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. 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.
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. 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.
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. Sklar  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)
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 (Nagel 1961, p. 355 note). 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. 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.
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. Again, see Sklar (1993) for a detailed discussion of some of these problems.
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 exhibiting the roles of the terms is by A. B. Pippard .) 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.
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. 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. Schaffner explicitly 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
(1) The primitive terms of T′* are associated with various terms of T via bridge laws or reduction functions
(2) T′* is derivable from T when supplemented by these bridge laws
(3) T′* corrects T in that it makes more accurate predictions than does T′
(4) 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.
It is clear that work must be done to explicate the intuitive notion of strong analogy playing a role in this model of reduction. See William C. Wimsatt (1976) for some suggestions along these lines.
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 (1967, 1976). 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.
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.
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. 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 A-fibers 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).
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. 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.
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. 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. 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. 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).
Reduction in the Other Direction
There is an interesting terminological ambiguity that infects the term reduction as it is typically used in the philosophical literature and as it is used in the physics literature. Philosophers will discuss the reduction of thermodynamics to statistical mechanics, the reduction of classical mechanics to quantum mechanics, and the reduction of the ray theory of light to the wave theory. The succeeded theories are all reduced to their successors. Physicists, when they talk about theory reduction at all, tend to put things the other way around. They will say that statistical mechanics reduces to thermodynamics in the limit as the number of degrees of freedom goes to infinity. They will say that quantum mechanics reduces to classical mechanics in some kind of correspondence limit. Furthermore, they will say that the wave theory reduces to the ray theory in the limit as the wavelength of light approaches zero. That there are such different senses of intertheoretic reduction was first noted by Thomas Nickles in the paper "Two Concepts of Intertheoretic Reduction" (1973).
Interestingly, the physicists' sense of reduction appeals to limiting relations between the pair of theories and is not concerned with derivation in logical/syntactic sense that has primarily concerned philosophers following in Nagel's footsteps. In other words, there is no explicit concern, say, with the derivation of the laws of thermodynamics from the laws of statistical mechanics. Instead, the interest is in the potential emergence of those laws as some sort of mathematical limit is asymptotically approached. Thus, while the philosophical tradition focuses on the schema according to which theory T′ reduces to theory T just in case one can derive (and thereby explain) the laws of T′ from the laws of T, this other sense of reduction focuses on a schema of the following form (1):
in which theory T reduces to T′ in the regime where a parameter ∈ appearing in theory T takes on a limiting value. For instance, quantum mechanics contains a constant (Planck's constant) that plays no role in classical mechanics. As a result, one may be motivated to study the limit of quantum mechanics in which Planck's constant approaches zero. This is a kind of correspondence limit.
The two schemas are related to one another at least in the following way. Should the equality in (1) hold for two theories T and T′, then it is reasonable to expect that the laws of T′ are derivable from those of T. That is to say, it is likely that one will be able to find the appropriate connections that will allow something like a Schaffner-style neo-Nagelian reduction. On the contrary, if the equality in 1fails to obtain for the pair of theories, then such neo-Nagelian reduction will not be possible. It will be impossible to form the relevant corrected reduced theory T′*.
One case for which the schema (1) does obtain is in the relationship between (certain aspects of) classical Newtonian mechanics (NM) and the special theory of relativity (SR). In the limit in which velocities are slow compared with the speed of light ((v /c ) → 0), SR reduces to NM. The limit exists and the formulas of SR smoothly (that is uniformly) approach those of NM.
However, far more often than not pairs of theories will be related to one another by so-called singular limits. Singular limits arise when the behavior as the limit is approached (no matter how small ∈ becomes) is qualitatively different from the behavior at the limit (when ∈ = 0). In such cases the equality in schema (1) fails to obtain: There will be no smooth approach of the formulas of theory T to those of theory T′.
In fact, it is fair to say that for most theory pairs of interest, schema (1) will fail. Important examples include those mentioned earlier: quantum mechanics and classical mechanics; wave theory and ray theory; and statistical mechanics and thermodynamics. Certain formulas in each of these theory pairs are related by singular limits, and it is best, perhaps, to give up on speaking of reductive relations between the theories or at least between those features characterized by the singularly related formulas (see Berry 1994, 2002; Batterman 1995, 2002).
It is important to stress that if this is correct, and so physicists' reductions are genuinely few and far between, this does not mean that there is no reason to study the singular limiting relationships between theories. In fact, the opposite is true. There is much of interest to study in the borderland between the theories. Michael V. Berry notes the importance of the failure of reduction due to singular limiting relations between the theories, "[M]any difficulties associated with reduction arise because they involve singular limits. These singularities have both negative and positive aspects: They obstruct smooth reduction of more general theories to less general ones, but they also point to a great richness of borderland physics between theories" (2001, p. 42).
The "great richness" of this borderland is fertile ground for studying certain aspects of emergence, a philosophical topic related to the failure of reduction. Emergent phenomena are typically taken to be novel in certain respects where this novelty is often understood as resulting from the failure of the more basic theory to explain or otherwise account for the phenomena. There has been considerable interest in the controversial issues surrounding the nature and existence of emergent phenomena. Two related approaches with the same starting point—the singular nature of limiting intertheoretic relations—are examined by Batterman (2002) and Alexander Rueger (2000a, 2000b). However, another related approach can be found in the work of Hans Primas (1998). For a different, more metaphysically motivated attempt to understand emergence, see Paul Humphreys (1997). This is currently an active area of research.
See also Copernicus, Nicolas; Fodor, Jerry A.; Galileo Galilei; Hempel, Carl Gustav; Kuhn, Thomas; Laws and Theories; Multiple Realizability; Nagel, Ernest; Newton, Isaac; Philosophy of Science; Properties; Scientific Theories.
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Robert W. Batterman (2005)
re·duc·tion / riˈdəkshən/ • n. 1. the action or fact of making a specified thing smaller or less in amount, degree, or size: talks on arms reduction | there had been a reduction in the number of casualties. ∎ the amount by which something is made smaller, less, or lower in price: special reductions on knitwear. ∎ the simplification of a subject or problem to a particular form in presentation or analysis: the reduction of classical genetics to molecular biology. ∎ Math. the process of converting an amount from one denomination to a smaller one, or of bringing down a fraction to its lowest terms. ∎ Biol. the halving of the number of chromosomes per cell that occurs at one of the two anaphases of meiosis. 2. a thing that is made smaller or less in size or amount, in particular: ∎ an arrangement of an orchestral score for piano or for a smaller group of performers. ∎ a thick and concentrated liquid or sauce made by boiling. ∎ a copy of a picture or photograph made on a smaller scale than the original. 3. the action of remedying a dislocation or fracture by returning the affected part of the body to its normal position. 4. Chem. the process or result of reducing or being reduced. 5. Phonet. substitution of a sound that requires less muscular effort to articulate: the process of vowel reduction.