Similar stories play out for the other examples. The possibility of discounting the physical reality of superfluous structure leaves the physical distinctness of the pairs in question. Among the natural pairs, consider the many observationally equivalent formulations of Newton's mechanics, each with a distinct inertial state of motion designated as the true, absolute state of rest. A Newtonian can insist that the formulations differ on a matter of fact: the true disposition of absolute rest. However the overwhelming modern consensus is that this is the wrong way to understand Newtonian mechanics. The absolute state of rest plays no role in the deduction of observational consequences and, because of its perfectly symmetrical entry into each theory, we have no basis in evidence to decide which inertial motion it coincides with. So it is routinely discarded as physically superfluous and all the formulations regarded as physically equivalent.
A similar analysis arises in the case of the standard formulation of Newtonian gravitation theory in terms of gravitational fields and the Cartan curved spacetime formulation. One could insist that the two are distinct in that the former posits a background set of inertial motions not present in the latter. Following the model of general relativity, it is now standard to assume that this background inertial structure is physically superfluous. Indeed, in the special case of homogenous cosmologies, symmetry arguments essentially similar to those used in the case of Newton's absolute space and Lorentz's ether makes insistence on the physical reality of the background inertial structure unsustainable. (See Norton, 1995.)
Consider quantum mechanics and Bohm's theory, setting aside again that they are not strictly observationally equivalent. Bohm's theory adds a definite, hidden position for the particle, always possessed by it at every moment, and our ignorance of its true value is expressed in a probability distribution. A Bohm theorist can insist that this is a physically real addition to the ontology, so that the Bohm theory is physically distinct from traditional quantum mechanics. A traditionalist can reply, however, that the particle position only becomes manifest at the moment of measurement, so that standard quantum mechanics can assert that the position and its probability distribution came to be at the moment of measurement. All a Bohm theorist has done is to project the position and associated probability distribution back in time to the initial set up--a superfluous addition since all the theoretical information needed to specify the actual measurement outcome is already fully encoded in the wave function. The debate over the proper attitude to take to the Bohm theory is current. I stress that I do not want to take a side in the debate here. All I want to point out is that there are readily available arguments for the physical superfluity of the additional structure posited by the Bohm theory, so the example is not an unequivocal instance for use in the induction to the underdetermination thesis.
Among the cultured pairs, the best known are the Poincaré-Reichenbach cases of multiple geometries. One could insist that Reichenbach's universal force field is a physically real field so that the different geometries are physically distinct. No one, neither Reichenbach nor his critics, wanted take the differences seriously, physically, and all regard the multiple accounts as simply variant presentations of the same facts. Reichenbach thought the examples demonstrated the conventionality of geometry and his critics thought they demonstrated the artificiality of universal force fields. (See Norton, 1992, §5.2.4) I am less sure of what the correct analysis is for the remaining cultured examples, observationally equivalent spacetimes and continua with and without reals. However exactly because the examples are highly contrived^{14}, I am not inclined to read much significance into them for the import of evidence in real science. A finitist in geometry might well want to read the addition of reals to rational continua as physically superfluous structure, so the two representations of continua would be merely variant formulations of the same physical facts. Alternatively, since the rationals fully fix the real in so far as the reals can be defined as Dedekind cuts of rationals, the example might illustrate what I shall call below "gratuitous impoverishment."
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