9. Indeterminacy in Classical Physics. Popper
questioned the absence, in principle, of indetermin-
acies, and in particular of u-indeterminacies, in classi-
cal physics. Calling a theory indeterministic if it asserts
that at least one event is not completely determined
in the sense of being not predictable in all its details,
Popper attempted to prove on logical grounds that
classical physics is indeterministic since it contains
u-indeterminacies (Popper, 1950). He derived this con-
clusion by showing that no “predictor,” i.e., a calculat-
ing and predicting machine (today we would say sim-
ply “computer”), constructed and working on classical
principles, is capable of fully predicting every one of
its own future states; nor can it fully predict, or be
predicted by, any other predictor with which it inter-
acts. Popper's reasoning has been challenged by G. F.
Dear on the grounds that the sense in which “self-
prediction” was used by Popper to show its impossibil-
ity is not the sense in which this notion has to be used
in order to allow for the effects of interference (Dear,
1961). Dear's criticism, in turn, has recently been
shown to be untenable by W. Hoering (Hoering, 1969)
who argued on the basis of Leon Brillouin's penetrating
investigations (Brillouin, 1964) that “although Popper's
reasoning is open to criticism he arrives at the right
conclusion.”
That classical physics is not free of u-indeterminacies
was also contended by Max Born (Born, 1955a; 1955b)
who based his claim on the observation that even in
classical physics the assumption of knowing precise
initial values of observables is an unjustified idealization
and that, rather, small errors must always be assigned
to such values. As soon as this is admitted, however,
it is easy to show that within the course of time these
errors accumulate immensely and evoke serious in-
determinacies. To illustrate this idea Born applied
Einstein's model of a one-dimensional gas with one
atom which is assumed to be confined to an interval
of length L, being elastically reflected at the endpoints
of this interval. If it is assumed that at time t = 0 the
atom is at x = x0 and its velocity has a value between
v0 and v0 + Δv0, it follows that at time t = L/Δv0,
the position-indeterminacy equals L itself, and our
initial knowledge has been converted into complete
ignorance. In fact, even if the initial error in the posi-
tion of every air molecule in a row is only one millionth
of a percent, after less than one micro-second (under
standard conditions) all knowledge about the air will
be effaced. Thus, according to Born, not only quantum
physics, but already classical physics is replete with
u-indeterminacies which derive from unavoidable
i-indeterminacies.
The mathematical situation underlying Born's
reasoning had been the subject of detailed investi-
gations in connection with problems about the stability
of motion at the end of the last century (Liapunov,
Poincaré), but its relevancy for the indeterminacy of
classical physics was pointed out only quite recently
(Brillouin, 1956).
Born's argumentation was challenged by von Laue
(von Laue, 1955), and more recently also by Margenau
and Cohen (Margenau and Cohen, 1967). As Laue
pointed out, the indeterminacy referred to by Born is
essentially merely a technical limitation of measure-
ment which in principle can be refined as much as
desired. If the state of the system is represented by
a point P in phase-space, observation at time t = 0
will assign to P a phase-space volume V0 which is larger
the greater the error in measurement. In accordance
with the theory it is then known that at time t = t1
the representative point P is located in a volume V1
which, according to the Liouville theorem of statistical
mechanics, equals V0. If, now, at t = t1 a measure-
ment is performed, P will be found in a volume V′1
which, if theory and measurement are correct, must
have a nonzero intersection D1 with V1. D1 is smaller
than V1 and hence also smaller than V0. To D1, as a
subset of V1, corresponds a subset of V0 so that the
initial indeterminacy, even without a refinement of the
immediate measurement technique, has been reduced.
Since this corrective procedure can be iterated ad
libidum and thus the “orbit” of the system defined with
arbitrary accuracy, classical mechanics has no un-
eliminable indeterminacies. In quantum mechanics, on
the other hand, due to the unavoidable interference
of the measuring device upon the object of measure-
ment, such a corrective procedure does not work; in
other words, the volume
V0 in phase-space cannot be
made smaller than
hn, where
n is the number of the
degrees of freedom of the system, and quantum-
mechanical indeterminacy is an irreducible fact. This
fundamental difference between classical and quantum
physics has its ultimate source in the different concep-
tions of an objective (observation-independent) physi-
cal reality.
