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# K. Popper - Logic scientific discovery

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of this problem—the fundamental problem of the theory of chance, as I shall call it. It will be shown in section 67 that this problem is connected with the ‘axiom of convergence’ which is an integral part 1 Waismann, Erkenntnis 1, 1930, p. 238, says: ‘There is no other reason for introducing the concept of probability than the incompleteness of our knowledge.’ A similar view is held by C. Stumpf (Sitzungsberichte der Bayerischen Akademie der Wissenschaften, phil.-hist. Klasse, 1892, p. 41). *I believe that this widely held view is responsible for the worst confusions. This will be shown in detail in my Postscript, chapters *ii and *v. some structural components of a theory of experience138 of the theory in its present form. But it is possible to ﬁnd a satisfactory solution within the framework of the frequency theory, after this axiom has been eliminated. It will be found by analysing the assump- tions which allow us to argue from the irregular succession of single occurrences to the regularity or stability of their frequencies. 50 THE FREQUENCY THEORY OF VON MISES A frequency theory which provides a foundation for all the principal theorems of the calculus of probability was ﬁrst proposed by Richard von Mises.1 His fundamental ideas are as follows. The calculus of probability is a theory of certain chance-like or random sequences of events or occurrences, i.e. of repetitive events such as a series of throws with a die. These sequences are deﬁned as ‘chance-like’ or ‘random’ by means of two axiomatic conditions: the axiom of convergence (or the limit-axiom) and the axiom of randomness. If a sequence of events satisﬁes both of these conditions it is called by von Mises a ‘collective’. A collective is, roughly speaking, a sequence of events or occur- rences which is capable in principle of being continued indeﬁnitely; for example a sequence of throws made with a supposedly indestruct- ible die. Each of these events has a certain character or property; for example, the throw may show a ﬁve and so have the property ﬁve. If we take all those throws having the property ﬁve which have appeared up to a certain element of the sequence, and divide their number by the total number of throws up to that element (i.e. its ordinal number in the sequence) then we obtain the relative frequency of ﬁves up to that element. If we determine the relative frequency of ﬁves up to every element of the sequence, then we obtain in this way a new sequence— the sequence of the relative frequencies of ﬁves. This sequence of frequencies is distinct from the original sequence of events to which it corresponds, 1 R. von Mises, Fundamentalsätze der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift 4, 1919, p. 1; Grundlagen der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift 5, 1919, p. 52; Wahrschein- lichkeit, Statistik, und Wahrheit (1928), 2nd edition 1936, English translation by J. Neyman, D. Sholl, and E. Rabinowitsch: Probability, Statistics and Truth, 1939; Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und theoretischen Physik (Vorlesungen über angewandte Mathematik 1), 1931. probability 139 and which may be called the ‘event-sequence’ or the ‘property- sequence’. As a simple example of a collective I choose what we may call an ‘alternative’. By this term we denote a sequence of events supposed to have two properties only—such as a sequence of tosses of a coin. The one property (heads) will be denoted by ‘1’, and the other (tails) by ‘0’. A sequence of events (or sequence of properties) may then be represented as follows: 0 1 1 0 0 0 1 1 1 0 1 0 1 0 . . . .(A) Corresponding to this ‘alternative’—or, more precisely, correlated with the property ‘1’ of this alternative—is the following sequence of relative frequencies, or ‘frequency-sequence’:2 O 1 2 2 3 2 4 2 5 2 6 3 7 4 8 5 9 5 10 6 11 6 12 7 13 7 14 . . . .(A′) Now the axiom of convergence (or ‘limit-axiom’) postulates that, as the event-sequence becomes longer and longer, the frequency-sequence shall tend towards a deﬁnite limit. This axiom is used by von Mises because we have to make sure of one ﬁxed frequency value with which we can work (even though the actual frequencies have ﬂuctuating values). In any collective there are at least two properties; and if we are given the limits of the frequencies corresponding to all the properties of a collective, then we are given what is called its ‘distribution’. The axiom of randomness or, as it is sometimes called, ‘the principle of the excluded gambling system’, is designed to give mathematical expres- sion to the chance-like character of the sequence. Clearly, a gambler would be able to improve his chances by the use of a gambling system 2 We can correlate with every sequence of properties as many distinct sequences of relative frequencies as there are properties deﬁned in the sequence. Thus in the case of an alternative there will be two distinct sequences. Yet these two sequences are derivable from one another, since they are complementary (corresponding terms add up to 1). For this reason I shall, for brevity, refer to ‘the (one) sequence of relative frequencies correl- ated with the alternative (α)’, by which I shall always mean the sequence of frequencies correlated with the property ‘1’ of this alternative (α). some structural components of a theory of experience140 if sequences of penny tosses showed regularities such as, say, a fairly regular appearance of tails after every run of three heads. Now the axiom of randomness postulates of all collectives that there does not exist a gambling system that can be successfully applied to them. It postulates that, whatever gambling system we may choose for selecting supposedly favourable tosses, we shall ﬁnd that, if gambling is con- tinued long enough, the relative frequencies in the sequence of tosses supposed to be favourable will approach the same limit as those in the sequence of all tosses. Thus a sequence for which there exists a gambling system by means of which the gambler can improve his chances is not a collective in the sense of von Mises. Probability, for von Mises, is thus another term for ‘limit of relative frequency in a collective’. The idea of probability is therefore applic- able only to sequences of events; a restriction likely to be quite unacceptable from a point of view such as Keynes’s. To critics objecting to the narrowness of his interpretation, von Mises replied by stressing the diﬀerence between the scientiﬁc use of probability, for example in physics, and the popular uses of it. He pointed out that it would be a mistake to demand that a properly deﬁned scientiﬁc term has to correspond in all respects to inexact, pre-scientiﬁc usage. The task of the calculus of probability consists, according to von Mises, simply and solely in this: to infer certain ‘derived collectives’ with ‘derived distributions’ from certain given ‘initial collectives’ with cer- tain given ‘initial distributions’; in short, to calculate probabilities which are not given from probabilities which are given. The distinctive features of his theory are summarized by von Mises in four points:3 the concept of the collective precedes that of prob- ability; the latter is deﬁned as the limit of the relative frequencies; an axiom of randomness is formulated; and the task of the calculus of probability is deﬁned. 51 PLAN FOR A NEW THEORY OF PROBABILITY The two axioms or postulates formulated by von Mises in order to deﬁne the concept of a collective have met with strong criticism— 3 Cf. von Mises, Wahrscheinlichkeitsrechnung, 1931, p. 22. probability 141 criticism which is not, I think, without some justiﬁcation. In particular, objections have been raised against combining the axiom of con- vergence with the axiom of randomness1 on the ground that it is inadmissible to apply the mathematical concept of a limit, or of con- vergence, to a sequence which by deﬁnition (that is, because of the axiom of randomness)