A similar reply might be given if we suppose our ignorance carried yet a step further. Instead of knowing that our die was loaded, and being ignorant only of the manner of its loading, we might be entirely ignorant whether it was loaded or not. The chances of a particular number turning up on the first throw would still be one-sixth. But the series to which this estimate would refer would neither be one composed of fair throws with a fair die, nor one composed of a series of throws with dice loaded at random, but one composed of a series of throws with dice chosen at random from a random collection of dice, loaded and not loaded!
It seems plain that we have no experimental knowledge of series piled on series after this fashion. Our conclusions about them are not based on observation, nor collected from statistics. They are arrived at a priori; and when the character of a series is arrived at a priori, the probability of a particular event belonging to it can be arrived at independently by the same method. No reference to the series is required. The reason we estimate the chances against any one of the six possible throws of a die as five to one under each and all of the suppositions we have been discussing is that under none of them have we any ground for thinking any one of the six more probable than another;—even though we may have ground for thinking that in a series of throws made with that particular die, some number, to us unknown, will in fact turn up with exceptional frequency.
The most characteristic examples, therefore, of problems in probability depend for their solution on a bold use of the “principle of sufficient reason.” We treat alternatives as equally likely when we cannot see any ground for supposing that one is more likely than another. This seems sensible enough; but how far may we carry this process of extracting knowledge from ignorance? An agnostic declines to offer any opinion on the being of God because it is a matter about which he professes to know nothing. But the universe either has a spiritual cause, or it has not. If the agnostic is as ignorant as he supposes, he cannot have any reason for preferring the first alternative to the second, or the second to the first. Must he, therefore, conclude that the chances of Theism are even? The man who knows this knows much. He knows, or may know, that God’s existence is slightly more probable than his own chance of winning a coup at Monte Carlo. He knows, or may know, the exact fraction by which the two probabilities differ. How, then, can he call himself an agnostic?
Every one must, I think, feel that such reasoning involves a misuse of the theory of probability. But is that misuse without some justification? The theory, unless I misread it, permits, or rather requires, us to express by the same fraction probabilities based on what is little less than complete knowledge, and probabilities based on what is little more than complete ignorance. To arrive at a clear conclusion, it seems only necessary to apply the “law of sufficient reason” to defined alternatives; and it is apparently a matter of perfect indifference whether we apply this law in its affirmative or its negative shape; whether we say “there is every reason for believing that such and such alternatives happen equally often,” or whether we say “there is no reason for thinking that one alternative happens more often than the other.” I do not criticise this method; still less do I quarrel with it. On the contrary, I am lost in admiration of this instrument of investigation, the quality of whose output seems to depend so little on the sort of raw material with which it is supplied.
III
My object, indeed, is neither to discuss the basis on which rests the calculus of probabilities—a task for which I own myself totally unfit—nor yet to show that a certain obscurity hangs over the limits within which it may properly be employed. I desire rather to suggest that, wherever those limits are placed, there lies beyond them a kind of probability yet more fundamental, about which the mathematical methods can tell us nothing, though it possesses supreme value as a “guide of life.”
Wherein lies the distinction between the two? In this: the doctrine of calculable probability (if I may so call it) has its only application, or its only assured application, within groups whose character is either postulated, or is independently arrived at by inference and observation. These groups, be they natural or conventional, provide a framework, marking out a region wherein prevails the kind of ignorance which is the subjective reflection of objective “randomness.” This is the kind of ignorance which the calculus of probabilities can most successfully transmute into knowledge: and herein lies the reason why the discoverers of the calculus found their original inspiration in the hazards of the gambling-table, and why their successors still find in games of chance its happiest illustrations. For in games of chance the group framework is provided by convention; perfect “randomness” is secured by fitting devices; and he who attempts to modify it is expelled from society as a cheat.
None of these observations apply to the kind of probability on whose importance I am now insisting. If calculable probability be indeed “common sense reduced to calculation,” intuitive probability lies deeper. It supports common sense, and it supplies the ultimate ground—be it secure or insecure—of all work-a-day practice and all scientific theory. It has nothing to do with “randomness”; it knows nothing of averages; it obeys no formal laws; no light is thrown on it by cards or dice; it cannot be reduced to calculation. How, then, is it to be treated? What place is it to occupy in our general scheme?
These are all important questions. But no answer to them can be given till we have pressed somewhat further the line of thought which the discussion in this present lecture has for a moment interrupted. Before I began this long parenthesis on the theory of chance, I was occupied with a most important example of a belief which possesses the highest degree of intuitive probability, but no calculable probability at all. I mean the belief in an independent physical universe. In the next lecture I shall resume the general thread of my argument, and consider another belief of the same kind which is not less—some would say even more—essential to natural science than the one with which I have already dealt. I mean a belief in the regularity of nature.