popper-logic-scientific-discovery

probability 157
which the corresponding sequence of relative frequencies is convergent.
This restriction amounts to introducing an axiom of convergence.
(The problems connected with this axiom will not be dealt
with until sections 63 to 66, since it turns out to be convenient to
discuss them along with the ‘law of great numbers’.)
Thus we shall be concerned only with mathematical sequences. Yet we
shall be concerned only with those mathematical sequences of which
we expect, or conjecture, that they approximate, as regards frequencies,
to empirical sequences of a chance-like or random character; for these are our main
interest. But to expect, or to conjecture, of a mathematical sequence
that it will, as regards frequencies, approximate to an empirical
one is nothing else than to frame a hypothesis—a hypothesis about the
frequencies of the empirical sequence. 1
The fact that our estimates of the frequencies in empirical random
sequences are hypotheses is without any influence on the way we may
calculate these frequencies. Clearly, in connection with finite classes, it
does not matter in the least how we obtain the frequencies from which
we start our calculations. These frequencies may be obtained by actual
counting, or from a mathematical rule, or from a hypothesis of some
kind or other. Or we may simply invent them. In calculating frequencies
we accept some frequencies as given, and derive other frequencies
from them.
The same is true of estimates of frequencies in infinite sequences.
Thus the question as to the ‘sources’ of our frequency estimates is
not a problem of the calculus of probability; which, however, does not
mean that it will be excluded from our discussion of the problems of
probability theory.
In the case of infinite empirical sequences we can distinguish two
main ‘sources’ of our hypothetical estimates of frequencies—that is to
say, two ways in which they may suggest themselves to us. One is an
estimate based upon an ‘equal-chance hypothesis’ (or equi-probability
hypothesis), the other is an estimate based upon an extrapolation of
statistical findings.
1 Later, in sections 65 to 68, I will discuss the problem of decidability of frequency hypotheses,
that is to say, the problem whether a conjecture or hypothesis of this kind can be
tested; and if so, how; whether it can be corroborated in any way; and whether it is
falsifiable. *Cf. also appendix *ix.