popper-logic-scientific-discovery

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some structural components of a theory of experience
being given a criterion of what is to be called ‘long’, we cannot know
when, or whether, we have reached an approximation to the
probability.
(3) How can we know that the desired approximation has in fact been
reached?
Although I believe that these objections are justified, I nevertheless
believe that we can retain the physicist’s definition. I shall support this
belief by the arguments outlined in the previous section. These showed
that probability hypotheses lose all informative content when they are
allowed unlimited application. The physicist would never use them in
this way. Following his example I shall disallow the unlimited application
of probability hypotheses: I propose that we take the methodological
decision never to explain physical effects, i.e. reproducible regularities, as accumulations of
accidents. This decision naturally modifies the concept of probability: it
narrows it.* 2 Thus objection (1) does not affect my position, for I do
not assert the identity of the physical and the mathematical concepts of
probability at all; on the contrary, I deny it. But in place of (1), a new
objection arises.
(1′) When can we speak of ‘accumulated accidents’? Presumably in
the case of a small probability. But when is a probability ‘small’? We
may take it that the proposal which I have just submitted rules out the
use of the method (discussed in the preceding section) of manufacturing
an arbitrarily large probability out of a small one by changing the
formulation of the mathematical problem. But in order to carry out the
proposed decision, we have to know what we are to regard as small.
In the following pages it will be shown that the proposed methodological
rule agrees with the physicist’s definition, and that the objections
raised by questions (1′), (2), and (3) can be answered with its
help. To begin with, I have in mind only one typical case of the application
of the calculus of probability: I have in mind the case of certain
reproducible macro effects which can be described with the help of
precise (macro) laws—such as gas pressure—and which we interpret,
or explain, as due to a very large accumulation of micro processes, such
* 2 The methodological decision or rule here formulated narrows the concept of
probability—just as it is narrowed by the decision to adopt shortest random-like sequences
as mathematical models of empirical sequences, cf. note *1 to section 65.