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| Reduction |
A cursory glance at the history of science reveals a continuous succession of scientific theories of various areas or domains. For example, since ancient times theories of the cosmos have been proposed to account for the observed behavior of the heavenly bodies.
The geocentric Ptolemaic theory was, for instance, succeeded by the heliocentric theory of Copernicus. Another example concerns the nature of light. Corpuscular theories were succeeded by wave theories of light. Wave theories, in turn, have been followed by the quantum theories of electromagnetic radiation.
This entry concerns the nature of certain relations that may obtain between different pairs of theories in such sequences. A radical or extreme view of those relations is that of Thomas Kuhn. Kuhn (1970) famously argues that across scientific revolutions there is a radical disconnect between theories. One can find a similar argument in Paul K. Feyerabend (1962). On such a view, no rational relations can obtain between a theory and its predecessor.
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| Paul K. Feyerabend |
However, it is fair to say that most philosophers have held, contrary to this extreme position, that there are, indeed, interesting and contentful relations between various pairs of scientific theories. One such relationship is that of reduction. It is often claimed that successor theories reduce those that they succeed.
Such a relation may involve the idea that the successor or reducing theory explains or otherwise absorbs the successful features of the reduced theory. However, getting clear about exactly how the notion of reduction should be understood has been and continues to be a difficult philosophical problem.
This entry begins with a discussion of what may be called the received view of theory reduction and examines how that view has evolved as the result of various criticisms. Work on intertheoretic relations from 1997 through 2005 is then considered.
Nagelian Reduction
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| Nagelian Reduction |
The locus classicus for contemporary discussion is Ernest Nagel's presentation of a model for theory reduction in The Structure of Science (1961). Nagel takes reduction to be an explanatory relation between theories where explanation is understood to involve deductive logical relations between statements characterizing the explanans and the statement characterizing the explanandum in accordance with the Hempelian (Hempel 1965) deductive-nomological model.
Nagel holds that "[r]eduction ... is the explanation of a theory or a set of experimental laws established in one area of inquiry, by a theory usually though not invariably formulated for some other domain" (1961, p. 338). The idea here is that a theory T reduces a theory T' just in case one can derive (and thereby explain) the laws of T' from the laws of T.
Nagel realizes that for some intuitive cases of theory reduction such derivations would not be immediately possible. If the vocabulary of the reduced (succeeded) theory contains terms referring to entities or properties that are not mentioned in the vocabulary of the reducing (successor) theory, then it will be impossible to derive the laws of the reduced theory containing those terms from the laws of the reducing theory.
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| Lawrence Sklar |
Reductions involving theories with distinct vocabularies are called heterogeneous by Nagel. By contrast, homogeneous reductions are taken by him to be rather straightforward and unproblematic.
This view of homogeneous reductions is somewhat naive. Lawrence Sklar (1967) points out that homogeneous reductions, in fact, are rare. Instead, what one has typically is the derivation of an approximation to the reduced theory and not of the reduced theory itself.
An example discussed by both Nagel and Sklar concerns the homogeneous reduction of the Galilean theory of free fall to Newtonian mechanics and gravitational theory. Sklar notes that there really is no strict derivation of the Galilean theory, although no terms appear in the Galilean theory that do not also appear in Newton's theory.
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| philosophical literature |
The example of a heterogeneous reduction Nagel discusses is the apparent reduction of thermodynamics to statistical mechanics. This example has become paradigmatic of intertheoretic reduction in the general philosophical literature. (In actual fact, the reduction of thermodynamics to statistical mechanics is much more complex than Nagel's discussion allows. Lawrence Sklar [1993] provides a detailed discussion of various difficulties involved in the reduction of thermodynamics to statistical mechanics.)
Thermodynamics contains terms referring to properties such as temperature and entropy. Such terms are completely lacking in the vocabulary of statistical mechanics. To effect the (supposed) derivational reduction, one must connect these thermodynamic terms with terms occurring in the vocabulary of statistical mechanics.
Nagel introduces two necessary formal conditions for such heterogeneous reductions:
- Connectability: "Assumptions of some kind must be introduced which postulate suitable relations between whatever is signified by 'A' [a term appearing in the reduced but not the reducing theory, such as 'temperature'] and traits represented by theoretical terms already present in the primary [reducing] science."
- Derivability: "With the help of these additional assumptions, all the laws of the secondary [reduced] science, including those containing the term 'A,' must be logically derivable from the theoretical premises and their associated coordinating definitions in the primary [reducing] discipline." (1961, pp. 353–354)
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| relations of connectability |
The connectability requirement is vague as it stands. What is the exact nature of the required "suitable relations"? In the literature such relations of connectability are typically called bridge laws or bridging hypotheses and their status is a matter of debate. Nagel allows that such bridge laws need not have the form of universally quantified biconditionals for theory reduction to be possible.
They might, he holds, have the form of one-way conditionals. It is this possibility that renders the requirement of derivability not superfluous With the aid of bridge laws, Nagel thinks that the reducing theory would be able to fully explain the laws of the reduced theory.
However, even having universal biconditionals as bridge laws may not itself be sufficient for reduction. Many examples exist where correlatory laws may be established—where the biconditionals are true and apparently lawlike—yet, where nothing resembling reduction can take place.
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| Wiedemann-Franz |
Lawrence Sklar (1967) offers the example of the Wiedemann-Franz law expressing a correlation between the thermal conductive properties of a material and its electrical conductivity properties. Such a law does not allow one to reduce the theory of thermal conductive properties of the material to a theory of its electrical conductive properties. Something more than mere correlation is required.
That something more is usually taken to be some kind of empirically established identity claim. For example, the reduction of physical optics to the theory of electromagnetic radiation is accomplished by noting the identity of one class of entities—light waves—with (part of) another class— electromagnetic radiation.
As Sklar notes, "Light waves are not correlated with electromagnetic waves, for they are electromagnetic waves. There are not two classes of entities, but only one" (1967, p.120). Another classic example is the reduction of Mendelian genetics to molecular genetics via the identification of genes with DNA molecules.
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| DNA molecules |
The idea that the bridge laws must express necessary identifications between entities or classes of entities has much to recommend it. However, in many cases of apparent intertheoretic reduction such identity relations are not available.
In the paradigmatic case of the reduction of thermodynamics to statistical mechanics, one sees that terms such as temperature and entropy, occurring in thermodynamics but not in statistical mechanics, refer to properties possessed by thermodynamic systems.
Still, it is not at all clear what properties of statistical systems can be identified with the thermodynamic properties. For example, the standard claim that temperature is just (identical to) mean molecular kinetic energy is deeply problematic.
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| orthodox thermodynamics |
One way of emphasizing the difficulty here is in terms of questions about the meaning of terms appearing in the distinct theories. In orthodox thermodynamics, for example, the term entropy gets its meaning (on one view of how theoretical terms acquire meaning) at least in part by the role the term plays in the theory. (A classic presentation of orthodox thermodynamics explicitly exhibitingthe roles of the terms is by A. B. Pippard [1957].) One sees that such terms refer to unvarying and nonstatistical properties of systems.
Nevertheless, in the apparent reduction of thermodynamics to statistical mechanics the concept of entropy changes to one that explicitly allows for statistical variation and fluctuation. In what sense can one identify here? Feyerabend (1962), for one, takes this to be evidence that reduction (understood as Nagelian derivation with bridge laws) must fail.
NEO-NAGELIAN REDUCTION. In contrast to Feyerabend's (1962) pessimistic conclusion many philosophers hold that some sort of reductive relation still obtains even in the face of problems of heterogeneity. In fact, it is often noted that in the process of reducing one theory to another, the reduced theory gets emended.
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| statistical thermodynamics |
One sees textbooks with titles referring to statistical thermodynamics, indicating that the orthodox thermodynamic conceptions of entropy and temperature have been changed to allow for (observable and observed) fluctuations in those quantities.
The explicit recognition that the reduced theory is often changed as a result of reduction or attempted reduction takes one beyond the Nagelian conception of reduction as a relatively straightforward explanatory derivation.
Kenneth Schaffner's (1967, 1976) model of reduction deserves mention here as a sophisticated attempt to incorporate this aspect of theoretical change into a Nagelian-type framework. Kenneth Schaffnerexplicitly includes the corrected reduced theory in the model. On this view a theory T reduces a theory T' just in case there is a corrected version of the reduced theory, T'* such that
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| Kenneth Schaffner |
- The primitive terms of T'* are associated with various terms of T via bridge laws or reduction functions
- T'* is derivable from T when supplemented by these bridge laws
- T'* corrects T in that it makes more accurate predictions than does T'
- T' is explained by T in that T' and T'* are strongly analogous to one another, and T says why T' works as well as it does in its domain of validity.
Objections to Nagelian Reductions
A number of influential objections have been raised against Nagelian models of reduction. Most of these concern the possibility of providing the appropriate bridge laws. As a result they can be seen as telling also against more sophisticated models such as Schaffner's. Additionally, it has been objected that even if such bridge laws can be provided, there remains an explanatory question about their status as laws. Consider the second objection first.
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| Objections to Nagelian Reductions |
EXPLANATORY QUESTIONS ABOUT BRIDGE LAWS. In those cases where bridge laws express the identification of classes of entities, to ask why those bridge laws hold is to ask a question that can be trivially answered. The reason the bridge laws hold is because the entities in question are one and the same. "Why should I believe that light waves are electromagnetic radiation?" Answer: "They just are. Period, end of story."
By contrast, in cases where bridge laws express some kind of (perhaps, nomologically) necessary coextensivity between properties appearing in two theories, such a question may seem legitimate and answers may be hard to come by. Jaegwon Kim (1998) forcefully argues that this poses a serious problem for Nagelian reduction understood as attempting to effect an explanatory relation among pairs of theories.
Jaegwon Kim discusses the attempted Nagelian reduction of psychology (the science of the mental) to physical theory, say, neurophysiology. One can suppose that one discovers empirically a nomological correlation between being in pain and having one's C-fibers firing.
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| Jaegwon Kim |
A statement characterizing this correlation is taken to be a bridge law necessary for Nagelian reduction. In this case it seems reasonable to ask: "Can we understand why we experience pain when our C-fibers are firing, and not when our Afibers are firing? Can we explain why pains, not itches or tickles, correlate with C-fiber firings?" (Kim 1998, p. 95).
Kim's point is that if Nagelian reduction is supposed to provide an explanation of the reduced theory in terms of the reducing theory, then surely one must demand an explanation of the bridge laws employed in the explanatory derivation. "For it is the explanation of these bridge laws, an explanation of why there are just these mind-body correlations, that is at the heart of the demand for an explanation of mentality" (p. 96, emphasis in the original).
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| explanatory derivation |
MULTIPLE REALIZABILITY. A different argument due to Jerry Fodor (1974) has been used to block attempts at almost every Nagelian reduction of a given (special science) theory to more basic (physical) theory.
This argument has come to be called the multiple realization argument. It depends on the assumption that properties appearing in the special (to-be-reduced) science may have diverse and "wildly heterogeneous" realizers in the reducing physical theory.
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| physical theory |
As Fodor puts it, "The problem ... has been that there is an open empirical possibility that what corresponds to the natural kind predicates of a reduced science may be a heterogeneous and unsystematic disjunction of predicates in the reducing science" (p. 108).
Thus, to continue the psychology example so prevalent in the literature, pain—a property appearing in the science of psychology whose predicate (perhaps) appears in its laws—may be realized by distinct physical or neurophysiological properties in humans, in reptiles, and possibly even in inorganic robots.
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| psychology example |
This has the consequence no one neurophysiological state can be correlated or identified with the psychological property pain. In humans it may be C-fibers firing; in reptiles it may be D-fibers firing; and in robots it may be the activation of some particular integrated circuit.
The heterogeneous nature of the distinct realizers also makes it unlikely that a disjunction of those realizers will be a natural kind term in the reducing theory. Given this, and if laws relate natural kinds to natural kinds, it is unlikely that there can be anything lawlike about the bridge laws.
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| natural kinds |
This argument has been applied to many functionally defined properties such as being a thermostat or being a heart—properties that can be realized in many different ways in different systems or organisms.
One response, due to Kim (1992), to the realization argument is to note that while the argument may block a kind of global reduction of the special science to the lower-level physical theory, it may be possible to have (local) species or structure specific reductions.
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| specific reductions |
Thus, for instance, one might be able to locally reduce human pain to human neurophysiology, reptilian pain to reptilian neurophysiology, and robot pain to robot neurocircuitry. Kim (1998, chapter 4) develops an alternative functional model of reduction appropriate to this response.
Another approach (Batterman 2000) asks for an account of what makes the multiple realizability possible. Typically, multiple realizability is simply assumed and applied via the multiple realization argument to block Nagelian reductions. For instance, Fodor (1974) cites it simply as an open possibility.
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| simply assumed |
However, it seems reasonable to ask whether one can explain, from the point of view of the supposed reducing theory, the possibility of multiple realizability. If so, this may lead to a kind of explanation without reduction. Cases where such explanations are indeed possible can be found in the physics literature where attempts are made to explain surprising universal features of various systems.
Universality means identical or similar behavior in physically distinct systems and is, therefore, a term essentially synonymous with multiple realizability. For details about how such an approach to multiple realizability will go, see Robert W. Batterman (2000, 2002).
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| multiple realizability |