## 1. The Aims of Intertheoretic Reduction



Intertheoretic reductions play an outstanding role in physics, but the
aims of reduction are not always clear. Traditionally, perhaps due to
the influence of Ernest Nagel (1949, 1961, 1970), the reduction of a
theory \(T\_2\) to another \(T\_1\) was taken to amount to the
*explanation* of the reduced theory \(T\_2\) by the reducing
theory \(T\_1\). This idea is also present in some reductive programs
in physics. For instance, Rudolf Clausius, one of the first physicists
to formulate the Second Law of thermodynamics, believed that the
atomic theory might be helpful in explaining why entropy never
decreases for isolated systems (Klein 1973). Similarly, Ludwig
Boltzmann justified his own attempts to connect the Second Law of
Thermodynamics with the atomic theory as a way of giving a statistical
mechanical explanation of the Second Law of Thermodynamics (Boltzmann
1885; Klein 1973).



However, it would be a mistake to take explanation to be the only
epistemic function of intertheoretic reduction. Nickles (1973), for
instance, pointed out that intertheoretic reductions can often serve
to *justify* the success of the reduced theory. According to
Fletcher (2019), this is particularly important in cases in which the
reduction of a theory to another does not seek to eliminate the
reduced theory, but rather to retain it as a useful device for making
predictions. For instance, the reduction of Newtonian gravitation to
general relativity may be understood as giving a justification for why
Newtonian gravitation was so successful in the past; at the same time,
this reductive relation may be used to justify the further use of
Newtonian theory as a convenient tool for making predictions under
certain conditions. Torretti (1990) points out that in order to
legitimize the use of the reduced theory, there needs to be a
structural relationship between the reduced and reducing theories,
which assures that the reduced theory will be successful in all
relevant cases. It will be seen below
 (Section 2.3)
 that this requirement imposes an important restriction upon the
models of reduction that will be suitable to describe successful
reductions in physics.



Nickles (1973) also stresses that the previous success of the reduced
theory generally imposes constraints on the physical variables and
their mathematical relations in potential successor or reducing
theories. This means that the reduced theory can also work as a
*heuristic guide* in the development of the reducing theory.
Crowther (2018) has emphasized this epistemic role in the context of
the reduction of general relativity and quantum field theory to
quantum gravity. Feintzeig (2022) has also emphasized this role in the
alleged reduction of classical mechanics to quantum mechanics.



Apart from helping in the construction of the reducing theory, the
reduction of a theory to another is also said to play a role in the
*acceptance* or *consolidation* of the new theory
(Lakatos 1974 [1978]; Crowther 2018; Palacios 2022). Allegedly, the
ability of the reducing theory to recover the successful predictions
of the old (reduced) theory should be taken as an important argument
for the consolidation of the new (reducing) theory. An example of this
is the recovery of the predictions of optical theory by
Maxwell's electromagnetic theory of light, which contributed to
the acceptance of Maxwell's theory itself (see Worrall 1989 for
details of this reduction). Dizadji-Bahmani, Frigg, and Hartmann
(2011) develop a Bayesian analysis to show how reduction impacts not
only on the consolidation, but also on the *confirmation* of the new
theory.



The reduction of a theory to another can also contribute to
*unification* in the traditional sense in philosophy of
science, for when one reduces a theory to another, one also unifies
the domains of the two theories (Fletcher 2019; van Riel 2011).



Apart from the epistemic roles mentioned above, reduction also aims to
support the relative *fundamentality* of the reducing theory
with respect to the reduced theory. Establishing fundamentality is a
crucial function of reduction, since it can account for its asymmetric
character. Loosely speaking, fundamentality can be understood in the
sense that the reducing theory has some "advantage" over
the reduced theory or it is in some sense "better" than
the reduced theory. In the philosophical literature, one can
distinguish at least three different ways in which a theory \(T\_1\) is
said to be more fundamental than the other \(T\_2\). 


1. Empirical power: \(T\_1\) accounts for every observation within
the scope of \(T\_2\), as well as for new observations outside
\(T\_2\)'s domain (Lakatos 1974 [1978]; Callender 1999; Nagel
1961),
2. Aesthetic or epistemic virtues: \(T\_1\) is "simpler",
"more elegant" or more "systematized" than
\(T\_2\) (Torretti 1990),
3. Ontological supremacy: \(T\_1\) has the right ontology or at least
an ontology that is closer to the truth than the ontology of \(T\_2\)
(Butterfield 2011a).



There is a large discussion on fundamentality in the metaphysics of
science: for a review of this literature, see McKenzie (2019) and
Morganti (2020a, 2020b).



Although the above are very important roles of reduction, the list is
not exhaustive. Crowther (2020), for instance, refers to other
important roles that reduction may play in scientific theories.


## 2. Models of Intertheoretic Reduction



In the philosophy of science, different models have been suggested,
which aim to fulfill the above mentioned roles. In this section, I
will focus on the models that are more relevant for the reduction
between physical theories. For a more general discussion around
scientific reduction see the entry on
 scientific reduction.


### 2.1 Nagelian model of reduction



In the 1940s, Nagel (1949) famously argued that all reductions in
science could be described in terms of logical deductions. These ideas
were then developed in his 1961 seminal book "The Structure of
Science". For Nagel, there is a general way of characterizing
the main components and the logical structure of the different types
of reduction in science. According to him, every reduction can be
constructed as a series of statements, in which one of them, i.e., the
reduced theory, is the conclusion while the others, i.e., the reducing
theory and auxiliary assumptions, are the premises. In the case of
*homogeneous reductions*, in which all the technical terms in
the reduced theory \(T\_2\) are present in the reducing theory \(T\_1\),
the formal structure is straightforwardly that of a deductive
argument. In the case of *inhomogeneous reductions*, in which
at least one term in the reduced theory does not occur in the reducing
theory, the reducing theory needs to be supplemented by "rules
of correspondence" or "bridge laws" (BL), which
establish connections between the terms of the reducing theory and the
terms of the reduced theory. Once the reducing theory has been
adequately supplemented by bridge laws, inhomogeneous reductions, like
homogeneous reductions, embody the pattern of deductive arguments.
This idea can also be put in terms of *definitional extension*
of a theory (Butterfield 2011a). The core idea of Nagelian reduction
is that a theory \(T\_1\) reduces another theory \(T\_2\), if and only
if \(T\_2\) can be presented as a *definitional extension* of
\(T\_1\), which means that \(T\_2\) can be shown to be a sub-theory of
the augmented theory \(T\_1 \cup \BL\).



Thus, according to the Nagelian model, there are two main properties
that serve to give a general characterization of reductions in
science:


* Derivability: The laws of the reduced theory can be logically derived
from the laws of the (augmented) reducing theory plus auxiliary
assumptions.
* Connectability: In the case of inhomogeneous reductions, there are
bridge laws, which connect the vocabulary of the reducing theory with
the vocabulary of the reduced theory.



Nagel (1961) also pointed out that if the reduction is not trivial,
the bridge laws should constitute scientific hypotheses, which should
be susceptible of empirical confirmation or disconfirmation (see Nagel
1970; Dizadji-Bahmani et al. 2010; Schaffner 2012; Bangu 2011 for a detailed discussion on
the status of bridge laws).



It is important to note that Nagel's model of reduction has the
same structure as Hempel's deductive nomological (DN) model of
scientific explanation, according to which a law is explained by
another, if and only if it is a logical consequence of the other law
plus auxiliary assumptions (Hempel 1965) (for a discussion on the DN
model, see the entry
 scientific explanation).
 This general characterization of intertheoretic reductions in terms
of logical deduction has important virtues if we think of the goals of
intertheoretic reduction mentioned in the previous section. In fact,
the greatest advantage of understanding reduction in terms of logical
deduction is that logical deduction is truth-preserving. This means
that if \(T\_1\) and \(\BL\) as well as the auxiliary assumptions are
all true, then the reduced theory must also be true. In other words,
if we accept the validity of the new theory \(T\_1\), then on the basis
of the logical relation between \(T\_1 \cup \BL\) and \(T\_2\), we are
forced to accept the validity of \(T\_2\) under certain auxiliary
assumptions. Straightforwardly, this leads to the justification of the
use of \(T\_2\) as a useful device for making predictions and can also
lead to explain the success of \(T\_2\) from the perspective of
\(T\_1\). Furthermore, given that in most cases the reduced theory
\(T\_2\) has proven to be empirically successful, showing that \(T\_2\)
is a logical consequence of the augmented theory \(T\_1 \cup \BL\),
implies that in the augmented theory, one can recover all the
successful predictions of \(T\_2\). One can then use this structural
relation between \(T\_2\) and the augmented theory \(T\_1 \cup \BL\) as
an argument for the consolidation or acceptance of the new theory
\(T\_1\), which in general replaces the old theory \(T\_2\).



In spite of its advantages, the Nagelian model of reduction has been
challenged by a number of philosophers. In the next section, I will
address some of these criticisms.


### 2.2 Problems with the Nagelian model



One of the most important criticisms to the Nagelian model is that
even in the most paradigmatic cases of reduction in science, the
reduced theory cannot be strictly deduced from the reducing theory
plus bridge laws (Feyerabend 1962; Sklar 1967; Schaffner 1967, 2012).
Consider the example of the reduction of the Galilean law of a freely
falling body to the Newtonian theory, which is interpreted by Nagel (1961) as a
paradigmatic case of homogeneous reduction. It has been often pointed
out (e.g., Schaffner 1967; Sklar 1967; Torretti 1990) that even in
this case, the reduced theory cannot be derived from the reducing
theory, since the two theories are strictly speaking inconsistent. In
fact, while Galileo's law asserts that the acceleration of a
freely falling body near the earth's surface is constant,
Newtonian theory implies that acceleration varies with the distance
between the falling body and the earth's center of mass. This
means that, at most, what can be derived from the reducing theory is
an approximation to the reduced theory, and not the reduced theory
itself. In the 1970s, Nagel (1970) explicitly recognized the use of
approximations in intertheoretic reductions and argued that
approximations are not incompatible with his model, since they can
take part in the auxiliary assumptions needed to derive the reduced
theory (see van Riel 2011
and Sarkar 2015 for a more detailed discussion of the problem of
approximations in the Nagelian model).



Feyerabend (1962) also criticized the model by pointing out
contentious issues associated not only with the derivability
condition, but also with the connectability condition. His criticism
of the connectability condition came from his
"incommensurability thesis", according to which all
scientific vocabulary, including observational terms, is globally
"infected" by the theory in which it functions. However,
Nagel (1970) replied to these objections with an incisive criticism to
the incommensurability thesis.



Another criticism to the model is that it relies on a syntactic view
of theories. Moulines (2006) argues, for
instance, that a problem of the Nagelian model is that it conceives
reduction as a relationship between sets of statements. Instead,
he--together with collaborators--developed an approach to
reduction known as the "Structuralist Approach" (Balzer,
Moulines and Sneed 1987), which was compatible with the semantic view
of theories (see
 Section 2.7).
 A further issue that has been pointed out is that the fact that there
are no restrictions for the auxiliary assumptions makes this model too
liberal (Sarkar 1992). Another critique to the Nagelian model is that
the status of bridge laws is not clear (Sklar 1993). This is the question about
what kinds of statements bridge laws are. Although Nagel (1961)
deliberately allows for different interpretations of bridge law
statements, the problem of specifying the type of statements that can
qualified as a bridge law became important to cope with the multiple
realizability argument, to which we now turn.



One of the most important critiques to the Nagelian model has to do
with the multiple realizability thesis, which is the possibility that
the same property can be instantiated or realized by different types
of systems (Putnam 1967). The most common example of multiple
realizability is that of pain. Allegedly, pain is multiply realizable,
since it can be instantiated, for instance, in human's and
octopus' brains. For some philosophers, multiple realizability
poses a problem for Nagelian reduction, since it challenges the idea
of constructing bridge laws that identify a specific property in the
reduced theory with a corresponding property in the reducing theory
(Fodor 1974, 1997; Kitcher 1984). Although this objection was raised
firstly in the context of philosophy of mind, philosophers of physics
have also addressed a similar concern by looking at specific case
studies in physics. Sklar (1993: 352), for instance, argues that
temperature is multiply realizable and that this poses a problem for
the Nagelian reduction of thermodynamics to statistical mechanics.
More recently, Batterman (2000, 2002, 2018, 2021) has offered a
similar argument in the context of *universality*, which is the
common behavior observed among different systems when they undergo a
critical phase transition (see
 Section 2.6
 for more details about this example). According to him, a reductive
explanation of universality (which should be understood as multiple
realizability) fails, because it does not allow us to answer the
question about why different systems behave in the same way close to a
phase transition. Franklin (2019) replies to this objection by
defending a reductive explanation of universality. More generally,
Dizadji-Bahmani et al. (2010) and Butterfield (2011a) reply to the
objection of multiple realizability arguing that it does not pose a
problem for Nagelian reduction.



Although some philosophers of science have offered a recent defense of
the Nagelian model trying to deal with many of the aforementioned
problems (e.g., Butterfield 2011a; Dizadji-Bahmani et al. 2010; van
Riel 2011; Sarkar 2015), the issues that were raised soon after Nagel
suggested his model motivated the introduction of alternative models
of reduction and also modifications of the original Nagelian model. We
will discuss some of these models in the next sections.


### 2.3 Kemeny and Oppenheim's model



In order to offer a solution to the issues related with the
derivability condition in Nagel's model, Kemeny and Oppenheim
(1956) suggested an alternative general model of reduction, by
relinquishing the idea of linking the structures of the reduced and
the reducing theories. In their model, intertheoretic reduction is
defined indirectly relative to a set of observational data. More
precisely, the model proposes that a theory \(T\_2\) is reduced by
means of \(T\_1\) relative to observational data \(O\), iff (i) the
vocabulary of \(T\_2\) contains terms that are not part of the
vocabulary of \(T\_1\), (ii) any part of \(O\) explainable by means of
\(T\_2\) is explainable by \(T\_1\), and (iii) \(T\_1\) has a greater
systematic power than \(T\_2\), which means that \(T\_1\) has the
ability to predict as wide a range of phenomena as possible from as
little data as possible.



The main problem of this model is that it seems too weak to account
for many cases of reduction in science (Sklar 1967; Torretti 1990).
The weakness of this model is due to the fact that it does not impose
any structural relationship between \(T\_1\) and \(T\_2\). Indeed, it
only requires that the two theories make the same observational
predictions within the range of phenomena covered by \(T\_2\). The
problem is that without positing a structural link between \(T\_1\) and
\(T\_2\), the reduction does not allow one to justify the success of
\(T\_2\) on the basis of \(T\_1\). This means that one cannot use the
reduction of \(T\_2\) to \(T\_1\) to justify the use of \(T\_2\) under
certain conditions. Furthermore, without a structural relationship
between \(T\_1\) and \(T\_2\), one cannot explain \(T\_2\) on the basis
of \(T\_1\) and one cannot argue for the ontological fundamentality of
\(T\_1\) over \(T\_2\). It seems therefore that the only epistemic goals
of the Kemeny and Oppenheim's model can be (i) to help
consolidate \(T\_1\), by showing that this new theory makes the same
(or similar) predictions as the older theory \(T\_2\), and (ii) to
support the relative fundamentality of \(T\_1\) over \(T\_2\) in the
epistemic sense that \(T\_1\) has greater systematic power than
\(T\_2\). Note that cases of "reduction" that only lead to
the consolidation of the reducing theory are cases in which the
reduced theory is in fact replaced by the reducing theory and the
reduced theory is overthrown. An example is the reduction of the
phlogiston theory by the oxygen theory of combustion or the caloric
theory by the energetic theory of heat.


### 2.4 Generalized Nagel-Schaffner model



Schaffner (1967) suggested a revision of the Nagelian model of
reduction, which was supposed to give a more general characterization
of intertheoretic reductions than the original model, and also was
supposed to cope with some of the problems associated to the original
formulation of the Nagelian model. This revised model says that
reduction occurs *if and only if*:


1. All the primitive terms \(q\_1,\ldots,q\_n\) appearing in the
*corrected* reduced theory \(T\_2^\*\) appear in the reducing
theory \(T\_1\) (in the case of homogeneous reductions), or are
associated with one or more of \(T\_1\)'s terms with the help of
reduction functions, i.e., bridge laws.
2. \(T\_2^\*\) is derivable from \(T\_1\), when \(T\_1\) is conjoined with
bridge laws.
3. \(T\_2^\*\) corrects \(T\_2\) in the sense that it provides more accurate
experimentally verifiable predictions than \(T\_2\) in almost all
cases, indicates why \(T\_2\) was incorrect (e.g., crucial variables
are ignored), and also elucidates why \(T\_2\) worked as well as it
did.
4. \(T\_2\) should be explicable by \(T\_1\) in the sense that \(T\_1\)
yields \(T\_2^\*\) as a deductive consequence.
5. The relations between \(T\_2\) and \(T\_2^\*\) should be one of
*strong analogy*, which means that \(T\_2^\*\) bears a *close
similarity* to \(T\_2\), or produces numerical predictions which
are "very close" to \(T\_2\)'s.



Schaffner's contention was that this model should be thought of
as a "general reduction paradigm" of intertheoretic
reductions in science. He also tries to show that other approaches to
reduction, such as the Nagelian model and the Kemeny and
Oppenheim's model were just special cases of this generalized
model. In Schaffner (1977, 2012), he reiterates this idea by proposing
an even more general model, which he baptized as "The general
reduction replacement model". More recently, Dizadji-Bahmani et
al. (2010) argue that a slightly revised version of this model, which
they dubbed as "the Generalized Nagel-Schaffner Model"
(GNS), gives us the right analysis of intertheoretic reductions. An
important difference between the GNS model and Schaffner's
original proposal is that the GNS model (like Nagel's model)
emphasizes the role of auxiliary hypotheses, which do not play an
important role in the original formulation of Schaffner's model.
According to the GNS model, reduction is the deductive subsumption of
a corrected version of \(T\_2\), i.e., \(T\_2^\*\), under \(T\_1\), where
the deduction involves (i) first deriving a restricted version,
\(T\_1^\*\), of the reducing theory \(T\_1\) by introducing boundary
conditions and auxiliary assumptions, and then (ii) using bridge laws
to obtain \(T\_2^\*\) (Dizadji-Bahmani et al. 2010: 398).



An important example of the GNS reduction, interpreted as a
paradigmatic example of reduction by Schaffner himself, is the
reduction of physical optics to Maxwell's electromagnetic
theory. In fact, in order to achieve the reduction of physical optics
\(T\_2\) to the electromagnetic theory \(T\_1\), Sommerfeld (1950
[1954]) modified the laws of physical optics. In particular, on the
basis of Maxwell's equations, he modified Fresnel's famous
sine and tangent laws of the ratio of the relevant amplitudes of
incident, reflected, and refracted polarized light. In other words, he
constructed a modified version \(T\_2^\*\) of the original secondary
theory \(T\_2\), which was approximately derived from a low-level
theory \(T\_1\), which in this case corresponds to Maxwell's
electromagnetic theory. This modified version of Fresnel's laws
can be regarded as a close analogue of the original laws of physical
optics, in the sense that it produces predictions that are very close
to the predictions of physical optics. Dizadji-Bahmani et al. (2010)
also take the reduction of the Second Law of Thermodynamics to
statistical mechanics to be an instance of the GNS model. However,
Valente (2021) has questioned this alleged GNS reduction by
distinguishing between the reduction of the Second Law and the Minus
First Law (see Callender 1999; Frigg 2008; Robertson 2022; Sklar 1993; and Uffink
2001, for a philosophical discussion around the reduction of the
Second Law, see also the entry on
 philosophy of statistical mechanics).



Like the original Nagelian model, the GNS model has been the target of
criticisms. Although the model aims to offer a solution for the
problem of approximations, it does not necessarily cope with other
problems associated to the Nagelian model, such as the problem of
multiple realizability (see
 section 2.2).
 A specific problem of this model is that the meaning of strong
analogy appears to be too vague to allow us give a precise
characterization of reduction (Sarkar 1998: 173). Dizadji-Bahmani et
al. (2010) address this and other problems of the GNS model and try to
offer a compelling reply to each of them.



Independently of whether the above-mentioned problems can be
successfully solved or not for some particular case studies, the
question remains about the generality of the GNS model or possible
revised versions of it. Although some authors have interpreted this
model as a general model of reduction in physics (Dizadji-Bahmani et
al. 2010), others have defended a more pluralistic approach to
reduction (Nickles 1973; Torretti 1990; Palacios 2023)


### 2.5 Limiting reduction



In contrast to Nagel (1961) and Schaffner (1974), Nickles (1973) rejects the
idea that there is a single model of reduction that is capable to
capture the logical structure of reductions in science.



In his analysis, Nickles distinguishes between two different models of
reduction attached to different scientific functions or purposes:
reduction1, which corresponds to the Nagelian model, and
reduction2, which consists in the recovery of one theory
from another by applying a set of limiting operations or other
appropriate transformations. According to his view, whereas
reduction1 amounts to the *explanation* of a theory
by another and leads to *ontological economy*,
reduction2 has a *heuristic* and
*justificatory* role (heuristic in the sense that it helps in
the construction of the new theory and justificatory in the sense that
it helps explaining why the previous theory was successful in the
past). For him, the different models are also associated to different
types of reduction. In fact, reduction1 serves to account
for "domain-combining reductions", in which there are two
different theories describing the same range of phenomena at different
levels of description. On the other hand, reduction2 serves
to account for "domain-preserving" reductions, in which
one theory is a successor of the other. Domain-combining and
domain-preserving have been sometimes understood in terms of
"synchronic" and "diachronic reductions"
respectively (Dizadji-Bahmani et al. 2010)



Since the Nagelian model has been discussed in the previous sections,
I will focus now on Nickles' notion of reduction2.
Nickles' (1973: 197) reduction2 can be characterized as
follows:



* **reduction2** Let \(O\_i\) be a set of
intertheoretic operations, then a theory \(T\_2\) reduces2
another \(T\_1\) iff \(O\_i(T\_1) \to T\_2\), where the arrow represents
"mathematical derivation" understood in a broad sense
including not only logical deduction but also limiting operations and
approximations of many kinds.




It it important to note that, for Nickles, in reduction2
one says that the *successor theory* \(T\_1\) reduces to the
*previous theory* \(T\_2\) under certain mathematical
operations. This inversion in the direction of the reduction with
respect to the philosophical terminology, in which generally it is the
previous theory the one that reduces to the successor theory, is
motivated by the way in which physicists talk about reduction. This is
the reason why this model is sometimes named as the
"physicist's concept of reduction" (e.g., Batterman
2002). However, since this difference in the direction of reduction is
not important to understand this approach to reduction, from now on I
will stick to the philosophers' jargon.



Nickles himself recognizes that in order for reduction2 to
hold, it is necessary that the mathematical operations performed on
\(T\_1\) make physical sense. Although he is not explicit about what he
means by "physical sense", one can interpret this
constraint as signifying that after applying a set of mathematical
operations on \(T\_1\), the resulting (limit) theory can still describe
realistic behavior (Butterfield 2011a, 2011b; Palacios 2018, 2019,
2020). For example, taking the limit of a constant of nature to zero
may result in a (limit) theory that does not account for realistic
behavior unless this limit is adequately explained. Similarly taking
the limit of a parameter such as temperature or the number of
particles in a system to infinity may also be illegitimate if these
limits are not adequately justified.



A special case of reduction2 is *limiting
reduction*, which refers to cases in which the transformations
correspond to mathematical limits. Based on Nickles' account,
Palacios (2019: 625) defines limiting reduction as follows:



* **Limiting reduction2** Let \(Q^1\) denote a relevant quantity of
\(T\_1\), \(Q^2\) a relevant quantity of \(T\_2\), then a quantity
\(Q^2\) of \(T\_2\) *limiting reduces* to a corresponding
quantity \(Q^1\) of \(T\_1\) iff (i) \(\lim\_{x \to \infty} Q\_x^1 =
Q^2\) (where \(x\) represents a parameter appearing in \(T\_1\)) and
(ii) the limiting operation makes physical sense.




A prototypical example of limiting reduction is the reduction of the
classical equations of motion to relativistic equations in the
Newtonian limit. As it is well known, in the theory of special
relativity physical quantities such as the energy and momentum of a
moving body with a rest mass \(m\_0\) can be expressed in terms of the
so-called Lorentz factor \(\gamma\): 

 
\[
\begin{align}
E & =\gamma m\_0c^2\\
p &=\gamma m\_0v\\
\end{align}
\]


Arguably, when the Lorentz factor goes to 1, one can recover the
values of the classical counterparts of these quantities defined,
respectively, as \(E=m\_0c^2\) and \(p=m\_0v\). Since the Lorentz factor
takes the form 

 
\[
\gamma (v) = \frac{1}{\sqrt{1-(v/c)^2}},
\]


one can make this expression go to 1 in the limit \((v/c)^2
\longrightarrow 0\), where \(c\) is the speed of light and \(v\) the
velocity of a moving body in a given inertial frame. Although there
are different ways of interpreting this limit (Fletcher 2019), the most common
interpretation is as an approximation of the velocity of bodies that
are moving slowly compared to the speed of light (Palacios & Valente 2021). This can be seen
more clearly by noticing that the Lorentz factor \(\gamma\) can be
expanded into a Taylor series:

 
\[
\begin{aligned}
\gamma (v) &= \frac{1}{\sqrt{1-(v/c)^2}} \\
&= \sum\_{n=0}^\infty \left( \frac{v}{c} \right)^{2n} \prod\_{k=1}^n \left(
\frac{2k-1}{2k} \right) \\
& = 1 + \frac{1}{2} \left( \frac{v}{c} \right)^2
+ \frac{3}{8} \left( \frac{v}{c} \right)^4 +
\frac{5}{16} \left( \frac{v}{c} \right)^6 + \cdots\\
\end{aligned}\]


If one considers only the first term of this Taylor expansion, one
will recover the exact values of the classical quantities of interest,
and this will constitute just an approximation of the velocity \(v\)
of the objects for which \(v \ll c\). Taking more terms into account
will give results that are more accurate from the relativistic point
of view, but that depart from the classical values. According to
Batterman (1995, 2002) this is a prototypical case of a
"regular" limit, which is different from a
"singular" limit, which will be discussed in
 Section 2.7.
 Norton (2012) also proposes a way of making precise a distinction
between *idealizations* and *approximations*, which
allegedly reflects common usage. According to him, idealizations
involve the appeal to fictional systems, whereas approximations do not
necessarily involve the existence of such systems. He then argues that
cases like the Newtonian limit are idealizations that can be trivially
demoted to approximations. According to him, idealizations that can be
demoted to approximations, or approximations that do not constitute
idealizations, do not pose a problem for intertheory reduction (see
also Palacios & Valente 2021 and Frigg 2022: Ch. 11 for a
discussion of this topic). Fletcher (2019) analyses in detail the
limiting reduction of general relativity to Newtonian mechanics, and also
explains how matter, energy and spacetime geometries differ from
relativistic and classical spacetimes.



If we understand limiting reduction as a reduction between functions
or equations representing physical quantities, then it is clear that
this type of reduction can be combined with other models of reduction,
such as the Nagelian or the GNS model, in order to achieve the
reduction of a theory to another. Nickles (1973) recognizes this and
explicitly says that in some cases limiting reduction may help clarify
the analogous relationship between \(T\_2\) and \(T\_2^\*\) in the
Schaffner model:



> 
> 
> There is no reason to think that reductions2 can in all
> cases take over the work of Schaffner's analogy relation, but
> when it can we shall have \(T\_1\) reducing1 \(T\_2\)
> approximatively by reducing1 \(T\_2^\*\), which in turn
> reduces2 to \(T\_2\). (Nickles 1973: 195)
> 
> 
> 



Palacios (2019, 2022) further elaborates this idea and argues that in
many cases, reduction1 and limiting reduction (or
reduction2) may be combined to achieve important epistemic
goals of reduction. She takes the case of the reduction of the
thermodynamics of phase transitions to statistical mechanics to be a
paradigmatic case in which the Nagelian model and limiting reduction
are successfully combined (see
 Section 2.6).


### 2.6 The Problem of Singular Limits



In physics, Michael Berry (2002) has stressed the importance of
"singular limits", which he interprets as limits that
connect physical theories involving concepts that are qualitatively
very different. Although Berry suggests that singular limits are
compatible with reduction, Batterman (e.g., 1995, 2002, 2005,
2011) and Rueger
(2000a, 2000b, 2004), among
others, have argued that limiting reduction, and more generally,
intertheoretic reduction, fails in cases involving such limits. More
precisely, Batterman (2002: 18-19) defines singular limits as
cases in which the behavior in the limit is of a *fundamentally
different character* than the nearby solutions one obtains
"on the way to the limit", that is, as \(\epsilon \to
\infty\) (or 0, depending on the limit that we are considering), where
\(\epsilon\) is a fundamental parameter. In other words, these are
cases in which the following schema--called the
"R-schema"--fails: 

 
\[
\lim\_{\epsilon \to \infty} T\_1 = T\_2.
\]


Cases in which the solution of the relevant equations in \(T\_1\)
smoothly approach the solutions of the corresponding equations in
\(T\_2\) are cases in which the R-Schema holds. For Batterman, these
cases should be interpreted as *regular limits* and they do not
pose a problem for reduction. On the other hand, he argues that cases
of singular limits are incompatible with limiting reduction
 (Section 2.5).



In this discussion, the use of the thermodynamic limit in the case of
phase transitions has been taken as one of the most paradigmatic cases
of singular limits. Phase transitions are sudden changes of a thermal
system from one state into another, occurring for instance when liquid
water changes to vapour due to an increase of temperature. According
to Batterman (2002, 2009, 2011), the thermodynamic limit used in the theory of phase
transitions is a case of a singular limit. The reason can be briefly
described as follows. According to thermodynamics, which deals with
the behaviour of thermal systems from a macroscopic point of view,
phase transitions occur when the function representing the derivatives
of the free energy is discontinuous. Such a discontinuity matches with
the observed data, since sudden transitions from one phase to another
appear to take place abruptly. Now, when attempting to recover the
same phenomena within statistical mechanics, which describes thermal
systems at the microscopic level as being composed by a very large
number \(N\) of molecules, one faces a mathematical impossibility;
namely, if \(N\) is finite, the function representing the derivative
of the free energy remains continuous no matter how large \(N\) is.
Instead, one can recover the sought-after discontinuity by taking the
thermodynamic limit, which prescribes that both the number of
molecules \(N\) and the volume \(V\) of the system go to infinity
while its density remains fixed (Goldenfeld 1992; Kadanoff 2009).



Bangu (2019) offers a recent defense of Batterman's position
based on a distinction between data and phenomena. On the other hand,
many philosophers have replied to Batterman's argument by giving
a deflationary interpretation of the thermodynamic limit in the case
of phase transitions, according to which phase transitions are a
successful case of intertheory reduction. This deflationary
interpretation mostly consists of showing, by appealing to toy
examples or computer simulations, that approximately the same behavior
that can be observed at the limit can also be observed on the way to
the limit (e.g., Ardourel 2018; Callender 2001; Butterfield 2011b;
Butterfield & Bouatta, 2012; Lavis et al. 2021; Feintzeig 2019; Kadanoff 2013; Norton
2012, 2014; Palacios 2019, 2022; Palacios & Valente 2021; Wu
2021). Shech (2013) also
discusses this problem and offers a different perspective on the issue
around the use of infinite limits in phase transitions.



Related to the case of continuous phase transitions, some
philosophers, most notably Batterman (2002, 2011) and Morrison (2012,
2015), have argued that the use of renormalization group methods in
the explanation of universal behavior is another instance of singular
limits. In a nutshell, it was observed that close to a critical phase
transition, disparate systems, as different as fluids and magnets,
display universal behavior, which means that they have the same
quantitative and qualitative behavior. According to Batterman and
Morrison, the explanation of universality on the basis of
renormalization group methods requires invoking the thermodynamic
limit. The reason is that this framework relies on finding non-trivial
fixed points, which can only be well-defined in the limit. Although
these arguments are persuasive, some philosophers have argued against
this conclusion by showing that universal behavior can also be
explained on the basis of finite systems (Franklin 2018; Palacios
2019, 2022; Saatsi & Reutlinger 2018; Wu 2021). Very recently, Ardourel and Bangu (2023)
have offered what they take to be a more nuanced perspective on this
matter. Another argument for the irreducibility of universal behavior
is associated with the problem of multiple realizability, which was
discussed in
 Section 2.5.



Apart from the case of classical phase transitions, other example that
has been extensively analyzed in the debate on singular limits is the
use of the thermodynamic limit in quantum phase transitions. In this
discussion, authors such as Liu and Emch (2005) and Ruetsche (2011)
argue that taking the thermodynamical limit is indispensable for a
quantum mechanical description of the phenomena; on the other hand,
other authors, such as Landsman (2013) and Fraser (2016), have
defended a deflationary view towards the limit.



Another putative example of a singular limit is the van der Pol
oscillator. Whereas Rueger (2000a, 2000b, 2004) argues that this a
case in which a singular limit is involved, Wayne (2012) gives a
deflationary interpretation of the limit. Finally, there is a similar
discussion on the use of infinite limits in the explanation of
symmetry-breaking phenomena. Some authors, most notably Morrison
(2012), have given arguments in the same direction as Batterman to
argue that the limits involved in spontaneous symmetry breaking are
*necessary*. On the other hand, Fraser (2016), Landsman
(2013), and Wallace (unpublished [OIR])
 have suggested that, although a deflationary explanation of the use
of the thermodynamic limit in the derivation of spontaneous symmetry
breaking is less straightforward than the one given in the case of
first-order phase transitions, such an explanation can be given.


### 2.7 The Structuralist Approach



A model of reduction that has been largely ignored in the
Anglo-American discussion is the model of Balzer, Moulines, and Sneed
(Balzer et al. 1987). This model is part of the so-called
"structuralist" program and is characterized by the use of
set-theoretical predicates. As such, it is associated with a semantic
view of theories, in which a theory is not viewed as a class of
statements or propositions, but rather as a class of models. In
addition, this approach has the advantage of giving a precise
reconstruction of intertheoretic relations and of offering a precise
notion of intertheoretic approximation.



According to the structuralist approach, to say that a domain \(A\) is
reducible to domain \(B\) is to say that theory \(T\_A\) has an
*ontological reductive link* to theory \(T\_B'\), where
"ontological reductive link" is a kind of intertheoretical
relation. Each theory is determined by a class of
models \(M\), each of which has a particular structure and satisfies
the same set of axioms. Even if there are different specific
formulations of reductive relations within the structuralist
framework, the common core of the structuralist program, can be
summarized as follows (Moulines 2006).



For kinds \(A\) and \(B\_1,\ldots,B\_n\), \(A\) is *reducible* to
\(B\_1,\ldots,B\_n\), only if there are theories \(T\) and \(T'\) such
that:


1. \(A\) appears as a basic domain within the models of \(T\);
2. analogously for the \(B\_i\)'s with respect to \(T'\);
3. the field of experience \(F\) subsumed by \(T\) (i.e., the field of
application of \(T\) we are interested in) is a subfield of the field
corresponding to \(T'\);
4. there is an *ontological reductive link* \(\phi\) from \(T\) to
\(T'\) relating A to \(B\_i\);
5. \(T\) is *nomologically reducible* to \(T'\).



According to this approach, there is an ontological reductive link
from \(A\) to \(B\_i\), if \(A\) "comes out of" \(B\_i\) by
successively applying set theoretical operations, such as Cartesian
products. Nomological reducibility means in this framework that those
models of \(T\) that subsume the interesting field of experience \(F\)
are correlated (through the ontological reductive link) to *actual
models of specializations* of \(T'\) (structures satisfying the
fundamental laws of \(T'\)). In particular, models that subsume the
same field of experience as in \(T\).



In contrast to the original Nagelian model, this approach does not
require that all concepts of \(T\) are definable in terms of \(T'\).
Only the relationship between \(A\) and \(B\_i\) matters and only in
those models that subsume the field \(F\) we are interested in.



In addition, this model does not require semantic
predicate-by-predicate connections (in the sense of Nagel's
bridge laws), nor the deducibility of statements, although it is
compatible with them. In this sense, this model is weaker than the
Nagelian model of reduction. Another advantage of the model is that if
one introduces an appropriate topology, it can give a precise
characterization of approximative reductions, which cannot only serve
to account for cases of limiting reduction but also for other cases of
approximate reductions (Balzer et al. 1987; Moulines 1980, 1984).



One can see then that the structuralistic account of reduction opens
the possibility of discussing matters of strong analogy and
approximative reduction on a less informal level than other models of
reduction; because of that it has the potential of improving the
approximative Nagelian model and Nickles' model of
reduction2. The problem is then shifted to the task of
showing that some of the interesting cases of reduction fit into this
account. Although the model has led to detailed reconstructions of
particular examples of reductions in physics, such as the reduction of
rigid body mechanics to Newtonian particle mechanics (Sneed 1971) and Kepler's planetary
theory to Newtonian particle mechanics (Moulines 1980), it is not at
all clear whether all relevant reductions in physics can be
reconstructed in this way (Torretti 1990).



Another limitation of this model is that it may appear too weak,
since, at least in the original formulation, it did not specify the
sort of transformation \(\phi\) that relates the domains in the two
theories. Therefore, it allows for cases that are not generally
considered to be successful (ontological) reductions. For example, it
could account for cases in which there is an *ad hoc*
mathematical relationship between the domains of two theories that are
completely unrelated (Moulines 1984; Schaffner 1967). A possible
solution to this problem comes from specifying the sort of
relationship admissible in successful reductions. For instance,
Moulines (1984) has argued that in order to have a real reduction at the
ontological level, the sets of physical individuals (constituting the
domain of a theory) need to be in a biunivoque correspondence. In
other words, the ontological reductive link needs to be a one-to-one
function.



Another approach to reduction, closely related to the structuralist
approach, is the "New Wave" approach, championed mostly by
philosophers of mind, such as Churchland (1979, 1985, 1990) and Bickle
(1996, 1998). This approach to reduction is more liberal than the
Nagelian and structuralist models and allows for a larger variety of
relations between the reduced and reducing theories. The two main
criticisms to this approach are that (i) it fails to distinguish
between cases of replacement and genuine reduction (Endicott 1998,
2001) and (ii) that is not entirely new (Endicott 1998;
Dizadji-Bahmani et al. 2010).
For more details of these models, see the entry
 scientific reduction.


### 2.8 Contemporary Approaches



Nickles' concept of limiting reduction and the structuralist
model inspired contemporary topological approaches to reduction (e.g.,
Scheibe 1997; Fletcher
2016, 2020). The goal of these approaches is mainly to make precise
the notion of approximation involved in limiting relations. In other
words, to make explicit the sense in which the reduced theory
approximates the reducing theory. A similar approach has been
developed by Landsman (2007, 2017), who has focused mainly on the
reduction of the classical theory to quantum theories in the classical
limit \(\hbar \to 0\), by employing the tools of the \(C^\*\)-algebra
framework (see Feintzeig 2020, 2022 for a discussion of the
interpretation of the formalism in this approach).



The contemporary literature in philosophy of physics has also improved
our understanding of some basic aspects of intertheory reduction.
Contrary to many approaches to intertheory reduction in physics, which
take intertheory reduction to be the "global" derivation
of one theory from another, Rosaler (2015) argues that reductions in
physics normally have a "local" character. In this weaker
form of intertheory reduction, the reductive relation is not between
complete theories, but rather between models of a single system.
Furthermore, he has also argued that reductive relations should be
better interpreted as *a posteriori* relations (Rosaler 2019).
More precisely, he argues that reduction not only depends on the
structural relation between the theories to be compared, but also on
empirical facts about where the theories succeed at describing real
systems. Reduction should be understood, therefore, as an *a
posteriori* relation.



Also, in the last few years, more and more attention has been paid to
the functional approach to reduction in the philosophy of physics
(e.g., Baker 2021; Esfeld & Sachse 2011; Knox 2014 [OIR], 2019; Lam &
Wuthrich 2018, 2020; 
 Butterfield & Gomes 2022,
2023; Robertson 2022; Lorenzetti 2023).
 Roughly speaking, functional reduction describes inter-theoretic
reduction as a relationship between upper-level realizers and
lower-level realizers. For instance, finding a statistical mechanical
realizer that plays the functional role of thermodynamic temperature
or thermodynamic free energy allows for establishing a reductive
relation between thermodynamics and statistical mechanics. The
functional approach to reduction originally elaborated by Lewis'
(1972) and Kim (1984), is
usually not interpreted as an alternative to existing models of
reduction, but instead it is thought to be embedded in traditional
models of reduction, since it can play a role, for example, in
interpreting the bridge laws in the Nagelian model. Butterfield and
Gomes (2023) take functionalism to be an improvement of Nagelian
reduction. Lorenzetti (ms - see Other Internet Resources),
 on the other hand, distinguishes between two types of functional
reduction, one that elaborates on Lewis's account (1972) and is embedded within
Nagelian reductionism, and the other that adopts a semantic view and
is associated with the structuralist approach to reduction. Yet other
authors discussing functionalism in physics (Baker 2021; Knox 2019) have made the relationship between functionalism and reduction
less explicit.


## 3. Reduction, Supervenience and Emergence



Discussions around inter-theory reductions usually overlap with
discussions on supervenience and emergence. In fact, in the
philosophical literature, especially in the philosophy of mind,
supervenience has often been associated with nonreductive physicalism
(Hellman & Thompson 1975; Davidson 1970; List & Menzies 2009),
which denies the reduction of mental properties to physical
properties. Similarly, in philosophy of science, emergence has been
frequently associated to a failure of inter-theory reduction
(Batterman 2002; Rueger 2000a, 2004). This section will analyse the
extent to which these terms are intrinsically related, or should be
considered as logically independent.


### 3.1 Supervenience with or without reduction



Informally, supervenience means that there cannot be two events (or
theories) alike in all low-level properties without being identical in
all their high-level properties. Consequently, two events cannot
differ with respect to the high-level properties without differing in
some lower-level properties. Kim (1984) distinguishes between weak
supervenience, strong supervenience and global supervenience. He
defines weak supervenience as:



> 
> 
> \(A\) **weakly supervenes** on \(B\) if and only if
> necessarily for any \(x\) and \(y\) if \(x\) and \(y\) share all
> properties in \(B\) then \(x\) and \(y\) share all properties in
> \(A\); that is, indiscernibility with respect to \(B\) entails
> indiscernibility with respect to \(A\). (1984: 158)
> 
> 
> 



In contrast to weak supervenience, which holds in a particular world,
strong supervenience holds across worlds and is defined as
follows:



> 
> 
> \(A\) **strongly supervenes** on \(B\) just in case,
> necessarily, for each \(x\) and each property \(F\) in \(A\), if \(x\)
> has \(F\), then there is a property \(G\) in \(B\) such that \(x\) has
> \(G\), and necessarily if any \(y\) has \(G\), it has \(F\). (1984:
> 165)
> 
> 
> 



Finally, there is global supervenience, which he proves to be
equivalent to strong supervenience:



> 
> 
> \(A\) **globally supervenes** on \(B\) just in case
> worlds that are indiscernible with respect to \(B\)
> ("B-indiscernible", for short) are also
> \(A\)-indiscernible. (1984: 168)
> 
> 
> 



According to Kim (1990), not all these types of supervenience imply
reduction understood in the Nagelian sense. But if strong and global
supervenience hold, it may be possible to reduce the supervenient to
the subvenient due to strong connectability. On the other hand, if
strong supervenience fails it should not be possible to obtain a
Nagelian reduction between the two. Butterfield (2011a) develops this
idea further, showing that under certain assumptions, global
supervenience collapses into Nagelian reduction, understood as
definitional extension. This equivalence between global supervenience
and reduction rests on the validity of Beth's theorem (1959), which requires, among other
assumptions, that the language of the theories is first-order,
finitary (i.e., has finitely many predicates) and extensional. Hellman
and Thompson (1977), who also discuss Beth's theorem, question
the plausibility of these assumptions and give three examples of
supervenience without reduction. One of these examples concerns the
relationship between thermodynamics and statistical mechanics, which
they consider to be a case of supervenience without reduction. They
claim that one of the reasons for the failure of reduction (as
definitional extension) in this case is that the microscopic mechanics
lacks some properties of the thermodynamic theory, which means that
reduction is blocked.



More recently, List (2019) and Dewar (2019) have reanalyzed the relationship between
supervenience and reduction. List argues, in a similar vein as Hellman
and Thompson, that there can be supervenience without reduction.
Dewar, on the other hand, focuses on the topological constrains that
we need to impose to first-order theories in order to obtain
reducibility. For a more detailed discussion on the concept of
supervenience and its relationship with emergence and reduction, see
the entry on
 supervenience.


### 3.2 On the compatibility between emergence and reduction



Sometimes, emergence is associated to the failure of intertheoretic
reduction. In his *Devil in the Details* (Batterman 2002), for
instance, Batterman defines emergent behavior as 



> 
> 
> a result of the singular nature of the limiting relationship between
> the finer and coarser theories that are relevant to the phenomenon of
> interest. (2002: 121) 
> 
> 
> 



He interprets this notion of emergence in an ontological sense,
suggesting that the singular behavior captured by a singular limit
denotes real behavior, which is ontologically distinct from the
behavior that appears in finite systems. Later (e.g., Batterman 2011),
he seems to move towards an epistemological conception, in which
emergence is understood as an epistemic failure, meaning that certain
concepts of the phenomenological theory cannot be predicted or
explained in terms of the microscopic theory. Rueger (2000a, 2004)
defends a similar view, defining emergence as a failure of
intertheoretic reduction and in particular of limiting reduction.



These conceptions are in line with the traditional conception in
philosophy of mind that emergence is an alternative position with
respect to the reduction of mental states to physical states (Broad
1925; Davidson 1970;
Beckermann, Flohr & Kim 1992; Kim 1998).



However, it should be noted that there is a recent tendency among
philosophers of physics to think of emergence and reduction as being
compatible (e.g., Butterfield 2011a, 2011b; Butterfield & Bouatta
2012; de Haro 2019,
Crowther 2015; Norton 2014, Palacios 2022). Although this alleged
compatibility between emergence and reduction is not new and can be
traced back to Nagel (1961) and Wimsatt (1976), the systematic
analysis done by Butterfield (2011a, 2011b) led many contemporary
philosophers of physics to endorse this position. The compatibility
defended by Butterfield and endorsed soon after by Norton (2012),
among others, relies on specific definitions of emergence and
reduction. Butterfield and Norton, for instance, understand emergence
as a relationship between scales and argue that this conception of
emergence is compatible with intertheory reduction in the Nagelian
sense, which describes a relationship between different theories. They
focus on several examples, where the case of phase transitions is the
most notable one. Crowther (2015) analyses this compatibility for the
case of quantum gravity. Palacios (2022) focuses on the case of
critical phase transitions and defends a compatibility between two notions of
emergence, which she dubbed as "few-many" and
"coarse-grained" emergence. Here intertheory reduction is
understood as a combination between the Nagelian model and limiting
reduction.



It is important to point out, however, that this alleged compatibility
between emergence and reduction only holds for what philosophers know as *weak emergence*. In contrast to *strong
emergence*, which denies *microphysicalism*, weak emergence
does not deny microphysicalism, namely the thesis that all natural phenomena
are ultimately constituted and metaphysically determined by
fundamental physical entities. For a more detailed discussion on weak
and strong emergence, see the entry on
 emergent properties.


## 4. Reduction and Reductionism



So far, we have focused on intertheory reductions that may hold
between two theories or parts of theories. Reductionism, in contrast,
is the much stronger claim that ultimately all sciences are reducible
to one basic science, usually fundamental physics. I will now discuss
the status of this project in light of three contemporary discussions
in physics and the philosophy of physics.


### 4.1 The Theory of Everything



The theory of everything is the idea sometimes attributed to Einstein
(1940) and Hawking
(1984) that there will be
an ultimate theory of the universe constituting a set of equations
capable of describing all phenomena that have been observed, or that
will ever be observed (Gribbin 2009; Laughlin & Pines 2000; 't Hooft et al. 2005).
Even if this idea may sound appealing and pragmatically valuable, many
physicists have questioned the feasibility of this project. Laughlin
and Pines (2000), for instance, have argued that the equations of
conventional nonrelativistic quantum mechanics, which may be the best
candidate for the theory of everything, have not been solved exactly
when the number of particles exceeds around ten. According to them,
this problem is not associated to the present state of the art, since
not even a future super computer could carry out this calculation due
to a catastrophe dimension. They also point out that even for other
candidates for theories of everything, one would need to use
approximate calculations for deriving the behavior of larger systems.
They say that it is through such calculations that we have learned why
atoms, for instance, have the size they do and why chemical bonds have
the length and strength they do. In early 1970s, Philip Anderson
(1972), in his seminal paper "More is Different", made a
similar point that such calculations were computationally
intractable:



> 
> 
> Starting with the fundamental laws and a computer, we would have to do
> two impossible things--solve a problem with infinitely many
> bodies, and then apply the result to a finite system--before we
> synthesized this [macroscopic] behavior. (1972: 395)
> 
> 
> 



Barrow (2007), Hartle (2003), 't Hooft (2017) have recently discussed the status
of the theory of everything. Although Barrow and Hartle have a rather
negative view towards this project, 't Hooft offers a more
optimistic perspective, although by adopting a weaker interpretation
of what is meant by "theory of everything."



Importantly, the skepticism about the so called "Theory of
Everything" should neither be taken as undermining intertheory
reduction, which allows for approximations, nor reductionism, which is
the idea that all theories converge to the most fundamental one.
Anderson (1972) himself realizes this when he distinguishes between
"reductionism" and "constructionism". For him,
reductionism is "the ability to reduce everything to simple
fundamental [higher-energy/lower-level] laws" (1972: 393).
Constructionism, on the other hand, corresponds to "the ability
to start from those [fundamental] laws and reconstruct the
universe" (ibid). One can see then that it is the latter rather
than the former that is closer to the original idea of the theory of
everything. One of the most important theses in Anderson's paper
is that the "reductionist hypothesis does not by any means imply
a 'constructionist' one"(ibid). In fact, he believes
in the truth of reductionism, but denies constructionism. However,
there is an ambiguity in Anderson's paper on whether he argues
against constructionism in principle or in practice. In some passages,
he can be interpreted as defending a failure of constructionism in
principle, yet in others a failure in practice. Luu and Meissner
(2020), for instance, take him to argue for the latter, whereas Ellis
(2020) takes him to defend the former (For a detailed analysis on
Anderson's paper, see Mainwood 2006 and Humphreys 2015).



Interestingly, Steven Weinberg, who has sometimes been taken to defend
the theory of everything, seems to argue for reductionism and not
constructionism. In fact, in his book *Facing Up: Science and Its
Cultural Adversaries*, he explains his idea of reductionism as
follows:



> 
> 
> Relying on this intuitive idea that different scientific
> generalizations explain others, we have a sense of direction in
> science. These are arrows of scientific explanation, which thread
> through the space of all scientific generalizations. Having discovered
> many of these arrows, we can now look at the pattern that has emerged,
> and we noticed a remarkable thing: perhaps the greatest scientific
> discovery of all. These arrows seem to converge to a common source!
> Start anywhere in science and, like an unpleasant child, keep asking
> "why"? You will eventually arrive at the level of the very
> small. (Weinberg 2001: 17-18)
> 
> 
> 



Weinberg reminds us that in the 1920s the lowest level that was known
was quantum mechanics. By the 1970s was the quantum field theory of
particles and he had the hope that the next level would be a theory of
superstrings, which was not completed at that time, and is still not
complete. Importantly, for Weinberg the truth of the thesis of
reductionism does not depend on the extent of human knowledge or on
the current state of physics. This is because he endorses an
*ontological* conception of reductionism, which he calls
"objective reductionism", according to which reductionism
is a feature of the world itself. In the philosophy of physics
literature, this concept corresponds to
"microphysicalism", which, as mentioned above, is the
thesis that everything is ultimately constituted by the same
fundamental physical entities (whatever they are) (e.g., Bain 2013a;
Huttemann 2004; Palacios 2022).


### 4.2 Effective Field Theories



A contemporary research program, which has sometimes been taken as the
realization of the reductionist approach (Giudice 2008), is the
Effective Field Theories (EFTs) approach.



Huggett and Weingard (1995) define an "Effective Field
Theory" as a theory that "effectively captures everything
relevant [at a given scale]" (1995: 172). EFTs play an essential
role in many different areas of physics, ranging from particle physics
and condensed matter physics to inflationary cosmology. Moreover, it
is common among physicists to think of all our current physical
theories, including the Standard Model, as low-energy EFTs of some
unknown fundamental theory (Luu & Meissner 2020, 2021; Rivat &
Grinbaum 2020; Wells 2012)



Usually, EFTs are constructed through a process in which the degrees
of freedom are eliminated (or integrated out) from a high-energy or
short-length scale theory by using a technique called
"Renormalization group methods" (Bain 2013b; Georgi 1993;
Huggett & Weingard 1995). An example of this approach is the
construction of the low-energy theory describing the behavior of the
quantum-Hall liquid from a corresponding high-energy theory, which is
achieved by interacting out the electron degrees of freedom (Bain
2013a; Shech & McGivern 2021). Another approach consists of going
from low-energy EFTs to high-energy EFTs, by adding new terms
(coupling-constants) in order to account for high-energy effects
(details in Crowther 2015; Hartmann 2001; Manohar 2020). In this case,
one constructs an effective higher-energy theory on the basis of a
well-known lower-energy theory. A candidate example for this is the
construction of a high-energy theory on the basis of the standard
model (see Bain 2013a, Georgi 1993, Manohar 1997 for a more detailed
discussion of this example).



In practice, arriving at a lower-energy EFT from the high-energy EFT
is not a simple task, and external guidance such as experimental
results are needed in order to find an adequate transformation that
keeps the free energy and the relevant physics approximately invariant
(Bain 2013a, 2013b; Crowther 2015; Luu and Meissner 2020). So, *in practice*,
the construction of the low-energy theory from the high-energy theory
is not just a matter of *deducing* the former from the latter.
Instead, a phenomenological investigation of the system is often
required in order to assure that the transformation will maintain the
relevant physics.



This apparent lack of deducibility in EFTs has played an important
role in the discussion on reductionism. For instance, Cao and Schweber
(1993) take it as a motivation for defending a pluralistic and
antireductionist picture of the world, in which the world is supposed
to be arranged into an infinite hierarchy of autonomous domains, each
level having its own ontology and its own fundamental laws. This
anti-reductionist picture relies, on the one hand, on the idea that
the low-energy theory cannot be simply derived from the high-energy
theory without appealing to low-energy empirical information. On the
other hand, it rests on "the decoupling theorem," which is
a theorem that demonstrates that high-energy contributions have
negligible effect on low-energy regimes. A similar anti-reductionist
view has recently been defended by Ellis (2020).



Hartmann (2001) criticizes Cao and Schweber's line of reasoning,
by pointing out that the decoupling theorem is valid only under
restrictive conditions (for instance it requires the high-energy
theory to be perturbatively renormalizable). He also argues that, in
many cases, distinct EFTs can be actually related by means of deductive
relations and bridge laws, hence they can be understood as instances
of Nagelian reduction. Castellani (2002) makes a similar point,
arguing that even if not in practice, the deduction of lower-energy
EFTs from higher energy EFTs is possible *in principle*. In
fact, she says that we normally have to appeal to empirical input,
because we do not have a complete renormalizable theory at infinitely
short distances. If we had such a theory, we would be able to
construct lower-level EFTs at any distance in a totally systematic way
(see also Georgi 1993):



> 
> 
> The EFT schema, by allowing definite connections between the theory
> levels, provides an argument against the basic antireductionist claim
> of the scientists' debate. A reconstruction (the way up) is not
> excluded, even though it may have to be only in principle. In this
> sense, EFT does not represent a vindication of Anderson's 1972
> views, as has been claimed. (Castellani 2002: 15)
> 
> 
> 



Bain (2013a, 2013b) is more skeptical about the possibility of
deriving low-energy EFTs from higher energy EFTs without empirical
input. He points out that in realistic cases, the low-energy dynamical
equations and degrees of freedom are formally distinct from those of
the high-energy theory. For him the difference between the two
theories and their degrees of freedom is substantial enough that the
two theories generally admit different ontologies and cannot strictly
speaking be considered to be "a part" of one another.



Rivat and Grinbaum (2020) agree that usually there are important
formal differences between the lower-level and higher-level EFTs,
however they contend that these differences are not incompatible with
Nagelian reduction. In fact, they stress that Nagelian reduction does
not require the two theories to be formally identical and admits
approximations and even the appeal to empirical input in the form of
auxiliary assumptions:



> 
> 
> [T]he relation between two EFTs involve approximations and heuristic
> reasoning: e.g., performing a saddle point approximation when
> integrating out heavy fields or taking the zero mass limit of some
> light field. Bain claims that these features make the task of
> formulating the relation between two theories in terms of a
> mathematical derivation difficult, if not impossible. Here, again,
> successful Nagelian reductions are relatively permissive to the use of
> intermediary approximations, assumptions, and heuristic reasoning. The
> ultimate goal of a Nagelian reduction is to explain away the success
> of the low-energy theory by means of the high-energy theory, not to
> provide a strict derivation relying only on the resources of the
> high-energy theory. (Rivat & Grinbaum 2020: 13)
> 
> 
> 



Butterfield (2014) also considers the EFT approach to be compatible
with Nagelian reduction and interprets it as a family of Nagelian
reductions. Despite these arguments, there is currently no consensus
as to whether the EFT framework is compatible with intertheory
reduction or not. For instance, recently, there has been a heated
discussion between Luu and Meissner (2020, 2021), who take EFTs
to be compatible with reductionism, and Ellis (2020), who argues that
EFTs imply an anti-reductionist picture of the world.



A related discussion concerns the notion of *naturalness* in
quantum field theory. For a discussion around this topic, see, e.g.,
Franklin (2020), Wallace (2019) and Williams (2015).


### 4.3 Multiscale Modeling



Another philosophical discussion, which has proved relevant for
the general topic on reduction in physics is the debate on multiscale
models. According to some philosophers (e.g., Batterman 2013b, 2021; Batterman & Green
2021; Wilson 2017; Bursten 2018), the fact that scientific modeling
usually requires appealing to features and properties of different
mesoscale levels, undermines the view that the world is sharply
divided into distinct levels, in particular, the microscopic and the
macroscopic. Moreover, according to them this undermines the
reductonist idea that the higher (phenomenological) level can be
ultimately derived from the next lower (microscopic) level. McGivern
(2008) has also pointed out that multiscale models do not fulfill
Kim's (1998) minimal condition for intertheory reduction,
according to which all high-level properties must be identifiable with
distinct micro-level properties.



Multiscale models have proved to be very useful in modeling systems in
several disciplines, such as fluid mechanics, aerodynamics and
geophysics. Although philosophers involved in this debate succeed in
emphasizing the importance of mesoscale levels in a variety of
case-studies, such as nanoscale crack propagation in silicon and
hydrodynamic models, the extent to which the existence of multiscale
models undermines reductionism is still contentious. Franklin
(unpublished
 [OIR]),
 for instance, argues that the antireductionist interpretation of
multiscale models relies on a very narrow conception of reductionism
associated to a bottom-up methodology. According to him, if we focus
instead on *reductive explanations*, which account for the
stability of various features at larger scales and explain why
bottom-up methodological reductionism fails, we will conclude that the
existence of multiscale models does not undermine reductionism. A
similar argument is given by Franklin (2019) in the context of
renormalization group explanations. Although Franklin's
arguments are persuasive, it would be interesting to investigate
whether specific models of intertheoretic reduction, such as the
Nagelian model, can be made compatible with the concrete multiscale
models examined in the literature. Butterfield and Bouatta (2016)
argue, for instance, that renormalization group explanations can be
indeed interpreted as Nagelian reductions, but it remains to be seen
whether such an interpretation can be given to the other examples
examined in the discussion around multiscale models.