by W. K. Clifford, in his Elements of Dynamic, left unhappily incomplete.3 § 5. There is another science necessary to exact measurement, and perhaps the most elementary of all, which has still to be mentioned, the science of Calculation. This also has its root in perceptual data, though in a manner very different from geometry and kinematic. Its root percept is that of Number, a percept acquired indeed by means of conception, but acquired from perception simply, not necessarily from the complex perception of an external or spatial world. However obviously objects may be offered to perception separately, as for instance a tree, a mountain, a flash of lightning, a thunderclap, the stars, the five fingers of the hand, and so on, yet they are only recognised as units, that is, counted, by an act of attention noticing the fact of their separation from their context. Some perceptual difference is necessary to originate the perception, but it need not be more than such a difference as is necessarily involved in perception itself. Man must in fact have learnt to count, before reaching the perception of such complex objects as are several of those just enumerated, that is to say, during the process of learning to perceive them as single objects, a process which has been already analysed in Book I. Number is therefore entirely independent of spatial extension, so far as what is essential to it is concerned; though it is also evident that, when the mode of attention called counting has become habitual, 3 Elements of Dynamic. Part I., Kinematic. Two vols. 1878 and 1887. Macmillan. BOOK II. § 5. Number. BOOK II. § 5. Number. there is no content of consciousness to which it cannot be applied.1 Counting the number of given empirical and discrete objects, such as the successive sounds of a bell, or a flock of sheep, or a heap of coins, means observing whether, for every act of counting one on our part, there is or is not, in the series or the aggregate to be counted, a single definitely marked object to correspond. In other words, it means observing the correspondence of the series or the aggregate, in point of number, to some abstract number which is known and recorded by a name or symbolic figure, and which has become part of the furniture of memory previously to being used as a standard of number for the given discrete and empirical objects which are counted by means of it. Moreover it must be noticed, that it is only similar objects which can be counted together in this way; that is to say, we must abstract from their differences in point of kind, before we can count them. Say, for instance, that the flock of sheep consists of white sheep and black sheep. To count the flock, we abstract from this difference, just as in 1 On the point that number is not necessarily dependent on space-perceptions, I am glad to find myself in accordance with my friend M. F. Pillon, in that valuable series of articles entitled À propos de la Notion du Nombre, which he contributed at intervals to different Nos. of La Critique Philosophique, from June 1882 to Jan. 1885. See particularly No. 25 (Douzième Année) 21 Juillet, 1883, and No. 27 (same year) 4 Août, 1883. The persons must be few who could read these closely reasoned articles without instruction and profit. They are besides recommended to English readers by containing much acute criticism of the views held by J. S. Mill, Dr. Alex. Bain, and Mr. Herbert Spencer, on the subject of our perception of sequence and co-existence. Nevertheless I am unable to accept the theory of the distinguished author as a whole, since I find myself at variance with the initial assumption which he lays at its basis, namely, that our perception of Number is partly but essentially due to an a priori idea or form in the mind, which he calls a Category, and to which he attributes a similar nature and an equal rank with those a priori forms, ideas, or categories, from which he holds our perception of space positions and time successions to be ultimately derived. See more particularly No. 39, for Oct. 27, 1883, pp. 202-203, and pp. 206-207. M. Pillon, in fact, is strictly faithful to that modified form of Kantianism, which owes its being to the powerful mind of M. Ch. Renouvier. counting either the white or the black, we abstract from differences which distinguish one white sheep from another, and one black sheep from another. So also we may count the sounds of a bell and the sheep of a flock together; but only on condition of abstracting from their differences, and treating them simply as so many separate percepts or experiences. Sameness of kind in the objects to be counted is therefore a condition of their being counted at all, that is, of their being compared, in point of number alone, with the scheme or series of abstract numbers, which we bring with us when we count them. The first question, therefore, with regard to Number is, how originally, or in the first instance, we obtain the idea of number or numbers in the abstract; what is the meaning of the word one; or what the origin of counting or numbering, previously to applying it to count given empirical and discrete objects, of which we know beforehand that they exist in some number or other. For it is Number in this strict and abstract sense, not Number as exemplified by given empirical and discrete objects, whether in time alone or in time and space together, which is the object-matter of Calculation. Nothing is given to us in original experience ready counted. Differences in perception are originally given, and are ultimate data of perception and of experience; but we have to count them for ourselves, and count them after having observed them as differences, and perceptions as differents. In what, then, does counting originally and essentially consist? How come we in the first instance to speak the words one, two, BOOK II. CH. I. § 5. Number BOOK II. three, and so on, with a meaning; or again, as we may also express it, to notice the fact of number in Number. perceptions? § 5. Now apart from the source just mentioned, namely, differences in perception, there is but one other positively known source from which counting or numbering can spring. It is the act or acts of attention to differences in perception, or to perceptions as different, with a view to knowing more about them than the simple perception of them as different tells us. It is an act the same in point of kind as that in which conception and logical thought originate, namely, the act of selective attention for the purpose of knowing something more of an occurring content of consciousness, (though of course neither the act nor its purpose are recognised as what they are, in the earliest instances of their performance). It is therefore subject to the same ultimate law of thought as the act is in which thought originates; namely, the law of which the so-called Postulates of Logic,-Identity, Contradiction, and Excluded Middle,—are the expression. The difference is, that, in the act of counting or numeration, the attention is directed, not to a sequel generally, whatever it may be, not to similarities or dissimilarities of any or all kinds in the content next to be experienced, but solely to one single kind of circumstance in that content, namely, the place in the time-stream of consciousness which different contents will occupy, abstracting both from the particular nature and the particular duration of those contents. The act which originates numeration, therefore, though identical in kind with the act which origi nates conception and thought, is not co-eval with that act, but on the contrary presupposes it. Acts of conception and thought, with their content, but in their simplest shape, are its object-matter, are the experiences in which it for the first time observes the feature, or the fact, that they are divisions of the time-stream, and modify its course; or in other words, are the experiences of which it first observes explicitly, that they contain (unobserved by themselves) the distinction of one part of the time-stream from another, and in fact break it up ideally into different portions, irrespective of the kind or quality of the contents belonging to those different portions. This attention to acts of thought in their simplest shape, with conscious abstraction from the qualities of their contents, is the act of counting or numeration in its simplest shape; is the act in and by which counting or numeration originates, and the words one, two, three, etc., are first uttered as the expression of a meaning. It is itself an act of thought, but it is not an act of thought in its simplest shape. It is an act supervening upon acts of thought in their simplest shape, an act of attention perceiving them as abstract divisions of the time-stream, apart from the differences in the quality of the contents which thereby they also divide. In numeration, therefore, the acts of dividing the time-stream of consciousness, abstracting from all particular differences in its content, are that to which we attend, or in other words are the things counted; and thus these acts and the order in which they occur become themselves the units with which Calculation is originally and essentially con |