Intertheoretical relations

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# Intertheoretical relations

### Introduction

Typical examples for the study of intertheoretical relations involve theories and their successors in the turn of scientific history e.g. Newtonian physics and special relativity or ray and wave optics. Another textbook example is the relation between thermodynamics and statistical mechanics.

The key concepts to discuss the relation alleged to exists between these different theories (or properties) are reduction'', emergence'' and supervenience''. In the following we give a brief account of these concepts and their relation.

### Reduction, supervenience and emergence

The notion of theory reduction is still playing an important role in the debate on scientific change and scientific explanation. An early and influential treatment was given by Nagel (1961, Ch. 11). Nagel essentially translated the deductive-nomological model of explanation (Hempel and Oppenheim 1948) into the context of intertheoretical relations. In order to reduce a secondary theory, T_s, on a primary theory, T_p, two formal conditions have to be met. First a condition of connectibility'', i.e. the requirement to translate the descriptive terms of T_s which are absent in T_p into the T_p-language (If the secondary theory has no terms which are not already contained in the vocabulary of the primary theory Nagel speaks of a homogeneous'' reduction. He takes this to be an unproblematic case.). Whether these linkage is a matter of convention or a physical hypothesis depends, according to Nagel, on the specific context (p. 354). The second condition of derivability'' demands the derivation of the laws of T_s from the ones of the primary theory. In the spirit of logical empiricism, Nagel characterized T_s, T_p, and the bridge principles syntactically, i.e. as sets of statements or propositions. This logical relation of derivability is apparently related to a purely cumulative picture of scientific progress. However, Nagel states also informal conditions that distinguish trivial reductions from noteworthy scientific achievements. Most important the primary science should be empirically supported and able to reveal new and unexpected connections between the laws of the secondary science. Surprisingly Nagel does suggest that the secondary science may become modified and corrected in the turn of the reduction:

"In general therefore, for a reduction to mark a significant intellectual advance, it is not enough that previously established laws of the secondary science be represented within the theory of the primary diciplin. The theory must also be fertile in usable suggestions for developing the secondary science, and must yield theorems referring to the latter's subject matter which augment or correct its currently accepted body of laws." (Nagel 1961, p.360)

This view has been the target of criticism in the philosophy of science since its appearance. Some constructive'' criticism was directed against the highly idealized picture of the Nagelian account and led to more sophisticated models of reduction (e.g. Schaffner 1967, 1969 Nickles 1973, Hooker 1981). Other critics claimed that this model is inappropriate when dealing with cross-border reductions for special sciences'' (e.g. Fodor 1974), like biology or psychology (According to the multiple realization'' argument it is not possible to establish proper reduction relation between e.g. mental and physical states. Here the prevalent view seems to be an anti-reductionist physicalism who claims that mental properties and facts supervene'' on physical states. However, since we are only dealing with intertheory relations in physics we will ignore this intricate debate.).

Criticism which was directed against the very precondition of the reduction program was raised by Feyerabend (1962), Kuhn (1962) and others. Kuhn (1962) argued, that e.g. the classical and the relativistic concept of mass are radically different:

But the physical referents of these Einsteinian concepts are by no means identical with those of the Newtonian concepts that bear the same name. (Newtonian mass is conserved and Einsteinian is convertible with energy[...]) (Kuhn 1962, p.102)

Such an incommensurability'' is apparently problematic for establishing a deductive relation between the corresponding theories. Kuhn (together with Lakatos and Toulmin) has drawn the attention to the historical process of theory development and gave rise to what some have called an anti-positivistic turn in the philosophy of science (Bayertz 1980). Here the scientific development is rather viewed as a mechanism of theory replacement (a discontinuity, e.g. the famous paradigm shift'' according to Kuhn) with no simple (e.g. logical) relation between successive theories

In particular Kuhn (1977) states with respect to scientific theories:

Theories, as the historian knows them, cannot be decomposed into constituent elements for purpose of direct comparison either with nature or with each other. (Kuhn 1977, p.19)
Kuhn frequently uses the term emergence'' in connection with the appearance of a new theory. However, presumably not in strict terminological sense.

Nagel did reply to some of this criticism (Nagel 1970) by claiming, that the alleged context dependence of theoretical terms is an exaggerated view'', given that theories are not quite the monolithic structure'' as Feyerabend takes them to be (918ff). However, a different line of defense was taken by Kenneth Schaffner, who developed a model of reduction which incorporates some form of theory replacement as well.

Kenneth Schaffner modified the Nagelian reduction scheme to account for this challenge. Schaffner (1967) emphasized what was anticipated in the Nagel quote above (Nagel 1961, p. 360), namely that typically the secondary theory becomes corrected in the turn of the reduction process. Thus the actual reduction relation holds between the primary theory and a {\em modified} theory (E.g. Callender (2001) argues for thermodynamics not to be taken too seriously''. For example a reduction to statistical mechanics fails if one does not allow for entropy fluctuations i.e. considers a modified version of thermodynamics. In Schaffner (1969) it is suggested that even the primary science is modified in the turn of reduction.), T_s^{\prime}. Schaffner (1992) has developed this model further into the General Reduction-Replacement (GRR) model''. According to him this weakened notions of reduction

(...) allow the continuum'' ranging from reduction as subsumption to reduction as explanation of the experimental domain of the replaced theory. (Schaffner 1992, p. 320)
However, in order to call T_s^{\prime} a modified version of T_s a relation of strong analogy'' is assumed. Admittedly, Schaffner provided little guidance as to what counts as a strong analogy. Thomas Nickles (1973) argued that in many instances these analogies could be understood mathematically as limit relations.

Thus Nickles distinguishes between a so-called reduction_1 which corresponds roughly to Nagel's (homogeneous) reduction and the reduction_2'' type which employs the notion of limit'' in order to reduce a successor theory to its predecessor. It thereby inverts the usual concept of reduction (i.e. in reduction_1 the move is from specific to general, while in reduction_2 from general to specific). While reduction_1 is domain combining, reduction_2 is domain preserving at the limit (inter- and intra-level reduction according to Wimsatt 1976). According to Nickles these two types of reduction have different goals: while reduction_1 is unifying and explanatory, reductions_2 serves a justificatory and heuristic role.

Most important the reductions_2 is not meant to establish a strict deductive (i.e. logical) relation between the corresponding theories: ...[But] rather than agree with Nagel's critics that we find no reduction here, I prefer to recognize certain of these important nonderivational intertheoretic relationships as a distinct type of reduction.'' (Nickles 1973 p.956).

The textbook example for such a reduction_2 is the relation between special relativity and classical mechanics in the limit (v/c)^2 → 0. However, the mathematical physicists Sir Michael Berry noted with respect to this example, that

(...) this simple state of affairs is an exceptional situation. Usually, limits of physical theories are not analytic: they are singular, and the emergent phenomena associated with reduction are contained in the singularity. (Berry 1994, p.599)
In such cases there is no smooth reduction relation between the corresponding theories e.g. between wave and ray optics in the λ/a → 0 limit (with the wavelength λ and a a typical linear dimension of the system). Berry argues, that this points to a great richness of borderland physics between the theories''(Berry 2001, p.4). Interestingly this is not taken as evidence against reduction per se. With a small side blow against philosophers Berry states:
What follows should not be misconstrued as antireductionist. On the contrary, I am firmly of the view [...] that all the sciences are compatible and that details links can be, and are being, forged between them. But of course the links are subtle, and my emphasis will be on a mathematical aspect of theory reduction that I regard as central, but which cannot be captured by the purely verbal arguments employed in philosophical discussions of reduction. (Berry 2001, p.4)
Berry (1994) ends his paper by expressing the hope that his ideas will benefit from the attention of philosophers''. Especially Batterman (Batterman 1995, 1997, 2002, 2005) took up this ideas and developed a view on intertheoretical relations based on asymptotic reasoning'' (Batterman 2002). He suggests, that certain phenomena (e.g. a rainbow) are not fully explainable either in terms of the finer wave theory or in terms of the ray theory alone. Instead, aspects of both theories are required for a full understanding of these emergent'' phenomena. This is indeed a failure of reduction (at least in the specific meaning of reductive explanation) and supports some sort of emergence''. A similar view is expressed by Primas (1998). However, emergence'' is a subtle concept and has no generally accepted definition. With respect to Batterman's notion of emergence O'Connor and Wong (2002) remark:
Note that no claim is made concerning their ontological novelty or impact upon the fundamental physical dynamics. Rather, it is a point about the adequacy of the would-be reducing theories: while all the phenomena may be grounded in',or contained by', the reducing theory, the theory itself is unable to capture or explain the distinctive nature of the phenomena.
The different versions of emergence roughly share the idea that emergent entities (properties or substance) arise' out of more fundamental entities and yet are novel' or irreducible' with respect to them'' (O'Connor and Wong (2002)). Another way to characterize emergence is simply by a denial of reduction (R-emergence) or a denial of supervenience (Supervenience may be characterized as an ontic relation between structures, i.e. sets of entities together with properties and relations among them. A structure S_A is said to supervene on an other, say S_B, if the A-entities are composed of B-entities and the properties and relations of S_A are are determined by properties and relations of S_B. It should be noted that neither does reduction entails supervenience nor the other way around.) (S-emergence) (see Howard 2003, p.3ff).

A different classification is suggested by Bishop (2004). He investigates the relation between \qm\ and molecular chemistry, especially whether the notion of molecular structure'' finds a firm quantum foundation. In order to capture the intertheoretic relation in this case he offers a unifying classification scheme for reduction, contextual emergence, supervenience and strong emergence. According to him these relations hold if more fundamental'' entities (i.e. properties or descriptions) provide (i) necessary and sufficient conditions (reduction), (ii) necessary but not sufficient conditions (contextual emergence), (iii) sufficient but not necessary conditions (supervenience) or (iv) neither necessary nor sufficient conditions (strong emergence) for less fundamental'' entities.

### The non-statement view

A different way to met Kuhn's challenge for intertheoretical relations in general and theory reduction in particular was offered by the so called non-statement view'' on theories. Here a theory is not construed as a set of sentences (or partially interpreted axiomatic system). There are several alternative views which go by the names semantic view'', model-theoretic view'' or structuralist view'' of theories. These approaches differ in many detailed respects but share the common view that a theory should be thought of not as a set of statements describing the world but as a class of structures that are approximately isomorphic, under suitable interpretations, to parts of the world'' (Grandy 1992, p.217). This is often summarized by the slogan that a theory is viewed as a class of models''. Some important exponents of this views are van Fraassen (1980), Suppe (1977), Suppes (1967), Sneed (1971, 1976), Stegmüller (1979), Ludwig (1978) and Scheibe (1997/1999).

With respect to the statement view van Fraassen (1980, p.56) makes the following radical claim:

Perhaps the worst consequence of the syntactical approach was the way it focused attention on philosophical irrelevant technical questions. [...] The main lesson of twentieth century philosophy of science may well be this: no concept which is essentially language-dependent has any philosophical importance at all.

In (Scheibe, 1997, 1999) he provides a theory of reduction'' as approach to the unity of physics. While rooted in the structuralistic tradition he is rather pragmatic. E.g. Scheibe qualifies the above statement by Fraassen as typical philosophical exaggeration'' (1997, p.46) with the intention to awaken from the dogmatic slumber''. Scheibe remarks:

It is not understandable why the areas of philosophically interesting and uninteresting matters should coincide with the language-dependent and independent phenomena, [...]. A priori it is more likely that there are questions of philosophical interest which concern language and others which do not. (Scheibe, 1997, p.46)
Scheibe calls it highly misleading that the new approach (or one version of it) goes by the name semantic view'' since to do without any objectification means actually to abandon any semantics in the closer meaning of this term. According to Scheibe the same allegation applies to Sneed's concept of theories as sets of models. Here the focus is on the models while their origin is neglected (Scheibe, 1997, p.47). Scheibe's ambition is to perform this return to the object level'' (Scheibe, 1997, p.47). According to him the key is not to construct an all embracing concept of reduction but to develop a set of elementary reduction concepts which need to be recursively combined. Given that Scheibe does not aim at a fixed concept of reduction, case studies figure prominently in this program since in turn of these studies further types of reduction relation can be discovered.

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