Fig. 3.
If we plot these equal increments of loudness as the dependent variable S in a graph, and the amplitude of the vibrations of the atmosphere as the independent variable E, we can obtain the following curve. If we investigate this we shall find that a certain relation exists between the “values” of the sensation and the values of the stimuli that correspond to them; a regular increase in the loudness of the sensation corresponds to a regular increase in the logarithms of the strength of the stimuli. Let S = the sensation, E the stimulus, and C and Q constants; then
| S = C log | E | ; |
Q |
Thus we decompose our stream of consciousness into a series of quantitatively different and qualitatively different things, upon each of which we confer independent existence. We attribute to these different aspects of our consciousness extension, but the extension is due only to our analysis; for the qualities of pitch, loudness, colour, odour, etc., which we disentangle from each other, did not exist apart from each other, any more than do the sine and cosine curves into which we decompose an arbitrarily drawn curved line. The multiplicity of our consciousness is intensive, like the multiplicity that we see to exist in the abstract number ten. This number stands for a group of things, but its multiplicity is intensive and only exists because we are able to subdivide anything in thought to an indefinite extent. Now, so far we have only separated what we agree to regard as the elemental parts of our general perception of the environment, but it is to be noted that we have not given to these elements anything like spatial extension.
We may, if we like, regard our intuition of space as that of an indefinitely large, homogeneous, empty medium which surrounds us and in which we may, in imagination, place things. So regarded it is difficult to see in what way our notion of space differs from our idea of “nothing,” a pseudo-idea incapable of analysis, except into the idea of something which might be somewhere else. The more we think about it the more we shall become convinced that space, that is the “form” of space, represents our actual or potential modes of motion, that is, our powers of exertional activity. Space, we say, has three dimensions; in all our analysis of the universe, and of the activities that we can perceive in it, this idea of movement in three dimensions, forward and backward, up and down, and right and left, occurs; and we have to recognise that in it there is something fundamental, as fundamental as the intuitive knowledge that we possess of the direction of right and left. It is because we can move in such a way that any of our motions, no matter how complex, can be resolved into the components of backward and forward, right and left, and up and down, these directions all being at right angles to each other, that we speak of our movements as three-dimensional ones. Our geometry is founded, therefore, on concepts derived from our modes of activity; and there is nothing in the universe, apart from our own activity, that makes this the only geometry possible to us. Euclidean geometry does not depend on the constitution of the external universe, but on the nature of the organism itself.
There is a little Infusorian which lives, in its adult phase, on the surface of the spherical ova of fishes. These ova float freely in sea water, and the Infusorian crawls on their surfaces, moving about by means of ciliary appendages. It does not swim about in the water, but adheres closely to the surface of the ovum on which it lives. Let us suppose that it is an intelligent animal and that it is able to construct a geometry of its own; if so, this geometry would be very different from our own.
It would be a two-dimensional geometry, for the animal can move backward and forward, and right and left, but not up and down; it is a stereotropic organism, as Jacques Loeb would say, that is, it is compelled by its organisation to apply its body closely to the surface on which it lives. But its two-dimensional geometry would, on this account, be different from ours. Our straight lines are really the directions in which we move from one point to another point in such a way as to involve the least exertion; they are the shortest distances between two points, and if we deviate from them we exert a greater degree of activity than if we had moved along them. For us there is only one straight line that can be drawn between two points, but this is not necessarily true for our Infusorian, and its straight line need not be the shortest distance between two points. It might be either the longest or the shortest distance between the points, for the latter can always be placed on a great circle passing through the two points and the poles of the egg, and in moving from a point on which it is placed the animal could reach the other point by moving in two directions, just as we could go round the earth along the equator by moving to the east or to the west. Therefore the straight line of the Infusorian would be not only a scalar quantity but a vector quantity, that is, it would represent, not only a quantity of energy, but a quantity of energy that has direction. For us only one straight line can be drawn between two given points, but this limitation would not exist in the two-dimensional geometry of a curved surface. Suppose that the two points are situated on a great circle and that they are exactly 180° apart; then the Infusorian could move from one pole to another pole along an infinite number of straight lines or meridians all of which had a different direction, but all of which were of the same length; that is to say, in this geometry an infinite number of straight lines can be drawn between the same two points. Again, its triangles might be different from ours; our triangles are figures formed by drawing straight lines between three points, and on a plane surface the sum of the angles of the triangle are together equal to two right angles, though on a curved surface they may be greater or less than two right angles. But our Infusorian could not imagine a triangle in which the sum of the angles was not greater than two right angles, for all its figures would be drawn on a convex surface.
Our three-dimensional geometry depends, therefore, on our modes of activity and the concepts with which it operates; points, straight lines, etc. are conceptual limits to those modes of activity. We can imagine a straight line only as a direction along which we can move without deviating to the right or the left, or up or down. But even if we draw such a line on paper with a fine pencil the trace would still have some width, and we can imagine ourselves small enough to be able to deviate to the right or the left within the width of the line drawn on the paper. We might make a very small mark on the paper, but no matter how small this mark is it would still have some magnitude; otherwise we should be unable to see it. If the straight line had no width and the point no magnitude they would have no perceptual existence. Our perceptual triangles are not figures, the angles of which are necessarily equal to two right angles. If we drive three walking sticks into a field and then measure the angles between them by means of a sextant we shall find that the sum is nearly 180°, but in general not that amount. If we stick a darning needle into the heads of each of the walking sticks and then remeasure the angles by means of a theodolite we shall obtain values which are nearer to that of two right angles, but we should not, except by “accident,” obtain exactly this value. We do not, therefore, get the “theoretical” result, and we say this is because of the errors of our methods of observation; but why do we suppose that there is such a theoretical result from which our observations deviate, if our observations themselves do not in general give this ideal result? We might accumulate a great series of measurements of the angles of our triangle, and we should then find that these results would tend to group themselves symmetrically round a certain value which would be 180°. Some of the results would be considerably less than the ideal, and some of them would be considerably more; but these relatively great deviations would be small in number and most of the results would be a very little less than 180° or a very little more, and there would be as many which would be a little less as those that were a little more. We should have formed a “frequency distribution”[2] with its “mode” at 180°.
But by “reasoning” about the “properties” of these lines and triangles in plane two-dimensional space, we should arrive at the conclusion that the angles of a triangle were equal to 180°, and neither more nor less. We should then think of a straight line as still a path along which we move in imagination, and a path which still has some width. But we imagine the width of the path to become less and less, so that, even if we imagine ourselves to become thinner and thinner, we should be unable to deviate either to the right or left in moving along the path, because the thinner we make ourselves the thinner becomes also the path. We imagine our intuition of a deviation to the right or left becoming keener and keener, so that, no matter how small the deviation we should still be able to appreciate it by the extra exertion which it would involve. We think of a point as a little spot, and we think of ourselves as being very small indeed, so that we can move about on this spot. But we can reduce the area of the spot more and more, until it becomes “infinitesimally” small; and at the same time we think of ourselves as becoming smaller and smaller, so that we can still move about on the spot. But we think of the area of the spot as becoming so small that no matter how small we make ourselves we are unable to move on it.