But in what different tones must we speak of mathematics! Mill, if I remember rightly, said it was as full of mysteries as theology. But while the value of theology for knowledge is disputed, the value of mathematics for knowledge is indisputable. Its triumphs can be appreciated by the most foolish, they appeal to the most material. If they seem sometimes lost to ordinary view in the realms of abstract infinities, they do not disdain to serve us in the humbler fields of practice. They have helped mankind to all the greatest generalisations about the physical universe: and without them we should still be fumbling over simple problems of practical mechanics, entangled in a costly and ineffectual empiricism. But while we thank the mathematician for his aid in conquering Nature, we envy him his powers of understanding her. Though he deals, it would seem, entirely with abstractions, they are abstractions which, at his persuasion, supply the key to the profoundest secrets of the physical universe. He holds the clues to mazes where the clearest intellect, unaided, would wander hopelessly astray. He belongs to a privileged caste. I intend no serious qualification of this high praise when I add that, as regards the immediate subject of this lecture, I mean Probability, mathematicians do not seem to have given ignorant inquirers like myself all the aid which perhaps we have a right to ask. They have treated the subject as a branch of applied mathematics. They have supplied us with much excellent theory. They have exercised admirable skill in the solution of problems. But I own that, when we inquire into the rational basis of all this imposing superstructure, their explanations, from the lay point of view, leave much to be desired. “Probability,” says an often-quoted phrase of Butler, “is the guide of life.” But the Bishop did not define the term; and he wrote before the theory of probability had attained to all its present dignities. Neither D’Alembert nor Laplace had discussed it. Quetelet had not applied it to sociology, nor Maxwell to physics. Jevons had not described it as the “noblest creation of the intellect.” It is doubtful whether Butler meant by it exactly what the mathematicians mean by it, and certain that he did not suspect any lurking ambiguity in the expression. Nor, indeed, would the existence of such ambiguity be commonly admitted by any school of thought. The ordinary view is that the theory of probabilities is, as Laplace described it, “common sense reduced to calculation.” That there could be two kinds of probability, only one of which fitted this description, would be generally regarded as a heresy. But it is a heresy in which I myself believe; and which, with much diffidence, I now propose to defend. II