popper-logic-scientific-discovery

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some structural components of a theory of experience
which the validity of the special multiplication theorem followed, and
with it that of the first binomial formula. In order to make the transition
to the limit, and to obtain Bernoulli’s theorem, it is only necessary
to assume that we may make n as large as we like. From this it can be
seen that Bernoulli’s theorem is true, approximately, even for finite
sequences, if they are n-free for an n which is sufficiently large.
It seems therefore that the deduction of Bernoulli’s theorem does
not depend upon an axiom postulating the existence of a frequency
limit, but only on ‘absolute freedom’ or randomness. The limit concept
plays only a subordinate rôle: it is used for the purpose of applying
some conception of relative frequency (which, in the first instance, is
only defined for finite classes, and without which the concept of nfreedom
cannot be formulated) to sequences that can be continued
indefinitely.
Moreover, it should not be forgotten that Bernoulli himself deduced
his theorem within the framework of the classical theory, which contains
no axiom of convergence; also, that the definition of probability
as a limit of frequencies is only an interpretation—and not the only
possible one—of the classical formalism.
I shall try to justify my conjecture—the independence of Bernoulli’s
theorem of the axiom of convergence—by deducing this theorem
without assuming anything except n-freedom (to be appropriately
defined).* 1 And I shall try to show that it holds even for those mathematical
sequences whose primary properties possess no frequency limits.
Only if this can be shown shall I regard my deduction of the law of
great numbers as satisfactory from the point of view of the epistemologist.
For it is a ‘fact of experience’—or so at least we are sometimes
told—that chance-like empirical sequences show that peculiar
* 1 I still consider my old doubt concerning the assumption of an axiom of convergence,
and the possibility of doing without it, perfectly justified: it is justified by the developments
indicated in appendix iv, note *2, and in appendix *vi, where it is shown that
randomness (if defined by ‘shortest random-like sequences’) entails convergence which
therefore need not be separately postulated. Moreover, my reference to the classical
formalism is justified by the development of the neo-classical (or measure-theoretical)
theory of probability, discussed in chapter *iii of the Postscript; in fact, it is justified by
Borel’s ‘normal numbers’. But I do not agree any longer with the view implicit in the
next sentence of my text, although I agree with the remaining paragraphs of this section.