Reference: http://www.relativitybook.com/resources/Einstein_space.html
NOTE: Einstein’s statements are in black italics. My understanding follows in bold color italics.
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The psychological origin of the idea of space, or of the necessity for it, is far from being so obvious as it may appear to be on the basis of our customary habit of thought. The old geometers deal with conceptual objects (straight line, point, surface), but not really with space as such, as was done later in analytical geometry. The idea of space, however, is suggested by certain primitive experiences.
The idea of space is suggested by the concepts of locations and extensions, which are represented by points, lines, surfaces and volumes in geometry. These are abstractions of material dimensions. Further abstraction of space is dealt with in analytical geometry.
Suppose that a box has been constructed. Objects can be arranged in a certain way inside the box, so that it becomes full. The possibility of such arrangements is a property of the material object “box”, something that is given with the box, the “space enclosed” by the box. This is something which is different for different boxes, something that is thought quite naturally as being independent of whether or not, at any moment, there are any objects at all in the box. When there are no objects in the box, its space appears to be “empty”.
The space enclosed by a box is defined by the extensions of the box. This space may remain empty or be filled by material objects of lesser dimensions.
So far, our concept of space has been associated with the box. It turns out, however, that the storage possibilities that make up the box-space are independent of the thickness of the walls of the box. Cannot this thickness be reduced to zero, without the “space” being lost as a result? The naturalness of such a limiting process is obvious, and now there remains for our thought the space without the box, a self-evident thing, yet it appears to be so unreal if we forget the origin of this concept. One can understand that it was repugnant to Descartes to consider space as independent of material objects, a thing that might exist without matter. (At the same time, this does not prevent him from treating space as a fundamental concept in his analytical geometry.) The drawing of attention to the vacuum in a mercury barometer has certainly disarmed the last of the Cartesians. But it is not to be denied that, even at this primitive stage, something unsatisfactory clings to the concept of space, or to space thought of as an independent real thing.
By reducing the thickness of the walls of the box to zero we can make the box disappear. We are then left with an impression of the extensions of the box on a background. Einstein is calling this impression “space without the box”. The actual SPACE, however, is the background on which the internal dimensions of the box are projected. The background SPACE is like a “blank canvas” on which impressions of the box are “drawn”.
The ways in which bodies can be packed into space (e.g. the box) are the subject of three-dimensional Euclidean geometry, whose axiomatic structure readily deceives us into forgetting that it refers to realisable situations.
The axiomatic structure of Euclidean geometry basically applies to the three-dimensional impressions left by solid objects.
If now the concept of space is formed in the manner outlined above, and following on from experience about the “filling” of the box, then this space is primarily a bounded space. This limitation does not appear to be essential, however, for apparently a larger box can always be introduced to enclose the smaller one. In this way space appears as something unbounded.
Bounded space is the three-dimensional impression of material objects projected on background SPACE. By increasing these dimensions we may approach the impression of unbounded space.
I shall not consider here how the concepts of the three-dimensional and the Euclidean nature of space can be traced back to relatively primitive experiences.
Rather, I shall consider first of all from other points of view the rôle of the concept of space in the development of physical thought.
The key idea to understand here is that the concept of space is subjective as it is derived from the impressions of material objects that are not there. This is the thought of space as abstracted from physical reality.
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Earlier notes by Vinaire:
We cannot seem to think of space independent of material bounds. We may think of unbounded space as space far away from material bounds.
The concepts of Euclidean geometry (straight line, point, surface) derive from material objects being arranged in space. We may consider its axiomatic structure to be matter-centric. Descartes analytical geometry deals with abstract relationships that may move away from being matter-centric.
Experience seems to consist of relationships that extend from physical to conceptual. We may conceive of a dimension of abstraction in which such relationships exist. We may expand the idea of physical reality to the idea of “overall reality” that consists of all physical and conceptual relationships. The “overall reality” shall then impose the need that all relationships must form a logically consistent whole. The concepts of Euclidean geometry would have to be consistent with physical reality to be “real”.
The physical, conceptual and abstract relationships exist in some background. That background is space. However, when there are no relationships it does not make sense to conceive of a background for them. Thus space exists only when relationships exist.
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