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gether, as by their immediate comparison, and as it were juxta-position or application one to another, to perceive their agreement or disagreement, it is fain, by the intervention of other ideas (one or more, as it happens to discover the agreement or disagreement which it searches; and this is that which we call reaJoning. Thus the mind being willing to know the agreement or disagreement in bigness, between the three angles of a triangle and two right ones, cannot by an immediate view and comparing them do it; be. cause the three angles of a triangle cannot be brought at once, and be compared with any one or two angles ; and so of this the mind has no immediate, no intuitive knowledge. In this case the mind is fain to find out some other angles, to which the three angles of a tri. angle have an equality ; and finding those equal to two right ones, comes to know their equality to two right ones.
§ 3. Depends on Proofs. Those intervening ideas which serve to show the agreement of any two others, are called proofs; and where the agreement or disagreement is by this means plainly and clearly perceived, it is called demonstration, it being shown to the understanding, and the mind made see that it is so. A quickness in the mind to find out these intermediate ideas (that shall discover the agreement or disagreement of any other) and to apply them right, is, I suppose, that which is called fagacity.
§ 4. But not so easy. This knowledge by intervening proofs, though it be certain, yet the evidence of it is not altogether fo clear and bright, nor the affent so ready, as in intuitive knowledge. For though in demonstration, the mind does at last perceive the agreement or disagreement of the ideas it confiders, yet it is not without pains and attention: there must be more than one tranfient view to find it. A steady application and pursuit is required to this discovery; and there must be a progrefson by steps and degrees, before the mind can in this
way arrive at certainty, and come to perceive the agreement or repugnancy between two ideas that need proofs and the use of reason to show it.
Ś s. Not without precedent Doubt. ANOTHER difference between intuitive and demonstrative knowledge, is, that though in the latter all doubt be removed, when by the intervention of the intermediate ideas the agreement or difagreement is perceived; yet before the demonstration there was a doubt, which in intuitive knowledge cannot happen to the mind, that has its faculty of perception left to a degree capable of diftinct ideas, no more than it can be a doubt to the eye (that can distinctly fee white and black) whether this ink and this paper be all of a colour. If there be fight in the eyes, it will at first glimpse, without hesitation perceive the words printed on this paper different from the colour of the paper ; and so if the mind have the faculty of distinct percep. tions, it will perceive the agreement or disagreement of those ideas that produce intuitive knowledge. If the eyes have lost the faculty of seeing, or the mind of perceiving, we in vain inquire after the quickness of light in one, or clearnefs of perception in the other.
jó. Not so clear. IT is true, the perception produced by demonstration is also very clear, yet it is often with a great abatement of that evident luftre and full assurance, that always accompany that which I call intuitive ; like a face reflected by several mirrors one to another, where, as long as it retains the fimilitude and agreement with the obje&, it produces a knowledge ; but it is still in every successive reflection with a leffening of that perfect clearness and distinctness, which is in the first, till at last, after many removes, it has a great mixture of dimness, and is not at first sight so knowable, especially to weak eyes. Thus it is with knowledge, made out by a long train of proofs.
§ 7. Each step must have intuitive Evidence. Now, in every step reason makes in demonstrative know. lodge, there is an intuitive knowledge of that agrece
ment or disagreement it seeks with the next intermediate idea, which it uses as a proof; for if it were not so, that yet would need a proof ; since without the perception of such agreement or disagreement, there is no knowledge produced. If it be perceived by itself, it is intuitive knowledge ; if it cannot be perceived by itself, there is need of some intervening idea, as a common measure to show their agreement or disagreement: By' which it is plain that every step in reasoning that produces knowledge, has intuitive certainty ; which when the mind perceives, there is no more required, but to remember it to make the agreement or disagreement of the ideas; concerning which we inquire, vilble and certain. So that to make any thing a demonstration, it is necessary to perceive the immediate agreement of the intervening ideas, whereby the agreement or disagreement of the two ideas under examination (whereof the one is always the first, and the other the last in the account) is found. This intuitive perception of the agreement or disagreement of the intermediate ideas, in each step and progression of the demonstration, must also be carried exactly in the mind, and a man must be sure that no part is left out; which, because in long deductions, and the use of many proofs, the memory does not al. ways so readily and exactly retain; therefore it comes to pass, that this is more imperfect than intuitive. knowledge, and men embrace often falsehood for de. monstrations.
$ 8. Hence the mistake, Ex præcognitis et preconcelis. The neceílity of this intuitive knowledge, in each step of scientifical or demonstrative reasoning, gave occa. fion, I imagine, to that mistaken axion, that all rea. Soning was ex præcognitis et præconcellis; which how far it is mistaken, I hhall have occasion to show more at large, when I come to confider propofitions, and particularly those propositions which are called maxims; and to show that it is by a mistake, that they are supposed-to be the foundation of all our knowsledge and reasonings.
$9. Demonstration not limited to Quantity. It has been generally taken for granted, that mathematics alone are capable of demonstrative certainty ; but to have such an agreement or disagreement as may intuitively be perceived, being, as I imagine, not the privilege of the ideas of number, extension and figure alone ; it may poslibly be the want of due method and application in us, and not of sufficient evi, dence in things, that demonstration has been thought to have so little to do in other parts of knowledge, and been scarce so much as aimed at by any but mathematicians. For whatever ideas we have, wherein the mind can perceive the immediate agreement or disagreement that is between them, there the mind is capable of intuitive knowledge ; and where it can perceive the agreement or disagreement of any two ideas, by an intuitive perception of the agreement or disagreement they have with any intermediate ideas, there the mind is capable of demonstration, which is not limited to ideas of extension, figure, number, and their modes.
§ 10. Wby it has been so thought. The reafon why it has been generally sought for, and fupposed to be only in those, I imagine has been not only the general usefulness of those sciences, but because, in comparing their equality or excess, the modes of numbers have every the least difference very clear and perceivable ; and though in extension, every the least excess is not so perceptible, yet the mind has found out ways to examine and discover demonstratively the just equality of two angles, or extensions, or figures; and both these, i. e. numbers and figures, can be set down by visible and lasting marks, wherein the ideas under consideration are perfectly determined; which-for the molt part they are not, where they are marked only by names and words.
11. Bur in other fimple ideas, whose modes and differences are made and counted by degrees, and not quan, tity, we have not so nice and accurate a ditinction of their differences, as to perceive or find ways to measure their just equality, or the least differences. For those other simple ideas, being appearances or sensa. tions, produced in us by the fize, figure, number and motion of minute corpuscles fingly infenfible, their different degrees also depend upon the variation of some, or all of those causes ; which since it cannot be observed by us in particles of matter, whereof each is too subtle to be perceived, it is imposible for us to have any exact measures of the different degrees of these fimple ideas. For supposing the sensation of idea we name whiteness be produced in us by a certain number of globules, which having a verticity about their own centres, strike upon the retina of the eye, with a certain degree of rotation, as well as progresfive swiftness, it will hence easily follow, that the more the superficial parts of any body are so ordered, as to refled the greater number of globules of light, and to give them that proper rotation which is fit to produce this sensation of white in us, the more white will that body appear, that from an equal space sends to the retina the greater number of such corpuscles, with that peculiar sort of motion. I do not say, that the nature of light consists in very small round glo. bules, nor of whiteness in such a texture of parts, as gives a certain rotation to these globules when it reflects them ; for I am not now treating physically of light or colours : But this, I think, I may say, that I cannot (and I would be glad any one would make intelligible that he did) conceive how bodies without us can any ways affect our senses, but by the imme. diate contact of the sensible bodies themselves, as in tafting and feeling, or the impulse of some insensible particles coming from them, as in seeing, hearing, and smelling; by the different impulse of which parts, caused by their different size, figure and motion, the variety of sensations is produced in us.