(Assembled from disclosures made by space people in various communications.)
As a first consideration let us consider what space appears to be to us. We have no doubts about its volumetric nature although we have no senses by which we can observe this condition. We conclude that space is volumetric by the nature of things which we find in it. What would it be like if everything were removed from space? Would we even have space?
Can we conceive of a space structure completely devoid of matter or energy in any form; real empty space? What would such space be like? If we were disembodied entities located in such a space, how could we identify our position or describe where or how we were? What sort of yardstick could we use? These and many more similar questions must be faced squarely by those seeking understanding.
When one has satisfied himself by actually doing it that he can conceive of space with absolutely nothing in it, and is not too terrified of his creation, he is then in a position to take the next step, namely to find his way about in it.
An arbitrary decision can be made to refer all concepts to where the investigator conceives himself to be, thereby establishing a point. He can next conceive of an adjacent point, thereby establishing a line. By turning in various directions he can establish the concepts of surfaces and volumes. But no matter what he does after that, the investigator cannot add more concepts to the space itself. He therefore concludes that what he instinctively felt soon after he first conceived the empty space, that it was volumetric, and nothing more, is correct.
The foregoing is actually an exercise in mental gymnastics, but as physical exercise is necessary for body-building, so are these exercises necessary for the building of understanding. It is absolutely necessary to satisfy oneself on these points before going on to the next concept.
The next step is to conceive of a way of getting about in this empty space, and to realize that one has done so when the operation is complete. The concept of an adjacent position, or point, is a good approach, and here again the operation becomes one of mental gymnastics, and a lot of manipulation and practice is necessary to get the feel of the situation.
As one gains in understanding of the properties of space, the various geometries become evident, and it becomes increasingly obvious that a wide variety can be made to work, if certain basic parameters are admitted. However, since our concern is primarily with our space and our universe, we will want to select the geometry that best fits our experience.
Ordinary Euclidean or rectilinear geometry is quite familiar and comprehensible to us, and we can understand easily how it can be applied to space as we conceive it. We can understand a sideways, forwards-backwards, up-down concept, or expressed mathematically, and x, y and z axis. Also, we experience no great difficulty in conceiving of these three axes as converging at right angles to a single point and extending outward therefrom to enormous distances. We can even introduce the idea of infinity in any direction as being somewhat beyond the farthest distance in which we have any interest.
In our rectilinear concept we can conceive of such a thing as a straight line, although we might be hard put to define it, since the concept is in itself axiomatic. However, if we understand what a straight line actually is in our concept and we are sure that others with whom we communicate also have the same understanding, we can use it as a real datum point in our appreciation of our universe. This point is of particular significance in what is to follow, as it is one of the few solid anchors we have on which to fasten our understanding. Let us never lose sight of this concept of a straight line, as entirely distinct from the behavior of matter or energy.
Our rectilinear concept at once validates our Euclidean geometry. Furthermore, it removes any limitations which might be imposed on it by either great or small distances. It provides us with a clear-cut framework within which we can think our way about in space. We should satisfy ourselves that this concept actually is a necessary and sufficient condition for this purpose, although we remember that its selection was arbitrary and that other geometries probably would work just as well. However, since we have made our decision to use the rectilinear concept, we must be prepared to stick to our decision unless and until proved wrong. As a matter of fact, any geometry can be used, and will work within the limiting parameters of its definitions. It is only when extended beyond these limits that corrections become necessary, but even with the corrections the geometry itself does not become invalid, only the things we expect it to do.