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Below, I argue that a completely systematic approach to reduction does not generally succeed in physics, and support this view with two examples: the re- duction of Newtonian mechanics to special relativity (by ‘special relativity,’ I mean Lorentz covariant classical dynamics) and the reduction of classical me- chanics to quantum mechanics.

Completely Systematic Approaches to the NM-SR Reduction

If a completely systematic type-2 reduction of NM to SR is probably not possi- ble, as argued in section 1.2.1, might a completely systematic type-1 reduction be? That is, for all physical systems whose behavior can be approximated by some Newtonian model, is it possible to provide a single general derivation of the corresponding image laws of NM that applies to all such systems? I claim that one cannot, and offer an example to illustrate my point.

Consider an SR model of an idealised system consisting of two masses con- nected by a spring that is sufficiently stiff and sufficiently compressed that the total system contains enough potential energy to make the rest mass of the total system substantially exceed the sum of the individual constituent masses; the spring itself is assumed to have effectively zero rest mass. (While perhaps not very realistic, one can at least in principle model such a system in SR.) Assume that the center of mass the combined system behaves in Newtonian fashion and that the centers of mass of each of the masses do as well. An explanation as to why this is so will have to include some explanation as to why the energy stored

clamp). For, if the potential energy (which is comparable to the rest energies of the two masses) were converted to kinetic energy, the bodies would fly off at some substantial fraction of the speed of light and non-Newtonian effects would become apparent. Consider also a system whose total rest mass is the same as the total mass of the earlier system, but this time in which the mass is due entirely to the rest masses of its constituents, and not to the contributions of internal energy resulting from their interactions or motion. In such a case, the explanation as to why such a system behaves in Newtonian fashion will be of a different nature, since there is no need to explain why potential energy is not being released. In this case, the fact that the body isn’t travelling too close to the speed of light (relative to some fixed inertial frame) and that there aren’t excessively strong external forces acting on it should suffice to account for its Newtonian behavior. Thus, in each case, system-specific details concerning the nature of the internal binding forces (or lack thereof) within the bodies will play some role in the explanation of Newtonian behavior.

To take a more realistic example, consider a heavy atomic nucleus that fol- lows an approximately Newtonian trajectory in a cyclotron (say, the circular trajectory of a moving charge in a constant magnetic field). The explanation as to why the trajectory of the nucleus is approximately Newtonian will de- pend in part on an explanation as to why the binding energy of the nucleus is not released, causing fragments of the nucleus to fly off at some significant portion of the speed of light. This explanation will depend on details of the internal constitution and binding of the nucleus in question and, in particular, on its half-life. Again, system-specific details play some role in the explana- tion of Newtonian behavior, precluding a single reduction that encompasses all Newtonian systems.

By demonstrating the need to invoke system-specific details when attempting provide a complete explanation of Newtonian behavior in different systems, these examples illustrate that a reduction of NM to SR that is both complete and completely general - or, to keep to my terminology, completely systematic

- does not exist. However, given that the NM-SR reduction ought to have been at the outset the most likely example of completely systematic reduction, this particular case does not bode well for the widspread applicability of completely systematic approaches to reduction.

Having demonstrated that no completely systematic account of the reduction of NM to SR, either of type 1 or type 2, is available, I will now turn to my second example, which also will be the focus of Chapter 2: the reduction of classical to quantum mechanics.

Completely Systematic Approaches to the CM-QM Reduction

If a completely systematic type-2 reduction of CM to QM is probably not possi- ble, as argued in section 1.2.1, might a completely systematic type-1 reduction be? That is, for all physical systems whose behavior can be approximated by some classical model, is it possible to provide some overarching demonstration of the approximate accuracy of classical models that encompasses all of such systems? The answer again is patently no, but the arguments to this effect depend to some degree on what one thinks counts as a successful application of classical mechanics.

There is potentially a wide range of things one could mean by the term ‘classical.’ From the point of view of quantum theory, the least stringent no- tion of classicality that we can adopt is apparent definiteness of the values of variables like position and momentum, absent any dynamical constraints on their evolution. As I argue later on, decoherence with respect to an appropri- ate pointer basis, combined with a solution to the measurement problem, will suffice to reproduce this attribute of classicality. However, people usually mean more than just apparent definiteness when they speak of the empirical success of classical mechanics; they also mean that certain dynamical constraints on the evolution of position and momentum variables are satisfied by the systems in question. Such dynamical constraints, in turn, come in different varieties. For

Law of Motion, without placing any constraints on the allowable force laws that may appear in this law. Such behavior will include, among other things, be- havior involving contact forces and friction such as the simple pendulum, mass on a spring, or normal force systems with and without friction. Also involving contact forces are the equations of classical fluid dynamics, such as the Navier- Stokes equation, which are derived from Newton’s Second Law (combined with additional assumptions affecting the form of the contact forces involved). On the other hand, one could restrict one’s notion of classical behavior further to systems described in terms of conservative forces, which can be characterised in terms of a simple classical potential - for example, the mass on a spring and the simple pendulum. And one could even further restrict one’s attention to classical behavior which consists only of behavior that can be described in terms of fundamental force laws, such as electromagnetism, in which the conservative forces arise from fields rather than contact forces.

These different classes of systems, which encompass different models of clas- sical behavior, will, of course, have different quantum mechanical underpinnings. For classical systems involving a fundamental force law, the potential that ap- pears in the classical equation of motion will be the same as the one appearing in the quantum equation of motion. On the other hand, for classical systems which involve contact forces or friction, the classical potential, if one exists, the potential appearing the Schrodinger equation of the underlying quantum model will be extremely complicated and different from the potential that appears in the underlying microscopic quantum equations (it will likely only match the potential employed in the classical model in some average sense). To be more specific, consider two distinct systems described by the classical model of the harmonic oscillator: the first a mass on a spring and the second an electric charge moving in a tube bored through an axis of a uniform spherical charge distribution (in which case the electric field will vary linearly with distance from the center of the sphere)9_{. In the second case, the classical potential generated}

9_{I assume that the charge is sufficiently massive that energy losses through radiation can}

by the electric field will be the same potential that appears in the underlying quantum model of the charge’es behavior. In the first case, the fact the that one can employ a harmonic oscillator potential to describe the motion of the block is something that needs to be explained in terms of the complex microscopic constitution of the spring - at the microscopic level, this potential will be wildly fluctuating on the length scale of the atoms making up the spring.

These two applications of the same classical model must be reduced sepa- rately, and no complete reduction can be given that encompasses both. Thus, a completely general, completely systematic, type-1 reduction of classical models of macroscopic systems cannot be given, especially if one adopts a relatively inclusive construal of what counts as classical behavior. Consideration of de- tails specific to the system in question, or to the class of systems into which it falls, will be required for a totally comprehensive reduction of the classical to the quantum model of that system.

1.6.3

Completely Piecemeal Approaches to Reduction

The failure of completely systematic approaches to reduction, either of type 1 or of type 2, might lead one to take the view that reductions must be performed in type-1, piecemeal fashion. A quote, again from David Wallace, suggests such a piecemeal approach:

Crucially: this ‘reduction,’ on the instantiation model, is a local affair: it is not that one theory is a limiting case of anotherper se, but that,in a particular situation, the ‘reducing’ theory instantiates the ‘reduced’ one. Consider the first example above, for instance. The reason that classical mechanics is applicable to the planets of the Solar System is not because of somegeneral [italics mine] result that classical mechanics is a limiting case of quantum mechanics. Rather, the particular system [italics mine] under consideration - the solar system - is such that some of its properties approximately instantiate a classical-mechanical dynamical system. Others do not, of course: it is not that the solar system is approximately classical, it is that it (or a certain subset of its degrees of freedom) instantiate an approximately classical system.

and macroeconomics at the top: rather, it is a patchwork of domain- relative instantiations [110].

According to Wallace, the fact that certain quantum systems approximately instantiate classical Newtonian systems - or rather, that, for certain systems, quantum models of those systems instantiate certain Newtonian models of those same systems - is not something that can be accounted for systematically by some general mathematical result, either involving a limit or, he seems also to suggest, of any other form, but something that must be explained on a piecemeal basis.

However, as I now argue, it is still possible to retain some measure of gen- erality in our accounts of reduction, and we can do better than to provide reductions in a totally piecemeal, case-by-case fashion. Indeed, it would be surprising if the rather striking formal results relating the dynamical and kine- matical structures of superseded and superseding theories in physics did not have somefairly widespread relevance to actual instances of reduction; to say, for instance, that the result that limv

c→0

1 q

1−v2

c2

= 1, and the fact that plenty of relativistic equations return Newtonian ones as a result, has no widespread relevance to the emergence of NM behavior from SR is (for reasons which I take to be self-evident) utterly implausible. Limits or other general formal results, while they do not in themselves constitute complete reductions, often do lend significant insight into the mechanisms and principles that relate the high- and low-level theory models of many or all such systems.

Piecemeal Reduction vs. the Pluralism of Cartwright and Dupre

Wallace’s piecemeal approach to reduction may call to mind the pluralism of Cartwright and Dupre, who, like Wallace, take pains to underscore the patch- work nature of regularities described by scientific theories and to characterise these regularities as islands of order in a much vaster sea of irregularity. Yet the patchwork of Wallace’s view differs dramatically in certain crucial respects from

that of Cartwright and Dupre. Wallace’s patchwork respects the hierarchical distinction between high- and low- level theories, with theories in physics at the bottom and those in chemistry, biology, and psychology successively further up; moreover, it respects the reducibility, in the sense specified by Wallace’s concept of instantiation, of high- to low- level theories. Cartwright and Dupre’s plural- ist view, on the other hand, is strongly anti-reductionist in that it denies the reducibility of higher- to lower- level theories, and moreover opposes the very distinction between high- and low- level theories [23], [32]. Thus, while both views commonly acknowledge the patchwork nature of scientific regularities, this is simply a reflection of the fact that both strive to grapple - in very differ- ent ways - with the same fact about about the way in which science describes nature.

As a consequence of the differences just cited, the reductionist and pluralist accounts may be further distinguished in terms of the way they characterise the relationship between thedomains of different theories. In Wallace’s patchwork, the domains of higher-level theories are contained in those of low-level theories; that is, systems that instantiate the laws or models of a higher-level theory also instantiate those of a lower-level theory (though the reverse is not generally true). Cartwright and Dupre’s patchwork imposes no requirement that the domain of a high-level theory, say in biology, be contained in that of some low-level theory in physics. In this sense, the various patches making up the patchwork on the pluralist view are on more level footing than they are on the reductionist view; the domains of theories in physics and in biology are simply different on the pluralist view and the former are not required to contain the latter; in fact, the pluralist view explicitly requires that this kind of containment does not occur. Thus, Wallace’s view will typically ascribe much larger domains to low-level theories than will the pluralist view of Cartwright and Dupre, since Wallace’s view requires these domains to subsume those of higher-level theories while Cartwright and Dupre’s denies this subsumption.

work nature of scientific regularities can in part be traced back to a difference in the degree to which they condone extrapolation from the observed success of scientific theories in the carefully controlled contexts where they are often tested, to their applicability in the much vaster, and typically much more com- plex, world outside of these contexts. Wallace, along with most of the scientific and philosophical communities, accepts the legitimacy of such extrapolations, implicitly taking them as a natural induction on the base of data extracted from experiments. For instance, while most of what is regarded as the confirm- ing evidence for the Standard Model of particle physics is drawn from scattering experiments performed in particle accelerators, physicists typically assume that these laws also apply in contexts highly remote from particle accelerator exper- iments, such as occur in efforts to describe the evolution of the early universe.

Cartwright and Dupre, on the other hand, regard such inferences as far too cavalier and argue that we ought to be more reserved in our extrapolations. Cartwright, for instance, claims that the theories of physics and the other sci- ences apply only ceteris paribus, under the carefully tuned conditions under which they are typically tested. Thus her view goes beyond mere skepticism about claims, for instance, that biological systems fall within the domain of the Standard Model or any potential successor to the Standard Model; it centers on an explicit denial of such claims. Thus, she argues that the patchwork of scientific practice most strongly supports a metaphysical picture in which it is not just the contexts in which we can apply our theories to make predictions that are disjointed, but nature itself.

In summary, one must not overlook the fact that the piecemeal approach to reduction is just that - an approach toreduction- and thus, unlike Cartwright’s view, will generally support the idea of higher-level regularities being reduced to lower-level ones. The kind of diversity that is supported by the piecemeal approach to reduction is only diversity in the sense of a single underlying theory providing different explanations of different high-level phenomena, though in contrast to Cartwright and Dupre’s pluralism, these different explantions may

be given within the same underlying theoretical framework; Wallace’s piecemeal approach to reduction is thus entirely compatible with the idea that there is a single underlying, universal theory of fundamental physics, and that the many disparate patches of higher-level regularity all fall within the domain of this one theory. Cartwright’s view is explicitly anti-reductionist and anti-unfication, and the diversity that is suggested by Cartwright’s view is diversity at a much deeper, metaphysical level, since it not only suggests different explanations for different higher-level regularities, but ultimately that the need for different explanations reflects a world that is dappled not only in terms of our ability to discern patterns in high-level phenomena, but fundamentally.