popper-logic-scientific-discovery

probability 201
at times may indeed occur to us whilst we are engaged in studying
the laws pertaining to micro events, but which can never be derived
from these laws. Frequency estimates form a special class of hypotheses:
they are prohibitions which, as it were, concern regularities in
the large. 1 Von Mises has stated this very clearly: ‘Not even the
tiniest little theorem in the kinetic theory of gases follows from
classical physics alone, without additional assumptions of a statistical
kind.’ 2
Statistical estimates, or frequency statements, can never be derived
simply from laws of a ‘deterministic’ kind, for the reason that in order
to deduce any prediction from such laws, initial conditions are needed.
In their place, assumptions about the statistical distribution of initial
conditions—that is to say specific statistical assumptions—enter into
every deduction in which statistical laws are obtained from micro
assumptions of a deterministic or ‘precise’ character.* 1
It is a striking fact that the frequency assumptions of theoretical
physics are to a great extent equal-chance hypotheses, but this by no means
implies that they are ‘self-evident’ or a priori valid. That they are far from
being so may be seen from the wide differences between classical
statistics, Bose-Einstein statistics, and Fermi-Dirac statistics. These show
1
A. March well says (Die Grundlagen der Quantenmechanik 1931, p. 250) that the particles of a
gas cannot behave ‘ . . . as they choose; each one must behave in accordance with the
behaviour of the others. It can be regarded as one of the most fundamental principles of
quantum theory that the whole is more than the mere sum of the parts’.
2 Von Mises, Über kausale und statistische Gesetzmässigkeiten in der Physik, Erkenntnis 1, 1930, p. 207
(cf. Naturwissenschaften 18, 1930).
* 1 The thesis here advanced by von Mises and taken over by myself has been contested
by various physicists, among them P. Jordan (see Anschauliche Quantentheorie, 1936,
p. 282, where Jordan uses as argument against my thesis the fact that certain forms of
the ergodic hypothesis have recently been proved). But in the form that probabilistic
conclusions need probabilistic premises—for example, measure-theoretical premises into
which certain equiprobabilistic assumptions enter—my thesis seems to me supported
rather than invalidated by Jordan’s examples. Another critic of this thesis was Albert
Einstein who attacked it in the last paragraph of an interesting letter which is here
reprinted in appendix *xii. I believe that, at that time, Einstein had in mind a subjective
interpretation of probability, and a principle of indifference (which looks in the
subjective theory as if it were not an assumption about equiprobabilities). Much later
Einstein adopted, at least tentatively, a frequency interpretation (of the quantum
theory).