## 1. Other structuralisms



The term 'structuralism' is used with different meanings
and therefore it seems appropriate to mention other
'structuralisms' and to explain how 'structuralism
in physics' is related to them. If you check the entry 'structuralism (disambiguation)' in Wikipedia,
you will be informed that there is a spectrum of
'structuralisms' in 11 different areas, including:


* linguistics [F. de Saussure (1857-1913)],
* anthropology [C. Levi-Strauss (1908-2009)],
* mathematics [N. Bourbaki (1935-), collective
pseudonym],
* philosophy of science [J. D. Sneed (1938-), W.
Stegmuller (1923-1991)].



Here we have mentioned some prominent representatives in brackets. All
types of structuralism share a common conviction about the role
of structures in their respective disciplines, but at first glance
they show little similarity. Nevertheless, there are connections and
mutual influences between the different structuralisms. It goes beyond
the scope of this entry to examine these influences in more detail.
For the relations between anthropologic and mathematical structuralism
see Aubin (1997). As mentioned, we will understand
'structuralism in physics' as a special case of
'structuralism in philosophy of science'. There are close
connections with mathematical structuralism, which we will discuss in
more detail in the main part of this entry. To illustrate these
connections here we only mention the full title of Stegmuller
(1979a): The structuralist view of theories, A possible analogue of
the Bourbaki programme in physical science.



Right now, we are taking the interim balance that 'Structuralism
in Physics' is part of an intellectual movement mainly in the
20th century and, compared to other structuralisms, represents a
rather late contribution.


## 2. Common traits



The three programs mentioned in the preamble share the following
characteristics and convictions:


* A metatheory of science requires a kind of formalization different
from that already employed by scientific theories themselves.
* The structuralistic program yields a framework for the rational
reconstruction of particular theories.
* A central tool of formalization is Bourbaki's concept of
"species of structures", as described in Bourbaki
(1986).
* Among the significant features of theories to be described are:
	+ Mathematical structure
	+ Empirical claims of a theory
	+ Function of theoretical terms
	+ Role of approximation
	+ Evolution of theories
	+ Intertheoretic relations


## 3. The problem of theoretical terms



A physical theory \(T\) consists, among other things, of a group of
laws which are formulated in terms of certain concepts. But an
apparent circularity arises when one considers how the laws of \(T\)
and the concepts acquire their content, because each seems to acquire
content from the other -- the laws of \(T\) acquire their content
from the concepts used in the formulation of the laws, while the
concepts are often "introduced" or "defined"
by the group of laws as a whole. To be sure, if the concepts can be
introduced independently of the theory \(T\), the circularity does not
appear. But typically every physical theory \(T\) requires some new
concepts which cannot be defined without using \(T\) (we call the
latter "\(T\)-theoretical concepts"). Is the apparent
circularity concerning the laws and the T-theoretical concepts a
problem? Some examples will help us assess the threat.


### 3.1 An example



As an example, consider the theory \(T\) of classical particle
mechanics. For simplicity we will assume that kinematical concepts,
such as the positions of particles, their velocities and accelerations
are given independently of the theory as functions of time. A central
statement of \(T\) is Newton's second law, \(\bF= m\ba\), which
asserts that the sum \(\bF\) of the forces exerted upon a particle
equals its mass \(m\) multiplied by its acceleration \(\ba\).



While we customarily think of \(\bF=m\ba\) as an empirical assertion,
there is a real risk that it turns out merely to be a definition or
largely conventional in character. If we think of a force merely as
"that which generates acceleration" then the force \(\bF\)
is actually defined by the equation \(\bF=m\ba\). We have a particle
undergoing some given acceleration \(\ba\), then \(\bF=m\ba\) just
defines what \(\bF\) is. The law is not an empirically testable
assertation at all, since a force so defined cannot fail to satisfy
\(\bF=m\ba\). The problem gets worse if we define the (inertial) mass
\(m\) in the usual manner as the ratio \(|\bF|/|\ba\)|. For now we are
using the one equation \(\bF=m\ba\) to define two quantities \(\bF\)
and \(m\). A given acceleration \(\ba\) at best specifies the ratio
\(\bF/m\) but does not specify unique values for \(\bF\) and \(m\)
individually.



In more formal terms, the problem arises because we introduced force
\(\bF\) and mass \(m\) as \(T\)-theoretical terms that are not given
by other theories. That fact also supplies an escape from the problem.
We can add extra laws to the simple dynamics. For example, we might
require that all forces are gravitational and that the net force on
the mass \(m\) be given by the sum \(\bF=\Sigma\_i \bF\_i\) of all
gravitational forces \(\bF\_i\) acting on the mass due to the other
masses of the universe, in accord with Newton's inverse square
law of gravity. (The law asserts that the force \(\bF\_i\) due to
attracting mass \(i\) with gravitational mass \(m\_{gi}\) is \(Gm\_g
m\_{gi} \boldsymbol{r}\_i / r\_{i}^3\), where \(m\_g\) is the
gravitational mass of the original body, \(\boldsymbol{r}\_i\) the
position vector of mass \(i\) originating from the original body, and
\(G\) the universal constant of gravitation.) That gives us an
independent definition for \(\bF\). Similarly we can require that the
inertial mass \(m\) be equal to the gravitational mass \(m\_g\). Since
we now have independent access to each of the terms \(\bF\), \(m\) and
\(\ba\) appearing in \(\bF=m\ba\), whether the law obtains is
contingent and no longer a matter of definition.



Further problems can arise, however, because of another
\(T\)-theoretical term that is invoked implicitly when \(\bF=m\ba\) is
asserted. The accelerations \(\ba\) are tacitly assumed to be measured
in relation to an inertial system. If the acceleration is measured in
relation to a different reference system, a different result is
obtained. For example, if it is measured in relation to a system
moving with uniform acceleration \(\ba\), then the measured
acceleration will be \(\ba' = (\ba - \ba)\). A body not acted on by
gravitational forces in an inertial frame will obey \(0=m\ba\) so that
\(\ba=0\). The same body in the accelerated frame will have
acceleration \(\ba' = -\ba\) and be governed by \(-m\ba = m\ba'\). The
problem is that the term \(-m\ba\) behaves just like a gravitational
force; its magnitude is directly proportional to the mass \(m\) of the
body. So the case of a gravitation free body in a uniformly
accelerated reference system is indistinguishable from a body in free
fall in a homogeneous gravitational field. A theoretical
underdetermination threatens once again. Given just the motions how
are we to know which case is presented to
 us?[1]
 Resolving these problems requires a systematic study of the relations
between the various \(T\)-theoretical concepts, inertial mass,
gravitational mass, inertial force, gravitational force, inertial
systems and accelerated systems and how they figure in the relevant
laws of the theory \(T\).



Similar problems arise in the formulation of almost all fundamental
physical theories.


### 3.2 Structuralistic solutions of the problem of theoretical terms



There are various ways to cope with this problem. One could try to
unmask it as a pseudo-problem. Or one could try to accept the problem
as part of the usual way science works, albeit not in the clean manner
philosophers would like it. The structuralistic programs, however,
agree that this is a non-trivial problem to be solved and devise
meta-theoretical machinery to enable its solution. They further agree
on dividing the vocabulary of the theory \(T\) into \(T\)-theoretical
and \(T\)-non-theoretical terms, the latter being provided from
outside the theory.


#### 3.2.1 Sneed's solution



In the Sneedean approach the "empirical claim" of the
theory is formulated by using an existential quantifier for the
\(T\)-theoretical terms (i.e., in terms of the "Ramsey
sentence" for \(T)\). In our above example, Newton's law
for gravitational forces would be reformulated as: "There exist
an inertial system and constants \(G, m\_i, m\_{gi}\) such that for each
particle the product of its mass times its acceleration equals the sum
of the gravitational forces as given above." This removes the
circularity but leaves open the question of content. Here the
structuralists a la Sneed would argue that the empirical claim
of the theory \(T'\) has to contain all the laws of the theory as well
as higher-order laws, called "constraints". In our
example, the constraints would be statements such as "all
particles have the same inertial and gravitational masses and the
gravitational constant assumes the same value in all models of the
theory." The theory would thereby acquire more content and
become non-vacuous.


#### 3.2.2 Ludwig's solution



Although Ludwig's meta-theoretical framework is slightly
different, the first part of his solution is essentially equivalent to
the above one. On the other hand, he proposes a stronger program
("axiomatic basis of a physical theory") which proceeds by
considering an equivalent form \(T\)\* of a theory \(T\) in which all
\(T\)-theoretical concepts are eliminated by explicit definitions.
This seems to be at variance with older results about the
non-definability of theoretical terms, but a closer inspection removes
the apparent contradiction. For example, the concept of
"mass" may be non-definable in a theory dealing only with
single orbits of a mechanical system, but definable in a theory
containing all possible orbits of that system.



However, to formulate the axiomatic basis of a real theory, not just a
toy model, is a non-trivial task and typically requires one or two
books; see the examples Ludwig (1985, 1987) and Schmidt (1979).


### 3.3 The measurement problem



Both programs address the further problem of how to determine the
extension, e.g., the numerical values, of a theoretical term from a
given set of observational data. We will call this the
"measurement problem", not to be confounded with the
well-known measurement problem in quantum theory. Typically the
measurement problem has no unique solution. Rather the values of the
theoretical quantities can only be measured within a certain degree of
imprecision and using auxiliary assumptions which, although plausible,
are not confirmed with certainty. In the above Newton example one
would have to use the auxiliary assumption that the trajectories of
the particles are twice differentiable and that other forces except
the gravitational forces can be neglected. For a recent critical
examination of the solution to the measurement problem within
Sneed's approach with detailed examples from astronomy see
Gahde (2014).


### 3.4 Measurement and approximation



The feature of imprecision and approximation plays a prominent
role in the structuralistic programs. In the context of the
measurement problem, imprecision seems to be a defect of the theory
which impedes the exact determination of the theoretical quantities.
However, imprecision and non-uniqueness is crucial in the context of
evolution of theories and the transition to new and
"better" theories. Otherwise the new theory could in
general not encompass the successful applications of the old theory.
Consider for example the transition of Kepler's theory of
planetary motion to Newton's and Einstein's theories:
Newtonian gravitation theory and general relativity replace the Kepler
ellipses with more complicated curves. But these should still be
consistent with the old astronomical observations, which is only
possible if they don't fit exactly into Kepler's theory
.


## 4. Problems of reduction


### 4.1 Reduction relation between theories



Part of the structuralistic program is the definition of various
intertheoretic relations. Here we will concentrate on the relation(s)
of "reduction", which play an important role in the
philosophical discourse as well as in the work of the physicists,
albeit not under this name. Consider a theory \(T\) which is
superseded by a better theory \(T'\). One could use \(T'\) in order to
understand some of the successes and failures of \(T\). If there is
some systematic way of deriving \(T\) as an approximation within
\(T'\), then \(T\) is "reduced" to or by \(T'\). In this
case, \(T\) is successful where it is a good approximation to \(T'\)
and \(T'\) is successful. On the other hand, in situations where
\(T'\) is still successful but \(T\) is a poor approximation to
\(T'\), \(T\) will fail. For example, classical mechanics should be
obtained as the limiting case of relativistic mechanics for velocities
small compared with the velocity of light. This would explain why
classical mechanics was, and is still, successfully applied in the
case of small velocities but fails for large (relative)
velocities.



As mentioned, the investigation of such reduction relations between
different theories is part of the every-day work of theoretical
physicists, but usually they do not adopt a general concept of
reduction. Rather they intuitively decide what has to be shown or to
be calculated, depending on the case under consideration. Here the
work of the structuralists could lead to a more systematic approach
within physics, although there does not yet exist a generally
accepted, unique concept of reduction.


### 4.2 Reduction and incommensurability



Another aspect is the role of reduction within the global
picture of the development of physics. Most physicists, but not all,
tend to view their science as an enterprise which accumulates
knowledge in a continuous manner. For example, they would not say that
classical mechanics has been disproved by relativistic mechanics, but
that relativistic mechanics has partly clarified where classical
mechanics could be safely applied and where not. This view of the
development of physics has been challenged by some philosophers and
historians of science, especially by the writings of T. Kuhn and P.
Feyerabend. These scholars emphasize the conceptual discontinuity or
"incommensurability" between reduced theory \(T\) and
reducing theory \(T'\). The structuralistic accounts of reduction now
opens the possibility of discussing these matters on a less informal
level. The preliminary results of this discussion are different
depending on the particular program.


### 4.3 Ludwig's account



In the writings of Ludwig there is no direct reference to the
incommensurability thesis and the corresponding discussion. But
obviously his approach implies the most radical denial of this thesis.
His reduction relation is composed of two simpler intertheoretic
relations called "restriction" and
"embedding". They come in two versions, exact and
approximate. Part of their definitions are detailed rules of
translation of the non-theoretic vocabulary of \(T'\) into that of
\(T\). Hence commensurability, at least on the non-theoretical level,
is insured by definition. The problem is then shifted to the task of
showing that some of the interesting cases of reduction, which are
discussed in the context of incommensurability, fit into
Ludwig's definition. Unfortunately, he gives only one
extensively worked-out example of reduction, namely thermodynamics vs.
quantum statistical mechanics, in Ludwig (1987). Incommensurability of
theoretical terms could probably be more easily incorporated in
Ludwig's approach since it could be traced back to the
difference between the laws of \(T\) and \(T'\).


### 4.4 Sneed's account



The relation between incommensurability and the Sneedean reduction
relation is to some extent discussed in Balzer *et al*. (1987,
chapter VI.7). The authors consider an exact reduction relation as a
certain relation between potential models of the respective theories.
More interesting for physical real-life examples is the approximate
version which is obtained as a "blurred exact reduction"
by means of a subclass of an empirical uniformity on the classes of
potential models. The Kepler-Newton case is discussed as an example of
approximate reduction. The discussion of incommensurability suffers
from the notorious difficulties of explicating such notions as
"meaning preserving translation". There is an interesting
application of the interpolation theorem of meta-mathematics which
yields the result that, roughly speaking, (exact) reduction implies
translation. However, the relevance of this result is questioned in
Balzer *et al*. (1987, 312 ff). Thus the discussion eventually
ends up as inconclusive, but the authors admit the possibility of a
spectrum of incommensurabilities of different degrees in cases of
pairs of reduced/reducing theories.


### 4.5 Scheibe's account



Scheibe in his (1999) also explicitly refers to the theses of Kuhn and
Feyerabend and gives a detailed discussion. Unlike the other two
structuralistic programs, he does not propose a fixed concept of
reduction. Rather he suggests a lot of special reduction relations
which can be combined appropriately to connect two theories \(T\) and
\(T'\). Moreover, he proceeds by means of extensive real-life case
studies and considers new types of reduction relations if the case
under consideration cannot be described by the relations considered so
far. Scheibe concedes that there are instances of incommensurability
which make it difficult to find a reduction relation in certain cases.
As a significant example he mentions the notions of an
"observable" in quantum mechanics on the one hand, and in
classical statistical mechanics on the other hand. Although there are
maps between the respective sets of observables, Scheibe considers
this as a case of incommensurability, since these maps are not Lie
algebra homomorphisms, see Scheibe (1999, 174).



Summarizing, the structuralistic approaches are capable of discussing
the issues of reduction and incommensurability and the underlying
problems on an advanced level. Thereby these approaches have a chance
of mediating between disparate camps of physicists and
philosophers.


## 5. Three structuralist programs



In this section we will describe more closely the particular programs,
their roots and some of the differences between them.


### 5.1 Sneed's program


#### 5.1.1 History and general traits



This program has been the most successful with respect to the
formation of a "school" attracting scholars and students
who adopt the approach and work on its specific problems. Hence most
of the structuralistic literature concerns the Sneedean variant.
Perhaps this is partly also due to the circumstance that only
Sneed's approach is intended to apply (and has been applied) to
other sciences and not only physics.



A more comprehensive account of the historical roots of structuralism
in philosophy of science can be found in Bolinger (2016), although
this book is not yet translated into English. The seminal book was
Sneed (1971) which presented a meta-theory of physics in the
model-theoretical tradition connected with P. Suppes, B. C. van
Fraassen, and F. Suppe. This approach was adopted and popularized by
the German philosopher W. Stegmuller (1923-1991), see e.g.,
Stegmuller (1979b) and further developed mainly by his disciples.
In its early days the approach was called the "non-statement
view" of theories, emphasizing the role of set-theoretical
tools as opposed to linguistic analyses. Later this aspect was
considered to be more of practical importance than a matter of
principle, see Balzer *et al*. (1987, 306 ff). Recently, H.
Andreas (2014) and G. Schurz (2014) have proposed two slightly
different frameworks that reconcile semantical and syntactical
formulations of Sneed's program. Nevertheless, the almost
exclusive use of set-theoretic tools remains one of the characteristic
stylistic features of this program and one that distinguishes it
conspicuously from the other programs.


#### 5.1.2 Central notions of Sneed's program



According to Moulines, in Balzer and Moulines (1996, 12-13), the
specific notions of the Sneedean program are the following. We
illustrate these notions by simplified examples, inspired by Balzer
*et al*. (1987), which are based on a system of \(N\) classical
point particles coupled by springs satisfying Hooke's law. For a
recent introduction into the basic concepts see also H. Andreas and F.
Zenker (2014).


* \(M\_p\): A class of potential models (the theory's
conceptual framework.


> 
> [One potential model contains a set of particles, a set of springs
> together with their spring constants, the masses of the particles, as
> well as their positions and mutual forces as functions of time.]
>
* \(M\): A class of actual models (the theory's empirical
laws).


> 
> [\(M\) is the subclass of potential models satisfying the
> system's equation of motion. ]
>
* \(\langle M\_p,M\rangle\): A model-element (the absolutely
necessary portion of a theory)
* \(M\_{pp}\): A class of partial potential models (the
theory's relative non-theoretical basis).


> 
> [One partial potential model contains only the particles'
> positions as functions of time, since the masses and forces are
> considered as \(T\)-theoretical.]
>
* \(C\): A class of constraints (conditions connecting different
models of one and the same theory).


> 
> [The constraints say that the same particles have the same masses and
> the same springs have the same spring constants.]
>
* \(L\): A class of links (conditions connecting models of different
theories).


> 
> [Among the conceivable links are:
> 
> 
> 	+ Links to the theory of classical spacetime
> 	+ Links to the theory of weights and balances, where mass ratios can
> 	be measured
> 	+ Links to theories of elasticity, where spring constants can be
> 	calculated]
>
* \(A\): A class of admissible blurs (degrees of approximation
admitted between different models).


> 
> [The functions occurring in the potential models are complemented by
> suitable error bars. These may depend on the intended applications,
> see below.]
>
* \(K = \langle M\_p,M,M\_{pp}, C,L,A\rangle\): A core (the
formal-theoretical part of a theory)
* \(I\): The domain of intended applications ("pieces of the
world" to be explained, predicted or technologically
manipulated).


> 
> [This class is open and contains, for example
> 
> 
> 	+ systems of small rigid bodies, connected by coil springs or rubber
> 	bands
> 	+ any vibrating mechanical system in the case of small amplitudes,
> 	including almost rigid bodies consisting of \(N\) molecules]
>
* \(T = \langle K,I\rangle\): A theory-element (the smallest unit to
be regarded as a theory).
* \(\sigma\): The specialization relation between theory-elements.


> 
> [\(T\) could be a specialization of similar theory-elements with more
> general force laws, e.g., including friction and/or time-dependent
> external forces. One could also imagine more abstract force laws which
> fix only some general properties such as
> "action=reaction". \(T\) in turn could be specialized to
> theory-elements of systems with equal masses and/or equal spring
> constants. ]
>
* \(N\): A theory-net (a set of theory-elements ordered by
\(\sigma\) -- the "typical" notion of a theory).


> 
> [An obvious theory-net containing our example of a theory-element is
> CPM = "classical particle mechanics", conceived as a
> network of theory-elements essentially ordered by the degree of
> generality of its force laws.]
>
* \(E\): A theory-evolution (a theory-net "moving"
through historical time).


> 
> [Special interesting new force laws could be discovered in the course
> of time, e.g., the Toda chain in 1967, as well as new applications of
> known laws.]
>
* \(H\): A theory-holon (a complex of theory-nets tied by
"essential" links).


> 
> [It is difficult to think of examples which are smaller than \(H =\)
> all physical theory-nets.]
>


### 5.2 Ludwig's program


#### 5.2.1 History and general traits



Gunther Ludwig (1918-2007) was a German physicist mainly
known for his work on the foundations of quantum theory. In Ludwig
(1970, 1985, 1987), he published an axiomatic account of quantum
mechanics, which was based on the statistical interpretation of
quantum theory. As a prerequisite for this work he found it necessary
to ask "What is a physical theory?" and developed a
general concept of a theory on the first 80 pages of his (1970). Later
this general theory was expanded into the book Ludwig (1978). A recent
elaboration of Ludwig's program can be found in Schroter
(1996).



His underlying "philosophy" is the view that there are
real structures in the world which are "pictured" or
represented, in an approximate fashion, by mathematical structures,
symbolically \(\boldsymbol{PT} = \boldsymbol{W} (-) \boldsymbol{MT}\).
The mathematical theory \(\boldsymbol{MT}\) used in a physical theory
\(\boldsymbol{PT}\) contains as its core a "species of
structure" \(\Sigma\). This is a meta-mathematical concept of
Bourbaki which Ludwig introduced into the structuralistic approach.
The contact between \(\boldsymbol{MT}\) to some "domain of
reality" \(\boldsymbol{W}\) is achieved by a set of
correspondence principles \((-)\), which give rules for translating
physical facts into certain mathematical statements called
"observational reports". These facts are either directly
observable or given by means of other physical theories, called
"pre-theories" of \(\boldsymbol{PT}\). In this way a part
\(\boldsymbol{G}\) of \(\boldsymbol{W}\), called "basic
domain" is constructed. But it remains a task of the theory to
construct the full domain of reality \(\boldsymbol{W}\), that is, the
more complete description of the basic domain that also uses
\(\boldsymbol{PT}\)-theoretical terms.


#### 5.2.2 Typical features of Ludwig's program



Superficially considered, this concept of theory shows some similarity
to neo-positivistic ideas and would be subject to similar criticism.
For example, the discussion of the so-called
'theory-laden' character of observation sentences casts
doubts on such notions as "directly observable facts".
Nevertheless, the adherents of the Ludwig approach would probably
argue for a moderate form of observationalism and would point out
that, within Ludwig's approach, the theory-laden character of
observation sentences could be analyzed in detail.



Another central idea of Ludwig's program is the description of
intra- and inter-theoretical approximations by means of "uniform
structures", a mathematical concept lying between topological
and metrical structures. Although this idea was later adopted by the
other structuralistic programs, it plays a unique role within
Ludwig's meta-theory in connection with his finitism. He
believes that the mathematical structures of the infinitely large or
small, *a priori*, have no physical meaning at all; they are
preliminary tools to approximate finite physical reality. Uniform
structures are vehicles for expressing this particular kind of
approximation.


#### 5.2.3 Ludwig's interpretation of quantum mechanics



We have already explained that for Ludwig the framework for the
reconstruction of physical theories was actually only a tool to
develop his interpretation of quantum mechanics.



It is no surprise that there are close relationships between the two
enterprises. We only mention the fact that the reconstruction of
theoretical terms by other terms that are more easily accessible is
particularly urgent when the theoretical terms refer to the
microscopic domain. This explains in particular why Ludwig is a
supporter of a statistical interpretation of quantum mechanics,
because more advanced interpretations such as the
single-particle-state interpretation of the wave function, in his
opinion, have no axiomatic basis. In the current debate on the
interpretation of quantum mechanics, the statistical interpretation
(or ensemble interpretation) plays only a marginal role and is,
moreover, usually attributed to L. E. Ballentine (1970). The Wikipedia
entry on the 'ensemble interpretation' does not mention
Ludwig at all.



It would be premature, however, to deny Ludwig any influence on the
development of quantum theory. There are some achievements, like the
generalization of observables to POV measures, see Busch et al (2016),
which are well known, e.g., in the community practicing quantum
information theory, and which finally go back to Ludwig. Usually, the
standard reference for these generalizations is not Ludwig but his
pupil K. Kraus, see Kraus (1983). Finally it should be mentioned that
Ludwig's axiomatics of quantum mechanics has been revived by new
mathematical results, see Casinelli and Lahti (2016).


#### 5.2.4 Ludwig's late work



One year before his death Ludwig, together with Gerald Thurler,
published a revised and simplified edition of Ludwig (1990) with the
title "A new foundation of physical theories". This work
cannot be used as a textbook but it is a remarkable document of the
central themes of his approach and his general views on physics. The
book clearly shows that Ludwig's main concern is about
scientific realism, i.e., about the question of how hypothetical
objects and relations occurring within a successful theory acquire the
status of physical reality. Entities which cannot claim this status
are dubbed as "fairy tales" throughout the book. Examples
of fairy tales in quantum theory are hidden variables and, perhaps
surprising for some readers, also the single-particle-state
interpretation (in contrast to the ensemble interpretation fostered by
Ludwig).



Among the new concepts and tools developed in Ludwig/Thurler (2006)
are the following:


* Physical observations are first translated into sentences of an
auxiliary mathematical theory containing only finite sets, and, in a
second step, approximately embedded into an idealized theory. By this
maneuver the authors accentuate the contrast between finite physical
operations and mathematical assumptions involving infinite sets.
* Inaccuracy sets and unsharp measurements are always considered
right from the start and not introduced later as in previous versions
of the Ludwig program.
* The "basic domain" of a theory is now that part of the
"application domain" where the theory is successfully
applied, up to a certain degree of inaccuracy.
* The complicated terminology concerning various kinds of hypotheses
in Ludwig (1990) is radically reduced to a small number of cases
including fuzzy hypotheses.
* The problem of unsharp indirect measurements is reformulated in an
elegant way which yet should be scrutinized by means of case
studies.


#### 5.2.5 Summary



Generally speaking, Ludwig's program is, in comparison to those
of Sneed and Scheibe, less descriptive and more normative with respect
to physics. He developed an ideal of how physical theories should be
formulated rather than reconstructing the actual practice. The
principal worked-out example that comes close to this ideal is still
the axiomatic account of quantum mechanics, as described in Ludwig
(1985, 1987).


### 5.3 Scheibe's program



The German philosopher Erhard Scheibe (1927-2010) has published
several books and numerous essays on various topics of philosophy of
science; see, for example, Scheibe (2001). He has often commented on
the programs of Sneed and Ludwig, such as in his "Comparison of
two recent views on theories", reprinted in Scheibe (2001,
175-194). Moreover, he published one of the earliest case
studies of approximate theory reduction; see Scheibe 2001
(306-323) for the 1973 case study.



In his books on "reduction of physical theories," Scheibe
(1997, 1999) developed his own concept of theory, which to some extent
can be considered an intermediate position between those of Ludwig and
Sneed. For example, he conveniently combines the model-theoretical and
syntactical styles of Sneed and Ludwig, respectively. Since his main
concern is reduction, he does not need to cover all the aspects of
physical theories that are treated in the other approaches. As already
mentioned, he proposes a more flexible concept of reduction that is
open to extensions arising from new case studies.



A unique feature of Scheibe's approach is the thorough
discussion of almost all the important cases of reduction considered
in the physical literature. These include classical vs.
special-relativistic spacetime, Newtonian gravitation vs. general
relativity, thermodynamics vs. kinetic theory, and classical vs.
quantum mechanics. He essentially arrives at the conclusion of a
double incompleteness: the attempts of the physicists to prove
reduction relations in the above cases are largely incomplete
according to their own standards, as well as according to the
requirements of a structuralistic concept of reduction. But this
concept is also not complete, Scheibe argues, since, for example, a
satisfactory understanding of "counter-factual" limiting
processes such as \(\hslash \rightarrow 0\) or \(c\rightarrow \infty\)
has not yet been developed. Bolinger in his (2016) gives a fairly
general account of the structuralistic programme with special emphasis
on Scheibe's work.


### 5.4 Interactions between the three structuralist programs



As already noted, the programs of Ludwig and Sneed have been
independently developed in the 1970s, whereas Scheibe's program,
at least partially, originated from a critical review of these two
programs. But this is only a coarse description. Additionally, there
have been numerous mutual interactions between the three programs that
influenced their later elaborations. Evidence for this interaction is
provided, besides various pertinent acknowledgements in books and
articles, by the following observations.


* Balzer, Moulines and Sneed in their (1987) introduce the concepts
of "species of structures" and "uniform
structures" that play a central role in Ludwig (1970,
1978) and are not yet contained in Sneed (1971).
* Vice versa, Ludwig in his (1990) added a section 9.3 on theory
nets (Theorienetze) citing respective works of Balzer and
Moulines.
* In his late (2006) Ludwig on p.3 refers to the work of Scheibe
"because of the many similarities". Later on p.107 he
mentions a "discussion through letters" with Scheibe. This
correspondence has been secured by B. Falkenburg and is waiting for a
scientific edition.


## 6. Summary



We have sketched three structuralistic programs which have been
developed since the 1970s in order to tackle problems in philosophy of
physics, some of which are relevant also for physics itself. Any
program which employs a weighty formal apparatus in order to describe
a domain and to solve specific problems has to be scrutinized with
respect to the economy of its tools: to what extent is this apparatus
really necessary to achieve its goals? Or is it concerned mainly with
self-generated problems? We have tried to provide some arguments and
material for the reader who ultimately has to answer these questions
for him- or herself.