the explanation is necessarily related to validity of the assertion
of an isomorphic relationship between phenomenal system and mathematical model. This is
similar to the standard view, in which the validity of an explanation is necessarily dependent
upon the validity of the laws of the theory to which an appeal is being made. Not surprisingly,
therefore, confirmation is central to the empirical enterprise in both cases. The assertion of an
isomorphism, is the foundational link between a theory and a phenomenal system. Confirmation
grounds the assertion. Confirmation is complex and multilayered.
(^20) This test was devised by the British scientist, mathematician and philosopher of science,
Karl Pearson, in 1900 (his positivist philosophy of science is largely contained in Pearson
[1892], the chi-squared test is found Pearson [1990]).X^2 =
P
(observed quantity — expected
quantity)^2 /(expected quantity). The quantity is often a frequency.
(^21) Another technique for determining goodness of fit is to compare a multidimensional graphic
representation of a theoretically predicted distributional landscape and distributional landscape
of observed results. Hartl [2000, pp.88-94] provides an excellent example of this with respect to
diffusion approximations of random genetic drift.
(^22) For an excellent mathematical presentation of regression analysis see [Sen and Srivastava,