« ForrigeFortsett »
quaintance with mathematics is sufficient to produce the most complete conviction, that whatever is universally true in that science, must be true of necessity; and, therefore, that a universal, and a necessary truth, are, in the language of mathematicians, synonymous expressions. If this view of the matter be just, the evidence afforded by mathematical induction must be allowed to differ radically from that of physical; the latter resolving ultimately into our instinctive expectation of the laws of nature, and consequently, never amounting to that demonstrative certainty, which excludes the possibility of anomalous exceptions.
I have been led into this train of thinking by a remark which La Place appears to me to have stated in terms much too unqualified;—“Que la marche de Newton, dans la découverte de la gravitation universelle, a été exacte ment la même, que dans celle de la formule du binome." When its recollected, that, in the one case, Newton's conclusion related to a contingent, and in the other to a necessary truth, it seems difficult to conceive, how the logical procedure which conducted him to both should have been exactly the same. In one of his queries, he has (in perfect conformity to the principles of Bacon's Jogic) admitted the possibility, that “God may vary the laws of nature, and make worlds of several sorts, in several parts of the universe."
“ At least, (he adds) I see pothing of contradiction in all this."* Would Newton have expressed himself with equal scepticism concerning the universality of his binomial theorem; or admitted the possibility of a single exception to it, in the indefinite
progress of actual involution? In short, did there exist the slightest shade of difference between the degree of his assent to this inductive result, and that extorted from him by a demonstration of Euclid ?
Although, therefore, the mathematician, as well as the natural philosopher, may, without any blameable latitude of expression, be said to reason by induction, when he draws an inference from the known to the unknown, yet it seems indisputable, that, in all such cases, he rests his conclusions on grounds essentially distinct from those which form the basis of experimental science.
The word analogy, too, as well as induction, is common to physics and to pure mathematics. It is thus we speak of the analogy running through the general properties of the different conic sections, with no less propriety than of the analogy running through the anatomical structure of different tribes of animals. In some instances, these mathematical analogies are collected by a species of induction ; in others, they are inferred as consequences from more general truths, in which they are included as particular cases. Thus, in the curves which have just been mentioned, while we content ourselves (as many elementary writers have done)* with deducing their properties from mechanical descriptions on a plane, we rise experimentally from a comparison of the propositions which have been separately demonstrated with respect to each curve, to more comprehensive theorems, applicable to all of them; whereas, when we begin with considering them in their common origin, we have it in our power to trace from
L'Hospital, Simson, &c.
the source, both their generic properties, and their specific peculiarities. The satisfaction arising from this last view of the subject can be conceived by those alone who have experienced it; although I am somewhat doubtful whether it be not felt in the greatest degree by such as, after having risen from the contemplation of particular truths to other truths more general, have been at last conducted to some commanding station, where the mutual connexions and affinities of the whole system are brought, at once, under the range of the eye. Even, however, before we have reached this vantage-ground, the contemplation of the analogy, considered merely as a fact, is pleasing to the mind; partly, from the mysterious wonder it excites, and partly from the convenient generalization of knowledge it affords. To the experienced mathematician this pleasure is farther enhanced, by the assurance which the analogy conveys, of the existence of yet undiscovered theorems, far more extensive and luminous than those whick have led him by a process so indirect, so tedious, and comparatively so unsatisfactory, to his general conclusions.
In this last respect, the pleasure derived from analogy in mathematics, resolves into the same principle with that which seems to have the chief share in rendering the analogies among the different departments of nature so interesting a subject of speculation. lo both cases, a powerful and agreeable stimulus is applied to the curiosity, by the encouragement given to the exercise of the inventive faculties, and by the hope of future discovery, which is awakened and cherished. As the analogous properties (for instance) of the conic sections, point to some general theorems of which they are corollaries ; so the analogy between the phenomena of Electricity and those of Gal
vanism irresistibly suggests a confident, though vague anticipation of some general physical law comprehending the phenomena of both, but differently modified in its sensible results by a diversity of circumstances. * Indeed, it is by no means impossible, that the pleasure we receive even from those analogies which are the foundation of poetical metaphor and simile, may be found resolvable, in part, into the satisfaction connected with the supposed discovery of truth, or the supposed acquisition of knowledge: the faculty of imagination giving to these illusions a momentary ascendent over the sober conclusions of experience: and gratifying the understanding with a flattering consciousness of its own force, or at least with a consolatory forgetfulness of its own weakness.
of certain misapplications of the words Experience
and Induction in the phraseology of Modern Science. Illustrations from Medicine and from Political Economy.
In the first Section of this Chapter, I endeavoured to point out the characteristical peculiarities by which the Inductive Philosophy of the Newtonians is distinguished from the hypothetical systems of their predecessors: and which entitle us to indulge hopes with respect to the permanent stability of their doctrines, which might be regarded as chimerical, if, in anticipating the future his
See Note (Y.)
tory of science, we were to be guided merely by the analogy of its revolutions in the ages that are past.
In order, however, to do complete justice to this argument, as well as to prevent an undue extension of the foregoing conclusions, it is necessary to guard the reader against a vague application of the appropriate terms of inductive science to inquiries which have not been rigore ously conducted, according to the rules of the inductive logic.
From a want of attention to this consideration, there is a danger, on the one hand, of lending to sophistry or to ignorance the authority of those illustrious names whose steps they profess to follow; and, on the other, of bringing discredit on that method of investigation, of which the language and other technical arrangements hare been thus perverted.
Among the distinguishing features of the new logic, when considered in contrast with that of the schoolmen, the most prominent is the regard which it professes to pay to experience, as the only solid foundation of human knowledge. It may be worth while, therefore, to consider, how far the notion commonly annexed to this word is definite and precise; and whether there may not sometimes be a possibility of its being employed in a sense more general and loose, than the authors who are looked up to as the great models of inductive investigation understood it to convey, *
* As the reflectio:s which foliow are entirely of a practical nature, I shall express myself
' (as far as is consistent with a due regard to precision) ngreeably to the modes of speaking in common use; without affecting a scrupulous attention to some speculative distinctions, whicb, however