7. Probability and Statistics Soviet Essays - Sheynin, Oscar

successes of physical statistics. Beginning with Galileo and Newton, mathematicians neverheld the principle of causality in special esteem. For us, much more important are thefunctional dependences or equations in several magnitudes allowing to determine any ofthem given the other ones and supposing that the elements not included in the equation donot influence the value of the magnitude sought. The so-called laws of nature, as for examplethe law of inertia or the Newton law of universal gravitation, whose cause he did not think fitto seek for, are expressed by dependences of such a type. Einstein united both these laws andnow they comprise the highest synthesis of the general theory of relativity, although weremain as far as Newton was from knowing their cause.[6] The new contemporary stage in the development of scientific thought is characterizedby the need to introduce the notion of probability into the statements of the elementary lawsof nature. And since we do not inquire into the cause of the law of inertia which is a propertyof the four-dimensional Minkowski space, I think that we may just as well assume, as acharacteristic of the isotropy of space, the law expressing that, when occurring far from theattracting masses, inertial movement can with equal probabilities take place within each oftwo equal angles.In a similar way, the postulate on the existence of independent magnitudes and phenomena,without which no general law of nature can be formulated, and lacking which the verycategory of causality would have become senseless, must be precisely stated in the languageof the theory of probability. For example, the independence of two points moving under theirown momenta is expressed by equal probabilities of all the values of the angle between theirvelocities 4 .Such are the initial assumptions of statistical mechanics, which, in the kinetic theory ofgases, led Maxwell to his well-known law of distribution of molecular velocities. They arefully corroborated experimentally by their corollaries emerging in accord with the calculus ofprobability. The paradox of the irreversibility of heat processes which are caused byreversible molecular movements can be fully explained by two equivalent suppositions, – bythe Jeans hypothesis of molecular chaos and by the Ehrenfest quasi-ergodic hypothesis, bothof them consistently advancing the principle of continuity.The study of the proper motions of stars revealed that the hypothesis of the isotropy of thestellar space in the above sense should be altered owing to the existence of an asymmetricfield of attraction. Nevertheless, Eddington and Charlier, in developing Poincaré’s idea,established interesting similarities between molecular and stellar motions. Indeed, theisotropic Maxwellian law of the distribution of velocities is replaced in the latterphenomenon by an analogous Schwarzschild ellipsoidal law of distribution of stellarvelocities which rather well conforms to astronomical observations.Thus, in general, along with the laws of nature according to which, under certainconditions and arbitrary circumstances of any other kind, the occurrence of a definite resultA in all trials is necessary, we also admit such laws which do not always lead to theoccurrence of the event A given the corresponding conditions . However, in this second case,no matter what are the other circumstances, all trials are characterized by some uniformity ofthe link between and A that we express by stating that conditions determine theprobability of the event A.The Mendelian law of heredity is of such a kind. It states that, when interbreeding hybridsof some definite species, for example, of lilac beans, with each other, the probability of theappearance of white beans (that is, of individuals belonging to one of the pure races) is 1/4. Itis out of order to dwell here on the genetic foundations of the Mendelian theory, or, ingeneral, on the various justifications that can guide a researcher when he assumes that in alltrials of a certain type the probability of the occurrence of the event A is one and the same.