The Logical Structure of Mathematical Physics by Joseph D. Sneed

Author:Joseph D. Sneed
Language: eng
Format: epub, pdf
Publisher: Springer Netherlands, Dordrecht

© Springer Science+Business Media Dordrecht 1979

Joseph D. SneedThe Logical Structure of Mathematical PhysicsA Pallas Paperback3510.1007/978-94-009-9522-2_7

CHAPTER VII

Identity, Equivalence and Reduction

Joseph D. Sneed1

(1)Dept. of Philosophy, State University of New York at Albany, USA

In this chapter we will attempt to use our understanding of the logical structure of the empirical claims in theories of mathematical physics–the account developed in the first five chapters–to clarify some other questions about these theories. First, we will attempt to say, as precisely as we can, just what a theory of mathematical physics is. That is, we will attempt to give some general, and precise characterization of theories of mathematical physics. Once we have developed this characterization, we will employ it to investigate the properties of two relations–equivalence and reduction–that are commonly alleged to hold between some theories of mathematical physics. In the course of this discussion, we will have occasion to examine the Lagrangian and Hamiltonian formulations of particle mechanics as examples of theories of mathematical physics that are, in some sense, equivalent to the Newtonian formulation of particle mechanics. We shall also examine rigid body mechanics as an example of a theory which reduces to particle mechanics.

The first questions we want to consider are roughly these: (a) what sort of thing is a theory of mathematical physics; and (b) how does one determine that there are two distinct theories of mathematical physics being considered, rather than just one? Thus far, we have been provisionally committed to the view that scientific theories are sets of statements (cf. Claim (A), Chapter I). Further, we have seen that, for theories of mathematical physics, there appears to be one statement that plays a central role in the theory. We say that sentences of form (5) appeared to be adequate, in general, for making such statements, though for some theories one might be able to use sentences of form (3), (2) and even (1). These statements were central to the theories under consideration in that, roughly speaking, they expressed the entire empirical content of the theory. All empirical claims of the theory could be regarded as consequences of these statements. Following our provisional commitment, it might seem natural to identify each theory of mathematical physics with the statement of this sort associated with it. There are however, implicit in the preceding discussion, several reasons for believing that this is not an accurate account of our intuitive understanding of ‘theory of mathematical physics’.

To make these reasons explicit, and to suggest a plausible alternative view, consider the sentence (5). We may distinguish three entities associated with this sentence: (i) the sentence itself; (ii) the mathematical structure referred to in the sentence; and (iii) the statement made by the sentence. The mathematical structure referred to in the sentence is essentially the extensions of the predicates appearing in the sentence. We may think of this structure as being characterized by the predicate appearing in the sentence and as being used to make the statement made by the sentence. This can be made more precise and will be shortly.

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