Reduction Templates

Although it is not possible to give a completely general, systematic account of how a superseded theoryTh reduces to a superseding theoryTl, it is often pos- sible to retain a significant degree of generality in explanations of why certain systems modelled in Tl also may be approximately modelled in Th, and there- fore to do better than to approach theory reduction on a completely piecemeal basis. The degree of generality that can be retained in such explanations de- pends strongly on the two theories in question, though in most if not all of the cases mentioned in the first paragraph of the Introduction to this chapter, is substantial.

Given that any system in the class is described both by a high-level model

Mh and a low level model Ml, the deductive portion of the reduction consists of deriving the image model M∗

h from Ml. Systems with models in Th can be paritioned into classes, such that reductions toTl of theTh models of systems in the same class follow the same ‘template,’ and reductions of systems in dif- ferent classes follow different templates. A template is an incomplete proof, or outline, of the basic steps and principles and mechanisms that are involved in deriving the image modelM_h∗ fromMl. A given template may take for granted

individual system in question, or that are merely plausible conjectures awaiting proof. Systems within a given reduction template may differ substantially in their Tl description, though at a certain intermediate level of detail, the basic outlines of the reasoning that explains their approximate Th behavior are the same.

To a degree, the separation into reduction classes of systems inTh’s domain is arbitrary, and depends in particular on how detailed the template associated with the class is. Ultimately, the most detailed possible explanation of why a particular Tl system exhibitsTh regularities will take account of things like the exact state of the system and its exact microscopic constitution; such an approach amounts essentially to cranking the relevant initial conditions through the appropriate equations of motion and then reading the information relevant to the Th level of description off from it. At this level, each template amounts to a completely detailed, rigorous proof of reduction, and each separate system has its own reduction template and is the sole member of its reduction class. However, such explanations in practice are never given because we are not ca- pable of gaining this kind of detailed information about the systems we care about. Moreover, they tend to obscure the general principles and mechanisms at work across a wide range of instances of reduction. If, on the other hand, one is willing to sacrifice some detail in the template by making certain general, plausible assumptions about the system in question (which one can go back and try to prove rigorously later if one likes), then one may gain some insight into these mechanisms and principles and where they fit in to the overall scheme of the full reduction. Templates significantly reduce the labor involved in explain- ing particular instances of reduction by compartmentalising the derivation of the image model on the one hand into those parts that require specific reference to the system in question, and, on the other hand, those parts of the derivation that are uniform across the reduction class.

Thus, a template-based approach to reduction is one in which a variety of explanations of the approximateThbehavior of a particular system, varying ac-

cording to level of detail, are possible. Explanations provided by more general, less detailed templates will apply across a correspondingly larger reduction class of systems; however, the derivations of the model of the systems in the reduc- tion class may take a number of claims for granted, since proving these claims would require considering the particular details of different systems within the class. On the other hand, less general, more detailed templates will tend to take the particularities of individual systems into account, and therefore only apply over smaller reduction classes; the reduction classes of these more detailed tem- plates should be subsets of the reduction classes corresponding to less detailed templates for the same system.

Both kinds of templates, general and detailed, are necessary to a full under- standing of reduction: the former because it illustrates the general principles and mechanisms at work in the reduction of theTh to theTl models of a wide range of systems, and the latter because it ‘fills in’ or completes the more general template with a detailed demonstration of the assumptions taken for granted in the more general template. Thus, accounts of reduction provide the most in- sight not when they are given exclusively at the finest level of detail, nor when they are given exclusively at the greatest level of generality, but rather when they are given in stages, with earlier stages corresponding to a template at the greatest level of generality and later stages corresponding to progressively more detailed templates, whose reduction classes narrow at each step to account for more details specific to the system under investigation. Each stage can be seen as combing over the same deductive path betweenTl andTh, but each time in progressively finer detail, so that both the broad outlines and the fine details of the explanation can be understood.

Below, I return briefly to the two examples of the CM-QM reduction and the NM-SR reduction, and suggest in broad terms how a template-based approach might be applied to them.

Reduction Templates for the NM-SR Reduction

I argued above that the reduction of Newtonian mechanics to special relativ- ity cannot be given a systematic formulation, either of type 1 or type 2. On the other hand, I claim that we can do better than to explain reductions of Newtonian behavior to SR on a totally piecemeal, case-by-case basis.

Earlier, I argued that Newtonian systems with significant internal energy (e.g., binding energy comparable to their rest energy) will require a different explanation for their Newtonian behavior than Newtonian systems that do not have significant internal energy; on a template-based approach, these two sets of systems would belong to distinct reduction classes, where the reduction template for the former class will include some account of why this energy is not released. However, at a less detailed level of explanation, in which the rest of mass of composite bodies is taken as an assumption rather than as something to be explained, and the internal structure of these bodies not considered, these two sets of systems might belong to the same reduction class and the reduction of their NM models to their SR models follow the same reduction template.

Reduction Templates in the CM-QM Reduction

As in the case of the NM-SR reduction, I argued above that in the case of the CM-QM reduction, systematic reductions of type 1 and 2 are not available.

As in the NM-SR case, the failure of totally systematic accounts does not require us to consider all classical systems on a case-by-case basis. The largest, most general reduction class for the quantum-classical reduction might be simply those systems which appear to have definite values for properties such as position and momentum. As I argue below, decoherence combined with a solution to the measurement problem should suffice to explain the classicality - construed in this broad sense - of these systems.

However, we might identify further subclasses of this reduction class: in par- ticular, the subclass of systems which approximately obey Newtonian equations of motion. Then we might identify further subclasses of this class corresponding

to Newtonian systems involving contact forces and those involving fundamental forces. In the latter case the potential appearing in classical equations is the same as the potential appearing in the underlying quantum equations, while in the latter the two potentials are different (assuming these contact forces even admit description in terms of a classical potential). Proceeding further, we can distinguish subclasses of the ‘contact force’ reduction class corresponding to fluid systems exhibiting regularities like the Navier-Stokes equation (which is derived from Newton’s second law and thus counts as an instance of it ap- plication) and to systems like the simple pendulum. The microscopic origins of the phenomenological force laws employed in these two sets of cases will be substantially different. On the other hand, we can also identify subclasses of the ‘fundamental force’ reduction class corresponding, say, to gravity and elec- tromagnetism.

At the most fine-grained level, there will be a reduction class for every sys- tem and every state of every system (for some states may yield classical behavior while others don’t). Yet, as was suggested above, if one proceeds immediately to this level of description without first providing the more general reduction templates, the general principles and mechanisms which underlie the emergence of many instances of classical behavior, such as decoherence, Ehrenfest’s the- orem, and the compounding of micro into macro degrees of freedom, will be completely obscured.

In the following chapter, I will illustrate the application of a template-based approach to the CM-QM reduction, in particular by focusing on the reduction class consisting of classical systems involving only basic force laws like the grav- itational and electrostatic forces.

Template-based Reduction and the Patchwork Nature of Higher-Level Regularities

ture of physical laws: namely, that both view higher-level regularities as islands of order in a much vaster sea of irregularity. Yet, on further analysis, the fact they share this should come as no surprise, for both views are simply making an effort to come to terms with what is, after all, a fact about science thatany account of higher level regularities in science must come to grips with. Scien- tific theories do often tend to operate in isolated patches; the laws of genetics do apply to living organisms, but not to the inorganic matter and energy and space and time that make up most of the cosmos; the Standard Model works well at predicting the results of scattering experiments, but none of us can use it to predict the population dynamics of badgers in the UK; the laws of circuit theory work well at predicting the currents and voltages in a circuit, but not for predicting the scattering cross sections for hadron collisions. No understanding of the relation between different sets of scientific regularities, whether they are characterized by different theories within a single science or theories between different sciences, can be plausible without somehow accommodating this fact. The template-based approach to reduction, I believe, accommodates the di- versity and patchwork nature of higher-level regularities more effectively than does the piecemeal approach to reduction. Like the piecemeal approach, the template-based view accommodates the diversity of higher-level regularities by providing different explanations where necessary for different patches of regu- larity, but all within the framework of the same fundamental theory; on the template-based view the different explanations correspond to different reduc- tion templates. Where the template-based approach surpasses the piecemeal approach is in the fact that the template-based approach provides a frame- work that facilitates the identification and emphasis of the general principles and mechanisms that apply across different patches of higher-level regularity, thereby enabling a more general, systematic understanding of the particular reductions under consideration than can be achieved by following the purely piecemeal approach.

Reduction Templates and Multiple Realisation

Template-based reduction is ideally suited to accommodate the fact of multiple realisation. On a DS approach, multiple realisability corresponds to the fact that more than one model of a theoryTlcan serve to reduce to the same model of Th, or that there is more than one bridge map through which reduction can occur between a single pair of models Ml and Mh, or that more than one low-level state xl _{may instantiate the same high-level state x}h _{(I discuss} multiple realisation in DS reduction further in the concluding chapter). The image modelsM_h∗ corresponding to these distinct realisations may be different, and so the templates for deriving these image models will be distinct at some fine-grained level of detail. However, it may happen that these distinct, more fine-grained templates, may have a sufficient number of steps in common that they can be seen as refinements of the same more general template.