popper-logic-scientific-discovery

probability 175
behaviour which I have described as ‘quasi-convergent’ or ‘statistically
stable’. (Cf. section 61.) By recording statistically the behaviour of long
segments one can establish that the relative frequencies approach
closer and closer to a definite value, and that the intervals within
which the relative frequencies fluctuate become smaller and smaller.
This so-called ‘empirical fact’, so much discussed and analysed, which
is indeed often regarded as the empirical corroboration of the law of
great numbers, can be viewed from various angles. Thinkers with
inductivist leanings mostly regard it as a fundamental law of nature,
not reducible to any simpler statement; as a peculiarity of our world
which has simply to be accepted. They believe that expressed in a
suitable form—for example in the form of the axiom of
convergence—this law of nature should be made the basis of the theory
of probability which would thereby assume the character of a
natural science.
My own attitude to this so-called ‘empirical fact’ is different. I am
inclined to believe that it is reducible to the chance-like character of the
sequences; that it may be derived from the fact that these sequences are
n-free. I see the great achievement of Bernoulli and Poisson in the field
of probability theory precisely in their discovery of a way to show that
this alleged ‘fact of experience’ is a tautology, and that from disorder in
the small (provided it satisfies a suitably formulated condition of nfreedom),
there follows logically a kind of order of stability in the
large.
If we succeed in deducing Bernoulli’s theorem without assuming an
axiom of convergence, then we shall have reduced the epistemological
problem of the law of great numbers to one of axiomatic independence,
and thus to a purely logical question. This deduction would also
explain why the axiom of convergence works quite well in all practical
applications (in attempts to calculate the approximate behaviour of
empirical sequences). For even if the restriction to convergent
sequences should turn out to be unnecessary, it can certainly not be
inappropriate to use convergent mathematical sequences for calculating
the approximate behaviour of empirical sequences which, on
logical grounds, are statistically stable.