The category of substance was as we saw a feature of any space-time which arose from the relation between the elements of Space and Time in it to one another. Existence was the occupation of a space-time. Substance was the persistence of a space in its time or the occupation of a space by a duration. Causality and reciprocity were relations of substances. The categories to which we now come, quantity and intensity, or, to follow Kant's terms, extensive and intensive quantity, also arise from various essential relations within Space-Time of Space and Time to one another. As regards nomenclature, I shall follow Mr. Russell in using quantity as the concrete term and magnitude as its corresponding abstract term. Magnitude is to quantity as universality is to universal or causality to the concrete relation of cause and effect. Thus quantities may be equal to one another but their magnitudes are not equal but identical. In ordinary practice the term intensity is used indifferently, I think, for intensive quantity and its magnitude. So too magnitude and extensive quantity are commonly used convertibly. But while any magnitude may be greater or less than another magnitude, it is convenient to be able to describe equality of quantity by a distinguishing phrase, identity of magnitude. Moreover when in what follows quantity is used by itself, it stands for extensive quantity.
As before, quantity and intensity are not concepts which can be applied to spaces and times, but they are features of things which are complexes of space-time because of certain characters belonging to any space-time. Extensive quantity is the occupation of any space by its time or rather the occurrence of any space in its time, or what is the same thing, the occupation of any time by its space. Space as so occupied is a length or area or volume. Time as so occupied is a duration. Of two spaces generated by the same motion the greater space occurs in the greater time, and the greater time occupies the greater space. More or less of a motion is more or less extension in space or time, the space traced out being in correspondence with the time in which it is traced; and this is extensive quantity—that is the crude or initial character which the thought of quantity represents. Quantity is thus equivalent to the bare fact that Space is swept out in Time, or that Time is occupation of Space.
Intensity or intensive quantity, on the other hand, is the occurrence of various spaces in the same time, or what is the same thing, the occupation of the same space by different times. The simplest case is the velocity of a simple motion. The same time occupies a greater or less space according as the motion is fast or slow; or the same space occurs in a greater or less time, according as the motion is slow or fast. A less simple but still simple case is the intensive quantity of a sound. If the pitch remains unaltered the louder sound has the greater amplitude of vibration; more space being contained in the same time of vibration. Thus while extensive quantity is the fact that a space is occupied by its time, whatever that time is, intensive quantity is the fact that Time may be filled by Space and Space by Time unequally.
The ground of this distinction is that a space (or a time) is both a whole and also a continuum of parts. Considered as a whole, a space is traced out by its time and more time means more space, by what we are accustomed to call, with the use of numerical notions, the addition of space to space. That is to say, when two spaces are compared, for instance two lengths, the one space covers the extent of the other and something more. But a space is also a continuum and infinitely divisible. Now two spaces, say two lengths, may be traversed in the same time, for owing to the continuity of Space and of Time there is a one-to-one correspondence between the points of the two unequal lengths of space and between them and the time which is also a continuum. Thus intensity is a relation of Space to Time in virtue of the continuity or infinite divisibility of each, which secures that the time being the same it may be filled with any extension of space; and the space being the same it may be filled with any extension of time. Extensive quantity is an affair of addition; intensive quantity is an affair of concentration, or in numerical language of division.1
Thus extensive quantity belongs to existents so far as the space and time of their space-time vary together; they have intensive quantity so far as one or other remaining constant the other varies. In Kant's language, in extensive quantity the idea of the parts makes that of the whole possible; in intensive quantity the idea of the whole makes the parts possible. It follows from this that one quantity may be added to or subtracted from another; it is but a matter of the shorter or longer generation of the two quantities. But an intensity cannot be subtracted from another nor added to it. All that we can do is to have a series of intensities which can (again in Kant's language) decrease from any given intensity downwards to zero; or increase from zero upwards to a given intensity; as when hot water cools and its temperature decreases in intensity continually, or as when the note of a tuning-fork dies away in loudness. An intensity is not increased by adding to it a fresh intensity; but only the additional stimulus, increased by a measurable extensive dose, brings about a condition of intensity which is unitary and has more of intensive quantity than the intensity with which it is compared. Psychologists have often urged this point in respect of the intensity of sensations, that the sensational intensity (for we are not concerned with whether sensations are extensive) is something complete and single and that it is unmeaning to add or subtract the intensities of sensations. Hence extensive quantity is directly measurable, for extensities may be correlated directly with numbers and this constitutes measurement.2 But intensities are not measurable directly but only indirectly. That is, we can make a scale of intensities beginning with some one arbitrary intensity as a standard, and arranging the others at various distances from this standard, and we can measure in this way the distances of intensities from one another. Thus the intensity of temperature is measured by the numbers on the scale of a thermometer. In dealing with sensations we may arrange intensities in a scale where each sensation appears to sense to be equally removed from its predecessor on the scale. So stars are arranged in order of their magnitude, when the star of the first magnitude is as much brighter than one of the second as that in turn is brighter than one of the third.3
Intensities are thus indirectly measurable by correlation with what is directly measurable. It is therefore incorrect to maintain that because intensities are unitary, they are not measurable at all. For measure depends on correlation with the series of numbers and this correlation is possible even in the case of intensities. What is true is that ‘more or less’ means different things in the case of extensive and intensive quantity. Intensities are more or less as being further or nearer from a standard intensity. They constitute therefore a class whose members are primarily ordinal and are a series. The class of extensive quantities may be arranged ordinally, but the ordinal arrangement is secondary, for extensities differ not merely by unlikeness but by actual distance in space or time. Intensities are intrinsically ordinal and are secondarily correlated with numbers, whether with the arbitrary divisions on a thermometer, or, as in the case of sensations, in the experiments which attest the law of Weber, with the extensive measures of their stimuli.
The intensity of sensations, that is of processes of sensing, is a particular case of the categorial character, intensity, at a highly developed stage of empirical doctrine, existence. We have been concerned with the category itself as applicable to finite existence at every stage, and have tried to trace it to its root in the relation of Space and Time within Space-Time. This account of the matter is so closely allied to Kant's difficult but famous doctrine of the ‘Anticipations of Perception,’ that it may be worth while to pause for a moment for a word of comparison. Kant established once for all the difference between intensive and extensive quantity, and the debt which psychology in particular owes him in this matter has been too little acknowledged. But his purpose was not psychological. Since there is in sensation, which is empirical, a filling of the moment of time with an intensity which cannot be regarded as made up of parts by successive addition, Kant urged that there must be in the object intensive quantity or degree. For since Time cannot be perceived by itself, that is without something which occurs in it; and much less therefore the filling of a Time with various intensities of sensation; there must be in experience itself something to account for this awareness of the filling of time. This ‘degree’ in the quality of an experience is not itself empirical, that is, in our phrase, it is not one of those characters which vary from bit to bit of experience but is pervasive. It must therefore be referred to the mind itself; it is one of those elements of objective experience whose non-empirical character Kant recognises by such reference. In this way the mind ‘anticipates experience’ by the axiom that any perception must have some degree (or intensive quantity) or other. From our point of view, the non-empirical element in experience is not referable to the mind but to Space-Time itself and it has nothing to do with anticipation at all and nothing specially to do with perception. But in essentials I have been following him. Only, Kant seems unable to give a satisfactory account of the reason of intensive quantity. He contrasts with extensive quantity the intensive filling of Time by sensation, but he can only explain this by reference to the empirical fact that a given intensity of sensation can decrease to zero in time. It is true that the sound falls away in loudness in a lapse of time, but this is only the empirical consequence of the filling of the moment of time from which the fall of intensity began; and there is no definite connection established, if any can be, between the lapse of time needed for the vanishing of the sound and the intensity of the sensation. As we have seen, that intensity is to be explained by the connection of Time with Space.
Just as intensive quantity depends upon Space-Time itself and not upon mind, so and more obviously does extensive quantity. Quantity for Kant arises in the process whereby the mind traverses in time an extension in space, so that we apprehend quantity in the act of adding homogeneous parts to one another. Quantity is in this sense the work of the mind. For us Space-Time is sufficient of itself. For Space-Time containing a moving principle, Time, generates quantity. No mind is needed for the “composition” of Space, nor could Time, as Kant himself so often urges, help mind to the composition of Time without Space. Space-Time therefore does the work of itself without making an appeal to mind.
An excellent illustration of the difference between extensive and intensive quantity is provided by a problem which arises in psychology or psychophysics in connection with the estimate of just perceivable differences of length of lines as measured by the eye. With lines of moderate length, the just perceivable difference follows Weber's law and is approximately a constant fraction of the length. But when the differences of length are larger we tend to equate not fractional but absolute differences, e.g. the difference of 5 and 7 inches seems equal to that of 10 and 12 inches, not to that of 10 to 14 inches, as it should if Weber's law held. H. Ebbinghaus, from whom I borrow this account (Psychologie, vol. i. § 45, pp. 504–5, ed. 1 Leipzig, 1902) explains the reason very clearly. When the difference of length is very small we compare the two lengths taken altogether, measuring by the movement sensations of the eye; and we compare two impressions which have different strength or intensity. But when the differences are larger, we tend to superpose one line on the other and find out the actual difference by subtraction. Thus in the second case we compare the lines as extensive quantities; in the first, we are as it were considering the lengths intensively. There is an apparent contradiction here with the statement of the text that extensive quantity arises out of the relation of the time to the space in spaces taken as wholes; whereas here we say that in taking the lines intensively we take them as wholes; but a little reflection shows that the contradiction is only apparent.
B. Russell, Principles of Mathematics, p. 176.
The measurement of intensities as an arrangement of unitary intensities according to their intervals is admirably explained by H. Ebbinghaus, Psychologic, ed. 1, vol. i. Bk. I. § 6, pp. 60 ff. Cf. also Introduction to E. B. Titchener's Exp. Psych., Quantitative (Instructors' Manual), New York, 1905.