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- 1. The Philosopher’s Sense of Reduction
- 2. The Physicist’s Sense of Reduction
- 3. Intertheory Relations
- Bibliography
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*T*reduces*T*just in case the laws of*T*are derivable from those of*T*.

Showing how these derivations are possible for "paradigm" examples of intertheoretic reduction turns out to be rather difficult.

Nagel distinguishes two types of reductions on the basis of whether
or not the vocabulary of the reduced theory is a subset of the
reducing theory. If it is---that is, if the reduced theory
*T* contains no descriptive
terms not contained in the reducing theory *T*, and the terms
of
*T* are understood to have
approximately the same meanings that they have in *T*, then
Nagel calls the reduction of
*T* by *T*
"homogeneous." In this case, while the reduction may very well be
enlightening in various respects, and is part of the "normal
development of a science," most people believe that there is nothing
terribly special or interesting from a philosophical point of view
going on here. (Nagel, 1961, p. 339.)

Lawrence Sklar (1967, p. 110--111) points out that, from a
historical perspective, this attitude is somewhat naive. The number of
actual cases in the history of science where a genuine homogeneous
reduction takes place are few and far between. Nagel, himself, took as
a paradigm example of homogeneous reduction, the reduction of the
Galilean laws of falling bodies to Newtonian mechanics. But, as Sklar
points out, what actually can be derived from the Newtonian theory are
*approximations* to the laws of the reduced Galilean theory. The
approximations, of course, are strictly speaking *incompatible*
with the actual laws and so, despite the fact that no concepts appear
in the Galilean theory that do not also appear in the Newtonian theory,
there is no deductive derivation of the *laws* of the one from
the *laws* of the other. Hence, strictly speaking, there is no
reduction on the deductive Nagelian model.

One way out of this problem for the proponent of Nagel-type reductions is to make a distinction between explaining a theory (or explaining the laws of a given theory) and explaining it away. (Sklar, 1967, pp. 112--113) Thus, we may still speak of reduction if the derivation of the approximations to the reduced theory’s laws serves to account for why the reduced theory works as well as it does in its (perhaps more limited) domain of applicability. This is consonant with more sophisticated versions of Nagel-type reductions in which part of the very process of reduction involves revisions to the reduced theory. This process arises as a natural consequence of trying to deal with what Nagel calls "heterogeneous" reductions.

The task of characterizing reduction is more involved when the
reduction is heterogenous---that is, when the reduced theory contains
terms or concepts that do not appear in the reducing theory. Nagel
takes, as a paradigm example of heterogeneous reduction, the (apparent)
reduction of thermodynamics, or at least some parts of thermodynamics,
to statistical
mechanics.^{[1]}
For instance, thermodynamics contains the
concept of temperature (among others) that is lacking in the reducing
theory of statistical mechanics.

Nagel notes that "if the laws of the secondary science [the reduced
theory] contain terms that do not occur in the theoretical assumptions
of the primary discipline [the reducing theory] ... , the logical
derivation of the former from the latter is *prima facie*
impossible." (Nagel, 1961, p. 352) As a consequence, Nagel introduces
two "necessary formal conditions" required for the reduction to take
place:

*Connectability*. "Assumptions of some kind must be introduced which postulate suitable relations between whatever is signified by ‘A’ [the term to be reduced, that is, an element of the vocabulary of theory*T*] 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 science, including those containing the term ‘A,’ must be logically derivable from the theoretical premises and their associated coordinating definitions in the primary discipline." (Nagel, 1961, pp. 353--354)

The connectability condition brings with it a number of interpretive
problems. Exactly what is, or should be, the status of the "suitable
relations," often called bridge "laws" or bridge hypotheses? Are they
established by linguistic investigation alone? Are they factual
discoveries? If the latter, what sort of necessity do they involve? Are
they identity relations that are contingently necessary or will some
sort of weaker relation, such as nomic coextensivity, suffice? Much of
the philosophical literature on reduction addresses these questions
about the status of the bridge
laws.^{[2]}

The consideration of certain examples lends plausibility to the idea,
prevalent in the literature, that the bridge laws should be
considered to express some kind of identity relation. For instance,
Sklar notes that the reduction of the "theory" of physical optics to
the theory of electromagnetic radiation proceeds by
*identifying* one class of entities -- light waves -- with
(part of) another class -- electromagnetic radiation. He says
"... the place of correlatory laws [bridge laws] is taken by
empirically established *identifications* of two classes of
entities. Light waves are not correlated with electromagnetic waves,
for they *are* electromagnetic waves." (Sklar, 1967, p. 120)
In fact, if something like Nagelian reduction is going to work, it is
generally accepted that the bridge laws should reflect the existence
of some kind of synthetic identity.

Kenneth Schaffner calls the bridge laws "reduction functions." He
too notes that they must be taken to reflect synthetic identities
since, at least initially they require empirical support for their
justification. "Genes were not discovered to be DNA via the analysis of
*meaning*; important and difficult empirical research was
required to make such an identification." (Schaffner, 1976. pp.
614--615)

Now one problem facing this sort of account was forcefully presented
by Feyerabend in "Explanation, Reduction, and Empiricism."
(Feyerabend, 1962) Consider the term "temperature" as it functions in
classical thermodynamics. This term is defined in terms of Carnot
cycles and is related to the strict, nonstatistical second law as it
appears in that theory. The so-called reduction of classical
thermodynamics to statistical mechanics, however, fails to identify or
associate *nonstatistical* features in the reducing theory,
statistical mechanics, with the nonstatistical concept of temperature
as it appears in the reduced theory. How can one have a genuine
reduction, if terms with their meanings fixed by the role they play in
the reduced theory get identified with terms having entirely different
meanings? Classical thermodynamics is not a statistical theory. The
very possibility of finding a reduction function or bridge law that
captures the concept of temperature and the strict, nonstatistical,
role it plays in the thermodynamics seems impossible.

The plausibility of this argument, of course, depends on certain views about how meaning accrues to theoretical terms in a theory. However, just by looking at the historical development of thermodynamics one thing seems fairly clear. Most physicists, now, would accept the idea that our concept of temperature and our conception of other "exact" terms that appear in classical thermodynamics such as "entropy," need to be modified in light of the alleged reduction to statistical mechanics. Textbooks, in fact, typically speak of the theory of "statistical thermodynamics." The very process of "reduction" often leads to a corrected version of the reduced theory.

In fact, Schaffner and others have developed sophisticated Nagelian
type schemas for reduction that explicitly try to capture these
features of actual theory change. The idea is explicitly to include
in the model, the "corrected reduced theory" such as statistical
thermodynamics. Thus, Schaffner (1976, p. 618) holds that *T*
reduces
*T* if and only if
there is a corrected version of
*T*, call it
*T** such that

- The primitive terms of
*T** are associated via reduction functions (or bridge laws) with various terms of*T*. *T** is derivable from*T*when it is supplemented with the reduction functions specified in 1.*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*indicates why*T*works as well as it does in its domain of validity.

Much work clearly is being done here by the intuitive conception of
"strong analogy" between the reduced theory
*T* and the corrected
reduced theory
*T**. In some cases, as
suggested by Nickles and Wimsatt, the conception of strong analogy
may find further refinement by appeal to what was referred to as the
"physicists" sense of reduction.

Philosphical theories of reduction would have it that, say, quantum
mechanics reduces classical mechanics through the derivation of the
laws of classical physics from those of quantum physics. Most
physicists would, on the other hand, speak of quantum mechanics
reducing to classical mechanics in some kind of correspondence limit
(e.g., the limit as Planck’s constant
(*h*/2) goes to zero). Thus, the
second type of intertheoretic reduction noted by Nickles fits the
following schema:

HereSchema R: lim_{0}T_{f}=T_{c}

One must take the equality here with a small grain of salt. In those
situations where **Schema R** can be said to hold, it is
likely not the case that every equation or formula from
*T*_{f} will yield a corresponding equation
of *T*_{c} .

Even given this caveat, the equality in **Schema R** can hold
only if the limit is "regular." In such circumstances, it can be
argued that it is appropriate to call the limiting relation a
"reduction." If the limit in **Schema R** is singular, however,
the schema fails and it is best to talk simply about intertheoretic
relations.

One should understand the difference between regular and singular
limiting relations as follows. If the solutions of the relevant
formulae or equations of the theory *T*_{f}
are such that for small values of they *smoothly* approach the solutions of the
corresponding formulas in *T*_{c}, then
**Schema R** will hold. For these cases we can say that
the "limiting behavior" as
0
equals the "behavior in the limit" where
= 0.
On the other hand, if the behavior in the limit is of a
*fundamentally different character* than the nearby solutions
one obtains as
0,
then the schema will fail.

A nice example illustrating this distinction is the following:
Consider the quadratic equation *x*^{2} +
*x*
9 = 0.
Think of as a
small expansion or perturbation parameter. The equation has two roots
for any value of
as
0.
In a well-defined sense, the solutions to this
quadratic equation as
0
smoothly approach solutions to
the "unperturbed" ( = 0) equation
*x ^{2}* +

A paradigm case where a limiting reduction of the form **R**
rather straightforwardly holds is that of classical Newtonian
particle mechanics (NM) and the special theory of relativity (SR). In
the limit where
(*v*/*c*)^{2}0,
SR reduces to
NM. Nickles says "epitomizing [the intertheoretic reduction of SR to
NM] is the reduction of the Einsteinian formula for momentum,

wherep=m_{0}v/ (1 (v/c)^{2})

This is a regular limit---there are no singularities or "blowups" as
the asymptotic limit is approached. As noted one way of thinking
about this is that the exact solutions for small but nonzero values
of || "smoothly [approach] the
unperturbed or zeroth-order solution
[
set identically equal to zero] as
0."
In the case where the limit is *singular* "the exact solution for
= 0 is *fundamentally different in
character* from the ‘neighboring’ solutions obtained in
the limit
0."
(Bender and Orszag, 1978, p. 324)

In the current context, one can express the regular nature of the limiting relation in the following way. The fundamental expression appearing in the Lorentz transformations of SR , can be expanded in a Taylor series as

1/(1(and so the limit is analytic. This means that (at least some) quantities or expressions of SR can be written as Newtonian or classical quantities plus an expansion of corrections in powers of (v/c)^{2}) = 1 1/2 (v/c)^{2}1/8 (v/c)^{4}1/16 (v/c)^{6}...

Examples like this have led some investigators to think of limiting
relations as forming a kind of new rule of inference which would allow
one to more closely connect the physicists’ sense of reduction with
that of the philosophers’. Fritz Rohrlich, for example, has argued that
NM reduces (in the philosophers’ sense) to SR because the
*mathematical framework* of SR reduces (in the physicists’
sense) to the *mathematical framework* of NM. The idea is that
the mathematical framework of NM is "rigorously derived" from that of
SR in a "derivation which involves limiting procedures." (Rohrlich,
1988, p. 303) Roughly speaking, for Rohrlich a "coarser" theory is
reducible to a "finer" theory in the philosophers’ sense of being
rigorously deduced from the latter just in case the mathematical
framework of the finer theory reduces in the physicists’ sense to the
mathematical framework of the coarser theory. In such cases, we will
have a systematic explication of the idea of "strong analogy" to which
Schaffner appeals in his model of philosophical reduction. The
corrected theory
*T** in this context is the
perturbed Newtonian theory as expressed in the Taylor expansion given
above. The "strong analogy" between Newtonian theory
*T* and the corrected
*T** is expressed by the
existence of the *regular* Taylor series expansion.

As noted the trouble with maintaining that this relationship between
the philosophical and "physical" models of reduction holds generally
is that far more often than not the limiting relations between the
theories are *singular* and not regular. In such situations,
**Schema R** fails to hold. Paradigm cases here include the
relationships between classical mechanics and quantum mechanics, the
ray theory of light and the wave theory, and thermodynamics and
statistical mechanics of systems in critical states.

It seems reasonable to expect something like philosophical reductions
to be possible in those situations where **Schema R** holds.
On the other hand, neither philosophical nor "physical" reduction
seems possible when the limiting correspondence relation between the
theories is singular. Perhaps in such cases it is best to speak
simply of intertheoretic relations rather than reductions. It is here
that much of philosophical and physical interest is to be found. This
claim and the following discussion should not be taken to be anything
like the received view among philosophers of science. Instead, it
reflects the views of the author.

Nevertheless, here is a passage from a recent paper by Michael Berry which expresses a similar point of view.

Even within physical science, reduction between different levels of explanation is problematic--indeed, it is almost always so. Chemistry is supposed to have been reduced to quantum mechanics, yet people still argue over the basic question of how quantum mechanics can describe the shape of a molecule. The statistical mechanics of a fluid reduces to its thermodynamics in the limit of infinitely many particles, yet that limit breaks down near the critical point, where liquid and vapour merge, and where we never see a continuum no matter how distantly we observe the particles . . . . The geometrical (newtonian) optics of rays should be the limit of wave optics as the wavelength becomes negligibly small, yet . . . the reduction (mathematically similar to that of classical to quantum mechanics) is obstructed by singularities ... .

My contention ... will be that many difficulties associated with reduction arise because they involvesingular limits. These singularities have both negative and positive aspects: they obstruct the smooth reduction of more general theories to less general ones, but they also point to a great richness of borderland physics between theories. (Berry, forthcoming, p. 3)

When **Schema R** fails this is because the mathematics of the
particular limit
(0) is singular. One can ask what, physically, is
responsible for this mathematical singularity. In investigating the
answer to this question one will often find that the mathematical
blow-up reflects a physical impossibility. For instance, if
**Schema R** held when *T*_{f}
is the wave theory of light and *T*_{c} is
the ray theory (geometrical optics), then one would expect to recover
rays in the shortwave limit
0
of the wave theory. On the ray theory, rays are the carriers of
energy. But in certain situations families of rays can focus on
surfaces or lines called "caustics." These are not strange estoteric
situations. In fact, rainbows are, to a first approximation,
described by the focusing of sunlight on these surfaces following its
refraction and reflection through raindrops. However, according to
the ray theory, the intensity of the light on these focusing surfaces
would be *infinite*. This is part of the physical reason for
the mathematical singularities.

One is led to study the asymptotic domain in which the parameter
in **Schema R**
approaches 0. In the example above, this is the short wavelength
limit. Michael Berry (1980,1990, 1994a, 1994b) has done much research
on this and other asymptotic domains. He has found that in the
asymptotic borderlands between such theories there emerge phenomena
whose explanation requires in some sense appeal to a third
intermediate theory. The emergent structures (the rainbow itself is
one of them) 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 fact calls into question certain received views about the
nature of intertheoretic relations. The wave theory, for example, is
surely the fundamental theory. Nevertheless, these considerations seem
to show that that theory is itself explanatorily deficient. There are
phenomena within its scope whose explanations require reference to
structures that exist only in the superseded, *false*, ray
theory. A similar situation arises in the asymptotic domain between
quantum mechanics and classical mechanics where Planck’s constant can
be considered asymptotically small.

- Batterman, R.W., 1991, "Chaos, quantization, and the correspondence
principle",
*Synthese*, 89: 189-227. - -------, 1993, "Quantum chaos and semiclassical mechanics", in
*PSA 1992*, volume 2, pages 50-65. Philosophy of Science Association. - -------, 1995, "Theories between theories: Asymptotic limiting
intertheoretic relations",
*Synthese*, 103: 171-201. - -------, forthcoming,
*The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence*. Oxford University Press, New York. - Bender, C.M., and Orszag, S.A., 1978,
*Advanced Mathematical Methods for Scientists and Engineers*. McGraw-Hill, New York. - Berry, M.V., 1990, "Beyond rainbows",
*Current Science*, 59/(21-22): 1175-1191. - -------, 1991, "Asymptotics, singularities and the reduction of
theories", in Dag Prawitz, Brian Skyrms, and Dag Westerståhl,
editors,
*Logic, Methodolog and Philosophy of Science, IX: Proceedings of the Ninth International Congress of Logic, Methodology and Philosophy of Science, Uppsala, Sweden, August 7-14, 1991*, volume 134 of*Studies in Logic and Foundations of Mathematics*, pages 597-607, Amsterdam, 1994. Elsevier Science B. V. - -------, 1994, "Singularities in waves and rays", in
R. Balian, M. Kléman, and J. P. Poirier (eds),
*Physics of Defects (Les Houches, Session XXXV, 1980)*, pages 453-543, Amsterdam, 1994. North-Holland. - -------, forthcoming, "Chaos and the Semiclassical Limit of Quantum Mechanics (Is the Moon There When Somebody Looks?)", in
*Proceedings of the CTNS-Vatican Conference on Quantum Physics and Quantum Field Theory*, in press (preprint in PDF format) - Berry, M.V., and Upstill, C., 1980, "Catastrophe optics:
Morphologies of caustics and their diffraction patterns", in
E. Wolf (ed),
*Progress in Optics*, volume XVIII, pages 257-346, Amsterdam, 1980. North-Holland. - Feyerabend, P.K., 1962, "Explanation, reduction, and empiricism", in
H. Feigl and G. Maxwell, (eds),
*Minnesota Studies in the Philosophy of Science*, volume 3, pages 28-97. D. Reidel Publishing Company. - Nagel, E., 1961,
*The Structure of Science*. Routledge and Kegan Paul, London. - Nickles, T., 1973, "Two concepts of intertheoretic reduction",
*The Journal of Philosophy*, 70/7: 181-201. - Rohrlich, F., 1988, "Pluralistic ontology and theory reduction in the
physical sciences",
*The British Journal for the Philosophy of Science*, 39: 295-312. - Schaffner, K. 1976, "Reductionism in biology: Prospects and problems",
in R.S. Cohen,
*et al.*(eds),*PSA 1974*, pages 613-632. D. Reidel Publishing Company. - Sklar, L., 1967, "Types of inter-theoretic reduction",
*The British Journal for the Philosophy of Science*, 18: 109-124. - -------, 1993,
*Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics*. Cambridge University Press, Cambridge. - Wimsatt, W. C., 1976, "Reductive Explanation: A Functional Account", in A. C. Michalos, C. A. Hooker, G. Pearce, and R. S. Cohen, eds., PSA-1974 (Boston Studies in the Philosophy of Science, volume 30) Dordrecht: Reidel, pp. 671-710.

- Berry, M.V., and Howls, C.J., 1993, "Infinity Interpreted ,

*First published: January 2, 2001*

*Content last modified: January 2, 2001*