Indeterminacy in Physics

1. Introduction. The idea of “indeterminacy”—
often also called loosely “uncertainty”—is widely used
in physics and is of particular importance for quantum
mechanics, but has rarely, if ever, been given a strict
explicit definition. As a consequence it is used in at
least three different, though related, meanings which
will be sharply distinguished in the present article. (a)
It may denote any type of acausal (accidental, contin-
gent, indeterministic) behavior of physical processes,
usually in the realm of microphenomena, implying
thereby a total or partial breakdown of the principle
of causality; (b) it may denote any type of unpredict-
behavior of such processes without necessarily
involving a renunciation of metaphysical causality; (c)
it may denote an essential limitation or imprecision of
measurement procedures for reasons to be specified by
a concomitant theory of measurement.

To avoid ambiguities we shall in what follows call
indeterminacy if used in the sense (a), acausal indeter-
minacy or briefly a-indeterminacy; if used in the sense
(b), u-indeterminacy; and if used in the sense (c),
i-indeterminacy. Clearly, a-indeterminacy implies, but
is not implied by, u-indeterminacy, and does not imply,
nor is implied by, i-indeterminacy; neither do the other
two imply their partners. If however predictability is
understood to refer exclusively to sharp values of
measurement results, i-indeterminacy implies u-inde-
terminacy. Furthermore, the validity of these concepts
may depend on the domain in which they are applied.
Thus a- or u-indeterminacies may be valid in micro-
physics but not in macrophysics.

2. Ancient Conceptions. In fact, the earliest known
thesis of indeterminacy restricted this notion to a
definite realm of applicability. According to Plato's
Timaeus (28D-29B) the Demiurge created the material
world after an eternal pattern; while the latter can
be spoken of with certainty, the created copy can be
described only in the language of uncertainties. In
other words, while the intelligible world, the realm
of ideas, is subject to strict laws, rigorous determi-
nations and complete predictability, the physical or
material world is not. However, even disregarding this
dichotomy of being, Plato's atomic theory admitted an
a-indeterminacy in the subatomic realm, whereas in
the world of atoms and their configurations to higher
orders determinacy was reinstated. “However strictly
the principle of mathematical order is carried through
in Plato's physics in the cosmos of the fixed stars as
well as in that of the primary elements,” writes an
eminent Plato scholar, “everything is indeterminate in
the realm below the order of the elementary atoms.
... What resists strict order in nature is due to the
indeterminate and uneven forces in the Receptacle”
(Friedländer, 1958). Indeed, for P. Friedländer Plato's
doctrine of the unintelligible subatomic substratum is
“an ancient anticipation of a most recent develop-
ment,” to wit: W. Heisenberg's uncertainty principle.
Still, whether such a comparison is fully justified may
be called into question.

An undisputable early example of indeterminacy, in
any case, is Epicurus' theory of the atomic “swerve”
(clinamen). Elaborating on Democritus' atomic theory
and his strict determinism of elementary processes,


Epicurus contended that “through the undisturbed void
all bodies must travel at equal speed though impelled
by unequal weights” (Lucretius II, lines 238-39),
anticipating thereby Galileo's conclusion that light and
heavy objects fall in the vacuum with the same speed.
Since consequently the idea that compounds are
formed by heavy atoms impinging upon light ones had
to be given up, “nature would never have created
anything.” To avoid this impasse, Epicurus resorted to
a device, the theory of the swerve, which some critics,
such as Cicero and Plutarch, regarded as “childish”;
others, like Guyau or Masson, as “ingenious.” “When
the atoms are travelling straight down through empty
space by their own weight, at quite unpredictable
times and place (incerto tempore incertisque locis), they
swerve ever so little from their course, just so much
that you can call it a change of direction” (Lucretius
II, lines 217-20). To account for change in the physical
world Epicurus thus saw it necessary to break up the
infinite chain of causality in violation of Leucippus'
maxim that “nothing occurs by chance, but there is
a reason and a necessity for everything.” This indeter-
minacy which, as the quotation shows, is both an
a-indeterminacy and a u-indeterminacy, made it possi-
ble for Epicurus to imbed a doctrine of free will within
the framework of an atomic theory.

In the extensive medieval discussions (Maier, 1949)
on necessity and contingency which were based, so far
as physical problems were concerned, on Aristotle's
Physics (Book II, Chs. 4-6, 195b 30-198a 13), the
existence of chance is recognized, but not as a breach
in necessary causation; it is regarded as a sequence of
events in which an action or movement, due to some
concomitant factor, produces exceptionally a result
which is of a kind that might have been naturally, but
was not factually, aimed at (Weiss, 1942). The essence
of chance or contingency is not the absence of a neces-
sary connection between antecedents and results, but
the absence of final causation. Absolute indeterminacy
in the sense of independence of antecedent causation
was exclusively ascribed to volitional decisions.

3. Indeterminacy as Contingency. With the rise of
Newtonian physics and its development, Laplacian
determinism gained undisputed supremacy. Only in the
middle of the nineteenth century did it wane to some
extent. One of the earliest to regard contingent events
in physics—an event being contingent if its opposite
involves no contradiction—as physically possible was
A. A. Cournot (Cournot, 1851; 1861). Charles Re-
nouvier, following Cournot, questioned the strict va-
lidity of the causality principle as a regulative deter-
minant of physical processes (Renouvier, 1864). A
philosophy of nature based on contingency was pro-
posed by Émile Boutroux, who regarded rigorous
determinism as expressed in scientific laws as an inade-
quate manifestation of a reality which in his opinion
is subject to radical contingency (Boutroux, 1874). The
rejection of classical determinism at the atomic level
played an important role also in Charles Sanders
Peirce's theory of tychism (Greek: tyche = chance)
according to which “chance is a basic factor in the
universe.” Deterministic or “necessitarian” philosophy
of nature, argued Peirce, cannot explain the undeniable
phenomena of growth and evolution. Another incon-
testable argument against deterministic mechanics was,
in his view, the incapability of the necessitarians to
prove their contention empirically by observation or
measurement. For how can experiment ever determine
an exact value of a continuous quantity, he asked, “with
a probable error absolutely nil?” Analyzing the process
of experimental observation, and anticipating thereby
an idea similar to Heisenberg's uncertainty principle,
Peirce arrived at the conclusion that absolute chance,
and not an indeterminacy originating merely from our
ignorance, is an irreducible factor in physical processes:
“Try to verify any law of nature, and you will find
that the more precise your observations, the more
certain they will be to show irregular departures from
the law. We are accustomed to ascribe these, and I
do not say wrongly, to errors of observation; yet we
cannot usually account for such errors in any anteced-
ently probable way. Trace their causes back far enough
and you will be forced to admit they are always due
to arbitrary determination, or chance” (Peirce, 1892).
The objection raised for instance by F. H. Bradley, that
the idea of chance events is an unintelligible concep-
tion, was rebutted by Peirce on the grounds that the
notion as such has nothing illogical in it; it becomes
unintelligible only on the assumption of a universal
determinism; but to assume such a determinism and
to deduce from it the nonexistence of chance would
be begging the question.

4. Classical Physics and Indeterminacy. The various
theses of indeterminacies in physics mentioned so far
have been advanced by philosophers and not by
physicists, the reason being, of course, that classical
physics, since the days of Newton and Laplace, was
the paradigm of a deterministic and predictable sci-
ence. It was also taken for granted that the precision
attainable in measurement is theoretically unlimited;
for although it was admitted that measurements are
always accompanied by statistical errors, it was
claimed that these errors could be made smaller and
smaller with progressive techniques.

The first physicist in modern times to question the
strict determinism of physical laws was probably
Ludwig Boltzmann. In his lectures on gas theory he
declared in 1895: “Since today it is popular to look


forward to the time when our view of nature will have
been completely changed, I will mention the possibility
that the fundamental equations for the motion of indi-
vidual molecules will turn out to be only approximate
formulas which give average values, resulting accord-
ing to the probability calculus from the interactions
of many independent moving entities forming the sur-
rounding medium” (Boltzmann, 1895). Boltzmann's
successor at the University of Vienna, Franz Exner,
proposed in 1919 a statistical interpretation of the
apparent deterministic behavior of macroscopic phe-
nomena which he regarded as resulting from a great
number of probabilistic processes at the sub-
microscopic level.

From a multitude of events... laws can be inferred which
are valid for the average state [Durchschnittszustand] of this
multitude whereas the individual event may remain un-
determined. In this sense the principle of causality holds
for all macroscopic occurrences without being necessarily
valid for the microcosm. It also follows that the laws of
the macrocosm are not absolute laws but rather laws of
probability; whether they hold always and everywhere
remains to be questioned; to predict in physics the outcome
of an individual process is impossible

(Exner, 1919).

In the same year Charles Galton Darwin, influenced
by Henri Poincaré's allusion toward a probabilistic
reformulation of physical laws and his doubts about
the validity of differential equations as reflecting the
true nature of physical laws (H. Poincaré, Dernières
), made the bold statement that it may “prove
necessary to make fundamental changes in our ideas
of time and space, or to abandon the conservation of
matter and electricity, or even in the last resort to
endow electrons with free will” (Charles Galton
Darwin, 1919). The ascription of free will to electrons—
clearly an anthropomorphic metaphorism for a- and
u-indeterminacies—was suggested by certain results in
quantum theory such as the unpredictable and appar-
ently acausal emission of electrons from a radioactive
element or their unpredictable transitions from one
energy level to another in the atom. In the early twen-
ties questions concerning the limitations of the sensi-
tivity of measuring instruments came to the forefront
of physical interest when, with no direct connection
with quantum effects, the disturbing effects of the
Brownian fluctuations were studied in detail (W.
Einthoven, G. Ising, F. Zernike). It became increasingly
clear that Brownian motion, or “noise” as it was called
in the terminology of electronics, puts a definite limit
to the sensitivity of electronic measuring devices and
hence to measurements in general. Classical physics,
it seemed, has to abandon its principle of unlimited
precision and to admit, instead, unavoidable i-indeter-
minacies. It can be shown that this development did
not elicit the establishment of Heisenberg's uncertainty
relations in quantum mechanics (Jammer [1966], p.

5. Indeterminacies in Quantum Mechanics. The
necessity of introducing indeterminacy considerations
into quantum mechanics became apparent as soon as
the mathematical formalism of the theory was estab-
lished (in the spring of 1927). When Ernst Schrödinger,
in 1926, laid the foundations of wave mechanics he
interpreted atomic phenomena as continuous, causal
undulatory processes, in contrast to Heisenberg's
matrix mechanics in which these processes were inter-
preted as discontinuous and ruled by probability laws.
When in September 1926 Schrödinger visited Niels
Bohr and Heisenberg in Copenhagen, the conflict be-
tween these opposing interpretations reached its climax
and no compromise seemed possible. As a result of this
controversy Heisenberg felt it necessary to examine
more closely the precise meaning of the role of
dynamical variables in quantum mechanics, such as
position, momentum, or energy, and to find out how
far they were operationally warranted.

First he derived from the mathematical formalism
of quantum mechanics (Dirac-Jordan transformation
theory) the following result. If a wave packet with a
Gaussian distribution in the position coordinate q, to
wit ψ(q) = const. exp [-q 2/22(Δq0 2], Δq being the half-
width and consequently proportional to the standard
deviation, is transformed by a Fourier transformation
into a momentum distribution, the latter turns out to
be ϕ(p) = const. exp [-p2/2(ℏ/Δq)2]. Since the corre-
sponding half-width Δp is now given by ℏ/Δq, Heisen-
berg concluded that Δq Δp ≈ ℏ or more generally, if
other distributions are used,

Δq Δp ≳ ℏ

This inequality shows that the uncertainties (or
dispersions) in position and momentum are reciprocal:
if one approaches zero the other approaches infinity.
The meaning of relation (1), which was soon called
the “Heisenberg position-momentum uncertainty rela-
tion,” can also be expressed as follows: it is impossible
to measure simultaneously both the position and the
momentum of a quantum-mechanical system with
arbitrary accuracy; the more precise the measurement
of one of these two variables is, the less precise is that
of the other.

Asking himself whether a close analysis of actual
measuring procedures does not lead to a result in
contradiction to (1), Heisenberg studied what has since
become known as the “gamma-ray microscopic exper-
iment.” Adopting the operational view that a physical
concept is meaningful only if a definite procedure is
indicated for how to measure its value, Heisenberg


declared that if we speak of the position of an electron
we have to define a method of measuring it. The elec-
tron's position, he continued, may be found by illumi-
nating it and observing the scattered light under a
microscope. The shorter the wavelength of the light,
the more precise, according to the diffraction laws of
optics, will be the determination of the position—but
the more noticeable will also be the Compton effect
and the resulting change in the momentum of the
electron. By calculating the uncertainties resulting
from the Compton effect and the finite aperture of the
microscope, the importance of which for the whole
consideration was pointed out by Bohr, Heisenberg
showed that the obtainable precision does not surpass
the restrictions imposed by the inequality (1). Similarly,
by analyzing closely a Stern-Gerlach experiment of
measuring the magnetic moment of particles, Heisen-
berg showed that the dispersion ΔE in the energy of
these particles is smaller the longer the time Δt spent
by them in crossing the deviating field (or measuring
ΔE Δt ≳ ℏ
It has been claimed that this “energy-time uncertainty
relation” had been implicitly applied by A. Sommer-
feld in 1911, O. Sackur in 1912, and K. Eisenmann
in 1912 (Kudrjawzew, 1965). Bohr, as we know from
documentary evidence (Archive for the History of
Quantum Physics, Interview with Heisenberg, Febru-
ary 25, 1963), accepted the uncertainty relations (1)
and (2), but not their interpretation as proposed by
Heisenberg. For Heisenberg they expressed the limita-
tion of the applicability of classical notions to micro-
physics, whether these notions are those of particle
language or wave language, one language being re-
placeable by the other and equivalent to it. For Bohr,
on the other hand, they were an indication that both
modes of expression, though conjointly necessary for
an exhaustive description of physical phenomena, can-
not be used at the same time. As a result of this debate
Heisenberg added to the paper in which he published
the uncertainty relations (Heisenberg, 1927) a “Post-
script” in which he acknowledged that an as yet un-
published investigation by Bohr would lead to a deeper
understanding of the significance of the uncertainty
relations and “to an important refinement of the results
obtained in the paper.” It was the first allusion to
Bohr's complementarity interpretation, often also
loosely called the “Copenhagen interpretation” of
quantum mechanics (Jammer [1966], pp. 345-61). Bohr
regarded the uncertainty relations whose derivations
(by thought-experiments) are still based on the de
Broglie-Einstein equations E = hv and p = h/λ, that
is, relations between particulate (energy E, momentum
p) and undulatory conceptions (frequency v, wavelength
λ), merely as a confirmation of the wave-particle
duality and hence of the complementarity interpre-
tation (Schiff, 1968).

6. Philosophical Implications of the Uncertainty
In their original interpretation, as we have
seen, the Heisenberg uncertainty relations express first
of all a principle of limited measurability of dynamical
variables (position, momentum, energy, etc.) of indi-
vidual microsystems (particles, photons), though ac-
cording to the complementarity interpretation their
significance is not restricted merely to such a principle
(Grünbaum, 1957). But even qua such a principle their
epistemological implications were soon recognized and
the relations became an issue of extensive discussions.
Heisenberg himself saw their philosophical import in
the fact that they imply a renunciation of the causality
principle in its “strong formulation,” viz., “If we know
exactly the present, we can predict the future.” Since,
now, in view of these relations the present can never
be known exactly, Heisenberg argued, the causality
principle as formulated, though logically and not re-
futed, must necessarily remain an “empty” statement;
for it is not the conclusion, but rather the premiss
which is false.

In view of the intimate connection between the statistical
character of the quantum theory and the imprecision of
all perception, it may be suggested that behind the statis-
tical universe of perception there lies hidden a “real” world
ruled by causality. Such speculation seems to us—and this
we stress with emphasis—useless and meaningless. For
physics has to confine itself to the formal description of
the relations among perceptions

(Heisenberg [1927], p. 197).

Using the terminology of the introductory section
of this article, we may say that Heisenberg interpreted
the uncertainties appearing in the relations carrying
his name not only as i-indeterminacies, but also as
a-indeterminacies, provided the causality principle is
understood in its strong formulation, and a fortiori also
as u-indeterminacies. His idea that the unascertainabil-
ity of exact initial values obstructs predictability and
hence deprives causality of any operational meaning
was soon hailed, particularly by M. Schlick, as a “sur-
prising” solution of the age-old problem of causality,
a solution which had never been anticipated in spite
of the many discussions on this issue (Schlick, 1931).

Heisenberg's uncertainty relations were also re-
garded as a possible resolution of the long-standing
conflict between determinism and the doctrine of free
will. “If the atom has indeterminacy, surely the human
mind will have an equal indeterminacy; for we can
scarcely accept a theory which makes out the mind
to be more mechanistic than the atom” (Eddington,


1932). The Epicurean-Lucretian theory of the “minute
swerving of the elements” enjoyed an unexpected re-
vival in the twentieth century.

The philosophical impact of the uncertainty rela-
tions on the development of the subject-object prob-
lem, one of the crucial stages of the interaction be-
tween problems of physics and of epistemology,
problems which still persist, was discussed in great
detail by Ernst Cassirer (Cassirer, 1936, 1937).

Heisenberg's interpretation of the uncertainty rela-
tions, however, became soon the target also of other
serious criticisms. In a lecture delivered in 1932
Schrödinger, who only two years earlier gave a general,
and compared with Heisenberg's formula still more
restrictive, derivation of the relations for any pair of
noncommuting operators, challenged Heisenberg's
view as inconsistent; Schrödinger claimed that a denial
of sharp values for position and momentum amounts
to renouncing the very concept of a particle (mass-
point) (Schrödinger, 1930; 1932). Max von Laue
charged Heisenberg's conclusions as unwarranted and
hasty (von Laue, 1932). Karl Popper challenged
Heisenberg with having given “a causal explanation
why causal explanations are impossible” (Popper,
1935). The main attack, however, was launched within
physics itself—by Albert Einstein in his debate with
Niels Bohr.

7. The Einstein-Bohr Controversy about Indeter-
Although having decidedly furthered the de-
velopment of the probabilistic interpretation of quan-
tum phenomena through his early contributions to the
photo-electric effect and through his statistical deriva-
tion of Planck's formula for black-body radiation,
Einstein never agreed to abandon the principles of
causality and continuity or, equivalently, to renounce
the need of a causal account in space and time, in favor
of a statistical theory; and he saw in the latter only
an incomplete description of physical reality which has
to be supplanted sooner or later by a fully deterministic
theory. To prove that the Bohr-Heisenberg theory of
quantum phenomena does not exhaust the possibilities
of accounting for observable phenomena, and is conse-
quently only an incomplete description, it would
suffice, argued Einstein correctly, to show that a close
analysis of fundamental measuring procedures leads to
results in contradiction to the uncertainty relations. It
was clear that disproving these relations means dis-
proving the whole theory of quantum mechanics.

Thus, during the Fifth Solvay Congress in Brussels
(October 24 to 29, 1927) Einstein challenged the cor-
rectness of the uncertainty relations by scrutinizing a
number of thought-experiments, but Bohr succeeded
in rebutting all attacks (Bohr, 1949). The most dramatic
phase of this controversy occurred at the Sixth Solvay
Congress (Brussels, October 20 to 25, 1930) where these
discussions were resumed when Einstein challenged the
energy-time uncertainty relation ΔE Δt ≳ ℏ with the
famous photon-box thought-experiment (Jammer
[1966], pp. 359-60). Considering a box with a shutter,
operated by a clockwork in the box so as to be opened
at a moment known with arbitrary accuracy, and re-
leasing thereby a single photon, Einstein claimed that
by weighing the box before and after the photon-
emission and resorting to the equivalence between
energy and mass, E = mc2, both ΔE and Δt can be
made as small as desired, in blatant violation of the
relation (2). Bohr, however (after a sleepless night!),
refuted Einstein's challenge with Einstein's own
weaponry; referring to the red-shift formula of general
relativity according to which the rate of a clock de-
pends on its position in a gravitational field Bohr
showed that, if this factor is correctly taken into ac-
count, Heisenberg's energy-time uncertainty relation
is fully obeyed. Einstein's photon-box, if used as a
means for accurately measuring the energy of the
photon, cannot be used for controlling accurately the
moment of its release. If closely examined, Bohr's
refutation of Einstein's argument was erroneous, but
so was Einstein's argument (Jammer, 1972). In any
case, Einstein was defeated but not convinced, as Bohr
himself admitted. In fact, in a paper written five years
later in collaboration with B. Podolsky and N. Rosen,
Einstein showed that in the case of a two-particle
system whose two components separate after their
interaction, it is possible to predict with certainty
either the exact value of the position or of the momen-
tum of one of the components without interfering with
it at all, but merely performing the appropriate meas-
urement on its partner. Clearly, such a result would
violate the uncertainty relation (1) and condemn the
quantum-mechanical description as incomplete (Ein-
stein, 1935). Although the majority of quantum-
theoreticians are of the opinion that Bohr refuted this
challenge also (Bohr, 1935), there are some physicists
who consider the Einstein-Podolsky-Rosen argument
as a fatal blow to the Copenhagen interpretation.

Criticisms of a more technical nature were leveled
against the energy-time uncertainty relation (2). It was
early recognized that the rigorous derivation of the
position-momentum relation from the quantum-
mechanical formalism as a calculus of Hermitian oper-
ators in Hilbert space has no analogue for the energy-
time relation; for while the dynamical variables q and
p are representable in the formalism as Hermitian
(noncommutative) operators, satisfying the relation
qp - pq = iℏ, and although the energy of a system
is likewise represented as a Hermitian operator, the
Hamiltonian, the time variable cannot be represented


by such an operator (Pauli, 1933). In fact, it can be
shown that the position and momentum coordinates,
q and p, and their linear combinations are the only
canonical conjugates for which uncertainty relations
in the Heisenberg sense can be derived from the oper-
ator formalism. This circumstance gave rise to the fact
that the exact meaning of the indeterminacy Δt in the
energy-time uncertainty relation was never unam-
biguously defined. Thus in recent discussions of this
uncertainty relation at least three different meanings
of Δt can be distinguished (duration of the opening time
of a slit; the uncertainty of this time-period; the dura-
tion of a concomitant measuring process c.f., Chyliński,
1965; Halpern, 1966; 1968). Such ambiguities led L. I.
Mandelstam and I. Tamm, in 1945, to interpret Δt
in this uncertainty relation as the time during which
the temporal mean value of the standard deviation of
an observable R becomes equal to the change of its
standard deviation: Δ̅R̅ = 〈Rt + Δt〉 - 〈Rt.
now, denotes the energy standard deviation of the
system under discussion during the R-measurement,
then the energy-time uncertainty relation acquires the
same logical status within the formalism of quantum
mechanics as that possessed by the position-momentum

A different approach to reach an unambiguous in-
terpretation of the energy-time uncertainty relation
had been proposed as early as 1931 by L. D. Landau
and R. Peierls on the basis of the quantum-mechanical
perturbation theory (Landau and Peierls, 1931; Landau
and Lifshitz [1958], pp. 150-53), and was subsequently
elaborated by N. S. Krylov and V. A. Fock (Krylov
and Fock, 1947). This approach was later severely
criticized by Y. Aharonov and D. Bohm (Aharonov and
Bohm, 1961) which led to an extended discussion on
this issue without reaching consensus (Fock, 1962;
Aharonov and Bohm, 1964; Fock, 1965). Recently at-
tempts have been made to extend the formalism of
quantum mechanics, as for instance by generalizing the
Hilbert space to a super-Hilbert space (Rosenbaum,
1969), so that it admits the definition of a quantum-
mechanical time-operator and puts the energy-time
uncertainty relation on the same footing as that of
position and momentum (Engelmann and Fick, 1959,
1964; Paul, 1962; Allcock, 1969).

8. The Statistical Interpretation of Quantum-
mechanical Indeterminacy.
If the ψ-function charac-
terizes the behavior not of an individual particle but
of a statistical ensemble of particles, as contended in
the "statistical interpretation" of the quantum-
mechanical formalism, then' obviously the uncertainty
relations, at least as far they derive from this formalism,
refer likewise not to individual particles but to statis-
tical ensembles of these. In other words, relation (1)
denotes, in this view, a correlation between the disper-
sion or "spread" of measurements of position, and the
dispersion or "spread" of measurements of momentum,
if carried out on a large ensemble of identically pre-
pared systems. Under these circumstances the idea that
noncommuting variables are not necessarily incompat-
ible but can be measured simultaneously on individual
systems would not violate the statistical interpretation.
Such an interpretation of quantum-mechanical
indeterminacy was suggested relatively early by
Popper (Popper, 1935). His reformulation of the un-
certainty principle reads as follows: given an ensemble
(aggregate of particles or sequence of experiments
performed on one particle which after each experiment
is brought back to its original state) from which, at
a certain moment and with a given precision Δq, those
particles having a certain position q are selected; the
momenta p of the latter will then show a random
scattering with a range of scatter Δp where ΔqΔp≳ℏ
and vice versa. Popper even thought, though errone-
ously as he himself soon realized, to have proved his
contention by the construction of a thought-experiment
for the determination of the sharp values of position
and momentum (Popper, 1934).

The ensemble interpretation of indeterminacy found
an eloquent advocate in Henry Margenau. Distin-
guishing sharply between subjective or a priori and
empirical or a posteriori probability, Margenau pointed
out that the indeterminacy associated with a single
measurement such as referred to in Heisenberg's
gamma-ray experiment is nothing more than a qualita-
tive subjective estimate, incapable of scientific verifi-
cation; every other interpretation would at once revert
to envisaging the single measurement as the constituent
of a statistical ensemble; but as soon as the empirical
view on probability is adopted which, grounded in
frequencies, is the only one that is scientifically sound,
the uncertainty principle, now asserting a relation
between the dispersions of measurement results, be-
comes amenable to empirical verification. To vindicate
this interpretation Margenau pointed out that, contrary
to conventional ideas, canonical conjugates may well
be measured with arbitrary accuracy at one and the
same time; thus two microscopes, one using gamma
rays and the other infra-red rays for a Doppler-experi-
ment, may simultaneously locate the electron and de-
termine its momentum and no law of quantum me-
chanics prohibits such a double measurement from
succeeding (Margenau, 1937; 1950). This view does not
abnegate the principle, for on repeating such measure-
ments many times with identically prepared systems
the product of the standard deviations of the values
obtained will have a definite lower limit.

Although Margenau and R. N. Hill (Margenau and


Hill, 1961) found that the usual Hilbert space formalism
of quantum mechanics does not admit probability
distributions for simultaneous measurements of non-
commuting variables, E. Prugovečki has suggested that
by introducing complex probability distributions the
existing formalism of mathematical statistics can be
generalized so as to overcome this difficulty. For other
approaches to the same purpose we refer the reader
to an important paper by Margenau and Leon Cohen,
and the bibliography listed therein (Margenau and
Cohen, 1967), and also to the analyses of simultaneous
measurements of conjugate variables carried out by E.
Arthurs and J. L. Kelly (Arthurs and Kelly, 1965),
C. Y. She and H. Heffner (She and Heffner, 1966),
James L. Park and Margenau (J. L. Park and Margenau,
1968). William T. Scott (Scott, 1968), and Dick H.
Holze and William T. Scott (Holze and Scott, 1968).
These investigations suggest the result that neither
single quantum-mechanical measurements nor even
combined simultaneous measurements of canonically
conjugate variables are, in the terminology of the
introduction, subject to i-indeterminacy, even though
they are subject to u-indeterminacy.

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

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 = Lv0,
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

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
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.


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[See also Atomism; Causation; Determinism; Entropy;