That to the apprehensions of uneducated men such metaphorical or analogical expressions should present the images and the things typified, inseparably combined and blended together, is not wonderful; but it is the business of the philosopher to conquer these casual associations, and, by varying his metaphors, when he cannot completely lay them aside, to accustom himself to view the phenomena of thought in that naked and undisguised state in which they unveil themselves to the powers of consciousness and reflection. To have recourse therefore to the analogies suggested by popular language, for the purpose of explaining the operations of the mind, instead of advancing knowledge, is to confirm and to extend the influence of vulgar errors. After having said so much in vindication of analogical conjectures as steps towards physical discoveries, I thought it right to caution my readers against supposing, that what I have stated admits of any application to analogical theories of the human mind. Upon this head, however, I must not enlarge farther at present. In treating of the inductive logic, I have studiously confined my illustrations to those branches of knowledge in which it has already been exemplified with indisputable success; avoiding, for obvious reasons, any reference to sciences in which its utility still remains to be ascertained. motion of the soul. This infuses a belief, that the mind of man is as a ball in motion, impelled and determined by the objects of sense, as necessarily as that is by the stroke of a racket.” (Principles of Human Knowledge.) III. Supplemental Observations on the words Induction and ANALOGY, as used in Mathematics. BEFORE dismissing the subjects of induction and analogy, considered as methods of reasoning in Physics, it remains for me to take some slight notice of the use occasionally made of the same terms in pure Mathematics. Although, in consequence of the very different natures of these sciences, the induction and analogy of the one cannot fail to differ widely from the induction and analogy of the other, yet, from the general bistory of language, it may be safely presumed, that this application to both of a common phraseology, has been suggested by certain supposed points of coincidence between the two cases thus brought into immediate comparison.* It has been hitherto, with a very few if any exceptions, the universal doctrine of modern as well as of ancient logicians, that“ no mathematical proposition can be proved by induction.” To this opinion Dr. Reid has given his * I bave already observed (See p. 335 of this volume) that mathematicians frequently avail themselves of that sort of induction which Bacon describes fó as proceeding by simple enumeration.” The induction, of which I am now to treat, has very little in common with the other, and bears a much closer resemblance to that recommended in the Novum Organon. sanction in the strongest terms; observing, that “ although in a thousand cases, it should be found by experience, that the area of a plane triangle is equal to the rectangle under the base and half the altitude, this would not prove that it must be so in all cases, and cannot be otherwise, which is what the mathematician affirms.' That some limitation of this general assertion is necessary, appears plainly from the well-known fact, that induction is a species of evidence on which the most scrupulous reasoners are accustomed, in their mathematical inquiries, to rely with implicit confidence; and which, although it may not of itself demonstrate that the theorems derived from it are necessarily true, is yet abundantly sufficient to satisfy any reasonable mind that they hold universally, It was by induction (for example) that Newton discovered the algebraical formula by which we are enabled to determine any power whatever, raised from a binomial root, without performing the progressive multiplications. The formula expresses a relation between the exponents and the co-efficients of the different terms, which is found to hold in all cases, as far as the table of powers is carried by actual calculation ;--from which Newton inferred, that if this table were to be continued in infinitum, the same formula would correspond equally with every successive power. There is no reason to suppose that he ever attempted to prove the theorem in any other way; and yet, there cannot be a doubt, that he was as firmly satisfied of its being universally true, as if he had examined all the different demonstrations of it which have since * Essays on the Intell. Powers, p. 615. 4to. edit. been given. * Numberless other illustrations of the same thing might be borrowed, both from arithmetic and geometry.t * « The truth of this theorem was long known only by trial in particular cases, and by induction from analogy ; nor does it appear that even Newton himself ever attempted any direct proof of it.” (Hutton's Mathematical Dictionary, Art. Binomial Theorem.) For some interesting information with respect to the bistory of this discovery, see the very learned Intro duction prefixed by Dr. Hutton to his edition of Sherwin's Mathematical Tables; and the second volume (p. 165) of the Scriptores Logarithmici, edited by Mr. Baron Maseres. + In the Arithmetica Infinitorum of Dr. Wallis, considerable use is made of the Method of Induction.“ A l'aide d'une induction babilement ménagée (says Montucla) et du fil de l'analogie dont il sçut toujours s'aider avec succès, il soumit â la géométrie une multitude d'objets qui lui avoient échappé jusqu'alors.” (Hist. des Mathem. Tome II. p. 299.) This innovation in the established forms of mathematical reasoning gave offence to some of his contemporaries; in particular, to M. de Fermat, one of the most distinguished geometers of the 17th century. The ground of his objection, however, it is worthy of notice) was not any doubt of the conclusions obtained by Wallis; but because he thought that their truth might hare been established by a more legitimate and elegant process. “Sa façon de demontrer, qui est fondée sur induction plutot que sur un raisonnement à la mode d'Archimede, fera quelque peine aux novices, qui veulent des syllogismes demonstratifs depuis le commencement jusqu'à la fin. Ce n'est pas que je ne l'approuve, mais toutes ses propositions pouvant être demontrées viâ ordinariú, legitimá, et Archimedaâ, en beaucoup moins de paroles, que n'en contient son livre, je ne sçai pas pourquoi il a préferé cette manière à l'ancienne, qui est plus convainquante et plus elegante, ainsi que j'espere lui faire voir à mon premier loisir." Lettre de M. de Fermat a M. le Chev. Kenelme Digby. (See Fermat's Varia Opera Mathematica, p. 191.) For Wallis's reply to these strictures, see his Algebra, Cap. Ixxix; and his Commercium Epistolicum. In the Opuscules of M. Le Sage, I find the following sentence quoted from a work of La Place, which I have not had an opportunity of seeing. The judgment of so great a master, on a logical question relative to his own sindics, is of peculiar value. “La methode d'induction, quoique excellente Into what principles, it may be asked, is the validity of such a proof in mathematics ultimately resolvable ?-To me it appears to take for granted certain general logical maxims; and to imply a secret process of legitimate and conclusive reasoning, though not conducted agreeably to the rules of mathematical demonstration, nor perhaps formally expressed in words. Thus in the instance mentioned by Dr. Reid, I shall suppose, that I have first ascertained experimentally the truth of the proposition in the case of an equilateral triangle ; and that I afterwards find it to hold in all the other kinds of triangles, whether isosceles or scalene, right-angled, obtuse-angled, or acuteangled. It is impossible for me not to perceive, that this property, having no connexion with any of the particular circumstances which discriminate different triangles from each other, must arise from something common to all triangles, and must therefore be a universal property of that figure. In like manner, in the binomial theorem, if the formula correspond with the table of powers in a variety of particular instances, (which instances agree in no other respect, but in being powers raised from the same binomial root,) we must conclude--and, I apprehend that our conclusion is perfectly warranted by the soundest logic,that it is this common property which renders the theorem true in all these cases, and consequently, that it must necessarily hold in every other. Whether, on the supposition that we had never had any previous experience of demonstrative evidence, we should have been led by the mere inductive process, to form the idea of necessary truth, may perhaps be questioned; but the slightest ac pour découvrir des verités générales, ne doit pas dispenser de les démontrer avec rigueur.” (Leçons données aux Ecoles Normales, Prem. Vol. p. 380.) |