Griinbaum's own article sets forth his views on the ontology of the curvature of empty space, especially in the geometrodynamics of Clifford and Wheeler. ...

This high-level study discusses Newtonian principles and 19th-century views on electrodynamics and the aether. Additional topics include Einstein's electrodynamics of moving bodies, Minkowski spacetime, gravitational geometry, time and causality, and other subjects. Highlights include a rich exposition of the elements of the special and general theories of relativity.

This article is concerned to formulate some open problems in the philosophy of space and time that require methods characteristic of mathematical traditions in the foundations of geometry for their solution. In formulating the problems an effort has been made to fuse the separate traditions of the foundations of physics on the one hand and the foundations of geometry on the other. The first part of the paper deals with two classical problems in the geometry of space, that of giving (...) operationalism an exact foundation in the case of the measurement of spatial relations, and that of providing an adequate theory of approximation and error in a geometrical setting. The second part is concerned with physical space and space-time and deals mainly with topics concerning the axiomatic theory of bodies, the operational foundations of special relativity and the conceptual foundations of elementary physics. (shrink)

We here present explicit relational theories of a class of geometrical systems (namely, inner product spaces) which includes Euclidean space and Minkowski spacetime. Using an embedding approach suggested by the theory of measurement, we prove formally that our theories express the entire empirical content of the corresponding geometric theory in terms of empirical relations among a finite set of elements (idealized point-particles or events) thought of as embedded in the space. This result is of interest within the general phenomenalist tradition (...) as well as the theory of space and time, since it seems to be the first example of an explicit phenomenalist reconstruction of a realist theory which is provably equivalent to it in observational consequences. The interesting paper "On the Space-Time Ontology of Physical Theories" by Ken Manders, Philosophy of Science, vol. 49, number 4, December 1982, p. 575-590, has significant affinities to this one. We both, in a sense, try to formally vindicate Leibniz's notion of a relational theory of space, by constructing theories of spatial relations among physical objects which are provably equivalent to the standard absolutist theories. The essential difference between our approaches is that Manders retains Leibniz's explicitly modal framework, whereas I do not. Manders constructs a spacetime theory which explicitly characterizes the totality of possible configurations of physical objects, using a modal language in which the notion of a possible configuration occurs as a primitive. There is no doubt that this is a more accurate realization of Leibniz's own conception of space than the embedding-based approach developed here. However, it also remains open to objections (such as those cited here from Sklar) on account of the special appeal to modal notions. Our approach here, by contrast, aims to avoid the special appeal to modal notions by giving directly a set of laws which are satisfied by a configuration individually, if and only if it is one of the allowable ones. One thus avoids the need for reference to possible but not actual configurations or objects, in the statement of the spacetime laws. We may then take this alternative set of laws as the actual geometric theory, and do away with the hypothetical entity called 'space'. Yet at the same time there is no invocation of modality, except in the ordinary sense in which every physical theory constrains what is possible. So that a relationalist is not forced to utilize a modal language (though Leibniz certainly does.). (shrink)

In formal theories of measurement meaningfulness is usually formulated in terms of numerical statements that are invariant under admissible transformations of the numerical representation. This is equivalent to qualitative relations that are invariant under automorphisms of the measurement structure. This concept of meaningfulness, appropriately generalized, is studied in spaces constructed from a number of conjoint and extensive structures some of which are suitably interrelated by distribution laws. Such spaces model the dimensional structures of classical physics. It is shown that this (...) qualitative concept corresponds exactly with the numerical concept of dimensionally invariant laws of physics. (shrink)

The material contained in the following translation was given in substance by Professor Hilbertas a course of lectures on euclidean geometry at the University of G]ottingen during the wintersemester of 1898-1899. The results of his investigation were re-arranged and put into the formin which they appear here as a memorial address published in connection with the celebration atthe unveiling of the Gauss-Weber monument at G]ottingen, in June, 1899. In the French edition, which appeared soon after, Professor Hilbert made some additions, (...) particularly in the concludingremarks, where he gave an account of the results of a recent investigation made by Dr. Dehn.These additions have been incorporated in the following translation.Geometry, like arithmetic, requires for its logical development only a small number ofsimple, fundamental principles. These fundamental principles are called the axioms ofgeometry. The choice of the axioms and the investigation of their relations to one anotheris a problem which, since the time of Euclid, has been discussed in numerous excellentmemoirs to be found in the mathematical literature.1 This problem is tantamount to thelogical analysis of our intuition of space. (shrink)

This work is concerned with the geometrical structure of flat spacetimes, especially Minkowski spacetime. In Part 1 we develop synthetic or axiomatic representations of these spacetime geometries, in analogy to classical synthetic spatial geometry. Following Hilbert, we take as primitive the relations of affine betweenness and congruence. We modify the Hilbert primitives slightly, to accommodate the distinction between space and time. Using these primitives, we give categorical axiomatizations for three classical spacetime geometries associated with classical mechanics and Maxwellian electrodynamics, and (...) for the Minkowski geometry of special relativity. Quite simple proofs of categoricity are obtained by appealing to previous work in foundations of geometry in the Hilbert tradition. We also give a direct synthetic development of the elements of Minkowski geometry from our axioms, in the manner of classical synthetic geometry. On the basis of our axiomatic representation of Minkowski geometry, we formulate a physical interpretation which is intended to give explicitly the physical content of the geometry when considered as a physical theory. ;In Part 2, we compare our formulation of Minkowski geometry with others known in the literature, in both physical and formal respects. The principal alternative formulations are the COORDINATE formulation, used in practical physics, and the various CAUSAL formulations of Robb, Reichenbach, and others. We also consider a representation due to Weyl, related to ours, which has received insufficient attention in the literature. The primary advantages which we claim for our approach are: As against the coordinate formulations, our formulation avoids the arbitrariness of coordinates, and the misleading concept of an "observe" as part of the theoretical basis for spacetime geometry. Our formulation shows explicitly that the entire physical content of Minkowski geometry may be expressed purely in terms of the physical properties and relations of physical objects. As against the causal formulations, our axiomatization has a significantly simpler formal structure; and our physical interpretation corresponds much more closely to the actual empirical basis for a theory of physical spacetime geometry, than does that of the causal formulations. (shrink)