K. Popper - Logic scientific discovery
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makes it possible to say of a given alternative A that it is approximately random; and it even allows us to deﬁne a degree of approximation. A more elaborate deﬁnition may be based upon the method (of maximizing my E-function) described under points 8 ﬀ. of my Third Note reprinted in appendix *ix. some structural components of a theory of experience152 begin with the segment of the ﬁrst n elements of α. Next comes the segment of the elements 2 to n + 1 of α. In general, we take as the xth element of the new sequence the segment consisting of the elements x to x + n − 1 of α. The new sequence so obtained may be called the ‘sequence of the overlapping n-segments of α’. This name indicates that any two consecutive elements (i.e. segments) of the new sequence overlap in such a way that they have n − 1 elements of the original sequence α in common. Now we can obtain, by selection, other n-sequences from a sequence of overlapping segments; especially sequences of adjoining n-segments. A sequence of adjoining n-segments contains only such n-segments as immediately follow each other in α without overlapping. It may begin, for example, with the n-segments of the elements numbered 1 to n, of the original sequence α, followed by that of the elements n + 1 to 2n, 2n + 1 to 3n, and so on. In general, a sequence of adjoining segments will begin with the kth element of α and its segments will contain the elements of α numbered k to n + k − 1, n + k to 2n + k − 1, 2n + k to 3n + k − 1, and so on. In what follows, sequences of overlapping n-segments of α will be denoted by ‘α(n)’, and sequences of adjoining n-segments by ‘αn’. Let us now consider the sequences of overlapping segments α(n) a little more closely. Every element of such a sequence is an n-segment of α. As a primary property of an element of α(n), we might consider, for instance, the ordered n-tuple of zeros and ones of which the segment consists. Or we could, more simply, regard the number of its ones as the primary property of the element (disregarding the order of the ones and zeros). If we denote the number of ones by ‘m’ then, clearly, we have m � n. Now from every sequence α(n) we again get an alternative if we select a particular m (m � n), ascribing the property ‘m’ to each element of the sequence α(n) which has exactly m ones (and therefore n − m zeros) and the property ‘m¯’ (non-m) to all other elements of α(n). Every element of α(n) must then have one or the other of these two properties. Let us now imagine again that we are given a ﬁnite alternative α with the primary properties ‘1’ and ‘0’. Assume that the frequency of the ones, αF″ (1), is equal to p, and that the frequency of the zeros, αF″ (0), probability 153 is equal to q. (We do not assume that the distribution is equal, i.e. that p = q.) Now let this alternative α be at least n−1-free (n being an arbitrarily chosen natural number). We can then ask the following question: What is the frequency with which the property m occurs in the sequence αn? Or in other words, what will be the value of α(n)F″(m)? Without assuming anything beyond the fact that α is at least n−1-free, we can settle this question1 by elementary arithmetic. The answer is contained in the following formula, the proof of which will be found in appendix iii: α(n) F″ (m) = nCmp mqn − m(1) The right-hand side of the ‘binomial’ formula (1) was given—in another connection—by Newton. (It is therefore sometimes called Newton’s formula.) I shall call it the ‘ﬁrst form of the binomial formula’.*1 With the derivation of this formula, I now leave the frequency theory as far as it deals with ﬁnite reference-classes. The formula will provide us with a foundation for our discussion of the axiom of randomness. 57 INFINITE SEQUENCES. HYPOTHETICAL ESTIMATES OF FREQUENCY It is quite easy to extend the results obtained for n-free ﬁnite sequences to inﬁnite n-free sequences which are deﬁned by a generating period (cf. section 55). An inﬁnite sequence of elements playing the rôle of the reference-class to which our relative frequencies are related may be 1 The corresponding problem in connection with inﬁnite sequences of adjoining seg- ments I call ‘Bernoulli’s problem’ (following von Mises, Wahrscheinlichkeitsrechnung, 1931, p. 128); and in connection with inﬁnite sequences of overlapping segments I call it ‘the quasi-Bernoulli problem’ (cf. note 1 to section 60). Thus the problem here discussed would be the quasi-Bernoulli problem for ﬁnite sequences. *1 In the original text, I used the term ‘Newton’s formula’; but since this seems to be rarely used in English, I decided to translate it by ‘binomial formula’. some structural components of a theory of experience154 called a ‘reference-sequence’. It more or less corresponds to a ‘collective’ in von Mises’s sense.*1 The concept of n-freedom presupposes that of relative frequency; for what its deﬁnition requires to be insensitive—insensitive to selection according to certain predecessors—is the relative frequency with which a property occurs. In our theorems dealing with inﬁnite sequences I shall employ, but only provisionally (up to section 64), the idea of a limit of relative frequencies (denoted by F′), to take the place of relative frequency in ﬁnite classes (F″). The use of this concept gives rise to no problem so long as we conﬁne ourselves to reference-sequences which are constructed according to some mathematical rule. We can always determine for such sequences whether the corresponding sequence of relative frequencies is convergent or not. The idea of a limit of relative frequencies leads to trouble only in the case of sequences for which no mathematical rule is given, but only an empirical rule (linking, for example the sequence with tosses of a coin); for in these cases the concept of limit is not deﬁned (cf. section 51). An example of a mathematical rule for constructing a sequence is *1 I come here to the point where I failed to carry out fully my intuitive programme— that of analysing randomness as far as it is possible within the region of ﬁnite sequences, and of proceeding to inﬁnite reference sequences (in which we need limits of relative frequencies) only afterwards, with the aim of obtaining a theory in which the existence of frequency limits follows from the random character of the sequence. I could have carried out this programme very easily by constructing, as my next step (ﬁnite) shortest n- free sequences for a growing n, as I did in my old appendix iv. It can then be easily shown that if, in these shortest sequences, n is allowed to grow without bounds, the sequences become inﬁnite, and the frequencies turn without further assumption into frequency limits. (See note *2 to appendix iv, and my new appendix *vi.) All this would have simpliﬁed the next sections which, however, retain their signiﬁcance. But it would have solved completely and without further assumption the problems of sections 63 and 64; for since the existence of limits becomes demonstrable, points of accumulation need no longer be mentioned. These improvements, however, remain all within the framework of the pure frequency theory: except in so far as they deﬁne an ideal standard of objective disorder, they become unnecessary if we adopt a propensity interpretation of the neo-classical (measure-theoretical) formalism, as explained in sections *53 ﬀ of my Postscript. But even then it remains necessary to speak of frequency hypotheses—of hypothetical estimates and their statistical tests; and thus the present section remains relevant, as does much in the succeeding sections, down to section 64. probability 155 the following: ‘The nth element of the sequence α shall be 0 if, and only if, n is divisible by four’. This deﬁnes the inﬁnite alternative 1 1 1 0 1 1 1 0 . . .(α) with the limits of the relative frequencies: αF′ (1) = 3/4; and αF′ (0) = 1/4. Sequences which are deﬁned in this way by means of a mathematical rule I shall call, for brevity, ‘mathematical sequences’.