Modern empiricism has been conditioned in large part by two dogmas. One is a belief in some fundamental cleavage between truths which are analytic, or grounded in meanings independently of matters of fact, and truth which are synthetic, or grounded in fact. The other dogma is reductionism: the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience. Both dogmas, I shall argue, are ill founded. One effect of abandoning them is, as (...) we shall see, a blurring of the supposed boundary between speculative metaphysics and natural science. Another effect is a shift toward pragmatism. (shrink)

Kant's question 'How are synthetic judgments a priori possible?' pre- cipitated the Critique of Pure Reason. Question and answer notwith- standing, Mill and others persisted in doubting that such judgments were possible at all. At length some of Kant's own clearest purported.

1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (...) . . . . . . . . . . . 7 1.3 Vector Fields, Integral Curves, and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Tensors and Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 The Action of Smooth Maps on Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.7 Derivative Operators and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.8 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.9 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1.10 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.11 Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.. (shrink)

The status of the geodesic principle in General Relativity has been a topic of some interest in the recent literature on the foundations of spacetime theories. Part of this discussion has focused on the role that a certain energy condition plays in the proof of a theorem due to Bob Geroch and Pong-Soo Jang [“Motion of a Body in General Relativity.” Journal of Mathematical Physics16(1) (1975)] that can be taken to make precise the claim that the geodesic principle is a (...) theorem, rather than a postulate, of General Relativity. In this brief note, I show, by explicit counterexample, that not only is a weaker energy condition than the one Geroch and Jang state insufficient to prove the theorem, but in fact a condition still stronger than the one that they assume is necessary. (shrink)

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In General Relativity (GR), it has been claimed that inertia receives a dynamical explanation. This is in contrast to the situation in other theories, such as Special Relativity, because the geodesic principle of GR can be derived from Einstein’s field equations. The claim can be challenged in different ways, all of which question whether the status of inertia in GR is physically different from its status in previous spacetime theories. In this paper I state the original argument for the claim (...) precisely, discuss the different objections to it and then propose a formulation that avoids the problems the original claim encounters. My conclusion is that one can say meaningfully that inertia is dynamically explained in GR. There are two senses in which the derivation of geodetic motion can be said to provide a (more) dynamical explanation of inertia in GR: it holds for any material test body that is a source of the gravitational field; and it is derivable without assuming inertial structures that are fixed independently of matter. (shrink)

It is often claimed that the geodesic principle can be recovered as a theorem in general relativity. Indeed, it is claimed that it is a consequence of Einstein's equation (or of the conservation principle that is, itself, a consequence of that equation). These claims are certainly correct, but it may be worth drawing attention to one small qualification. Though the geodesic principle can be recovered as theorem in general relativity, it is not a consequence of Einstein's equation (or the conservation (...) principle) alone. Other assumptions are needed to drive the theorems in question. One needs to put more in if one is to get the geodesic principle out. My goal in this short note is to make this claim precise (i.e., that other assumptions are needed). (shrink)

Isaac Newton's Scientific Method examines Newton's argument for universal gravity and his application of it to resolve the problem of deciding between geocentric and heliocentric world systems by measuring masses of the sun and planets. William L. Harper suggests that Newton's inferences from phenomena realize an ideal of empirical success that is richer than prediction. Any theory that can achieve this rich sort of empirical success must not only be able to predict the phenomena it purports to explain, but also (...) have those phenomena accurately measure the parameters which explain them. Harper explores the ways in which Newton's method aims to turn theoretical questions into ones which can be answered empirically by measurement from phenomena, and to establish that propositions inferred from phenomena are provisionally accepted as guides to further research. This methodology, guided by its rich ideal of empirical success, supports a conception of scientific progress that does not require construing it as progress toward Laplace's ideal limit of a final theory of everything, and is not threatened by the classic argument against convergent realism. Newton's method endorses the radical theoretical transformation from his theory to Einstein's. Harper argues that it is strikingly realized in the development and application of testing frameworks for relativistic theories of gravity, and very much at work in cosmology today. (shrink)

A four dimensional approach to Newtonian physics is used to distinguish between a number of different structures for Newtonian space-time and a number of different formulations of Newtonian gravitational theory. This in turn makes possible an in-depth study of the meaning and status of Newton's Law of Inertia and a detailed comparison of the Newtonian and Einsteinian versions of the Law of Inertia and the Newtonian and Einsteinian treatments of gravitational forces. Various claims about the status of Newton's Law of (...) Inertia are critically examined including these: the Law of Inertia is not an empirical law but a definition; it is not a law simpliciter but a family of schemata; it is a convention and gravitational forces exist only by convention; it is (or is not) redundant; the concepts it embodies can be dispensed with in favor of operationally defined entities; it is unique for a given theory. More generally, the paper demonstrates the importance of space-time structure for the philosophy of space and time and provides support for a realist interpretation of space-time theories. (shrink)

The use of multiple means of determination to “triangulate” on the existence and character of a common phenomenon, object, or result has had a long tradition in science but has seldom been a matter of primary focus. As with many traditions, it is traceable to Aristotle, who valued having multiple explanations of a phenomenon, and it may also be involved in his distinction between special objects of sense and common sensibles. It is implicit though not emphasized in the distinction between (...) primary and secondary qualities from Galileo onward. It is arguably one of several conceptions involved in Whewell’s method of the “consilience of inductions” (Laudan 1971) and is to be found in several places in Peirce. (From M. Brewer and B. Collins, eds., (1981); Scientific Inquiry in the Social Sciences (a festschrift for Donald T. Campbell), San Francisco: Jossey-Bass, pp. 123–162.). (shrink)

A theorem due to Bob Geroch and Pong Soo Jang ["Motion of a Body in General Relativity." Journal of Mathematical Physics 16, ] provides a sense in which the geodesic principle has the status of a theorem in General Relativity. I have recently shown that a similar theorem holds in the context of geometrized Newtonian gravitation [Weatherall, J. O. "The Motion of a Body in Newtonian Theories." Journal of Mathematical Physics 52, ]. Here I compare the interpretations of these two (...) theorems. I argue that despite some apparent differences between the theorems, the status of the geodesic principle in geometrized Newtonian gravitation is, mutatis mutandis, strikingly similar to the relativistic case. (shrink)

A theorem due to Bob Geroch and Pong Soo Jang [“Motion of a Body in General Relativity.” Journal of Mathematical Physics 16, ] provides the sense in which the geodesic principle has the status of a theorem in General Relativity. Here we show that a similar theorem holds in the context of geometrized Newtonian gravitation. It follows that in Newtonian gravitation, as in GR, inertial motion can be derived from other central principles of the theory.