popper-logic-scientific-discovery

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some structural components of a theory of experience
be regarded as a decisive argument, as a kind of experimentum crucis, in
favour of the statistical interpretation of the quantum theory.
76 AN ATTEMPT TO ELIMINATE METAPHYSICAL
ELEMENTS BY INVERTING HEISENBERG’S
PROGRAMME; WITH APPLICATIONS
If we start from the assumption that the formulae which are peculiar to
quantum theory are probability hypotheses, and thus statistical statements,
then it is difficult to see how prohibitions of single events could
be deduced from a statistical theory of this character (except perhaps in
the cases of probabilities equal to one or to zero). The belief that single
measurements can contradict the formulae of quantum physics seems
logically untenable; just as untenable as the belief that a contradiction
might one day be detected between a formally singular probability
statement αP k(β) = p (say, ‘the probability that the throw k will be a five
equals 1/6’) and one of the following two statements: k εβ (‘the throw
is in fact a five’) or k εβ - (‘the throw is in fact not a five’).
These simple considerations provide us with the means of refute any
of the alleged proofs which have been designed to show that exact
measurements of position and momentum would contradict the quantum
theory; or which have been designed, perhaps, to show that the
mere assumption that any such measurements are physically possible
must lead to contradictions within the theory. For any such proof must
make use of quantum-theoretical considerations applied to single particles;
which means that it has to make use of formally singular probability
statements, and further, that it must be possible to translate the
proof—word for word, as it were—into the statistical language. If we
do this then we find that there is no contradiction between the single
measurements which are assumed to be precise, and the quantum
theory in its statistical interpretation. There is only an apparent contradiction
between these precise measurements and certain formally singular
probability statements of the theory. (In appendix v an example
of this type of proof will be examined.)
But whilst it is wrong to say that the quantum theory rules out exact
measurements, it is yet correct to say that from formulae which are
peculiar to the quantum theory—provided they are interpreted