-dimensional sensory continuum.
Let us suppose, then, that we have conceived of a three-dimensional mathematical space, obtained as an abstraction from the three-dimensional space of common experience. Just as was the case with our sensory continuum, the mathematical continuum will be amorphous; no intrinsic metrics will be inherent in it, hence it will present us with no definite geometry. The definition of congruence, that is, of the equality of two spatial stretches and more generally of two volumes, and the identity of shapes and sizes, will remain as conventional as before; and it will be only after we have introduced measuring conventions into an otherwise indifferent mathematical space that Metrical Geometry as opposed to Analysis Situs will be possible. However, the discussion of these points will be reserved for the next chapters.
CHAPTER II
THE BIRTH OF METRICAL GEOMETRY
IN the preceding chapter, we mentioned that the concept of continuity was suggested by our sense experiences; and that our understanding of space as a three-dimensional continuum had arisen from a synthesis of our various sense impressions. In the present chapter, we shall be concerned more especially with the geometry of the space continuum; and we shall show how measurement in turn would appear to have arisen from experience, more particularly visual experience.
For this purpose, let us consider the case of a motionless observer, rooted to the earth ever since his birth—a species of man-plant. Viewing the world whence he stood, he would notice that whereas certain visual impressions manifested a property of what he would recognise as “permanence,” others would appear as squirming forms moving across his field of vision. In order to simplify this discussion, we shall omit to take into consideration any awareness of focussing efforts, on the part of our observer, as also any appreciation on his part of the convergence of his eyes. Under the circumstances, his visual perceptions would reveal a world of two dimensions, “up and down, right and left”; the third dimension with which we are all familiar, i.e., “away from and towards,” would be lacking. As a result, the squirming forms passing through his field of vision would be interpreted as betraying a two-dimensional world of changing forms, which would in no wise be connected with the existence of a third dimension. In particular, there would be no reason for him to attribute these changes to the variations in the distances of rigid material objects from his post of observation.
But suppose, now, that concomitant with the activity of his will, our observer were to become aware of certain muscular exertions that accompanied a variation in the shapes of those forms which had hitherto remained fixed and undeformed in his field of vision. In ordinary parlance, our observer would be displacing his post of observation, that is to say, “walking.” He would no longer remain fixed like a tree. As a result of these displacements, which he would end by recognising as such, not only would forms erstwhile fixed in shape appear to vary, but, vice versa, certain forms hitherto squirming could be made to maintain an unchanging appearance. Eventually, he would recognise that in those cases where variations of shape and size could be counteracted by suitable displacements of his post of observation, he had been observing rigid bodies varying their distances along a third dimension with respect to him. In this way, there would arise an understanding both of rigid bodies and of a third dimension. Furthermore, owing to his being able to repeat his experiments here as there, a realisation of the homogeneity of space would ensue.
We see, then, that the three-dimensional space of experience appears to have arisen as the result of a synthesis of private views, each one of which would be that of an observer unable to move from a certain fixed spot. This synthesis would be extremely complex; unfortunately we have no time to mention the various conditions that would have to be taken into consideration. Suffice it to say that our senses of sight, of touch, of muscular effort, of sound and of smell, to which should also be added the action of the semi-circular canals, would all play a part, dovetailing one into the other. Further considerations would also show that there is nothing mysterious in the fact that these various data should yield concordant results, rather than an incompatible set of conflicting spaces.
All we wish to point out is that by the physical space of experience, we do not merely wish to imply space with the objects located therein, such as it would appear from some definite point of observation. We do not mean the private vision in which rails converge and distant objects appear smaller; we mean a synthesis of these private perspectives, yielding us a common public space.
One private perspective with its converging rails taken by itself and considered without reference to other perspectives could not contain sufficient data to enable us to conceive of three-dimensional space, homogeneous and isotropic.[6] That this synthesis has been arrived at without the conscious effort of reason is granted. Nevertheless, though instinctive, the co-ordination of private experiences and perspectives is of great complexity; and it would not be impossible to conceive of this co-ordination as having followed other lines, just as an aggregate of books may be arranged in alphabetical order, or in order of size or of content, and so forth. With a change in our ordering relation we might have obtained a space of a greater number of dimensions. Undoubtedly, however, when account is taken of the facts of experience, the three-dimensional co-ordination is by far the simplest; hence there is no reason to be surprised at its having imposed itself with such force. These too brief indications must suffice for the present.