Page:Encyclopædia Britannica, Ninth Edition, v. 3.djvu/176

Rh contend for the possibility of reducing to these (with the help of definitions) the special principles of mathematics, commonly allowed to pass and do duty as axiomatic. Still others apply the name equally and in the same sense to the general principles of thought and to some principles of special science. In view of such differences of opinion as to the actual matter in question, it is not to be expected that there should be agreement as to the marks character istic of axioms, nor surprising that agreement, where it appears to exist, should often be only verbal. The charac ter of necessity, for example, so much relied upon for ex cluding the possibility of an experiential origin, may either, as by Kant, be carefully limited to that which can be claimed for propositions that are at the same time syn thetic, or may be vaguely taken (as too frequently by Leibnitz) to cover necessity of mere logical implication the necessity of analytic, including identical, propositions which Kant allowed to be quite consistent with origin in experience. The question being so perplexed, uo other course seems open than to try to determine the nature of axioms mainly upon such instances as are, at least practically, admitted by all, and these are mathematical principles. That propositions with an exceptional character of certainty are assumed in mathematical science is notorious ; that such propositions must be assumed as principles of the science, if it is to be at once general and demonstrative, is now conceded even by extreme experientialists ; while it is, farther, universally held that it is the exceptional character of the subject-matter of mathematics that renders possible such determinate assumption. What the actual principles to be assumed are, has, indeed, always been more or less disputed ; but this is a point of secondary importance, since it is possible from different sets of assumption to arrive at results practically the same. The particular list of proposi tions passing current in modern times as Euclid s axioms, like his original list of common notions, is open to objection, not so much for mixing up assertions not equally underiva- tive (as the ancient critics remarked), but for including two the 8th and 9th which are unlike all the others in being mere definitions (viz., of equals and of whole or part). Being intended as a body of principles of geometry in particular within the general science of mathematics, the modern list is not open to exception in that it adds to the pro positions of general mathematical import, forming Euclid s original list, others specially geometrical, provided the addi tions made are sufficient for the purpose. It does, in any case, contain what may be taken as good representative in stances of mathematical axioms both general and special; for example, the 1st. Things equal to the same are equal to one another, applicable to all quantity; and the 10th, Two straight lines cannot enclose a space, specially geometrical. (The latter has been regarded by some writers as either a mere definition of straight lines, or as contained by direct implication in the definition ; but incorrectly. If it is held to be a definition, nothing is too complex to be so called, and the very meaning of a definition as a principle of science is abandoned ; while, if it is held to be a logical implication of the definition, the whole science of geometry may as well be pronounced a congeries of analytic propositions. When straight line is strictly defined, the assertion is clearly seen to be synthetic.) Now of such propositions as the two just quoted it is commonly said that they are self-evident, that they are seen to be true as soon as stated, that their opposites are inconceivable; and the expressions are not too strong as descriptive of the peculiar certainty pertaining to them. Nothing, however, is thereby settled as to the ground of the certainty, which is the real point in dispute between the experiential and rational schools, as these have become determinately opposed since the time and mainly through the influence of Kant. Such axioms, according to Kant, being necessary as well as synthetic, cannot be got from experience, but depend on the nature of the knowing faculty ; being immediately synthetic, they are not thought discursively but apprehended by way of direct intuition. According to the experientialists, as represented by J. S. Mill, they are, for all their certainty, inductive generalisations from particular experiences ; only the experiences are peculiar (as already said) in being extremely simple and uniform, while the experience of space Mill does not urge the like point as regards number is farther to be distinguished from common physical experience in that it supplies matter for induction no less in the imaginative (representative) than in the presentative form. Mill thus agrees with Kant on a vital point in holding the axioms to be synthetic propositions, but takes little or no account of that which, in Kant s eyes, is their distinctive characteristic their validity as universal truths in the guise of direct intuitions or singular acts of percep tion, presentative or representative. The synthesis of subject and predicate, thus universally valid though, imme diately effected, Kant explains by supposing the singular presentation or representation to be wholly determined from within through the mind s spontaneous act, instead of being received as sensible experience from without ; to speak more precisely, he refers the apprehension of quantity, whether continuous or discrete, to &quot; productive imagina tion,&quot; and regards it always as a pure mental construction. Mill, who supposes all experience alike to be passively received, or, at all events, makes no distinction in point of original apprehension between quantity and physical quali ties, fails to explain what must be allowed as the specific character of mathematical axioms. Our conviction of their truth cannot be said to depend upon the amount of support ing experience, for increased experience (which is all that Mill secures and secures only for figured magnitude, without psychological reason given) does not make it stronger ; and, if they are conceded on being merely stated, which, unless they are held to be analytic proposi tions, amounts to their being granted upon direct inspection of a particular case, it can be only because the case, so decisive, is made and not found is constituted or con structed by ourselves, as Kant maintains, with the guarantee for uniformity and adequacy which direct construction alone gives. Still it does not therefore follow that the construc tion whereby synthesis of subject and predicate is directly made is of the nature described by Kant due to the activity of the pure ego, opposed to the very notion of sensible experience, and absolutely a priori. As we have a natural psychological experience of sensations passively received through bodily organs, we also have what is not less a natural psychological experience of motor activity exerted through the muscular system. Only by muscular movements, of which we are conscious in the act of perform ing them, have we perception of objects as extended and figured, and in itself the activity of the describing and circumscribing movements is as much matter of experience as is the accompanying content of passive sensation. At the same time, the conditions of the active exertion -and of the passive affection are profoundly different. While, in objective perception, within the same or similar movements, the content of passive sensation may indefinitely vary beyond any control of ours, it is at all times in our power to describe forms by actual movement with or without a content of sensation, still more by represented or imagined movement. Our knowledge of the physical qualities of objects thus becomes a reproduction of our mani fold sensible experience, as this in its variety can alone be reproduced, by way of general concepts ; our knowledge of their mathematical attributes is, first 