popper-logic-scientific-discovery

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some structural components of a theory of experience
result. For upon these assumptions it is possible to prove that the law of
great numbers is tautological. It is admissible and not selfcontradictory
(as has sometimes been asserted 6 ) to uphold the conclusion
that in an irregular sequence in which, as it were, anything may
happen at one time or another—though some things only rarely—a
certain regularity or stability will appear in very large sub-sequences.
Nor is this conclusion trivial, since we need for it specific mathematical
tools (the Bolzano and Weierstrass theorem, the concept of n-freedom,
and Bernoulli’s theorem). The apparent paradox of an argument from
unpredictability to predictability, or from ignorance to knowledge,
disappears when we realize that the assumption of irregularity can be
put in the form of a frequency hypothesis (that of freedom from aftereffects),
and that it must be put in this form if we want to show the
validity of that argument.
It now also becomes clear why the older theories have been unable
to do justice to what I call the ‘fundamental problem’. The subjective
theory, admittedly, can deduce Bernoulli’s theorem; but it can never
consistently interpret it in terms of frequencies, after the fashion of
the law of great numbers (cf. section 62). Thus it can never explain the
statistical success of probability predictions. On the other hand, the
older frequency theory, by its axiom of convergence, explicitly postulates
regularity in the large. Thus within this theory the problem of
inference from irregularity in the small to stability in the large does not
arise, since it merely involves inference from stability in the large
(axiom of convergence), coupled with irregularity in the small (axiom
of randomness) to a special form of stability in the large (Bernoulli’s
theorem, law of great numbers).* 4
The axiom of convergence is not a necessary part of the foundations
6 Cf., for instance, Feigl, Erkenntnis 1, 1930, p. 254: ‘In the law of great numbers an attempt
is made to reconcile two claims which prove on closer analysis to be in fact mutually
contradictory. On the one hand . . . every arrangement and distribution is supposed to be
able to occur once. On the other hand, these occurrences . . . are to appear with a
corresponding frequency.’ (That there is in fact no incompatibility here is proved by the
construction of model sequences; cf. appendix iv.)
* 4 What is said in this paragraph implicitly enhances the significance, for the solution of
the ‘fundamental problem’, of an objectively interpreted neo-classical theory. A theory of
this kind is described in chapter *iii of my Postscript.