SCIENCE AND PHILOSOPHY
Probability, Statistics and Truth. By Richard von Mises. (William Hodge. los. 6cL) IT has always been the function of philosophers to criticise and interpret the methods and conclusions of other disciplines. When philosophy ceased to be the servant of theology, its problems were borrowed from natural science. During the last two centuries Newtonian mechanics was the paradigm of natural science. Consequently, philosophers first asked them- selves the question, " How are Newton's principles of natural knowledge to be interpreted and justified? " Kant's philosophy purports to be a final demonstration of the universal va:idity of these principles ; he was satisfied that the world could not conceivably be described in any other terms, and that real knowledge could not be acquired by any other method. It was assumed that classical mechanics was the final revelation of the form of empirical knowledge, towards which the undeveloped sciences would approximate. This period of successful extroversion among scientists ended with the revolu- tion in microscopic physics, the new formalisation of mathe- matics, and the rise of the social sciences ; the subsequent confusion provided philosophy, which had been slowly dying of inanition, with the wealth of raw material which it needed. Professor von Mises points out that, between a revolution in science and the corresponding revolution in philosophy, there is always a time-lag ; philosophers tend to be preoccupied with the presuppositions of a terminology which has already been discarded. Ernst Mach was the first in the new tradition of scientific philosophy, which was popularised in England by Karl Pearson and Bertrand Russell ; its vitality is shown in the excellent series of monographs in which this book appeared in German ; it is to be hoped that other volumes in this series of Schriften zur Wissenschaftlichen Weltauffassung—for in- stance, Karl Popper's Logik der Forschung—will now be translated into English.
Professor von Mises has written an original book on a neglected subject which is short, systematic, and lucid. Although he discusses without superficiality some of the problems of statistical physics, he nowhere presupposes any knowledge of advanced mathematics in the reader. He argues that the definition of " probability " which is required is not a paraphrase which can be substituted for the word in its colloquial use, but an exact statement of how it must be used in quantitative science. He rejects the orthodox, Laplacean theory that probability is the ratio of the number of favourable cases to the total number of equally likely cases, on the ground that this equi-probability can itself only be determined empiri- cally. In opposition to Keynes, he denies that the probability. which we ascribe to onr statements depends on the degree of Our knowledge. Statements of probability in the exact sense must refer to an infinite class of possible experiments or re- petitive events, and never to any single particular event ; they state the limiting value of the relative frequency with which a particular result recurs in an indefinitely prolonged sequence of observations ; they are therefore ordinary empirical state- ments, and do not presuppose any particular limited amount of evidence.
For such statements to be possible there must be infinite classes of possible observations which conform to two condi- t"ons: first, obviously, that the relative frequency of a par- ticular result should converge towards a certain limiting value. Secondly, that this value should be indifferently the same for any sufficiently large sub-class within the given class. That
there are, in fact, classes of events which fulfil these two con- ditions we know from experience, particularly from our ex- perience of the failure of gambling systems in games of pure chance. Professor von Mises shows how in the calculus of probability we infer from one statement of frequency to another by arithmetical operations ; the Law of Large Numbers and Bernoulli's theorem are useful empirical pro- positions only if his frequency theory is presupposed. After a discussion of some problems in statistics and the theory of errors, he argues that statistical theories in physics are irre- concilable with strictly deterministic theories. In this bald summary I have omitted the fascinating illustrations of the many possible applications of probability—in the theory of Brownian movement, of radio-active discharge, in the gas- theory of Einstein, Bose and Fermi, in the Uncertainty Principle in quantum mechanics.
There are general logical and epistemological questions which Professor von Mises leaves unanswered. Must we not justify our belief that there are classe; of observations of the required kind by an argument from probability? What is the rela- tion of this very general existential proposition to the Principle of Indifference and the Uniformity of Nature? How can we show that there are no biassed sub-classes within the given class, except by an argument from probability? Can we significantly speak of the probability of a causal law? Is there no exact, derivative sense in which we can speak of the proba- bility of a single event? But it is many years since any philo- sopher has answered so many questions in the theory of probability.
STUART HAMPSHIRE.
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