By Casey J.

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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.

That only 'ultimately' the reference is to be conceived in this form, because we know that by means of the interposition of conceptual relations the reference may be rather remote" [72, p. 128]. An even stronger repudiation of the operationist account of the definition of "length" because of its failure to allow for the role of auxiliary theory is presented by K. R. Popper, who says: "As to the doctrine of operationalism—which demands that scientific terms, such as length . . should be defined in terms of the appropriate experimental procedure—it can be shown quite easily that all so-called operational definitions will be 14 For arguments supporting the conclusion that homocentric operationism is similarly dispensable and, in fact, unsuccessful in giving an account of the conceptual innovations of the special theory of relativity, see [35].

This does not seem to be in accordance with a continuum theory, and must lead to an attempt to find a purely algebraic theory for the description of reality. " GEOMETRY, CHRONOMETRY, AND EMPIRICISM 31 depends numerically on the units used and is thus arbitrary to within a constant factor, and (ii) in virtue of the lack of an intrinsic metric, sameness or change of the length possessed by a body in different places and at different times consists in the sameness or change respectively of the ratio (relation) of that body to the conventional standard of congruence.

### A treatise on the analytical geometry by Casey J.

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