popper-logic-scientific-discovery

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some structural components of a theory of experience
49 THE FUNDAMENTAL PROBLEM OF
THE THEORY OF CHANCE
The most important application of the theory of probability is to what
we may call ‘chance-like’ or ‘random’ events, or occurrences. These
seem to be characterized by a peculiar kind of incalculability which
makes one disposed to believe—after many unsuccessful attempts—
that all known rational methods of prediction must fail in their case.
We have, as it were, the feeling that not a scientist but only a prophet
could predict them. And yet, it is just this incalculability that makes us
conclude that the calculus of probability can be applied to these
events.
This somewhat paradoxical conclusion from incalculability to calculability
(i.e. to the applicability of a certain calculus) ceases, it is true,
to be paradoxical if we accept the subjective theory. But this way of
avoiding the paradox is extremely unsatisfactory. For it entails the view
that the probability calculus is not a method of calculating predictions,
in contradistinction to all the other methods of empirical science. It is,
according to the subjective theory, merely a method for carrying out
logical transformations of what we already know; or rather what we do
not know; for it is just when we lack knowledge that we carry out these
transformations. 1 This conception dissolves the paradox indeed, but it
does not explain how a statement of ignorance, interpreted as a frequency statement, can be
empirically tested and corroborated. Yet this is precisely our problem. How can
we explain the fact that from incalculability—that is, from ignorance—
we may draw conclusions which we can interpret as statements about
empirical frequencies, and which we then find brilliantly corroborated
in practice?
Even the frequency theory has not up to now been able to give a
satisfactory solution of this problem—the fundamental problem of the theory
of chance, as I shall call it. It will be shown in section 67 that this problem
is connected with the ‘axiom of convergence’ which is an integral part
1 Waismann, Erkenntnis 1, 1930, p. 238, says: ‘There is no other reason for introducing the
concept of probability than the incompleteness of our knowledge.’ A similar view is held
by C. Stumpf (Sitzungsberichte der Bayerischen Akademie der Wissenschaften, phil.-hist. Klasse, 1892,
p. 41). *I believe that this widely held view is responsible for the worst confusions. This
will be shown in detail in my Postscript, chapters *ii and *v.