In my publications I have followed a general policy of not duplicating material that is readily available in the textbooks, in order to conserve space for the new ideas that I am presenting. I therefore do not define terms that are in general use, commenting on the usage only where I have introduced some new concept, or have modified the meaning of a term. There was some confusion about my usage of the term “direction” originally, and I had occasion to discuss this matter in some of my publications. (See, for instance, Nothing But Motion, p. 48). These explanations apparently took care of the problem, as I have heard nothing about directions lately. It now appears that some misunderstandings also exist with respect to my use of the term “dimension.” Some comments on the usage of this term may therefore be helpful.
The dimensional situation is complicated by the fact that I necessarily have to use the term in its broadest sense, whereas it is more generally used with a very restricted meaning. From the general standpoint, “dimension” is a mathematical term that may be, but is not necessarily, capable of being represented in geometric form. An n-dimensional quantity is simply one that requires n independent numbers for definition. As one dictionary says, by way of illustration, “a²b²c is a term of five dimensions.” Within a certain limited range, dimensions of space may be represented in the conventional reference system, and because this usage is so common, the qualification “spatial” is commonly omitted. Thus we say that a cube is three-dimensional, meaning that it extends into three vectorial dimensions of space. But we also say that space is three-dimensional, and here we mean something different. We do not mean that space extends into three dimensions of space. That statement is an absurdity. What we mean is that three scalar magnitudes, or numbers are required in order to define a location in space.
The space of the conventional reference system is three-dimensional. But it takes all three of these spatial dimensions to represent one dimension of motion in space. Consequently, the present-day physicist, who does not recognize the existence of anything outside the reference system, deal only with one dimension of motion. The prevailing opinion, therefore, is that all real motion can be represented geometrically in the reference system. Where the theorists have to resort to multiple dimensions in order to explain some of the more difficult experimental results, an expedient that has become quite common since observation and measurement have penetrated into the smaller, faster, and more distant regions of the universe, they portray the extra dimensions as in some way unreal. Heisenberg, for example, characterizes the atom as existing in an “abstract multidimensional space,” whatever that means.
My finding is that the real physical universe extends beyond the one dimension of motion represented in the reference system. What I have done is to take the physicists' vague idea of multiple dimensions, and put it into concrete form. This was the key to the development of a complete and consistent physical theory. One of the requirements for a full understanding of that theory is a recognition that the dimensions of motion are mathematical. When I refer to dimensions in my works, this term has no geometrical connotations, except where so specified. Dimensions are scalar magnitudes, just numbers. Different phenomena involve different numbers of independent magnitudes. It follows that the number of dimensions with which we are concerned depends on the particular phenomenon with which we are dealing.
The first unit of motion, from the spatial zero to unit speed, the speed of light, is one-dimensional in space. The second unit is one-dimensional in time, but because we base our reference system on a spatial speed of zero, it appears in that reference system as a dimension of motion in space plus a dimension of of motion in time (to the extent that the reference system can respond to motion in time) from an inverse speed of unity to the temporal zero. On this linear basis, there are two dimensions of motion between zero spatial motion and zero temporal motion; that is, it takes two numbers, one representing the quantity of motion in space and one representing the quantity of motion in time, to express the total magnitude of the motion difference between these two zero levels. Here, then, in this simple situation, we already have a case where the number of dimensions is either one or two, depending on the nature of the phenomenon with which we are dealing; that is, whether it is something that we refer to a zero base, or something that is necessarily referred to the natural base at unity. This is not all. Further dimensions may be introduced into the same situation because the one-dimensional motion that I have been describing can be distributed over three dimensions, in a manner similar to the way in which radiation from a light source is distributed. This does not change the one-unit magnitude, as the cube of one is still one. But if the two-unit magnitude is so distributed it extends to 2/3, or 8, dimensions.
Inasmuch as our base is the spatial zero, a speed of three units adds a second dimension of motion in space to the two-unit combination. The result, three units of speed equivalent, measured from the spatial zero, is equal to three units of inverse speed equivalent, measured from the temporal zero. Beyond this neutral level, the motion as a whole converts to motion in time. But as long as the total speed remains below the neutral level, any motion in time that may exist acts as a modifier of the magnitude of the motion in space, rather than causing an actual change of position in time. This is easily understood on a mathematical basis. If a small negative number is added to a larger positive number, the result is simply a reduction in the magnitude of the positive number. The second dimension of motion is thus a motion in the spatial equivalent of time.
From the foregoing it can be seen that there are six dimensions of motion between the spatial zero and the temporal zero. The basic fact is that the universe is three-dimensional. Beyond this, the number of dimensions that have to be taken into consideration depends on the particular feature of the universe with which we are dealing. Of course, all this is very complicated compared to a simple three-dimensional coordinate system, and many individuals would like to put it into some simpler form. But we are dealing with nature, and nature does not accommodate itself to our preferences. Physical theory claims to be able to deal with all of the modern discoveries without going beyond the one dimension of motion that can be represented in a spatial coordinate system. Conventional physics has found it necessary to place the small-scale phenomena of the physical universe in a strange half-world, the “abstract multidimensional space” that Heisenberg refers to, a world that is populated by “virtual” particles and other entities that admittedly do not “exist objectively.” These ghostly denizens of the phantom sector of the physicists’ universe do not obey the normal physical laws or the rules of logic, and are governed by mysterious “forces” of which there is no physical evidence. When all this is taken into consideration, it can easily be seen that I am not increasing the complexity of physical theory. I am merely taking the metaphysical ideas that are too vague to be useful in practice, and putting them into concrete form. The universe is, in fact, complex, and if we want to understand it we will have to meet it on its own terms.