“Semper sub Sextis perdita Roma fuit.”
The utmost precautions will not provide against all contingencies. To avoid errors in important calculations, it is usual to have them repeated by different computers; but a case is on record in which three computers made exactly the same calculations of the place of a star, and yet all did it wrong in precisely the same manner, for no apparent reason.[179]
Summary of the Theory of Inductive Inference.
The theory of inductive inference stated in this and the previous chapters, was suggested by the study of the Inverse Method of Probability, but it also bears much resemblance to the so-called Deductive Method described by Mill, in his celebrated System of Logic. Mill’s views concerning the Deductive Method, probably form the most original and valuable part of his treatise, and I should have ascribed the doctrine entirely to him, had I not found that the opinions put forward in other parts of his work are entirely inconsistent with the theory here upheld. As this subject is the most important and difficult one with which we have to deal, I will try to remedy the imperfect manner in which I have treated it, by giving a recapitulation of the views adopted.
All inductive reasoning is but the inverse application of deductive reasoning. Being in possession of certain particular facts or events expressed in propositions, we imagine some more general proposition expressing the existence of a law or cause; and, deducing the particular results of that supposed general proposition, we observe whether they agree with the facts in question. Hypothesis is thus always employed, consciously or unconsciously. The sole conditions to which we need conform in framing any hypothesis is, that we both have and exercise the power of inferring deductively from the hypothesis to the particular results, which are to be compared with the known facts. Thus there are but three steps in the process of induction:—
(1) Framing some hypothesis as to the character of the general law.
(2) Deducing consequences from that law.
(3) Observing whether the consequences agree with the particular facts under consideration.
In very simple cases of inverse reasoning, hypothesis may seem altogether needless. To take numbers again as a convenient illustration, I have only to look at the series,
1, 2, 4, 8, 16, 32, &c.,