Abstract
Having examined the general features of Cartesian space and motion, and Newton’s famous criticism of this theory, we can now proceed to the analysis of the underlying theoretical, or structural, components of the theory of space and time presupposed in Newton’s argument. This investigation will not only determine the extent of the deficiencies, if any, in Descartes’ system, but it will also outline the necessary structural or theoretical remedies necessary to cure the Cartesian theory of its presumed deficiencies
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Endnotes
For a discussion of symmetry conditions, see, J. R. Lucas, Space, Time, and Causality ( Oxford: Oxford University Press, 1984 ), 120.
J. Earman, World Enough and Space-Time (Cambridge, Mass.,: MIT Press, 1989), 8. Much of the technical terminology and concepts will be drawn from Earman 1989.
For a nice discussion of these details on a non-technical level, see, J. D. Norton, “Philosophy of Space and Time”, in Introduction to the Philosophy of Science,eds. M. H. Salmon, et al. (Englewood Cliffs: Prentice Hall, 1992), 204.
Here, my terminology is adopted from, M. Friedman, Foundations of Space-Time Theories ( Princeton: Princeton University Press, 1983 ), 77.
Additionally, the transformations can also be conceived as a structure preserving mapping on spacetime itself which takes the “old” points to “new” points as viewed from the same coordinate system (deemed “active transformations”). In this essay, however, I will exclusively represent the “passive” formulation.
I. Newton, De Motu, in Unpublished Scientific Papers of Isaac Newton, trans. and eds. A. R. Hall and M. B. Hall ( Cambridge: Cambridge University Press, 1962a ).
See, H. Stein, “Some Philosophical Prehistory of General Relativity”, in Minnesota Studies in the Philosophy of Science, Vol. 8, eds. John Earman, et al. ( Minneapolis: University of Minnesota Press, 1977 ), 3–49.
L. Sklar, Space, Time, and Spacetime ( Berkeley: University of California Press, 1974 ), 204–205.
See, M. Wilson, “There’s a Hole and a Bucket, Dear Leibniz”, in Midwest Studies in Philosophy Vol. XVIII, Philosophy of Science, eds., P. A. French, T. E. Uehling, Jr., H. K. Wettstein ( Notre Dame, Ind.: U. of Notre Dame Press, 1993 ), 211.
See, for example; C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation ( San Francisco: W. H. Freeman, 1973 ), 48–50.
Once the appropriate reference frame has been located where all the components of the metric tensor located at a point vanish (for very small regions around the point), one can determine the unique inertial path (or shortest line—geodesic) which advances temporally forward of the point (connecting the other points which also lie close along the geodesic). See, for example, D. F. Lawden, An Introduction to Tensor Calculus, Relativity and Cosmology. 3rd ed. ( Chichester: John Wiley Sons, 1982 ), 108–110.
Nevertheless, this fact has not prevented modern relationalists from attempting to provide a relational basis for Newtonian mechanics via other means: e.g., J. B. Barbour and B. Bertotti, in “Gravity and Inertia in a Machian Framework,” 1977, Nuovo Cimento 38B: 1–27. Barbour and Bertotti utilize action-at-a-distance principles to overcome the limitations imposed by relationalist spacetimes. Yet, it remains unclear whether such spacetime models can effectively explain the phenomena of non-inertial motion, especially rotation. See, Earman, ibid., 89–96.
E. Mach, The Science of Mechanics. 9th edition (London: Open Court) 1942, although first published in 1883.
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Slowik, E. (2002). The Structure of Spacetime Theories. In: Cartesian Spacetime. International Archives of the History of Ideas / Archives Internationales d’Histoire des Idées, vol 181. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0975-0_3
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