IN the present chapter it is proposed to prove, that —
It is more probable that any law, at the knowledge of which we have arrived by observation, shall be subject to one of those violations which, according to Humes definition, constitutes a miracle, than that it should not be so subjected.
To show this, we may be allowed again to revert to the Calculating Engine: and to assume that it is possible to set the [149/150] machine, so that it shall calculate any algebraic law whatsoever: and also possible so to arrange it, that at any periods, however remote, the first law shall be interrupted for one or more times, and be superseded by any other law; after which the original law shall again be produced, and no other deviation shall ever take place.
Now, as all laws, which appear to us regular and uniform in their course, and to be subject to no exception, can be calculated by the engine: and as each of these laws may also be calculated by the same machine, subject to any assigned interruption, at distinct and definite periods; each simple law may be interrupted at any point by the temporary action of a portion of any one of all the other simple laws: it follows, that the class of laws subject to Interruption is far more extensive than that of laws which are uninterrupted. It is, in fact, infinitely more numerous. Therefore, the probability of any law with which we have become acquainted [150/151] by observation being part of a much more extensive law, and of its having, to use mathematical language, singular points or discontinuous functions contained within it, is very large.
Perhaps it may be objected, that the laws calculated by such an engine as I have referred to are not laws of nature; and that any deviation from laws produced by human mechanism does not come within Hume's definition of miracles. To this it may be answered, that a law of nature has been defined by Hume to rest upon experience, or repeated observation, just as the truth of testimony does. Now, the law produced by the engine may be arrived at by precisely the same means—namely, repeated observation.
It may, however, be desirable to explain further the nature of the evidence, on which the fact, that the engine possesses those powers, rests. [151/152]
When the Calculating Engine has been set to compute the successive terms of any given law, which the observer is told will have an apparent exception (at, for example, the ten million and twenty-third term,) the observer is directed to note down the commencement of its computations; and, by comparing these results with his own independent calculations of the same law, he may verify the accuracy of the engine as far as he chooses. It may then be demonstrated to him, by the very structure of the machine, that if its motion were continued, it would, necessarily, at the end of a very long time, arrive at the ten-millionth term of the law assigned to it; and that, by an equal necessity, it would have passed through all the intermediate terms.1 The inquirer is now desired to turn on the wheels with his own hand, until they are precisely in the same situation as they would have been had the engine itself gone on continuously, to the ten-millionth term. The machine is again put in motion, [152/153] and the observer again finds that each successive term it calculates fulfils the original law. But, after passing twenty-two terms, he now observes one term which does not fulfil the original law, but which does coincide with the predicted exception.
The continued movement now again produces terms according with the first law, and the observer may continue to verify them as long as he wishes. It may then be demonstrated to him, by the very structure of the machine, that, if its motion were continued, it would be impossible that any other deviation from the apparent law could ever occur at any future time.
Such is the evidence to the observer; and, if the superintendent of the engine were, at his request, to make it calculate a great variety of different laws, each interrupted by special and remote exceptions, he would have ample ground to believe in the assertion of its director, that he could so arrange the engine that any [153/154] law, however complicated, might be calculated to any assigned extent, and then there should arise one apparent exception; after which the original law should continue uninterrupted for ever.
Let us now consider the miracle alluded to by Hume—the restoration of a dead man to life. According to the definition of that author, our belief that such a fact is contrary to the laws of nature, arises from our uniform experience against it. Our personal experience is small: we must therefore have recourse to testimony; and from that we learn, that the dead are never restored to life; and, consequently, we have the uniform experience of all mankind since the creation, against one assigned instance of a dead man being so restored. Let us now find the numerical amount of this evidence. Assuming the origin of the human race to have been about six thousand years ago, and taking thirty years as the duration of a generation, we have — [154/155]
6000/30 = 200 generations
And allowing that the average population of the earth has been a thousand millions, we find that there have been born and have died since the creation,
200 x 1,000,000,000 = 2,000,000,000 individuals
Such, then, according to Hume, are the odds against the truth of the miracle: that is to say, it is found from experience, that it is about two hundred thousand millions to one against a dead man having been restored to life.
Let us now compare this with a parallel case in the calculations of the engine; let us suppose the number above stated to be a hundred million times as great, or that the truth of the miracles is opposed by a number of instances, expressed by twenty places of figures. [155/156]
The engine may be set to count the natural numbers — 1, 2, 3, 4, &c.; and it shall continue to fulfil that law, not merely for the number of times just mentioned, for that number is quite insignificant among the vast periods it involves; but the natural numbers shall follow in continual succession, until they have reached an amount which requires for its expression above a hundred million places of figures. If every letter in the volume now before the reader's eyes were changed into a figure, and if all the figures contained in a thousand such volumes were arranged in order, the whole together would yet fall far short of the vast induction the observer would then have had in favour of the truth of the law of natural numbers. The widest range of all the cycles of astronomy and geology combined, sink into insignificance before such a period. Yet, shall the engine, true to the prediction of its director, after the lapse of myriads of ages, fulfil its task, and give that one, the first and only exception to that time-sanctioned law. What would have [156/157] been the chances against the appearance of the excepted case, immediately prior to its occurrence? It would have had, according to Hume, the evidence of all experience against it, with a force myriads of times more strong than that against any miracle.
Now, let the reader, who has fully entered into the nature of the argument, ask himself this question :—Does he believe that such an engine has really been contrived, and what reasonable grounds has he for that belief?
The testimony of any single witness is small against such odds; besides, the witness may deceive himself. Whether he speaks truly, will be estimated by his moral character—whether he deceives himself, will be estimated by his intellectual character. The probability that such an engine has been contrived, will, however, receive great addition, when it is remarked, that mathematical—and, especially, geometrical evidence is, of all others, that in [157/158] which the fewest mistakes arise, and in which they are most readily discovered; and when it is added, that the fact of the invention of such an engine rests on precisely the same species of evidence as the propositions of Euclid, and may be deduced from the drawings with all the force of demonstration. Whether such an engine could be actually made in the present state of mechanical art, is a question of quite a different order: it must rest upon the opinions of those who have had extensive experience in that art. The author has not the slightest hesitation in stating his opinion to be, that it is fully within those limits.
This, however, is a question foreign to the nature of the argument, which might have been stated in a more abstract manner, without any reference to such an engine. As, however, the argument really arose from that machine, and as visible forms make a much deeper impression on the mind than any abstract reasonings, it has been stated in conjunction with that subject.
13 December 2008