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Additional info for Algebraic topology. Errata (web draft, Nov. 2004)
And in view of the conventionality of spatial congruence, we are entitled to metrize G and E either in the customary way or in other ways so as to describe E as a Euclidean GEOMETRY, CHRONOMETRY, AND EMPIRICISM 39 plane with a hemispherical hump R in the center and G as a Euclidean plane throughout. To assure the correctness of the latter nowcustomary descriptions, we need only decree the congruence of those respective intervals which our questioner called "really equal" as opposed to apparently equal in parts (1) and (2) of his question respectively.
Thus, once the above definition is given, its uniqueness is not a matter subject to regulation by semantical convention. If now we assume with Einstein as a fact of nature that light in vacuo is the fastest causal chain, then this postulate entails the nonuniqueness of the definition GEOMETRY, CHRONOMETRY, AND EMPIRICISM 23 of "topological simultaneity" given above and thereby also prevents topological simultaneity from being a transitive relation. On the other hand, if the facts of the physical world had the structure assumed by Newton, this nommiqueness would not arise.
May be seen from the following facts: (a) the 'operational' definition of length involves temperature corrections, and (b) the (usual) operational definition of temperature involves measurements of length" [66, p. * In pointing out earlier in §2 that the status of spatial and temporal congruence is decisively illuminated by Riemann's theory of continuous manifolds, I stated that this theory will not bear critical scrutiny as a characterization of continuous manifolds in general. To justify and clarify this indictment, we shall now see that continuity cannot be held with Riemann to furnish a sufficient condition for the intrinsic metric amorphousness of any manifold independently of the character of its elements.
Algebraic topology. Errata (web draft, Nov. 2004) by Hatcher A.