Chapter II (part 4)

But it can be measured in only three directions independent of one another—in length, breadth, and height; these directions we call the dimensions of space, and we say that our space has three dimensions: it is three-dimensional. By independent direction we mean in this case a line at right angles to another line. Our geometry (or the science of measurement of the earth , or matter in space) knows only three such lines, which are mutually at right angles to one another and not parallel among themselves. But why three only, and not ten or fifteen? This we do not know. And here is another very significant fact: either because of some mysterious property of the universe, or because of some mental p. 28 limitation, we cannot even imagine to ourselves more than three independent directions. But we speak of the universe as infinite, and because the first condition of infinity is infinity in all directions and in all possible relations, so we must presuppose in space an infinite number of dimensions: that is, we must presuppose an infinite number of lines perpendicular and not parallel to each other; and yet out of these lines we know, for some reason , only three. It is usually in some such guise that the question of higher dimensionality appears to normal human consciousness. Since we cannot construct more than three mutually independent perpendiculars, and if the three-dimensionality of our space is conditional upon this, we are forced to admit the indubitable fact of the limitedness of our space in relation to geometrical possibilities: though of course if the properties of space are created by some limitation of consciousness, then the limitedness lies in ourselves. No matter what this limitedness depends on, it is a fact that it exists. A given point can be the vertex of only eight independent tetrahedrons. Through a given point it is possible to draw only three perpendicular and not parallel straight lines. Upon this as a basis, we define the dimensionality of space by the number of lines it is possible to draw in it which are mutually at right angles one with another. The line upon which there cannot be a perpendicular, that is, another line , constitutes linear, or one-dimensional space. Upon the surface two perpendiculars are possible. This is superficial, or two-dimensional space. In "space" three perpendiculars are possible. This is solid, or three-dimensional space. The idea of the fourth dimension arose from the assumption that in addition to the three dimensions known to our geometry there exists still a fourth, for some reason unknown and inaccessible to us, i.e., that in addition to the three known to us, a mysterious fourth perpendicular is possible. This assumption is practically founded on the consideration p. 29 that there are things and phenomena in the world undoubtedly really existing , but quite incommensurable in terms of length, breadth and thickness, and lying as it were outside of three-dimensional space. ______ By really existing we understand that which produces definite action, which possesses certain functions, which appears to be the cause of something else. That which does not exist cannot produce any action, has no function, cannot be a cause. But there are different modes of existence. There is physical existence, recognized by certain sorts of actions and functions, and there is metaphysical existence, recognized by its actions and its functions. A house exists , and the idea of good and evil exists. But they do not exist in like manner. One and the same method of proof of existence does not suffice for the proof of the existence of a house and for the proof of the existence of an idea. A house is a physical fact , an idea is a metaphysical fact . Physical and metaphysical facts exist , but they exist differently. In order to prove the idea of a division into good and evil, i.e., a metaphysical fact, I have only to prove its possibility . This is already sufficiently established.