The most common version of empirical equivalence discussed by philosophers is the case of exact empirical equivalence for all models of two theories. The potential interest of such a scenario is evident: obviously there is no chance in any nomologically possible world that experimental progress will resolve the debate, while settling it on theoretical grounds might also be difficult. However, this scenario runs the risk that the two supposedly rival theories are in fact one and the same theory in different guises. Such identification was often made by those influenced by logical empiricism. A related weaker claim is made today by John Norton, namely, that for theories for which the
observational equivalence can be demonstrated by arguments brief enough to be included in a journal article . . . we cannot preclude the possibility that the theories are merely variant formulations of the same theory.
Norton evidently has in mind journal articles in the philosophy of science, not physics or some other science. While Norton aims to deny that the underdetermination of theories by data is generic and that philosophers’ algorithmic rivals carry much force, there are some serious candidates for underdetermination that arise from within real physics and that have not been discussed much, if at all, by philosophers. However, the examples available from real physics do seem sufficiently widespread and interesting that it might well frequently be the case that scientific, or rather physical, theories are permanently underdetermined by data…..
Apart from some exceptions of perhaps little physical importance (such as solutions of an Ashtekar formulation with a degenerate metric, for example), the various sets of variables for GTR (broadly construed in the fashion of physicists) are empirically equivalent in the sense that all or most solutions of one set of equations are suitably related (not always one-to-one) with solutions in other sets of variables. Physicists are generally not tempted to regard the resulting theory formulations as distinct theories, partly because their criteria for physical reality are attuned to this mathematical interrelation. Each description comes with an adequate recipe for distinguishing the physically meaningful from the descriptive fluff, and no further ontological questions are typically asked or answered. Physicists are also quite comfortable with a certain amount of vagueness or merely implicit specificity. For example, is a given energy condition, such as the weak energy condition, part of General Relativity or not? The answer to that question depends, at least, on whether ‘realistic’ matter fields satisfy the condition; but whether a certain kind of matter is realistic is malleable in light of both empirical factors (such as the apparent observation of dark energy in the late 1990s) and theoretical factors (such as recognition that seemingly tame matter fields or quantum fields violate an energy condition hitherto regarded as important). GTR for physicists is in effect a cluster of theories sharing a hard core including Einstein’s equations, while partially overlapping in including or failing to include various additional claims with various degrees of importance, not unlike a Lakatosian research program. Perhaps Arthur Fine (Arthur Fine-The shaky game_ Einstein, realism, and the quantum theory) would commend to philosophers the physicists’ approach, which sounds something like his Natural Ontological Attitude that there is no distinctively philosophical question about the real existence of entities employed in scientific theories, so neither realism nor anti-realism is an appropriate doctrine. Physicists typically assume some sort of mathematical equivalence as necessary and sufficient for two formulations to be the same theory (though strict equivalence is not always required).