The act of counting, (one, two, three, and so on,) is the foundation of all our knowledge of number. The intuition of the relations of number involves this act of counting; for, as we have just seen, the conception of number cannot be obtained in any other way. And thus the whole of theoretical arithmetic depends upon an act of the mind, and upon the conditions which the exercise of that act implies. These have been already explained in the last chapter.
5. But if the apprehension of number be accompanied by an act of the mind, the apprehension of rhythm is so still more clearly. All the forms of versification and the measures of melodies are the creations of man, who thus realizes in words and sounds the [144] forms of recurrence which rise within his own mind. When we hear in a quiet scene any rapidly-repeated sound, as those made by the hammer of the smith or the saw of the carpenter, every one knows how insensibly we throw these noises into a rhythmical form in our own apprehension. We do this even without any suggestion from the sounds themselves. For instance, if the beats of a clock or watch be ever so exactly alike, we still reckon them alternately tick-tack, tick-tack. That this is the case, may be proved by taking a watch or clock of such a construction that the returning swing of the pendulum is silent, and in which therefore all the beats are rigorously alike: we shall find ourselves still reckoning its sounds as tick-tack. In this instance it is manifest that the rhythm is entirely of our own making. In melodies, also, and in verses in which the rhythm is complex, obscure and difficult, we perceive something is required on our part; for we are often incapable of contributing our share, and thus lose the sense of the measure altogether. And when we consider such cases, and attend to what passes within us when we catch the measure, even of the simplest and best-known air, we shall no longer doubt that an act of our own thoughts is requisite in such cases, as well as impressions on the sense. And thus the conception of this peculiar modification of time, which we have called rhythm, like all the other views which we have taken of the subject, shows that we must, in order to form such conceptions, supply a certain idea by our own thoughts, as well as merely receive by senses, whether external or internal, the impressions of appearances and collections of appearances.
NOTE TO CHAPTER X.
I have in the last ten chapters described Space, Time, and Number by various expressions, all intended to point out their office as exemplifying the Ideal Element of human knowledge. I have called them Fundamental Ideas; Forms of Perception; Forms of Intuition; and perhaps other names. I might add yet other phrases. I might say that the properties of Space, Time, and Number are Laws of the Mind’s Activity in apprehending what is. For the mind cannot apprehend any thing or event except conformably to the properties of space, time, and number. It is not only that it does not, but it can not: and this impossibility shows that the law is a law of the mind, and not of objects extraneous to the mind.
It is usual for some of those who reject the doctrines here presented to say that the axioms of geometry, and of other sciences, are obtained by Induction from facts constantly presented by experience. But I do not see how Induction can prove that a proposition must be true. The only intelligible usage of the word Induction appears to me to be, that in which it is applied to a proposition which, being separable from the facts in our apprehension, and being compared with them, is seen to agree with them. But in the cases now spoken of, the proposition is not separable from the facts. We cannot infer by induction that two straight lines cannot inclose a space, because we cannot contemplate special cases of two lines inclosing a space, in which it remains to be determined whether or not the proposition, that both are straight, is true.
I do not deny that the activity of the mind by which it perceives objects and events as related according to the laws of space, time, and number, is awakened and developed by being constantly exercised; and that we cannot imagine a stage of human existence in which the powers have not been awakened and [146] developed by such exercise. In this way, experience and observation are necessary conditions and prerequisites of our apprehension of geometrical (and other) axioms. We cannot see the truth of these axioms without some experience, because we cannot see any thing, or be human beings, without some experience. This might be expressed by saying that such truths are acquired necessarily in the course of all experience; but I think it is very undesirable to apply, to such a case, the word Induction, of which it is so important to us to keep the scientific meaning free from confusion. Induction cannot give demonstrative proofs, as I have already stated in Book 1. C. i. [sect. 3], and therefore cannot be the ground of necessary truths.
Another expression which may be used to describe the Fundamental Ideas here spoken of is suggested by the language of a very profound and acute Review of the former edition. The Reviewer holds that we pass from special experiences to universal truths in virtue of ‘the inductive propensity—the irresistible impulse of the mind to generalize ad infinitum.’ I have already given reasons why I cannot adopt the former expression; but I do not see why space, time, number, cause, and the rest, may not be termed different forms of the impulse of the mind to generalize. But if we put together all the Fundamental Ideas as results of the Generalizing Impulse, we must still separate them as different modes of action of that Impulse, showing themselves in various characteristic ways in the axioms and modes of reasoning which belong to different sciences. The Generalizing Impulse in one case proceeds according to the Idea of Space; in another, according to the Idea of Mechanical Cause; and so in other subjects.