instead of
we should derive a different space; in this case, however, we have no mirror to show us what it looks like. Such a space is said to have one negative dimension and it has the peculiar property that in the figure derived from the right triangle of ordinary space the square of the “hypotenuse” equals the difference and not the sum of the squares of the other two sides, so that the length of a line may sometimes have to be represented by the square-root of a negative number, a “complex” number.
In considering what at first sight may appear to be fantastic statements made by this theory, it must be borne in mind that all our knowledge of the external universe comes through our sense-impressions, and our most confident statements about external things are really of the nature of inferences from these sense-impressions and, being inferences, liable to be wrong. So that if the theory says that a stone lying on the ground is not a simple three-dimensional object, and that its substance is not the same as its substance a moment before, the matter is one for due consideration and not immediate disbelief.
The idea that the universe extends in time as well as in space is not new, and fiction-writers have familiarized us with wonderful machines in which travellers journey in time and are present at various stages of the world’s history. This conception of the universe, to which the name “space-time” is usually applied, is adopted by the new theory and assigned the status of a physical reality.
The World Geometry
The fundamental creed of the new theory is that the space-time universe constitutes a true four-dimensional space of one negative dimension, this dimension being time. The variations from point to point of the direction-defining and length-defining magnitudes generate the geometrical properties of curvature, etc., and these are cognised by the human mind as physical phenomena: our sense-impressions are nothing more nor less than perceptions of the geometry of a fourspace. So instead of inferring from our sense-impressions the existence of matter, motion and the like as we are accustomed to do, we should with equal justice infer the existence of a geometrical fourspace. Thus it becomes necessary to prepare a dictionary in which the familiar things of our world are identified with those geometrical properties of the four-space which really constitute them, and in so doing parts of our geometrical knowledge assume the guise of new physical knowledge.
Through the fourspace our consciousness travels, cognising a changing three-dimensional section of it as it goes and thus giving rise to time. It becomes aware that the fourspace is pleated or folded along lines all running roughly in the same direction, and possibly because this is the easiest direction to follow, it travels along the lines. The direction of this motion is the negative dimension. Thus consciousness is always aware of the nearly constant forms of the cross-sections of the pleats along which it travels. These unvarying forms constitute matter: matter is the form of a section through a uniform pleat of the fourspace—a three-dimensional aspect of a four-dimensional curvature; so that in strict accuracy we should say that a stone is the shape or form of a changing section of a four-dimensional object, the complete object being a long fold in the fourspace. The physical interpretation of this conservation of form of the cross-section is that matter is conserved. It is thus seen that the conscious mind, by following these pleats, has so determined time that the law of the conservation of matter must hold. The mathematical treatment of the subject makes it clear that practically all other physical laws similarly follow as a direct result of this choice of time. The type of order prevailing in the physical universe, the laws of gravitation, heat, motion and the rest are not directly imposed by some external power, but are apparently chosen by mind itself.