Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

In cases where the enumeration is imperfect, Dr. Wallis afterwards observes, "That our conclusion can only amount "to a probability or to a conjecture; and is always liable to

be overturned by an instance to the contrary." He observes also, "That this sort of reasoning is the principal in"strument of investigation in what is now called experimental "philosophy; in which, by observing and examining parti"culars, we arrive at the knowledge of 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 philosophising pointed out in the Novum Organon. It coincides exactly with those empirical inferences from mere experience, of which Bacon entertained such slender hopes for the advancement of science. "Restat experientia mera; quæ si occurrat, casus; si quæsi"ta sit, experimentum nominatur. Hoc autem experientiæ “ genus nihil aliud est, quam mera palpatio, quali homines "noctu utuntur, omnia pertentando, si forte in rectam viam "incidere detur; quibus multo satius et consultius foret, diem "præstolari aut lumen accendere, deinceps viam inire. "contra, verus, experientia ordo primo lumen accendit, de"inde per lumen iter demonstrat, incipiendo ab experientia "ordinata et digesta, et minime præpostera aut erratica, atque "ex ea educendo axiomata, atque ex axiomatibus constitutis "rursus experimenta nova, quum nec verbum divinum in re❝rum massam absque ordine operatum sit."†

At

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

* 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 skil"fully arranged,) 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

very close affinity; inasmuch as it is on experience alone that every legitimate induction 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.

" 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 sufficient 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 deduci"ble from a general one, already known to us; and the latter demonstrating a gene"ral 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 general laws, are, in respect of number, if not infinite, at least incalculable and incomprehensible by our faculties.

* Nov. Org. Aph. cv. ciii.

SECTION III.

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.

I.

Preliminary Observations on the Analysis and Synthesis of the Greek Geometricians.

Ir appears from a very interesting relic of an ancient writer,* 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 exemplifications 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.t I do.

[ocr errors]

Preface to the seventh book of the Mathematical Collections of Pappus Alexan drinus. An extract from the Latin version of it by Dr. Halley may be found in Note (P.)

Auctore Matthæo

+ Propositiones Geometrica More Veterum Deinonstratæ. Stewart, S. T. P. Matheseos in Academia Edinensi Professore, 1763.

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 geometer ;-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 prin ciples 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.

[blocks in formation]

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 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 proposi tion 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

« ΠροηγούμενηΣυνέχεια »