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# K. Popper - Logic scientific discovery

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By contrast, a rule for constructing an empirical sequence would be, for instance: ‘The nth element of the sequence α shall be 0 if, and only if, the nth toss of the coin c shows tails.’ But empirical rules need not always deﬁne sequences of a random character. For example, I should describe the following rule as empirical: ‘The nth element of the sequence shall be 1 if, and only if, the nth second (counting from some zero instant) ﬁnds the pendulum p to the left of this mark.’ The example shows that it may sometimes be possible to replace an empirical rule by a mathematical one—for example on the basis of certain hypotheses and measurements relating to some pendulum. In this way, we may ﬁnd a mathematical sequence approximating to our empirical sequence with a degree of precision which may or may not satisfy us, according to our purposes. Of particular interest in our present context is the possibility (which our example could be used to establish) of obtaining a mathematical sequence whose various frequencies approximate to those of a certain empirical sequence. In dividing sequences into mathematical and empirical ones I am making use of a distinction that may be called ‘intensional’ rather than ‘extensional’. For if we are given a sequence ‘extensionally’, i.e. by listing its elements singly, one after the other—so that we can only know a ﬁnite piece of it, a ﬁnite segment, however long—then it is impossible to determine, from the properties of this segment, whether the sequence of which it is a part is a mathematical or an empirical sequence. Only when a rule of construction is given—that is, an ‘inten- sional’ rule—can we decide whether a sequence is mathematical or empirical. Since we wish to tackle our inﬁnite sequences with the help of the concept of a limit (of relative frequencies), we must restrict our investigation to mathematical sequences, and indeed to those for some structural components of a theory of experience156 which the corresponding sequence of relative frequencies is con- vergent. This restriction amounts to introducing an axiom of con- vergence. (The problems connected with this axiom will not be dealt with until sections 63 to 66, since it turns out to be convenient to discuss them along with the ‘law of great numbers’.) Thus we shall be concerned only with mathematical sequences. Yet we shall be concerned only with those mathematical sequences of which we expect, or conjecture, that they approximate, as regards frequencies, to empirical sequences of a chance-like or random character; for these are our main interest. But to expect, or to conjecture, of a mathematical sequence that it will, as regards frequencies, approximate to an empirical one is nothing else than to frame a hypothesis—a hypothesis about the frequencies of the empirical sequence.1 The fact that our estimates of the frequencies in empirical random sequences are hypotheses is without any inﬂuence on the way we may calculate these frequencies. Clearly, in connection with ﬁnite classes, it does not matter in the least how we obtain the frequencies from which we start our calculations. These frequencies may be obtained by actual counting, or from a mathematical rule, or from a hypothesis of some kind or other. Or we may simply invent them. In calculating frequen- cies we accept some frequencies as given, and derive other frequencies from them. The same is true of estimates of frequencies in inﬁnite sequences. Thus the question as to the ‘sources’ of our frequency estimates is not a problem of the calculus of probability; which, however, does not mean that it will be excluded from our discussion of the problems of probability theory. In the case of inﬁnite empirical sequences we can distinguish two main ‘sources’ of our hypothetical estimates of frequencies—that is to say, two ways in which they may suggest themselves to us. One is an estimate based upon an ‘equal-chance hypothesis’ (or equi-probability hypothesis), the other is an estimate based upon an extrapolation of statistical ﬁndings. 1 Later, in sections 65 to 68, I will discuss the problem of decidability of frequency hypoth- eses, that is to say, the problem whether a conjecture or hypothesis of this kind can be tested; and if so, how; whether it can be corroborated in any way; and whether it is falsiﬁable. *Cf. also appendix *ix. probability 157 By an ‘equal-chance hypothesis’ I mean a hypothesis asserting that the probabilities of the various primary properties are equal: it is a hypoth- esis asserting equal distribution. Equal-chance hypotheses are usually based upon considerations of symmetry.2 A highly typical example is the con- jecture of equal frequencies in dicing, based upon the symmetry and geometrical equivalence of the six faces of the cube. For frequency hypotheses based on statistical extrapolation, estimates of rates of mortality provide a good example. Here statistical data about mortality are empirically ascertained; and upon the hypothesis that past trends will continue to be very nearly stable, or that they will not change much—at least during the period immediately ahead—an extrapolation to unknown cases is made from known cases, i.e. from occurrences which have been empirically classiﬁed, and counted. People with inductivist leanings may tend to overlook the hypo- thetical character of these estimates: they may confuse a hypothetical estimate, i.e. a frequency-prediction based on statistical extrapolation, with one of its empirical ‘sources’—the classifying and actual counting of past occurrences and sequences of occurrences. The claim is often made that we ‘derive’ estimates of probabilities—that is, predictions of frequencies—from past occurrences which have been classiﬁed and counted (such as mortality statistics). But from a logical point of view there is no justiﬁcation for this claim. We have made no logical deriv- ation at all. What we may have done is to advance a non-veriﬁable hypothesis which nothing can ever justify logically: the conjecture that frequencies will remain constant, and so permit of extrapolation. Even equal-chance hypotheses are held to be ‘empirically derivable’ or ‘empiric- ally explicable’ by some believers in inductive logic who suppose them to be based upon statistical experience, that is, upon empirically observed frequencies. For my own part I believe, however, that in mak- ing this kind of hypothetical estimate of frequency we are often guided solely by our reﬂections about the signiﬁcance of symmetry, and by similar considerations. I do not see any reason why such conjectures should be inspired only by the accumulation of a large mass of induct- ive observations. However, I do not attach much importance to these 2 Keynes deals with such questions in his analysis of the principle of indiﬀerence. Cf. op. cit., Chapter IV, pp. 41–64. some structural components of a theory of experience158 questions about the origins or ‘sources’ of our estimates. (Cf. section 2.) It is more important, in my opinion, to be quite clear about the fact that every predictive estimate of frequencies, including one which we may get from statistical extrapolation—and certainly all those that refer to inﬁnite empirical sequences—will always be pure conjecture since it will always go far beyond anything which we are entitled to aﬃrm on the basis of observations. My distinction between equal-chance hypotheses and statistical extrapolations corresponds fairly well to the classical distinction between ‘a priori’ and ‘a posteriori’ probabilities. But since these terms are used in so many diﬀerent senses,3 and since they are, moreover, heavily tainted with philosophical associations, they are better avoided. In the following examination of the axiom of randomness, I shall attempt to ﬁnd mathematical sequences which approximate to random empirical sequences; which means that I shall be examining frequency-hypotheses.*2 58 AN EXAMINATION OF THE AXIOM OF RANDOMNESS The concept of an ordinal selection