popper-logic-scientific-discovery

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some structural components of a theory of experience
from the formally singular probability statements of the theory, as
probability statements in their turn, and again as formally singular, if
they apply to a single particle. They too must therefore be interpreted,
ultimately, as statistical assertions.
As against the subjective interpretation, ‘The more precisely we
measure the position of a particle the less we can know about its
momentum’, I propose that an objective and statistical interpretation of
the uncertainty relations should be accepted as being the fundamental
one; it may be phrased somewhat as follows. Given an aggregate of
particles and a selection (in the sense of a physical separation) of those
which, at a certain instant, and with a certain given degree of precision,
have a certain position x, we shall find that their momenta px will show
random scattering; and the range of scatter, ∆px, will thereby the
greater, the smaller we have made ∆x, i.e. the range of scatter or
imprecision allowed to the positions. And vice versa: if we select, or
separate, those particles whose momenta px all fall within a prescribed
range ∆px, then we shall find that their positions will scatter in a random
manner, within a range ∆x which will be the greater, the smaller
we have made ∆px, i.e. the range of scatter or imprecision allowed to the
momenta. And finally: if we try to select those particles which have
both the properties ∆x and ∆px, then we can physically carry out such a
selection—that is, physically separate the particles—only if both ranges
are made sufficiently great to satisfy the equation ∆x . ∆px � h
. This
objective interpretation of the Heisenberg formulae takes them as
asserting that certain relations hold between certain ranges of scatter;
and I shall refer to them, if they are interpreted in this way, as the
‘statistical scatter relations’.* 1
In my statistical interpretation I have so far made no mention of
* 1 I still uphold the objective interpretation here explained, with one important change,
however. Where, in this paragraph, I speak of ‘an aggregate of particles’ I should now
speak of ‘an aggregate—or of a sequence—of repetitions of an experiment undertaken
with one particle (or one system of particles)’. Similarly, in the following paragraphs; for
example, the ‘ray’ of particles should be re-interpreted as consisting of repeated experiments
with (one or a few) particles—selected by screening off, or by shutting out,
particles which are not wanted.
4π