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# K. Popper - Logic scientific discovery

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axiom of randomness—mainly a mathematical problem; and the complete elim- ination of the axiom of convergence—a matter of particular concern for the epistemologist.2 (Cf. section 66.) In what follows I propose to deal ﬁrst with the mathematical, and afterwards with the epistemological question. The ﬁrst of these two tasks, the reconstruction of the mathematical theory,3 has as its main aim the derivation of Bernoulli’s theorem— the ﬁrst ‘Law of Great Numbers’—from a modiﬁed axiom of randomness; 1 Waismann, Erkenntnis 1, 1930, p. 232. 2 This concern is expressed by Schlick, Naturwissenschaften 19, 1931. *I still believe that these two tasks are important. Although I almost succeeded in the book in achieving what I set out to do, the two tasks were satisfactorily completed only in the new appendix *vi. 3 A full account of the mathematical construction will be published separately. *Cf. the new appendix *vi. some structural components of a theory of experience142 modiﬁed, namely, so as to demand no more than is needed to achieve this aim. Or to be more precise, my aim is the derivation of the Binomial Formula (sometimes called ‘Newton’s Formula’), in what I call its ‘third form’. For from this formula, Bernoulli’s theorem and the other limit theorems of probability theory can be obtained in the usual way. My plan is to work out ﬁrst a frequency theory for ﬁnite classes, and to develop the theory, within this frame, as far as possible—that is, up to the derivation of the (‘ﬁrst’) Binomial Formula. This frequency theory for ﬁnite classes turns out to be a quite elementary part of the theory of classes. It will be developed merely in order to obtain a basis for discussing the axiom of randomness. Next I shall proceed to inﬁnite sequences, i.e. to sequences of events which can be continued indeﬁnitely, by the old method of introducing an axiom of convergence, since we need something like it for our discussion of the axiom of randomness. And after deriving and exam- ining Bernoulli’s theorem, I shall consider how the axiom of convergence might be eliminated, and what sort of axiomatic system we should be left with as the result. In the course of the mathematical derivation I shall use three diﬀerent frequency symbols: F″ is to symbolize relative frequency in ﬁnite classes; F′ is to symbolize the limit of the relative frequencies of an inﬁnite frequency-sequence; and ﬁnally F, is to symbolize objective probability, i.e. relative frequency in an ‘irregular’ or ‘random’ or ‘chance-like’ sequence. 52 RELATIVE FREQUENCY WITHIN A FINITE CLASS Let us consider a class α of a ﬁnite number of occurrences, for example the class of throws made yesterday with this particular die. This class α, which is assumed to be non-empty, serves, as it were, as a frame of reference, and will be called a (ﬁnite) reference-class. The number of elements belonging to α, i.e. its cardinal number, is denoted by ‘N(α)’, to be read ‘the number of α’. Now let there be another class, β, which may be ﬁnite or not. We will call β our property-class: it may be, for example, the class of all throws which show a ﬁve, or (as we shall say) which have the property ﬁve. probability 143 The class of those elements which belong to both α and β, for example the class of throws made yesterday with this particular die and having the property ﬁve, is called the product-class of α and β, and is denoted by ‘α.β’, to be read ‘α and β’. Since α.β is a subclass of α, it can at most contain a ﬁnite number of elements (it may be empty). The number of elements in α.β is denoted by ‘N(α.β)’. Whilst we symbolize (ﬁnite) numbers of elements by N, the relative frequencies are symbolized by F″. For example, ‘the relative frequency of the property β within the ﬁnite reference-class α’ is written ‘αF″(β)’, which may be read ‘the α-frequency of β’. We can now deﬁne αF″(β) = N(α.β) N(α) (Deﬁnition 1) In terms of our example this would mean: ‘The relative frequency of ﬁves among yesterday’s throws with this die is, by deﬁnition, equal to the quotient obtained by dividing the number of ﬁves, thrown yester- day with this die, by the total number of yesterday’s throws with this die.’*1 From this rather trivial deﬁnition, the theorems of the calculus of frequency in ﬁnite classes can very easily be derived (more especially, the general multiplication theorem; the theorem of addition; and the the- orems of division, i.e. Bayes’s rules. Cf. appendix ii). Of the theorems of this calculus of frequency, and of the calculus of probability in general, it is characteristic that cardinal numbers (N-numbers) never appear in them, but only relative frequencies, i.e. ratios, or F-numbers. The N- numbers only occur in the proofs of a few fundamental theorems which are directly deduced from the deﬁnition; but they do not occur in the theorems themselves.*2 *1 Deﬁnition 1 is of course related to the classical deﬁnition of probability as the ratio of the favourable cases to the equally possible cases; but it should be clearly distinguished from the latter deﬁnition: there is no assumption involved here that the elements of α are ‘equally possible’. *2 By selecting a set of F-formulae from which the other F-formulae can be derived, we obtain a formal axiom system for probability; compare the appendices ii, *ii, *iv, and *v. some structural components of a theory of experience144 How this is to be understood will be shown here with the help of one very simple example. (Further examples will be found in appendix ii.) Let us denote the class of all elements which do not belong to β by ‘β - ’ (read: ‘the complement of β’ or simply: ‘non-β’). Then we may write αF″(β) + αF″(β - ) = 1 While this theorem only contains F-numbers, its proof makes use of N- numbers. For the theorem follows from the deﬁnition (1) with the help of a simple theorem from the calculus of classes which asserts that N(α.β) + N(α.β - ) = N(α). 53 SELECTION, INDEPENDENCE, INSENSITIVENESS, IRRELEVANCE Among the operations which can be performed with relative frequen- cies in ﬁnite classes, the operation of selection1 is of special importance for what follows. Let a ﬁnite reference-class α be given, for example the class of but- tons in a box, and two property-classes, β (say, the red buttons) and γ (say, the large buttons). We may now take the product-class α.β as a new reference-class, and raise the question of the value of α.βF″ (γ), i.e. of the frequency of γ within the new reference-class.2 The new reference- class α.β may be called ‘the result of selecting β-elements from α’, or the ‘selection from α according to the property β’; for we may think of it as being obtained by selecting from α all those elements (buttons) which have the property β (red). Now it is just possible that γ may occur in the new reference-class, α.β, with the same relative frequency as in the original reference-class α; i.e. it may be true that α.βF″ (γ) = αF″ (γ) 1 Von Mises’s term is ‘choice’ (‘Auswahl’). 2 The answer to this question is given by the general division theorem (cf. appendix ii). probability 145 In this case we say (following Hausdorﬀ3) that the properties β and γ are ‘mutually independent, within the reference-class α’. The relation of independence is a three-termed relation and is symmetrical in the properties β and γ.4 If two properties β and γ are (mutually) independent within a reference-class α we can also say that the prop- erty γ is, within α, insensitive to the selection of β-elements; or perhaps that the reference-class α is, with respect to this property γ, insensitive to a selection according to the property β. The mutual independence, or insensitiveness, of β and γ within α could also—from the point of view of the subjective theory—be inter- preted as follows: If we are informed that a particular element of