Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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Dictionary of the History of Ideas | ||

*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

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.

Dictionary of the History of Ideas | ||