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The present paper surveys the three most prominent accounts in contemporary debates over how sound reduction should be executed. The classical Nagelian model of reduction derives the laws of the target-theory from the laws of the base theory plus some auxiliary premises (so-called bridge laws) connecting the entities of the target and the base theory. The functional model of reduction emphasizes the causal definitions of the target entities referring to their causal relations to base entities. The new-wave model of reduction deduces not the original target theory but an analogous image of it, which remains inside the vocabulary of the base theory. One of the fundamental motivations of both the functional and the new-wave model is to show that bridge laws can be evaded. The present paper argues that bridge laws—in the original Nagelian sense—are inevitable, i.e. that none of these models can evade them. On the one hand, the functional model of reduction needs bridge laws, since its fundamental concept, functionalization, is an inter-theoretical process dealing with entities of two different theories. Theoretical entities of different theories (in a general heterogeneous case) do not have common causal relations, so the functionalization of an entity—without bridge laws—can only be executed in the framework of its own theory. On the other hand, the so-called images of the new-wave account cannot be constructed without the use of bridge laws. These connecting principles are needed to guide the process of deduction within the base theory; without them one would not be able to recognize if the deduced structure was an image of the target theory.

Philosophers of science have offered different accounts of what it means for one scientific theory to reduce to another. I propose a more or less friendly amendment to Kenneth Schaffner’s “General Reduction-Replacement” model of scientific unification. Schaffner interprets scientific unification broadly in terms of a continuum from theory reduction to theory replacement. As such, his account leaves no place on its continuum for type irreducible and irreplaceable theories. The same is true for other accounts that incorporate Schaffner's continuum, for example, those developed by Paul Churchland, Clifford Hooker, and John Bickle. Yet I believe a more general account of scientific unification should include type irreducible and irreplaceable theories in an account of their partial reduction, specifically, when there is a reduction of their tokens. Thus I propose a “Reduction-Reception-Replacement” model wherein type irreducible and irreplaceable theories are accepted or received for the purpose of unifying domains of particulars. I also suggest a link between this kind of token reduction and mechanistic explanation.

Schaffner’s model of theory reduction has played an important role in philosophy of science and philosophy of biology. Here, the model is found to be problematic because of an internal tension. Indeed, standard antireductionist external criticisms concerning reduction functions and laws in biology do not provide a full picture of the limits of Schaffner’s model. However, despite the internal tension, his model usefully highlights the importance of regulative ideals associated with the search for derivational, and embedding, deductive relations among mathematical structures in theoretical biology. A reconstructed Schaffnerian model could therefore shed light on mathematical theory development in the biological sciences and on the epistemology of mathematical practices more generally. *Received November 2006; revised March 2009. †To contact the author, please write to: Philosophy Department, University of California, Santa Cruz, 1156 High St., Santa Cruz, CA 95064; e‐mail: rgw@ucsc.edu.

Examples of reduction outside of physics typically concern in principle possibilities; e.g., if we had a decent psychological theory of human behavior, we could reduce it to neurophysiology once we know more. However, in one instance, a reduction is actually well underway – the reduction of Mendelian genetics to molecular biology. Empirical and conceptual difficulties in setting out this reduction have led certain philosophers to modify the traditional logical empiricist analysis of theory reduction, first, to allow for necessary corrections and, second, to introduce a temporal dimension [reduction, genetics, theory, logical empiricism]. CiteULike Connotea Del.icio.us What's this?

Four current accounts of theory reduction are presented, first informally and then formally: (1) an account of direct theory reduction that is based on the contributions of Nagel, Woodger, and Quine, (2) an indirect reduction paradigm due to Kemeny and Oppenheim, (3) an "isomorphic model" schema traceable to Suppes, and (4) a theory of reduction that is based on the work of Popper, Feyerabend, and Kuhn. Reference is made, in an attempt to choose between these schemas, to the explanation of physical optics by Maxwell's electromagnetic theory, and to the revisions of genetics necessitated by partial biochemical reductions of genetics. A more general reduction schema is proposed which: (1) yields as special cases the four reduction paradigms considered above, (2) seems to be in better accord with both the canons of logic and actual scientific practice, and (3) clarifies the problems of meaning variance and ontological reduction.

Philosophical theories about reduction and integration in science are at variance with what is happenign in science. A realistic approach to science show that possibilities for reduction and integration are limited. The classical ideal of a unified science has since long been rejected in philosophy. But the current emphasis on interdisciplinary integration in philosophy and in science shows that it survives in a different guise. It is necessary to redress the balance, specifically in biology. Methodological analysis shows that many of the grand interdisciplinary theories involving biology actually represent pseudo-integration covered up by inappropriate, overgeneral concepts. Integrationism is not bad, but it must be kept within reasonable bounds. If the present analysis is appropriate, there will have to be fundamental changes in research strategy both in science and in the philosophy of science.

The discussion of theory reduction in genetics threatens to become more and more confused. The position taken is that before one tries to work out complicated reduction principles which might be applicable to broad areas of biology in their relationships to chemistry and physics, it would be better to attempt first to elucidate the internal structure of some limited biological theories in a formal way and to consider simple constructs for reduction between them. This proposal is elaborated with respect to the original Mendelian genetics, linkage genetics and fine-structure genetics, and their relationship to non-formalized molecular genetics.

Many studies of the unification of science focus on the theories of different disciplines. The model for integration is the theory reduction model. This paper argues that the embodiment of theories in scientists, and the institutions in which scientists work and the instruments they employ, are critical to the sort of integration that actually occurs in science. This paper examines the integration of scientific endeavors that emerged in cell biology in the period after World War II when the development of cell fractionation and electron microscopy made serious investigations of cell organelles possible. One surprising feature of such integration is that it generated further disintegration as the new institutions of cell biology separated the practitioners of the new discipline from other, closely related biological disciplines.

The applicability of Nagel's concept of theory reduction, and related concepts of reduction, to the reduction of genetics to molecular biology is examined using the lactose operon in Escherichia coli as an example. Geneticists have produced the complete nucleotide sequence of two of the genes which compose this operon. If any example of reduction in genetics should fit Nagel's analysis, the lactose operon should. Nevertheless, Nagel's formal conditions of theory reduction are inapplicable in this case. Instead, it is argued that genetics has been partially reduced to molecular biology in the sense of token-token reduction.

A classification of models of reduction into three categories — theory reductionism, explanatory reductionism, and constitutive reductionism — is presented. It is shown that this classification helps clarify the relations between various explications of reduction that have been offered in the past, especially if a distinction is maintained between the various epistemological and ontological issues that arise. A relatively new model of explanatory reduction, one that emphasizes that reduction is the explanation of a whole in terms of its parts is also presented in detail. Finally, the classification is used to clarify the debate over reductionism in molecular biology. It is argued there that while no model from the category of theory reduction might be applicable in that case, models of explanatory reduction might yet capture the structure of the relevant explanations.