popper-logic-scientific-discovery

probability 155
called a ‘reference-sequence’. It more or less corresponds to a
‘collective’ in von Mises’s sense.* 1
The concept of n-freedom presupposes that of relative frequency; for
what its definition requires to be insensitive—insensitive to selection
according to certain predecessors—is the relative frequency with which a
property occurs. In our theorems dealing with infinite sequences I shall
employ, but only provisionally (up to section 64), the idea of a limit of
relative frequencies (denoted by F′), to take the place of relative frequency in
finite classes (F″). The use of this concept gives rise to no problem so long
as we confine ourselves to reference-sequences which are constructed
according to some mathematical rule. We can always determine for such
sequences whether the corresponding sequence of relative frequencies
is convergent or not. The idea of a limit of relative frequencies leads to
trouble only in the case of sequences for which no mathematical rule is
given, but only an empirical rule (linking, for example the sequence
with tosses of a coin); for in these cases the concept of limit is not
defined (cf. section 51).
An example of a mathematical rule for constructing a sequence is
* 1 I come here to the point where I failed to carry out fully my intuitive programme—
that of analysing randomness as far as it is possible within the region of finite sequences,
and of proceeding to infinite reference sequences (in which we need limits of relative
frequencies) only afterwards, with the aim of obtaining a theory in which the existence
of frequency limits follows from the random character of the sequence. I could have
carried out this programme very easily by constructing, as my next step (finite) shortest nfree
sequences for a growing n, as I did in my old appendix iv. It can then be easily shown
that if, in these shortest sequences, n is allowed to grow without bounds, the sequences
become infinite, and the frequencies turn without further assumption into frequency
limits. (See note *2 to appendix iv, and my new appendix *vi.) All this would have
simplified the next sections which, however, retain their significance. But it would have
solved completely and without further assumption the problems of sections 63 and 64;
for since the existence of limits becomes demonstrable, points of accumulation need no
longer be mentioned.
These improvements, however, remain all within the framework of the pure frequency
theory: except in so far as they define an ideal standard of objective disorder, they
become unnecessary if we adopt a propensity interpretation of the neo-classical
(measure-theoretical) formalism, as explained in sections *53 ff of my Postscript. But even
then it remains necessary to speak of frequency hypotheses—of hypothetical estimates
and their statistical tests; and thus the present section remains relevant, as does much in
the succeeding sections, down to section 64.