popper-logic-scientific-discovery

if sequences of penny tosses showed regularities such as, say, a fairly
regular appearance of tails after every run of three heads. Now the
axiom of randomness postulates of all collectives that there does not
exist a gambling system that can be successfully applied to them. It
postulates that, whatever gambling system we may choose for selecting
supposedly favourable tosses, we shall find that, if gambling is continued
long enough, the relative frequencies in the sequence of tosses
supposed to be favourable will approach the same limit as those in the
sequence of all tosses. Thus a sequence for which there exists a gambling
system by means of which the gambler can improve his chances is not
a collective in the sense of von Mises.
Probability, for von Mises, is thus another term for ‘limit of relative
frequency in a collective’. The idea of probability is therefore applicable
only to sequences of events; a restriction likely to be quite unacceptable
from a point of view such as Keynes’s. To critics objecting to the
narrowness of his interpretation, von Mises replied by stressing the
difference between the scientific use of probability, for example in
physics, and the popular uses of it. He pointed out that it would be
a mistake to demand that a properly defined scientific term has to
correspond in all respects to inexact, pre-scientific usage.
The task of the calculus of probability consists, according to von Mises,
simply and solely in this: to infer certain ‘derived collectives’ with
‘derived distributions’ from certain given ‘initial collectives’ with certain
given ‘initial distributions’; in short, to calculate probabilities
which are not given from probabilities which are given.
The distinctive features of his theory are summarized by von Mises
in four points: 3 the concept of the collective precedes that of probability;
the latter is defined as the limit of the relative frequencies; an
axiom of randomness is formulated; and the task of the calculus of
probability is defined.
51 PLAN FOR A NEW THEORY OF PROBABILITY
The two axioms or postulates formulated by von Mises in order to
define the concept of a collective have met with strong criticism—
3 Cf. von Mises, Wahrscheinlichkeitsrechnung, 1931, p. 22.
probability 141