Abstract
There is a raft of issues that have to be taken into account when the background theory for a law is schematic.
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Notes
- 1.
We have already noted above a reason why the first law does not say what happens when the total force is zero. Rather, it says what happens when there are no forces on a body. We argued previously that the total force interpretation is not a correct expression of the law. Here is J.C. Maxwell’s similar way of representing the first law: “The first law tells us under what conditions there is no external force.” (p. 27) Matter and Motion, with notes and appendices by Sir Joseph Larmor. Original edition 1877. Dover Publications, Inc.,
- 2.
However, some authors such as J.C. Maxwell, and R. Feynman, use both kinds of representation.
- 3.
There is an interesting argument why the Second Law does not follow from the first that is due to I.B. Cohen and Anne Whitman, Isaac Newton, The Principia A New Translation, Preceded by a Guide to Newton’s Principia, by I. Bernard Cohen, with contributions by Michael Nauenberg, and George E. Smith, University of California Press, 1999, pp. 110–111. It is this: The forces considered in the second law are impulses, while the forces concerned in Newton’s discussion of the first law are continuous. Thus, Cohen concluded that the first law is not a special case of the second, since different kinds of force are involved. While there is certainly a difference between the two kinds of forces, there seems to be no explicit mention of the difference in Newton’s description of the two laws. If there is such an ambiguity in the occurrences of “force” in the First and Second Laws, why wasn’t that difference disambiguated explicitly?
- 4.
Matter and Motion, 1876, recent edition Dover Press.
- 5.
Newton thought that there could be various forces acting on a body when he said “Corollary I” A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately”, Sir Isaac Newton’s Mathematical Principles Of Natural Philosophy And His System Of The World, Tr. F. Cajori, University of California Press, 1947, p. 14.
- 6.
Thus we are in full agreement with J. Earman And J. Robert’s construal of Newton’s Law of Gravitation as holding even in the presence of other non-gravitational forces, in their paper Ceteris Paribus: There Is No Problem of Provisos”, Synthese, 118, 439–478, 1999, fn.14. This is in sharp contrast with N. Cartwright’s version of the Newtonian Law of Gravitation, How the Laws of Physics Lie, Oxford University press, 1983, according to which the law holds when only gravitational forces are acting on a body. That is hardly ever the case, because other forces are usually present. As Earman and Roberts put it, “the law is irrelevant to real world situations”. This is a good example of strange consequences sometimes following from strange assumptions.
- 7.
The one interesting objection to the Second law as specifying a condition for each of various forces that is known to me, was the proposal of the Eighteenth Century Croation physicist Roger J. Boscovich, Theoria philosophiae naturalis, Latin-English translation of the Venetian edition 1763. tr. by J.M. Child, Open Court Publishing Company, Chicago 1922. He declared that there was only one force, which varied with the distance between bodies, being sometimes repulsive and sometimes attractive, depending on the distance, but repulsive without limit when the bodies got closer beyond limit. J.C. Maxwell has an extended but dismissive discussion of Boscovich’s view in his article on the Atom reprinted in his Collected Papers, Dover Press.
- 8.
The use of schematic letters of sentences, predicates, and functions rather than variables of the appropriate type, allows them to be replaced by specific sentences, predicates and relations. This is not permitted if for example, “F” in our formulation of Newton’s Second Law were required to be a variable ranging over functions.
- 9.
This is not the only example of an important physical theory that has successful applications that are not among its logical consequences. Cf. S. Morgenbesser and A. Koslow, “Theories and their Worth”, Journal of Philosophy 2012, pp. 616–647.
- 10.
T.S. Kuhn, The Structure of Scientific Revolutions, Second Edition, Enlarged, volume 11 Number 2, The University of Chicago Press, Volume 11, Number 2, pp. 188–189. Thanks to an anonymous referee for calling my attention to this reference to Tom Kuhn’s comments on Newton’s Second Law.
- 11.
This sketch follows the elegant presentation of that result including some other familiar conservation laws given in L. D. Landau and E. M. Lifshitz (Mechanics, volume 1 of Course of Theoretical Physics, Tr. By J.B. Sykes and J.S. Bell, Pergamon Press, Addison-Wesley Publishing Co, Reading Mass, 1969), Chapters I and II. A more mathematical presentation of these results together with more advanced mathematical results can be found in V.I. Arnold’s Mathematical Methods of Classical Mechanics, Second Edition, 1989, Chapters 3 and 4.
- 12.
For a very lucid and thorough mathematical exposition, V.I. Arnold, Mathematical Methods of Classical Mechanics, Second edition, Springer 2010, Chapters 7, 8, and 9. For an illuminating discussion of the different ways in which radically different Hamiltonian functions can figure in a variety of physical and mathematical theories, cf. Terence Tao, “Hamiltonians”, in The Princeton Companion to Mathematics, Timothy Gowers, (ed.), J Barrow-Green and I Leader (associate eds.), 2008, III.35 pp. 215–216.
- 13.
The two quotations are from A. N. Kolgmogorov, Foundations of the Theory of Probability, second English Edition, translation N Morrison, added bibliography by A.T. Bharucha-Reid, Chelsea Ppublishing Company, New York, 1956. p. 1., Original German, 1933.
- 14.
We have noted above that Kolmogorov’s formulation of probability is schematic. There are formulations in which probabilities are assigned to events and other formulations where the assignment is to sentences. That possibility is permissible for schematic theories. Nevertheless the two versions are not equivalent. That important but overlooked point is made clearly by A. Hajek and C. Hitchcock in their introduction to The Oxford Handbook of Probability and Philosophy, Oxford University Press, 2016, p. 20. In the following section, we are using a formulation in which probabilities are assigned to sentences rather than events, for the simple reason that here we are interested in probabalistic-truth.
- 15.
This kind of modal operator as we mentioned in Chap. 11, fn. 2, is studied extensively in A. Koslow, A Structuralist Theory of Logic, Cambridge University Press, Cambridge 1992, Part IV, pp. 239–371, and “The implicational Nature of Logic: A Structuralist Account”, in European Review of Philosophy, The Nature of Logic, volume 4, edited by A. Varzi, 1999, CSLI Publications, Stanford, pp. 111–155, where it was simply called a modal.
- 16.
We assume here that S is closed under the classical logical operators.
- 17.
It is interesting to note that this modal implies its dual but not conversely. That is, for any A, T∗(A) ⇒ ¬T∗(¬A), but not conversely (unless the probability function p is trivial).
- 18.
Cf. the accounts of E. Nagel and R.B. Braithwaite in Chap. 8.
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Koslow, A. (2019). Schematic Theories, Subsumtion of Laws, and Non-accidental Generalizations. In: Laws and Explanations; Theories and Modal Possibilities. Synthese Library, vol 410. Springer, Cham. https://doi.org/10.1007/978-3-030-18846-7_12
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