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"universal truths."* All this is clearly and correctly expressed; but it must not be forgotten, that it is the language of a writer trained in the schools of Bacon and of Newton.
Even. however, the induction here described by Dr. Wallis, falls greatly short of the method of philosophizing pointed out in the Norum Organon. It coincides exactly with those empirical inferences from mere experience, of which Bacon entertained such slender hopes for the advancement of science. "(1) Restat experien"tia mera; quae si occurrat, casus; si quaesita sit, experi"mentum nominatur. Hoc autem experientiae genus nihil aliud "est, quam mera palpatio, quali homines noctu utuntur, omnia per"tentando, si forte in rectam viam incidere detur; quibus multo "satius et consultius foret, diem praestolari aut lumen accendere, "deinceps viam inire. At contra, verus experientiae ordo primo "lumen accendit, deinde per lumen iter demonstrat, incipiendo ab "experientia ordinata et digesta, et minime praepostera aut erra❝tica, atque ex ea educendo axiomata, atque ex axiomatibus con"stitutis rursus experimenta nova, quum nec verbum divinum in "rerum massam absque ordine operatum sit."
It is a common mistake, in the logical phraseology of the present times, to confound the words experience and induction as controvertible terms. There is, indeed, between them a very close affinity; in as much as it is on experience alone that every legitimate induc
* Institutio Logica.-See the Chapter De Inductione et Exemplo.
Nov. Org. Aph. lxxxii.
"Let it always be remembered, that the author who first taught this doctrine (that the true art of reasoning is nothing but a language accurately defined and skilfully ar• ranged.) had previously endeavoured to prove, that all our notions, as well as the signs by which they are expressed, originate in perceptions of sense; and that the principles on which languages are first constructed, as well as every step in their progress to perfection, all ultimately depend on inductions from observation; in one word, on experience merely."—Aristotle's Ethics and Politics by Gillies, Vol. I. pp. 94, 95.
In the latter of these pages, I observe the following sentence, which is of itself suffi cient to shew what notion the Aristotelians still annex to the word under consideration. "Every kind of reasoning is carried on either by syllogism or by induction; the former proving to us, that a particular proposition is true, because it is deducible from a general one, already known to us; and the latter demonstrating a general truth, because it holds in all particular cases."
It is obvious, that this species of induction never can be of the slightest use in the study of nature, where the phenomena which it is our aim to classify under their gene ral laws, are, in respect of number, if not infinite, at least incalculable and incomprehen. sible by our faculties.
(1) [There remains simple experience; which if it chance to occur, is called accident; if sought for, is denominated experiment. This kind of experience is mere feeling, such as men make use of in the dark, trying every thing, if by this means they may happen to hit upon the right path. Indeed they would act more advisedly and wisely, to wait for daylight, or to light a lamp, and then to attempt the road. The true order of experience takes another method; first, it kindles a light, then eans of this light points out the path; commencing with a regular and well digested course of experi ment, and not with what is absurd and erroneous; and thence deducing axioms, and from these axioms, when established, still new experiments; Even the divine Word did not operate upon chaos but in an appointed order.]
tion must be raised. The process of induction therefore presupposes that of experience; but, according to Bacon's views, the process of experience does by no means imply any idea of induction. Of this method, Bacon has repeatedly said, that it proceeds "by means of rejections and exclusions" (that is, to adopt the phraseology of the Newtonians, in the way of analysis) to separate or decompose nature; so as to arrive at those axioms or general laws, from which we may infer (in the way of synthesis) other particulars formerly unknown to us, and perhaps placed beyond the reach of our direct examination.*
But enough, and more than enough, has been already said to enable my readers to judge, how far the assertion is correct, that the induction of Bacon was well known to Aristotle. Whether it be yet well known to all his commentators, is a different question; with the discussion of which I do not think it necessary to interrupt any longer the progress of my work.
Of the Import of the Words Analysis and Synthesis, in the Language of Modern Philosophy.
As the words Analysis and Synthesis are now become of constant and necessary use in all the different departments of knowledge; and as there is reason to suspect, that they are often employed without due attention to the various modifications of their import, which must be the consequence of this variety in their application, it may be proper, before proceeding farther, to illustrate, by a few examples, their true logical meaning in those branches of science, to which I have the most frequent occasions to refer in the course of these inquiries. I begin with some remarks on their primary signification in that science, from which they have been transferred by the moderns to Physics, to Chemistry, and to the Philosophy of the Human Mind.
Preliminary Observations on the Analysis and Synthesis of the Greek Geometricians.
Ir appears from a very interesting relic of an ancient writer,f that, among the Greek geometricians, two different sorts of analysis were employed as aids or guides to the inventive powers; the one adapted to the solution of problems; the other to the demonstration of theorems. Of the former of these, many beautiful exemplifica
*Nov. Org. Aph. cv. ciii.
Preface to the seventh book of the Mathematical Collections of Pappus Alexandrinus. An extract from the Latin version of it by Dr. Halley may be found in note (P.)
tions have been long in the hands of mathematical students; and of the latter, (which has drawn much less attention in modern times) a satisfactory idea may be formed from a series of propositions published at Edinburgh about fifty years ago.* I do not, however, know, that any person has yet turned his thoughts to an examination of the deep and subtle logic displayed in these analytical investigations; although it is a subject well worth the study of those who delight in tracing the steps by which the mind proceeds in pursuit of scientific discoveries. This desideratum it is not my present purpose to make any attempt to supply; but only to convey such general notions as may prevent my readers from falling into the common error, of confounding the analysis and synthesis of the Greek Geometry, with the analysis and synthesis of the Inductive Philosophy.
In the arrangement of the following hints, I shall consider, in the first place, the nature and use of analysis in investigating the demonstration of theorems.-For such an application of it, various occasions must be constantly presenting themselves to every geo meter; when engaged, for example, in the search of more elegant modes of demonstrating propositions previously brought to light; or in ascertaining the truth of dubious theorems, which, from analogy, or other accidental circumstances, possess a degree of verisimilitude sufficient to rouse the curiosity.
In order to make myself intelligible to those who are acquainted only with that form of reasoning which is used by Euclid, it is necessary to remind them, that the enunciation of every mathematical proposition consists of two parts. In the first place, certain suppositions are made; and secondly, a certain consequence is affirmed to follow from these suppositions. In all the demonstrations which are to be found in Euclid's Elements (with the exception of the small number of indirect demonstrations,) the particulars involved in the hypothetical part of the enunciation are assumed as the principles of our reasoning and from these principles, a series or chain of consequences is, link by link, deduced, till we at last arrive at the conclusion which the enunciation of the proposition asserted as a truth. A demonstration of this kind is called a Synthetical demonstration.
Suppose now, that I arrange the steps of my reasoning in the reverse order; that I assume hypothetically the truth of the proposition which I wish to demonstrate, and proceed to deduce from this assumption, as a principle, the different consequences to which it leads. If, in this deduction, I arrive at a consequence which I already know to be true, I conclude with confidence that the principle from which it was deduced is likewise true. But if, on the other hand, I arrive at a consequence which I know to be false, I conclude, that the principle or assumption on which my reasoning
Propositiones Geometricae More Veterum Demonstratae. Auctore Matthaeo Stewart, S. T. P. Matheseos in Academia Edinensi Professore, 1763,
has proceeded is false also.-Such a demonstration of the truth or falsity of a proposition is called an Analytical demonstration.
According to these definitions of Analysis and Synthesis, those demonstrations in Euclid which prove a proposition to be true, by shewing, that the contrary supposition leads to some absurd inference, are, properly speaking, analytical processes of reasoning.-In every case, the conclusiveness of an analytical proof rests on this general maxim, That truth is always consistent with itself; that a supposition which leads, by a concatenation of mathematical deductions, to a consequence which is true, must itself be true; and that which necessarily involves a consequence which is absurd or impossible, must itself be false.
It is evident, that, when we are demonstrating a proposition with a view to convince another of its truth, the synthetic form of reasoning is the more natural and pleasing of the two; as it leads the understanding directly from known truths to such as are unknown. When a proposition, however, is doubtful, and we wish to satisfy our own minds with respect to it; or when we wish to discover a new method of demonstrating a theorem previously ascertained to be true; it will be found (as I already hinted) far more convenient to conduct the investigation analytically. The justness of this remark is universally acknowledged by all who have ever exercised their ingenuity in mathematical inquiries; and must be obvious to every one who has the curiosity to make the experiment. It is not, however, so easy to point out the principle on which this remarkable difference between these two opposite intellectual processes depends. The suggestions which I am now to offer appear to myself to touch upon the most essential circumstance; but I am perfectly aware that they by no means amount to a complete solution of the difficulty.
Let it be supposed, then, either that a new demonstration is required of an old theorem; or, that a new and doubtful theorem is proposed as a subject of examination. In what manner shall I set to work, in order to discover the necessary media of proof ?-From the hypothetical part of the enunciation, it is probable, that a great variety of different consequences may be immediately deducible ; from each of which consequences, a series of other consequences will follow At the same time, it is possible, that only one or two of these trains of reasoning may lead the way to the truth which I wish to demonstrate. By what rule am I to be guided in selecting the line of deduction which I am here to pursue? The only expedient which seems to present itself, is merely tentative or experimental; to assume successively all the different proximate consequences as the first link of the chain, and to follow out the deduction from each of them, till I, at last, find myself conducted to the truth which I am anxious to reach. According to this supposition, I merely grope my way in the dark, without rule or method: the object I am in quest of, may, after all my labour, elude my search; and even, if I should be so fortunate as to attain it, my
success affords me no lights whatever to guide me in future on a similar occasion.
Suppose now that I reverse this order, and prosecute the investi gation analytically; assuming (agreeably to the explanation already given) the proposition to be true, and attempting, from this supposition, to deduce some acknowledged truth as a necessary consequence. I have here one fixed point from which I am to set out; or, in other words, one specific principle or datum from which all my consequences are to be deduced; while it is perfectly immaterial in what particular conclusion my deduction terminates, provided this conclusion be previously known to be true. Instead, therefore, of being limited, as before, to one conclusion exclusively, and left in a state of uncertainty where to begin the investigation, I have one single supposition marked out to me, from which my departure must necessarily be taken; while, at the same time, the path which I follow, may terminate with equal advantage in a variety of differ ent conclusions. In the former case, the procedure of the understanding bears some analogy to that of a foreign spy, landed in a remote corner of this island, and left to explore, by his own sagacity, the road to London. In the latter case, it may be compared to that of an inhabitant of the metropolis, who wished to effect an escape, by any one of our sea-ports, to the continent. It is scarcely necessary to add, that as this fugitive,-should he happen, after reaching the coast, to alter his intentions,-would easily retrace the way to his own home; so the geometer, when he has once obtained a conclusion in manifest harmony with the known principles of his science, has only to return upon his own steps, (cæca regens filo vestigia) in order to convert his analysis into a direct synthetical proof.
A palpable and familiar illustration (at least in some of the most essential points) of the relation in which the two methods now described stand to each other, is presented to us by the operation of unloosing a difficult knot, in order to ascertain the exact process by which it was formed. The illustration appears to me to be the more apposite, that I have no doubt it was this very analogy, which suggested to the Greek geometers the metaphorical expressions of analysis and of solution, which they have transmitted to the philosophical language of modern times.
Suppose a knot, of a very artificial construction, to be put into my hands as an exercise for my ingenuity, and that I was required to investigate a rule, which others, as well as myself, might be able to follow in practice, for making knots of the same sort. If I were to proceed in this attempt according to the spirit of a geometrical synthesis I should have to try, one after another all the various experiments which my fancy could devise, till I had, at last, hit upon the particular knot I was anxious to tie. Such a process, however, would evidently be so completely tentative, and its final success would, after all, be so extremely doubtful, that common sense could not fail to suggest immediately the idea of tracing the knot through all the various complications of its progress, by cautiously undoing or unknitting each successive turn of the thread in a retrogade or