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Mathematical models and methods 533
with whose aid writers endeavour - usually on highly idealized lines - to express
real situations.
This twin tradition was mostly the work of professional mathematicians. It is
anything but dead today. It has, for example, an heir in game theory, which is
an attempt at a mathematical analysis of the rationality of decisions in compet-
itive situations. Similarly, in sociology, we have Rashevsky’s theory of imita-
tion, which analyses highly simplified imitation mechanisms. Though using the
most classical of mathematical tools this theory exhibits a degree of formal
complexity which is far above the elementary level. However, it figures situa-
tions showing so little correspondence with those of real life that it is almost
unanimously regarded as an exercise in pure mathematics, for which the social
reality is no more than a pretext. Of course, Rashevsky cannot be considered a
typical example.
Any theory, whatever it be, must have recourse to idealized and simplified
descriptions. The result is that the simplification can be overdone to a greater
or lesser extent, and the usefulness of the theories can vary. Hence the contrast
between game theory, which is a ‘live’ theory, and Rashevsky’s theory which we
can reckon is already forgotten after a brief life of IO years. Nevertheless,
Rashevsky’s case does serve the useful purpose of underscoring one basic
difficulty: the human and social sciences yield case after case of models con-
ceived on such idealized hypotheses that it is hard to see what scientific
function they can fulfil.
b. The second traditional approach is one whose products would now be
regarded as belonging to the domain of mathematical statistics. As we have
seen, this is a longstanding tradition which goes back to the last years of the
eighteenth century, since it is associated with the name of Laplace. Apart from
Laplace, the chief pioneers in this domain are - as we have seen - Verhulst,
Poisson and Lexis. This traditional approach is inductive: having observed that
the distribution of a given phenomenon presents a particular set of characteris-
tics, the investigator begins by working out mathematical expressions of the
hypothetical mechanisms which he guesses to be operative in the process, and
then inquires whether the distribution deducible from them is comparable to
that observed.
By way of example let us take Verhulst’s concept of logistic distribution,
worked out to explain the propagation phenomena of epidemics. It had been
observed that in drawing the graph of proportions of infected individuals as a
function of time elapsed, the result was a characteristic curve like a stretched-
Figure I