popper-logic-scientific-discovery

probability 191
probability, the physicist might perhaps offer something like a physical
definition of probability, on lines such as the following: There are certain
experiments which, even if carried out under controlled conditions,
lead to varying results. In the case of some of these experiments—those
which are ‘chance-like’, such as tosses of a coin—frequent repetition
leads to results with relative frequencies which, upon further repetition,
approximate more and more to some fixed value which we may
call the probability of the event in question. This value is ‘ . . . empirically
determinable through long series of experiments to any degree of
approximation’; 1 which explains, incidentally, why it is possible to
falsify a hypothetical estimate of probability.
Against definitions on these lines both mathematicians and logicians
will raise objections; in particular the following:
(1) The definition does not agree with the calculus of probability
since, according to Bernoulli’s theorem, only almost all very long segments
are statistically stable, i.e. behave as if convergent. For that reason,
probability cannot be defined by this stability, i.e. by quasi-convergent
behaviour. For the expression ‘almost all’—which ought to occur in the
definiens—is itself only a synonym for ‘very probable’. The definition is
thus circular; a fact which can be easily concealed (but not removed)
by dropping the word ‘almost’. This is what the physicist’s definition
did; and it is therefore unacceptable.
(2) When is a series of experiments to be called ‘long’? Without
idea of ‘probability hypotheses of first, second, . . . k th order’: a probability hypothesis
of second order, for example, is an estimate of the frequency with which certain frequencies
occur in an aggregate of aggregates. However, P. and T. Ehrenfest do not operate
with anything corresponding to the idea of a reproducible effect which is here used in a
crucial way in order to solve the problem which they expounded so well. See especially
the opposition between Boltzmann and Planck to which they refer in notes 247 f., and
which can, I believe, be resolved by using the idea of a reproducible effect. For under
appropriate experimental conditions, fluctuations may lead to reproducible effects, as
Einstein’s theory of Brownian movement showed so impressively. See also note *1 to
section 65, and appendices *vi and *ix.
1 The quotation is from Born-Jordan Elementare Quantenmechanik, 1930, p. 306, cf. also the
beginning of Dirac’s Quantum Mechanics, p. 10 of the 1st edition, 1930. A parallel passage
(slightly abbreviated) is to be found on p. 14 of the 3rd edition, 1947. See also Weyl,
Gruppentheorie und Quantenmechanik, 2nd edition, 1931, p. 66; English translation by H. P.
Robertson: The Theory of Groups and Quantum Mechanics, 1931, p. 74 f.