Until a more careful analysis of the situation is made it would seem therefore that there is some ground for the suspicion that the principle of relativity is involved in the possibility of giving to physical phenomena a _complete_ mathematical formulation, understanding "complete" to mean "including the descriptive background." CHAPTER III DETAILED CONSIDERATION OF VARIOUS CONCEPTS OF PHYSICS WE now begin our detailed consideration of the most important concepts of physics. It is entirely beyond the scope of this essay to attempt more than an indication of some of the most important matters. Neither is it to be expected that the parts of this discussion will always have a very close connection with each other; the purpose of the discussion is to aid in acquiring the greatest possible self-consciousness of the whole structure of physics. THE CONCEPT OF SPACE A logically satisfying definition of what we understand by the concept of space is doubtless difficult to give, but we shall not be far from the mark if we think of it as the aggregate of all those concepts which have to do with position. Position means position of something. The position of things is determined by some system of measurement; perhaps the simplest is that implied in a Cartesian coördinate system with its three measurements of length. Hence much of the essential discussion of space has already been given in connection with the concept of length. We have seen that measurements of length are made with physical measuring rods applied to some physical object. We cannot measure the distance between two points in empty space, because if space were empty there would be nothing to identify the position of the ends of the measuring rod when we move it from one position to the next. We see, then, from the point of view of operations that the framework of Cartesian geometry, often imagined in an ideal mathematical sense, is really a physical framework, and that what we mean by spatial properties is nothing but the properties of this framework. When we say that space is Euclidean, we mean that the physical space of meter sticks is Euclidean: it is meaningless to ask whether empty space is Euclidean. Geometry, therefore, in so far as its results are expected to apply to the external physical world, and in as far as it is not a logical system built up from postulates, is an experimental science. This view is now well understood and accepted, but there was a time when it was not accepted, but vigorously attacked; the change of attitude toward this question is symptomatic of a change of attitude toward many other similar questions.