popper-logic-scientific-discovery

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some structural components of a theory of experience
criticism which is not, I think, without some justification. In particular,
objections have been raised against combining the axiom of convergence
with the axiom of randomness 1 on the ground that it is
inadmissible to apply the mathematical concept of a limit, or of convergence,
to a sequence which by definition (that is, because of the
axiom of randomness) must not be subject to any mathematical rule or
law. For the mathematical limit is nothing but a characteristic property of the
mathematical rule or law by which the sequence is determined. It is merely a property
of this rule or law if, for any chosen fraction arbitrarily close to
zero, there is an element in the sequence such that all elements following
it deviate by less than that fraction from some definite value—
which is then called their limit.
To meet such objections it has been proposed to refrain from combining
the axiom of convergence with that of randomness, and to
postulate only convergence, i.e. the existence of a limit. As to the axiom
of randomness, the proposal was either to abandon it altogether
(Kamke) or to replace it by a weaker requirement (Reichenbach).
These suggestions presuppose that it is the axiom of randomness
which is the cause of the trouble.
In contrast to these views, I am inclined to blame the axiom of
convergence no less than the axiom of randomness. Thus I think that
there are two tasks to be performed: the improvement of the axiom of
randomness—mainly a mathematical problem; and the complete elimination
of the axiom of convergence—a matter of particular concern
for the epistemologist. 2 (Cf. section 66.)
In what follows I propose to deal first with the mathematical, and
afterwards with the epistemological question.
The first of these two tasks, the reconstruction of the mathematical
theory, 3 has as its main aim the derivation of Bernoulli’s theorem—
the first ‘Law of Great Numbers’—from a modified axiom of randomness;
1 Waismann, Erkenntnis 1, 1930, p. 232.
2 This concern is expressed by Schlick, Naturwissenschaften 19, 1931. *I still believe that
these two tasks are important. Although I almost succeeded in the book in achieving
what I set out to do, the two tasks were satisfactorily completed only in the new
appendix *vi.
3 A full account of the mathematical construction will be published separately. *Cf. the
new appendix *vi.