incarnation of the lusts, powers, and instincts of our animal nature .... yield to the wonder-working sway and sovereignty of benignant reason, represented by Prospero." Now reason rules no one: it only points the way to the fulfilment of one's strongest desires. That much neglected essayist, Mr. Bernard Shaw, has put this whole matter with inimitable skill. "The difference between Caliban and Prospero is not that Prospero has killed passion in himself whilst Caliban has yielded to it, but that Prospero is mastered by holier passions than Caliban's." Shaw goes on: "The ingrained habit of thinking of the propensities of which we are ashamed as 'our passions,' and our shame of them and our propensities to noble conduct as a negative and inhibitory department called generally our conscience [Mr. Sherman calls it the reason, Mr. More the inner check], leads us to conclude that to accept the guidance of our passions is to plunge recklessly into the insupportable tedium of what is called a life of pleasure.... Reactionists against the almost equally insupportable slavery of what is called a life of duty are nevertheless willing to venture on these terms.... No great harm is done beyond the inevitable and temporary excesses produced by all reactions; for, as I have said, the would-be wicked ones find, when they come to the point, that the indispensable qualification for a wicked life is not freedom but wickedness." Here are words which Mr. Sherman and the philosophers of the humanist school may ponder. At least, they will not retort that Mr. Shaw is a pseudo-scientist, for on this point he has the whole psychological world behind him. Finally we come to the third story of Mr. Sherman's edifice, the spiritual world, defined as "the plane of spiritual beings and the home of eternal ideas." I fear that science has sadly depopulated this world. As Mr. Sherman knows full well, "Al was this land fulfild of fayerye; I speke of manye hundred yeres ago! And alas, with the elves have gone the incubi and witches, the devils and angels, the cherubim and seraphim, until for most of us God on his sapphire throne is the solitary inhabitant of the spiritual world. And there are statistics to show that the most eminent scientists have dismissed this concept also. Mr. Sherman himself seems to reduce God to that benevolent abstraction, "a power not ourselves that makes for righteousness." But if the third floor front is almost vacant, there is yet “the home of eternal ideas" in the third floor back. Among these eternal verities, doubtless, are the elementary principles of conduct, which, Mr. Sherman declares in his introduction, "have been adequately tested and are now to be unequivocally accepted." When were the tests completed? In 1791, when Burke announced that no more discoveries were to be made in morality? Or within the last fifty years and one, since the Nation has upheld the changeless principles of idealism? Only twenty-five years ago the Nation's review of Tess of the D'Urbervilles came out flat-footed for a double standard of morality. Is this one of the tested principles of conduct to be unequivocally accepted, or has the perfected idealism been defined only within the last quarter of a century? The humanist critics, Messrs. Babbitt, More, and Sherman, seem as reluctant to ventilate their eternal ideas in categorical form as the Germans are in the matter of their war aims, and doubtless for the same reason: they would be starting a fight behind their own lines. The home of the old eternal verities is being prepared for evacuation, and we may look for the eventual passing of the third floor back. It has always been a taunt of the humanist critics that the romantic writers lived in an ivory tower, remote from the crowd and bustle of life, and dreamed of man as he never was and never could be. Does not Mr. Sherman's analysis of human nature reveal the humanists themselves dwelling in an ivory tower of academic contemplation, the walls lined with the orthodox classics, the vaults containing the latest authoritative copy of the Decalogue for the benefit of the inmates? When they wish to discover what is the nature of man, they take from the shelf Aristotle or Shakespeare or even Hooker, and without asking awkward questions imbibe reverently the inspired conjectures which once upon a time did duty for a systematic, experimental study of the mind. If we do not look to the ivory tower of romanticism for the best that has been thought and said in the world, why should we listen below the ivory tower of humanism for echoes of the rudimentary psychology and ethics of ancient Greece and Elizabethan England! CHARLES HEATON. ON THE CONCEPTION OF PROBABILITY. The desire for a mathematically and a philosophically sound introduction to the theory of probability is dictated by the importance of the field of applications, by the fact that the theory of probability is mathematics, and by the philosophical interest attached to the term probability. The applications of the theory were originally confined to problems in gambling, but they are now found in statistics, theory of error, statistical mechanics, insurance, etc. This expansion of the field of usefulness would by itself be a sufficient cause for minor changes in the fundamental conceptions and definitions, just as, for instance, in the steel industry the development in methods and use has been followed by changes in the standard definitions and specifications of iron and steel. To this comes that the introduction of new mathematical and logical methods, such as the axiomatic method, has furnished new view-points for the initial steps in the theory of probability. And the inquiry of philosophy into the nature of probability can never be expected to be answered absolutely and finally. In this way all of the relations of the theory to its applications and sources are likely to exert their influence on the fundamental part of this subject. The fact remains that it is extremely difficult to give a satisfactory definition of probability; quoting Poincaré, it is even hardly possible. These conditions taken together explain why the question "what is probability?" in spite of its long history is still alive, and this serves as an apology for the reappearance of the subject in this paper. The historical methods of introducing probability may be classified according to their relation to four principal methods. The first three of these will, for the purpose of orientation, be briefly mentioned. The discussion in connection with the fourth of the principal methods forms the main part of this paper. It presents a view-point which is thought to throw some new light on the question. The classification follows: 1. Bayes's definition based on the notion of mathematical expectation (espérance mathématique, mathematische Erwartung). 1 H. Poincaré, Calcul des probabilités, ed. 1912, p. 24. 2. The method involving the introduction of the notion of equally possible cases as a fundamental notion. 3. The axiomatic method. 4. The method which will here be called the statistical, which has been used by statisticians, and which is based primarily on the law of great numbers. 1. Bayes's ideas of probability are perhaps the oldest historically. At least according to Von Kries's interpretation, they are underlying, though not definitely formulated, in the classical correspondence between Pascal and Fermat. Bayes expressed those ideas in a definition. An example will explain the principle involved. Playing head and tail with two dollars at stake, with even chance of winning and losing, the value of the chance to win is reasonably estimated as one dollar. Then, according to Bayes's definition, the probability of winning would be expressed as one dollar divided by two dollars, that is 1/2. If in general the reasonable value of the chance-the mathematical expectation—of obtaining A is B then B/A is the probability. It seems evident that the modern student of applied mathematics can hardly expect to obtain the clearest notion of probability by way of Bayes's definition unless he has occupied himself extensively with gambling. It seems that only thereby may one develop the notion of the value of a chance, the notion of the mathematical expectation of gain, as a fundamental conception or as an idea of an existing tangible reality. The definition depends on a reference to this reality. Bayes's definition though interesting has now chiefly historical value. 2. The second principal method, found in many classical discussions, makes use of the expressions equally possible or equally probable. These expressions are taken from the ordinary spoken language, and their meaning remains essentially unchanged after they have been introduced and used in the mathematical theory. Illustrative examples are used among which one is predominant, or fundamental as far as it may be said to represent schematically the formation of any probability. It is the example of the bag containing balls of different color but otherwise the same. Assume p white balls in the bag out of a total of q, then there are p equally possible cases out of q in which one draws a white ball by taking one out. Then by the definition the probability is p/q. This gen 2 Von Kries, Die Prinzipien der Wahrscheinlichkeitsrechnung, 1886, p. 267. 3 Th. Bayes, "An Essay toward Solving a Problem in the Doctrine of Chances," Philosoph. Transactions, 1763, p. 370. eral method has its disadvantages. What is "equally possible"? To answer this additional explanations of great length have been deemed necessary at the various times. It is sufficient at this place to mention the classical works by Laplace, J. F. Fries, Lexis, Von Kries, and Bertrand, and besides, two more recent discussions by Lourié and Grelling. In spite of the high value of these works, in spite of the increased understanding of probability due to them, their inevitable extensiveness certainly makes them less accessible than is to be desired. And if the notion of the equally possible cases is maintained as the fundamental idea, it is not unlikely that further discussions on the same basis will be found necessary in the future. 3. The axiomatic method has been tried by Broggi and Bohlmann. Terms such as "event A," "probability of event A," "event A independent of B or excluding B," etc., are introduced in the axioms stating the two principal laws of combining probability. The axioms can be stated briefly, and logical rigidity as far as the mathematical theory itself is concerned may thus be obtained. When partially independent notions are to be introduced, such as, for instance, that of continuity of probabilities, then the necessary additional axioms are established without difficulty. Nevertheless, as mentioned by the originators of the axioms, the method solves the questions involved only in part. If one does not know what probability is before the axioms are stated the chance remains that one will not know it afterward either. The logical problem left is essentially that to which the main efforts of the previously mentioned extensive works were devoted. The axiomatic method then has its value as supplementary to other solutions. Between the axioms by themselves and reality there is no bridge. 4. The fourth and last principal method is the statistical method Laplace, Théorie analytique des probabilités, philosophical introduction from 2d ed. on. J. F. Fries, Versuch einer Kritik der Prinzipien der Wahrscheinlichkeitsrechnung, 1842. W. Lexis, Zur Theorie der Massenerscheinungen, 1877. Von Kries, Die Prinzipien der Wahrscheinlichkeitsrechnung, 1886. Bertrand, Calcul des probabilités, 1889 (introduction). S. Lourié, Die Prinzipien der Wahrscheinlichkeitsrechnung, 1910. K. Grelling, Die philosophischen Grundlagen der Wahrscheinlichkeitsrechnung, Abhandlungen der Friesschen Schule, 1910. 5 G. Bohlmann, Encyc. d. math. Wiss., Vol. I, Part II, Art. 1D 4b (19001904). U. Broggi, Die Axiome der Wahrscheinlichkeitsrechnung, Dissertation, Göttingen, 1907. G. Bohlmann, Die Grundbegriffe der Wahrscheinlichkeitsrechnung, Atti del 4. Congresso Internazionale dei Matematici, Roma, 6-11 Aprile 1908, Vol. III, 1909, pp. 244 etc. The point of view of the continuous probabilities is emphasized by L Bachelier in his Calcul des probabilités, 1912. |