popper-logic-scientific-discovery

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some structural components of a theory of experience
empirical sequences as chance-like or random of which we hypothetically
assume that they have no frequency limit. But what possible
action could we take in response to this warning? 4 What sort of considerations
or conjectures about the possible convergence or
divergence of empirical sequences should we indulge in or abstain
from, in view of this warning, seeing that criteria of convergence are
no more applicable to them than are criteria of divergence? All these
embarrassing questions 5 disappear once the axiom of convergence has
been got rid of.
Our logical analysis thus makes transparent both the form and the
function of the various partial requirements of the system, and shows
what reasons tell against the axiom of randomness and in favour of the
requirement of uniqueness. Meanwhile the problem of decidability
seems to be growing ever more menacing. And although we are not
obliged to call our requirements (or axioms) ‘meaningless’, 6 it looks
as if we were compelled to describe them as non-empirical. But does
not this description of probability statements—no matter what words
we use to express it—contradict the main idea of our approach?
67 A PROBABILISTIC SYSTEM OF
SPECULATIVE METAPHYSICS
The most important use of probability statements in physics is this:
certain physical regularities or observable physical effects are
4 Both the axiom of randomness and the axiom of uniqueness can properly be regarded
as such (intensional) warnings. For example, the axiom of randomness cautions us not to
treat sequences as random if we suppose (no matter on what grounds) that certain
gambling systems will be successful for them. The axiom of uniqueness cautions us
not to attribute a probability q (with q ≠ p) to a sequence which we suppose can be
approximately described by means of the hypothesis that its probability equals p.
5 Similar misgivings made Schlick object to the limit axiom (Die Naturwissenschaften 19,
1931, p. 158).
6 Here the positivist would have to recognize a whole hierarchy of ‘meaninglessnesses’.
To him, non-verifiable natural laws appear ‘meaningless’ (cf. section 6, and quotations in
notes 1 and 2), and thus still more so probability hypotheses, which are neither verifiable
nor falsifiable. Of our axioms, the axiom of uniqueness, which is not extensionally
significant, would be more meaningless than the meaningless axiom of irregularity,
which at least has extensional consequences. Still more meaningless would be the limit
axiom, since it is not even intensionally significant.