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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Science has taken over from pre-scientific thought the concepts space, time, and material object (with the important special case “solid body”) and has modified them and rendered them more precise. Its first significant accomplishment was the development of Euclidean geometry, whose axiomatic formulation must not be allowed to blind us to its empirical origin (the possibilities of laying out or juxtaposing solid bodies). In particular, the three-dimensional nature of space as well as its Euclidean character are of empirical origin (it can be wholly filled by like constituted “cubes”).
The subtlety of the concept of space was enhanced by the discovery that there exist no completely rigid bodies.
The axiomatic formulation of Euclidean geometry has brought precision to the concepts of space-time-event, which are an abstraction of material dimensions. But these material dimensions belong to bodies that are not totally rigid.
All bodies are elastically deformable and alter in volume with change in temperature. The structures, whose possible congruences are to be described by Euclidean geometry, cannot therefore be represented apart from physical concepts. But since physics after all must make use of geometry in the establishment of its concepts, the empirical content of geometry can be stated and tested only in the framework of the whole of physics.
In physics we study the elastic deformation of material bodies and the change in their volume with temperature. Such physical phenomena affects material dimensions. Hence it should be taken into account by the concepts of space-time-event.
In this connection atomistics must also be borne in mind, and its conception of finite divisibility; for spaces of sub-atomic extension cannot be measured up.
Atomistics also compels us to give up, in principle, the idea of sharply and statically defined bounding surfaces of solid bodies. Strictly speaking, there are no precise laws, even in the macro-region, for the possible configurations of solid bodies touching each other.
The atoms are not uniformly solid. They are made of frequency gradients from zero frequency of space to very high frequency of the nucleus of the atom. Thus material objects are not bound by sharply defined boundaries, and they do not exactly touch each other. This should also be taken into account by the concepts of space-time-event.
In spite of this, no one thought of giving up the concept of space, for it appeared indispensable in the eminently satisfactory whole system of natural science.
Mach, in the nineteenth century, was the only one who thought seriously of an elimination of the concept of space, in that he sought to replace it by the notion of the totality of the instantaneous distances between all material points. (He made this attempt in order to arrive at a satisfactory understanding of inertia).
Such minutiae in the concepts of space-time-event become important only when working with the concept of Inertia.
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Earlier notes by Vinaire:
The fundamental ideas in natural science have been there all this time. We are simply looking at them more closely to free them of filters (biases, prejudices, fixed ideas, assumptions and blind faith) and make them logically consistent with reality.
Euclidean Geometry assumes completely rigid solid bodies to come up with its axiomatic structure. But there are no completely rigid bodies. When physics uses geometry to set up its concepts, it must take care in this regard.
Consider the following.
(1) We cannot keep dividing matter infinitely. Division of matter ultimately seem to emit electromagnetic waves.
(2) We cannot measure spaces of sub-atomic extension. Points in space are approximations.
(3) In reality, sharply defined bounding surfaces do not exist. Interface of space with solids is blurred.
(4) There is no precise definition for solid bodies touching each other.
If there is no way to define the dimensions of solids precisely, then there cannot be a precise concept of space. We associate inertia with motion of material points. So we need to look closely at how we define “material point”.
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