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Mathematical models and methods 569
mathematical instrumentation characteristic of the beginnings of methodological
research. The same type of preoccupation seems as late as Hull, whose work
on learning is of considerable importance.43
We should however note that Hull's mathematicizations have already got a
good way beyond the stage of simply trying to fit a function to a curve: for
example, he introduces non-observable factors. Thus he postulates the existence
of a non-observable variable which he calls 'habit strength' (sH,) with the
assumption that it is linked to the number n of reinforcements by the relation
,H, = m(I-e- '"),
where m and i are positive constants. From this equation other equations are
developed. Thus Hull uses the results of certain experiments to assert that the
quantity rn is to be considered as a function of the reward (w) attached to
learning, which gives the new equation:
m = in' (I-e-'"),
where m' and k are new constants. He further postulates that the upper limit
of ,H,, that is rn, decreases as the interval of time t between the response and the
reinforcement increases, whence :
m' = rn" e - Jt.
These examples are enough to characterize Hull's theory from an epistemological
point of view. Firstly he deems it necessary to introduce non-observed
variables of which the observed quantities are indicators (thus, N is the indicator
of &). In so doing he is performing a process comparable to Fechner's on
Weber's law. Secondly the relations between the variables are given an analytical
form imposed a priori: as can be seen from the examples given, the chosen
function, in accordance with the tradition of learning theory, is exponential in
form. It wil furthermore be observed that the variable ,H, is conceived as an
average (average of performances). In other words, there is no question of
analysing the learning process in sequential terms. Finally, the situations analysed
are essentially acquirement situations.
The epistemology accepted by Bush, Estes, Burke and the like is different.
Again non-observable variables are introduced, but the mathematics does not
consist in the analytical expression of an observed curve. It is the actual learning
mechanisms, obviously hypothetical, which are expressed in the form of a
mathematical theory. The learning curve is a consequence of this theory.
To give an idea of the importance of this change, there follows a brief description
of Estes and Burke's mode144 which, just as much as Hull's theory, falls
within the acquisition experiment framework. The experiments which provide
the basis for Estes and Burke's formalization are Humphrey's, and their design
is as follows: the left-hand light lights up; the subject must then guess if the
right-hand one is going to light up as well. On the experimenter's side, he may
choose to 'reinforce' the response with varying frequency (for instance for o %,
25 % or 100 % of the 'queries' according to the experiment). Formally, the subject
has a choice between two responses, A, and A,. The experimenter chooses
between E, (reinforcement of A,, or a sign that A, is the correct response) and Ez