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115. Studies in the History of Statistics and Probability, Vol. 18

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115. Studies in the History of Statistics and Probability, Vol. 18 Studies in the History of Statistics and Probability Vol. 18 Compiled by Oscar Sheynin Berlin 2020 Contents I am the author of all the contributions listed below Notation I. Prehistory of the theory of probability, 1974 II. Poisson and statistics, 2012 III. Simon Newcomb as a statistician, 2002 IV. Mathematical treatment of astronomical observations, 1973 V. Gauss and geodetic observations, 1994 VI. Gauss, Bessel and the adjustment of triangulation, 2001 VII. The theory of probability. Definition and relation with statistics, 1998 [email protected] 2 Notation Notation S, G, n refers to downloadable file n placed on my website www.sheynin.de which is being diligently copied by Google (Google, Oscar Sheynin, Home). I apply this notation in case of sources either rare or those in my translation into English. L, M, R = Leningrad, Moscow, in Russian 3 I On the Prehistory of the Theory of Probability Arch. Hist. Ex. Sci., vol. 12, N. 2, 1974, pp. 97 – 141 1. Introduction Evidently, none of the traditional sciences busies itself about the accidental, says ARISTOTLE1, continuing that this (the accidental) none of the recognized sciences considers, but only sophistic‚ and repeats himself m other places2. However, this opinion is wide of the mark since neither does the modern theory of probability busy itself with chance, but rather with the laws of chance, with the probable2a. And ARISTOTLE describes rhetoric as an art of persuasion based on probabilities (§ 3.2). Moreover, reasoning on the probable abound in various sciences in antiquity. The study of this aspect of various sciences before the origin of the theory of probability (i. e., before the second half of the 17th century) is attempted in §§ 3 – 8 while § 2 is devoted mainly to ARISTOTLE and his notable commentator, THOMAS AQUINAS. § 9 is a short account of appropriate philosophical reasoning in the new time with a special reference to JAKOB BERNOULLI, and general conclusions are formulated in § 10. It proved difficult to incorporate the prehistory of the theory of errors into this article, and it has been dealt with separately3. 2. Randomness and Probability in Antique Philosophy 2.1. Prior to ARlSTOTLE. The period prior to ARISTOTLE is known partly, possibly even mostly, from his own account4. Thus (196a 10) Certainly the early physicists found no place for chance among the causes which they recognized. … This is strange, whether they supposed that there is no such thing as chance or whether they … omitted to mention it and that too when they sometimes used it, as Empedocles does. … He tells us that most of the parts of animals came to be by chance. There are some too who ascribe this heavenly sphere and all the worlds to spontaneity. … This statement might well cause surprise. For they are asserting that chance is not responsible for the existence or generation of animals and plants, nature or mind yet they assert that the heavenly sphere and the divinest of visible things arose spontaneously. Others, continues ARISTOTLE (196b 5), believe that chance is a cause, ... but that it is inscrutable to human intelligence. … Still others (195b 35) say that nothing happens by chance. In each case the unnamed person(s) is DEMOCRITUS, says the editor of ARISTOTLE. The ambiguity of DEMOCRITUS in his attitude towards chance, which ARISTOTL'E does not mention, seems genuine. This is proved by comparing the opinions of various ancient commentators, including PLATO. Indeed, some of them thought that, according to DEMOCRITUS, people made a fetish of 4 chance so as to conceal their own want of sense while others ascribed to him the explanation of everything by chance5. 2.2. ARISTOTLE. He was the first to attempt an explanation of chance. He included this explanation in the general context of his teaching of causes6, repeatedly mentioned chance, and also accidents and coincidents. An accident, says ARISTOTLE7, actually introducing this term into classical philosophy, is something which may possibly either belong or not belong to any one and the selfsame thing. As to chance (and change), these8 are characteristic of the perishable things of the earth. Some effects can be caused incidentally, i.e., by spontaneity and chance (198a 5); chance is opposed to mind and reason9 and its cause cannot be determined10. The products of chance and fortune are opposed to what is, or comes to be, always or usually. Similar assertions are made about accidents12, chance conjunctions13 and coincidences14. ARISTOTLE also distinguishes between chance and spontaneity15: Chance and what results from chance are appropriate to agents that are capable of good fortune and of moral action generally. Therefore necessarily chance is in the sphere of moral actions. The spontaneous on the other hand is found both in the lower animals and in many inanimate objects. Possibly the best account of ARISTOTLE’S concept of chance is given in a rather rare source by JULIENNE JUNKERSFELD16: by chance ARISTOTLE means something which takes place Occasionally; has the character of an end; is such that it might have been the object of a natural or of a rational appetite; was not in fact the object of any appetite but came into being by accident. As shown above, the circularity of this definition is indeed present in ARISTOTLE’S writings. JUNKERSFELD (p. 78) also mentions the essential difference between chance as understood by ARISTOTLE and in the modern sense: in the latter case chance is not connected with intention (or, rather, with its non-fulfilment)‚ but it seems that by referring only to COURNOT she overestimates his role in the formation of the concept of chance. I mention now three examples of chance events given by ARISTOTLE17 (JUNKERSFELD mentions a dozen): Digging a hole for a plant, someone finds a treasure (not a rusty nail, which can hardly be the object of a natural or of a rational appetite). ARISTOTLE calls this an accident in its first meaning (which actually is the same as chance). I notice that the difficulty of dividing unexpected events into remarkable and usual is here considerably less than in a similar division of outcomes of a random event in natural science needed for evaluating the probability of remarkable outcomes. A meeting of two persons which takes place by chance. Mistakes in the operations of nature which give rise to monstrosities. The first departure of nature from the type is that the offspring should become female instead of male; … as it is possible for the male sometimes not to prevail over the female … either through youth or age or some other such cause. 5 Thus ARISTOTLE attributes the birth of a female to chance, adding though that this is a natural necessity. Then, in general, for ARISTOTLE the outcome of a chance event depends on rather small changes in the chain (chains) of previous events. Of course, this is true only if his connection of chance with non-fulfilment of intention is disregarded. This general illustration is the same as was used by POINCARÉ18: a chance event takes place when, in conditions of an unstable equilibrium‚ very slight causes determine considerable effects. ARISTOTLE connects the chance occurrence of sex with natural necessity, i. e. with a definite (optimal) ratio of males and females for any given species. Though he does not elaborate, this seems to be the first statement connecting chance and necessity. ARISTOTLE’S writings also contain reasoning on the probable. A probability19 Is a generally approved proposition: what men know to happen or not to happen, to be or not to be, for the most part thus and thus ..., e. g. the envious hate. Formulating what actually is a rather weak corollary of the strong law of large numbers, ARISTOTLE20 notices that what is improbable does happen … therefore it is probable that improbable things will happen. On the other hand, ARISTOTLE understands that a very rare event either in games of chance or in nature is impossible (see § 5). He also describes rhetoric as an art of persuasion based on probabilities (§ 3.2) and, speaking about persuasion in poetry and anticipating ARCESILAS and CARNEADES, he even introduces a rudimentary scale of subjective probabilities21: a likely impossibility is always preferable to an unconvincing possibility. Several times ARISTOTLE mentions luck22 and fortune23. His understanding of these concepts is that they qualitatively express deviations from reasonable expectation, which is another qualitative notion. Even if expectation is understood as a numerical measure in the sense of the classical theory of probability, ARISTOTLE’S luck and fortune will still differ from the concepts used by Jakob BERNOULLI: although in one place ARISTOTLE (1361 b – 1362 a) holds that luck equally means luck accruing beyond expectation or escape from expected evil, elsewhere he (1207a 30) supposes that Good fortune would seem to consist to a greater extent and more properly in the obtaining of good … while the escaping of evil is a piece of good fortune indirectly. ARISTOTLE in fact qualifies his own opinion on the impossibility of a science of the accidental (§ 1) when he24 attributes strategy and navigation to matters involving art, but into which chance largely enters. However, he refutes his own qualification by saying (1247a 20) that in navigation not the cleverest are the most fortunate, but it is as in throwing dice. Does this really mean that there can be no science of navigation? A saner opinion is (PLATO25) that, though Chance is almost everything in the arts of the … pilot, and the physician (§ 6), and the general26 … yet in a storm there must surely be a great advantage in having the aid of the pilot’s art. 6 S. SAMBURSKY27 quotes SIMPLICIUS as saying that PLATO (and ARISTOTLE) called natural science the science of the probable (ecotologia).
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