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from my principles, in a variety of different ways; and nothing, "in general, appears to me more difficult, than to ascertain by which "of these processes it is really produced." * The same remark may (with a very few exceptions) be extended to every hypothetical theory which is unsupported by any collateral probabilities arising from experience or analogy; and it sufficiently shews, how infinitely inferior such theories are, in point of evidence, to the conclusions obtained by the art of the decypherer. The principles, indeed, on which this last art proceeds, may be safely pronounced to be nearly infallible.
In these strictures upon Hartley, I have endeavoured to do as much justice as possible to his general argument, by keeping entirely out of sight the particular purpose which it was intended to serve. By confining too much his attention to this, Dr. Reid has been led to carry, farther than was necessary or reasonable, an indiscriminate zeal against every speculation to which the epithet hypothetical can, in any degree, be applied. He has been also led to overlook the essential distinction between hypothetical inferences from one department of the Material World to another, and hypothetical inferences from the Material World to the Intellectual. It was with the view of apologizing for inferences of the latter description, that Hartley advanced the logical principle which gave occasion to the foregoing discussion; and therefore, I apprehend, the proper answer to his argument is this:-Granting your principle to be true in all its extent, it furnishes no apology whatever for the Theory of Vibrations. If the science of mind admit of any illustration from the aid of hypotheses, it must be from such hypotheses alone as are consonant to the analogy of its own laws. To assume, as a fact, the existence of analogies between these laws and those of matter, is to sanction that very prejudice which it is the great object of the inductive science of mind to eradicate.
I have repeatedly had occasion, in some of my former publications, to observe, that the names of almost all our mental powers and operations are borrowed from sensible images. Of this number are intuition; the discursive faculty; attention; reflection; conception; imagination; apprehension; comprehension; abstraction; invention; capacity; penetration; acuteness. The case is precisely similar with the following terms and phrases, relative to a different class of mental phenomena;-inclination; aversion; deliberation; pondering; weighing the motives of our actions; yielding to that motive
Dissertatio de Methodo. In the sentence immediately following, Des Cartes mentions the general rule which he followed, when such an embarrassinent occurred. "Hinc aliter me extricare non possum, quàm si rursus aliqua experimenta quaeram : quae talia sint, ut eorum idem non sit futurus eventus, si hoc modo quam si illo explicetur." The rule is excellent; and it is only to be regretted, that so few exemplifications of it are to be found in his writings.
[From this difficulty I am not able to extricate myself, otherwise, than by having recourse to other experiments, of such a nature, that their issue may not be the same, if un folded in the present rather than the former manner.]
which is the strongest ;-expressions (it may be remarked in passing) which, when employed without a very careful analysis of their import, in the discussion concerning the liberty of the will, gratuitously prejudge the very point in dispute; and give the semblance of demonstration, to what is, in fact, only a series of identical propositions, or a sophistical circle of words.*
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.
Supplemental Observations on the words INDUCTION and ANALOGY, as used in Mathe matics.
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 dif ferent 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 history of language, it may be safely presumed, that this application to both of a common phraseology,
*"Nothing (says Berkeley) seems more to have contributed towards engaging men in controversies and mistakes with regard to the nature and operations of the mind, than the being used to speak of those things in terms borrowed from sensible ideas. For example, the will is termed the 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.)
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 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 rect"angle 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."t
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 inferrred, 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 been given. Numberless other illustrations of the same thing might be borrowed, both from arithmetic and geometry.§
* I have already observed (see p. 197 of this volume) that mathematicians frequently avail themselves of that sort of induction which Bacon describes " as proceeding by sim "ple enumeration." The induction, of which I am now to treat, has very little in com mon with the other, and bears a much closer resemblance to that recommended in the Novum Organon.
Essays on the Intell. Powers, p. 615. 4to edit.
"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 history of this discovery, see the very learned Introduction prefixed by Dr. Hutton to his edition of Sherwin's Mathemati cal 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 habilement ménagée (says ontucla) et du fil de l'analogie dont il scut toujours s'aider avec succès, il soumit à la géométrie une multi31
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, obtuseangled, or acute angled. 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
tude 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 have 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 ordinariâ, legitimâ, et Ärchimedœâ, en beaucoup moins de paroles, que n'en contient son livre, je ne scai 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. lxxix.; and his Commercium Epistolicum.
In his 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 studies, is of peculiar value. "La methode d' induction, quoique excellente pour découvrir des vérités générales, ne doit pas dispenser de les démontrer avec rigueur." (Leçons données aux Ecoles Normales, Prem. Vol. p. 380.)
[By the aid of an induction ably managed, and of the clue of analogy which he well knew how to use with success, he submits to the proof of geometry a multitude of objects which had formerly escaped him.]
+[This method of demonstration, which is founded upon induction rather than a metbod of reasoning resembling that of Archimedes, will give some trouble to novices, who wish for demonstrative syllogisms from the beginning to the end. It is not because I do not approve of this method, but as all his propositions may be demonstrated in the ordinary, legitimate and Archimedaan way, in fewer words than his book contains, I know not why he has preferred this method to the ancient, which is more convincing and more elegant, as I hope to make him see at my earliest leisure. Letter of M. Fermat to Sir Kenelm Digby.]
[The method of induction, although excellent for the discovery of general truths, ought not to excuse us from demonstrating them rigorously.]
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 acquaintance 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;"(1) Que la marche de Newton, dans la découverte de la gravitation “universelle, a été exactement la même, que dans celle de la formule "du binome." When it is 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 logic) 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 nothing 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
* Query 31.
(1) [The process of Newton in his discovery of universal gravitation, has been exactly the same as in that of the formula of the Binomial Theorem.]