*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-*

able 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*

pensées), 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.

331).

**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(Δ*q*0
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/Δ*q*)2]. Since the corre-

sponding half-width Δ*p* is now given by ℏ/Δ*q*, Heisen-

berg concluded that Δ*q* Δ*p* ≈ ℏ or more generally, if

other distributions are used,

Δ*q* Δ*p* ≳ 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 Δ*E* in the energy of

these particles is smaller the longer the time Δ*t* spent

by them in crossing the deviating field (or measuring

device):

Δ*E* Δ*t* ≳ 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**

Relations. 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-**

minacy. 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 Δ*E* Δ*t* ≳ 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* = *mc*2, both Δ*E* and Δ*t* 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* = *i*ℏ, 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 Δ*t* 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 Δ*t* 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 Δ*t*

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: Δ̅*R*̅ = 〈*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

relation.
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 Δ*q*, 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 Δ*p* where Δ*q*Δ*p*≳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

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* = *x*0 and its velocity has a value between

*v*0 and *v*0 + Δ*v*0, it follows that at time *t* = *L*/Δ*v*0,

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 *V*0 which is larger

the greater the error in measurement. In accordance

with the theory it is then known that at time *t = t*1

the representative point *P* is located in a volume *V*1

which, according to the Liouville theorem of statistical

mechanics, equals *V*0. If, now, at *t = t*1 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 *D*1 with *V*1. *D*1 is smaller

than *V*1 and hence also smaller than *V*0. To *D*1, as a

subset of *V*1, corresponds a subset of *V*0 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 *V*0 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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MAX JAMMER

[See also Atomism; Causation; Determinism; Entropy;

Probability.]