following values so far as we can succeed in describing 65,536 256 19,729 figures. Fifth order, number expressed by 19,729 figures. Sixth order, number expressed by figures, to express the number of which figures would require about It may give us some notion of infinity to remember that at this sixth step, having long surpassed all bounds. of intuitive conception, we make no approach to a limit. Nay, were we to make a hundred such steps, we should be as far away as ever from actual infinity. It is well worth observing that our powers of expression rapidly overcome the possible multitude of finite objects which may exist in any assignable space. Archimedes showed long ago, in one of the most remarkable writings of antiquity, the Liber de Arena Numero, that the grains of sand in the world could be numbered, or rather, that if numbered, the result could readily be expressed in arithmetical notation. Let us extend his problem, and ascertain whether we could express the number of atoms which could exist in the visible universe. The most distant stars which can now be seen by telescopes-those of the sixteenth magnitude-are supposed to have a distance of about 33,900,000,000,000,000 miles. Sir W. Thomson has shown reasons for supposing that there do not exist more than from 3 x 1024 to 1026 molecules in a cubic centimetre of a solid or liquid substance.1 Assuming these data to be true, for the sake of argument, a simple calculation enables us to show that the almost inconceivably vast sphere of our stellar system if entirely filled with solid matter, would not contain more than about 68 × 1090 atoms, that is to say, a number requiring for its expression 92 places of figures. Now, this number would be immensely less than the fifth order of the powers of two. In the variety of logical relations, which may exist 1 Nature, vol. i. p. 553. between a certain number of logical terms, we also meet a case of higher combinations. We have seen (p. 142) that with only six terms the number of possible selections of combinations is 18,446,744,073,709,551,616. Considering that it is the most common thing in the world to use an argument involving six objects or terms, it may excite some surprise that the complete investigation of the relations in which six such terms may stand to each other, should involve an almost inconceivable number of cases. Yet these numbers of possible logical relations belong only to the second order of combinations. CHAPTER X. THE THEORY OF PROBABILITY. THE subject upon which we now enter must not be regarded as an isolated and curious branch of speculation. It is the necessary basis of the judgments we make in the prosecution of science, or the decisions we come to in the conduct of ordinary affairs. As Butler truly said, " Probability is the very guide of life." Had the science of numbers been studied for no other purpose, it must have been developed for the calculation of probabilities. All our inferences concerning the future are merely probable, and a due appreciation of the degree of probability depends upon a comprehension of the principles of the subject. I am convinced that it is impossible to expound the methods of induction in a sound manner, without resting them upon the theory of probability. Perfect knowledge alone can give certainty, and in nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowledge-knowledge mingled with ignorance, producing doubt, A great difficulty in this subject consists in acquiring a precise notion of the matter treated. What is it that we number, and measure, and calculate in the theory of probabilities? Is it belief, or opinion, or doubt, or knowledge, or chance, or necessity, or want of art? Does probability exist in the things which are probable, or in the mind which regards them as such? The etymology of the name lends us no assistance: for, curiously enough, probable is ultimately the same word as provable, a good instance of one word becoming differentiated to two opposite meanings. 1 Chance cannot be the subject of the theory, because there is really no such thing as chance, regarded as producing and governing events. The word chance signifies falling, and the notion of falling is continually used as a simile to express uncertainty, because we can seldom predict how a die, a coin, or a leaf will fall, or when a bullet will hit the mark. But everyone sees, after a little reflection, that it is in our knowledge the deficiency lies, not in the certainty of nature's laws. There is no doubt in lightning as to the point it shall strike; in the greatest storm there is nothing capricious; not a grain of sand lies upon the beach, but infinite knowledge would account for its lying there; and the course of every falling leaf is guided by the principles of mechanics which rule the motions of the heavenly bodies. Chance then exists not in nature, and cannot coexist with knowledge; it is merely an expression, as Laplace remarked, for our ignorance of the causes in action, and our consequent inability to predict the result, or to bring it about infallibly. In nature the happening of an event has been pre-determined from the first fashioning of the universe. Probability belongs wholly to the mind. This is proved by the fact that different minds may regard the very same event at the same time with widely different degrees of probability. A steam-vessel, for instance, is missing and some persons believe that she has sunk in mid-ocean; others think differently. In the event itself there can be no such uncertainty; the steam-vessel either has sunk or has not sunk, and no subsequent discussion of the probable nature of the event can alter the fact. Yet the probability of the event will really vary from day to day, and from mind to mind, according as the slightest information is gained regarding the vessels met at sea, the weather prevailing there, the signs of wreck picked up, or the previous condition of the vessel. Probability thus belongs to our mental condition, to the light in which we regard events, the occurrence or non-occurrence of which is certain in themselves. Many writers accordingly have asserted that probability is concerned with degree or quantity of belief. De Morgan says,1 "By degree of proba 1 Formal Logic, p. 172. bility we really mean or ought to mean degree of belief." The late Professor Donkin expressed the meaning of probability as "quantity of belief; " but I have never felt satisfied with such definitions of probability. The nature of belief is not more clear to my mind than the notion which it is used to define. But an all-sufficient objection. is, that the theory does not measure what the belief is, but what it ought to be. Few minds think in close accordance with the theory, and there are many cases of evidence in which the belief existing is habitually different from what it ought to be. Even if the state of belief in any mind could be measured and expressed in figures, the results would be worthless. The value of the theory consists in correcting and guiding our belief, and rendering our states of mind and consequent actions harmonious with our knowledge of exterior conditions. This objection has been clearly perceived by some of those who still used quantity of belief as a definition of probability. Thus De Morgan adds-"Belief is but another name for imperfect knowledge." Donkin has well said that the quantity of belief is "always relative to a particular state of knowledge or ignorance; but it must be observed that it is absolute in the sense of not being relative to any individual mind; since, the same information being presupposed, all minds ought to distribute their belief in the same way."1 Boole seemed to entertain a like view, when he described the theory as engaged with "the equal distribution of ignorance; but we may just as well say that it is engaged with the equal distribution of knowledge. 2 I prefer to dispense altogether with this obscure word | belief, and to say that the theory of probability deals with quantity of knowledge, an expression of which a precise. explanation and measure can presently be given. An event is only probable when our knowledge of it is diluted with ignorance, and exact calculation is needed to discriminate how much we do and do not know. The theory has been described by some writers as professing to evolve knowledge out of ignorance; but as Donkin admirably remarked, it is really "a method of avoiding the erection 1 Philosophical Magazine, 4th Series, vol. i. p. 355. 2 Transactions of the Royal Society of Edinburgh, vol. xxi. part 4. |