1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

demand to apply transfinite numbers but was rather reduced to usualmathematical induction.In the applied theory of probability the harmless imp turns out asharp horned devil who favours to corrupt meanly statisticalhomogeneity. So far as we keep to the concept of ensemble and checkthat homogeneity by available methods, we are able at least to revealin time the devilish dirty trick whereas, abandoning it, we whollysurrender ourselves to the devil’s rule and ought to be prepared forsurprises. Thus, from the point of view of modern probability theory,the boundary between science and magic is defined by the notion ofstatistical ensemble. It follows that inferences, derived by applyingthat theory when a statistical ensemble of experiments is lacking, hasno scientific certainty.Unlike the arsenal of magic, the tools of science must be entirelyjustified. However, when concluding that, for example, the error of aresult obtained from a single realization of a stochastic process issituated in the given interval with probability 0.95, we do not know towhat does that probability correspond, − to an ensemble of realizationswhich we ought to conjuncture by issuing from the single observedrealization so as to apply the notion of stochastic process?But all those other realizations are irrelevant and it is very easy toprovide examples of faulty inferences made when applying the theoryof probability in manufacturing, geology, etc where it is senseless todiscuss statistical ensembles. Historically, science emerged frommagic but treats it disdainfully and would wish to ignore it. However,we should not wholly yield to that temptation either.A representative of the constructive direction in mathematicsconsiders the usual mathematical analysis a magic. We should ratherdistinguish between white and black magic the latter connected withbeing subjectively unconscionable. At present, we can not ignorehonest attempts to apply probability theory when statistical ensemblesare lacking. I venture to forecast that something being magic todaywill become science tomorrow. It would have been unreasonable tokeep too strongly to the established concept of statistical homogeneity.However, here I will entirely hold on to that concept since nowadaysany other method of obtaining really plausible results is lacking.1.4. Summary. Thus, while perfection of experimenting is going onin one or another branch of science or technology, a special situationoften arises when statistical stability is present but complete stabilityof the results is impossible to achieve. The former is characterized bystability of the frequencies of the occurrences of the various eventsconnected with the experiment’s outcome.An exhausting check of such stability (statistical homogeneity,statistical ensemble) is impossible, but in many cases the presence of astatistical ensemble is sufficiently certain. According to modern ideas,these cases indeed comprise the field of scientific applications of theprobability theory.And still there exists a readily understood wish to apply it also inother cases in which the results of the experiments are not definite, butthe existence of a statistically homogeneous ensemble is impossible.For the time being, such applications belong to magic rather than11