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I am, at the same time, fully aware, that whoever, in treating of the human mind, aims to be understood, must lay his account with forfeiting, in the opinion of a very large proportion of readers, all pretensions to depth, to subtlety, or to invention. The acquisition of a new nomenclature is, in itself, no inconsiderable reward to the industry of those, who study only from motives of literary vanity; and, if D'Alembert's idea of this branch of science be just, the wider an author deviates from truth, the more likely are his conclusions to assume the appearance of discoveries. I may add, that it is chiefly in those discussions which possess the best claims to originality, where he may expect to be told by the multitude, that they have learned from him nothing but what they knew before.
The latitude with which the word metaphysics is frequently used, makes it necessary for me to remark,with respect to the foregoing passage from D'Alembert, that he limits the term entirely to an account of the origin of our ideas. “The generation
“ The generation of our ideas, he tells us, " belongs to metaphysics. It forms one of the principal objects, and perhaps ought to form the sole object of that science.”* -If the meaning of the word be extended, as it too often is in our language, so as to comprehend all those inquiries which relate to the theory and to the improvement of our mental powers, some of his observations must be understood with very important restrictions. What he has stated, however, on the inseparable connexion between perspicuity of style and soundness of investigation in metaphysical disquisitions, will be found to hold equally in every research to which that epithet can, with any color of propriety, be applied.
OF THE FUNDAMENTAL LAWS OF HUMAN BELIEF; OR THE PRI
MARY ELEMENTS OF HUMAN REASON.
The propriety of the title prefixed to this Chapter will, I trust, be justified sufficiently by the speculations which are to follow. As
espèce de réminiscence de ce que notre ame a déja su ; l'obscurité, quand il y en a, vient toujours de la faute de l'auteur, parce que la science qu'il se propose d'en. seigner n'a point d'autre langue que la langue commune. Aussi peut-on appliquer aux bons auteurs de metaphysique ce qu'on a dit des bons écrivains, qu'il n'y a personne qui en les lisant, ne croie pouvoir en dire autant qu'eux.
“ Mais si dans ce genre tous sont faits pour entendre, tous ne sont pas faits pour instruir. Le mérite de fair entrer avec facilité dans les esprits des notions vraies et simples, est beaucoup plus grand qu'on ne pense, puisque l'expérience nous prouve combien il est rare ; les saines idées métaphysiques sont des vérités communes que chacun saisit, mais que peu d'hommes ont le talent de développer ; tant il est difficile, dans quelque sujet que ce puisse être, de se rendre propre ce qui appartient à tout le monde.”—Elémens de Philosophie.
# " La génération de nos idées appartient à la métaphysique ; c'est un de ses objets principaux, et peut-être devroit elle s'y borner.”'-Ibid.
these differ, in some essential points, from the conclusions of former writers, I found myself under the necessity of abandoning, in various instances, their phraseology;—but my reasons for the particular changes which I have made, cannot possibly be judged of, or even understood, till the inquiries by which I was led to adopt them be carefully examined.
I begin with a review of some of those primary truths, a conviction of which is necessarily implied in all our thoughts and in all our actions ; and which seem, on that account, rather to form constituent and essential elements of reason, than objects with which reason is conversant. The import of this last remark will appear more clearly afterwards.
The primary truths to which I mean to confine my attention at present are, 1. Mathematical axioms: 2. Truths (or more properly speaking, laws of belief,) inseparably connected with the exercise of consciousness, perception, memory, and reasoning.--Of some additional laws of belief, the truth of which is tacitly recognized in all our reasonings concerning contingent events, I shall have occasion to take notice under a different article.
Of Mathematical Axioms.
I HAVE placed this class of truths at the head of the enumeration, merely because they seem likely, from the place which they hold in the elements of geometry, to present to my readers a more interesting and at the same time an easier subject of discussion, than some of the more abstract and latent elements of our knowledge, afterwards to be considered. In other respects, a different arrangement might perhaps have possessed some advantages, in point of strict logical method.
1. On the evidence of mathematical axioms it is unnecessary to enlarge, as the controversies to which they have given occasion are entirely of a speculative, or rather scholastic description ; and have no tendency to affect the certainty of that branch of science to which they are supposed to be subservient.
It must at the same time be confessed, with respect to this class of propositions (and the same remark may be extended to axioms in general,) that some of the logical questions connected with them continue still to be involved in much obscurity. In proportion 10 their extreme simplicity is the difficulty of illustrating or of describing their nature in unexceptionable language ; or even of ascertaining a precise criterion by which they may be distinguished from other truths which approach to them nearly. It is chiefly owing to this, that, in geometry, there are no theorems of which it is so difficult to give a rigorous demonstration, as those, of which persons unacquainted with the nature of mathematical evidence are apt to say, that they require no proof whatever. But the inconveniences arising from these circumstances are of trifling moment; occasioning, at the worst, some embarrassment to those mathematical writers, who are studious of the most finished elegance in their exposition of elementary principles; or to metaphysicians, anxious to display their subtilty upon points which cannot possibly lead to any practical conclusion.
It was long ago remarked by Locke, of the axioms of geometry, as stated by Euclid, that although the proposition be at first enunciated in general terms, and afterwards appealed to, in its particular applications as a principle previously examined and admitted, yet that the truth is not less evident in the latter case than in the former. He observes farther, that it is in some of its particular applications, that the truth of every axiom is originally perceived by the mind; and, therefore, that the general propsoition, so far from being the ground of our assent to the truths which it comprehends, is only a verbal generalization of what, in particular instances, has been already acknowledged as true.
The same author remarks, that some of these axioms “are no more than bare verbal propositions, and teach us nothing but the respect and import of names one to another. The whole is equal to all its parts : what real truth, I beseech you, does it teach us? What more is contained in that maxim, than what the signification of the word totum, or the whole, does of itself import? And he that knows that the word whole stands for what is made up of all its parts, knows very little less, than that “the whole is equal to all its parts.” And upon the same ground, I think, that this proposition, A hill is higher than a valley, and several the like, may also pass for maxims.
Notwithstanding these considerations, Mr. Locke does not object to the form which Euclid has given to his axioms, or to the place which he has assigned to them in his Elements. On the contrary, he is of opinion, that a collection of such maxims is not without reason prefixed to a mathematical system ; in order that learners, “having in the beginning perfectly acquainted their thoughts with these propositions made in general terms, may have them ready to apply to all particular cases as formed rules and sayings. Not that, if they be equally weighed, they are more clear and evident than the instances they are brought to confirm ; but that, being more familiar to the mind, the very naming of them is enough to satisfy the understanding." In farther illustration of this, he adds very justly and ingeniously, that “although our knowledge begins in particulars, and so spreads itself by degrees to generals; yet, afterwards, the mind takes quite the contrary course, and having drawn its knowledge into as general propositions as it can, makes them familiar to its thoughts, and accustoms itself to have recourse to them, as to the standards of truth and falsehood.”
But although, in mathematics, some advantage may be gained, without the risk of any possible inconvenience, from this arrangement of axioms, it is a very dangerous example to be followed in other branches of knowledge, where our notions are not equally clear and precise ; and where the force of our pretended axioms (to use Mr. Locke's words,)“ reaching only to the sound, and not to the signification of the works, serves only to lead us into confusion, mistakes and error.” For the illustration of this remark, I must refer to Locke.
Another observation of this profound writer deserves our attention, while examining the nature of axioms; "that they are not the foundations on which any of the sciences is built ; nor at all useful in helping men forward to the discovery of unknown truths."-(Book iv. chap. 7. sec. 11.-2. 3.) This observation I intend to illustrate afterwards, in treating of the futility of the syllogistic art. At present I shall only add to what Mr. Locke has so well stated, that, even in mathematics, it cannot with any propriety be said, that the axioms are the foundation on which the science rests; or the first principles from which its more recondite truths are deduced. Of this I have little doubt that Locke was perfectly aware ; but the mistakes which some of the most acute and enlightened of his disciples have committed in treating of the same subject, convince me that a farther elucidation of it is not altogether superfluous. With this view, I shall here introduce a few remarks on a passage in Dr. Campbell's Philosophy of Rhetoric, in which he has betrayed some misapprehensions on this very point, which a little more attention to the bints already quoted from the Essay on Human Understanding might have prevented. These remarks will, I hope, contribute to place the nature of axioms, more particularly of mathematical axioms, in a different and clearer light than that in which they have been commonly considered.
Of intuitive evidence, says Dr. Campbell, that of the following propositions may serve as an illustration : One and four make five. Things equal to the same thing are equal to one another. The whole is greater than a part; and, in brief, all axioms in arithmetic and geometry. These are, in effect, but so many expositions of our own general notions, taken in different views. Some of them are no more than definitions, or equivalent to definitions. To say, one and four make five, is precisely the same thing as to say, we give the name of five to one added to four. In fact, they are all in some respects reducible to this axiom, whatever is, is. I do not say they are deduced from it, for they have in like manner that original and intrinsic evidence, which makes them, as soon as the terms are understood, to be perceived intuitively. And, if they are not thus perceived, no deduction of reason will ever confer on
them any additional evidence. Nay, in point of time, the discovery of the less general truths has the priority, not from their superior evidence, but solely from this consideration, that the less general are sooner objects of perception to us. But I affirm, that though not deduced from that axiom, they may be considered as particular exemplifications of it, and coincident with it, inasmuch as they are all implied in this, that the properties of our clear and adequate ideas can be no other than what the mind clearly perceives them to be.
But, in order to prevent mistakes, it will be necessary further to illustrate this subject. It might be thought that, if axioms were propositions perfectly identical, it would be impossible to advance a step by their means, beyond the simple ideas first perceived by the
And it must be owned, if the predicate of the proposition were nothing but a repetition of the subject, under the same aspect, and in the same or synonymous terms, no conceivable advantage could be made of it for the furtherance of knowledge. Of such propositions, for instance, as these, -seven are seven, eight are eight, and ten added to eleven are equal to ten added to eleven, it is manifest that we could never avail ourselves for the improvement of science. Nor does the change of the term make any
alteration in point of utility. The propositions, twelve are a dozen, twenty are a score, unless considered as explications of the words dozen and score, are equally insignificant with the former. But when the thing, though in effect coinciding, is considered under a different aspect; when what is single in the subject is divided in the predicate and conversely; or when what is a whole in the one is regarded as a part of something else in the other; such propositions lead to the discovery of innumerable and apparently remote relations. One added to four may be accounted no other than a definition of the word five, as was remarked above. But when I say, 'Two added to three are equal to five, “ I advance a truth which, though equally clear, is quite distinct from the preceding. Thus, if one should affirm, 'That twice fifteen make thirty," and again, that 'thirteen added to seventeen make thirty,' nobody would pretend that he had repeated the same proposition in other words. The cases are entirely similar. In both cases, the same thing is predicated of ideas which, taken severally, are different. From these again result other equations, as "one added to four are equal to two added to three,' and “twice fifteen are equal to thirteen added to seventeen.
Now, it is by the aid of such simple and elementary principles, that the arithmetician and algebraist proceed to the most astonishing discoveries. Nor are the operations of the geometrician essentially different.”
I have little to object to these observations of Dr. Campbell, as far as they relate to arithmetic and to algebra ; for, in these sciences, all our investigations amount to nothing more than to a com