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Mathematical models and methods 541
A typical example of this attitude is to be found in Hull's efforts to express
his theory of learning in mathematical terms. Nevertheless, Hull's ideas on the
application of mathematics to the human or social sciences were close to those
of the psycho-physicists : he sought to express universal relations or discover
laws. Like Tarde in the matter of imitation phenomena, he saw in the exponen-
tial function the preferential expression of learning phenomena. The fact thus
remains that Hull's experiments were of the classical type, analogous to the
procedures already being used by the psycho-physicists.
However, over the last few years - to be precise since the initial efforts of
Bush and Estes - a mathematical theory of learning has been taking shape
which draws largely on the theory of stochastic processes. It is certainly no
exaggeration to say that this new line of development has completely revolu-
tionized the nature and principles of experimentation on the learning process.
The aim is no longer to verify that a hypothetical relation -e.g. between learning
time and the number of syllables memorized - has a particular form. The first
step is to describe the mechanism of the learning process as being hypothecized
in a particular situation. We thus get a stochastic process, whose mathematical
properties are then studied, with the effect that the learning curves - which
earlier theories attempted to describe by using a known function to express
them directly - are now deduced from a theory. In other words, it is no longer a
matter of adjusting a function to a curve, but of deducing the form of the curve
from hypothetical mechanisms expressed in a mathematical form.
There is no point in dwelling on the considerable epistemological consequences
which these changes involve: no longer is it believed that the phenomena of the
learning process follow simple laws like those postulated by Hull, for instance.
It is truer to say that they only follow these simple laws in particular cases.
Hence the principle, obligatory in learning theory henceforward, of expressing
mathematically the learning mechanisms corresponding to each specific learning
situation and deducing the corresponding learning curves from the mathematical
theory thus constructed.
This naturally implies a complete change in experimental research. In effect,
obedience to the rule of expressing learning mechanisms in a mathematical
formimplies therestriction of the study to comparatively simple situations, though
it can be anticipated that as research develops this technical limitation wil be
progressively overcome. Epistemologically the important point is that the
mathematization of learning theory imposes a formal framework on experimen-
tal research which it previously lacked.
These two examples - which are by no means of equal importance - show the
influence of a mathematical methodology on experimental research. Although
the characteristic situation of the modern psychology of the learning process
conforms to a model which is widespread in the natural sciences, it is impossible
to exaggerate its novelty in the human and social sciences. Modern learning
theory is surely one of those rare instances in which, in the same manner as in a
science like physics, experimental research is closely tied in with a mathematical
theory.
Of course, the three new functions of applied mathematics in the human and