popper-logic-scientific-discovery

questions about the origins or ‘sources’ of our estimates. (Cf. section
2.) It is more important, in my opinion, to be quite clear about the fact
that every predictive estimate of frequencies, including one which we
may get from statistical extrapolation—and certainly all those that refer
to infinite empirical sequences—will always be pure conjecture since it
will always go far beyond anything which we are entitled to affirm on
the basis of observations.
My distinction between equal-chance hypotheses and statistical
extrapolations corresponds fairly well to the classical distinction
between ‘a priori’ and ‘a posteriori’ probabilities. But since these terms
are used in so many different senses, 3 and since they are, moreover,
heavily tainted with philosophical associations, they are better
avoided.
In the following examination of the axiom of randomness, I shall
attempt to find mathematical sequences which approximate to random
empirical sequences; which means that I shall be examining
frequency-hypotheses.* 2
58 AN EXAMINATION OF THE AXIOM
OF RANDOMNESS
probability 159
The concept of an ordinal selection (i.e. of a selection according to
position) and the concept of a neighbourhood-selection, have both
been introduced and explained in section 55. With the help of these
concepts I will now examine von Mises’s axiom of randomness—the
principle of the excluded gambling system—in the hope of finding a
weaker requirement which is nevertheless able to take its place. In von
Mises’s theory this ‘axiom’ is part of his definition of the concept of a
collective: he demands that the limits of frequencies in a collective shall
be insensitive to any kind of systematic selection whatsoever. (As he
3 Born and Jordan, for instance, in Elementare Quantenmechanik, 1930, p. 308, use the first of
these terms in order to denote a hypothesis of equal distribution. A. A. Tschuprow, on the
other hand, uses the expression ‘a priori probability’ for all frequency hypotheses, in order to
distinguish them from their statistical tests, i.e. the results, obtained a posteriori, of empirical
counting.
* 2 This is precisely the programme here alluded to in note *1 above, and carried out in
appendices iv and *vi.