A Representational Approach to Reduction in Dynamical Systems

A Representational Approach to Reduction

in Dynamical Systems

Marco Giunti∗

Abstract

According to the received view, reduction is a deductive relation between two formal the-ories. In this paper, I develop an alternative approach, according to which reduction is arepresentational relation between models, rather than a deductive relation between theories;more specifically, I maintain that this representational relation is the one of emulation. Tosupport this thesis, I focus attention on mathematical dynamical systems and I argue that, asfar as these systems are concerned, the emulation relation is sufficient for reduction. I thenextend this representational model-based view of reduction to the case of empirically inter-preted dynamical systems, as well as to a treatment of partial, approximate, and asymptoticreduction.

1 Introduction

Traditionally, reduction has been analyzed in terms of a deductive relation between two formaltheories (Nagel 1961). Schaffner’s General Reduction Paradigm (1967; see also 1969, 1976, 1977,1993, 2006) was an early attempt to modify Nagel’s classic account, so as to accomodate caseswhere the reduced theory is, strictly speaking, false. The most comprehensive and detailed deduc-tivist account of reduction is Churchland and Hooker’s Imaging Approach (Churchland 1979, 1985;Hooker 1979, 1981, 2005), which can be seen as a creative development of Nagel’s basic insights,as well as a sensible departure from Nagel’s explicit tenets (Beckermann 1992; Bickle 1998, 2003;Endicott 1998; Marras 2002). Marras 2002 has convincingly argued that Kim’s FunctionalizingApproach to reduction (1998) is in fact a version of Nagel’s account; such a version is essentiallyequivalent to the Imaging Approach.

The main thrust of this paper is to advance an alternative view, according to which reductionis better conceived as a representational relation between two mathematical models M1 and M2,which grants the construction, within the representing model M1, of an isomorphic image of M2.1

Bickle’s New Wave Reduction (1998, ch. 3) is a version of the Imaging Approach by Church-land and Hooker in which (i) theories are construed as sets of models (semantically), rather thansets of sentences (syntactically), and thus (ii) reduction is not a deductive relation between formal

∗Dipartimento di Pedagogia, Psicologia, Filosofia, Universita di Cagliari. Via Is Mirrionis 1, 09123 Cagliari, Italy.giunti[at]unica.it

1The term “isomorphic image” is intended here in its rigorous mathematical sense. This is not the sense in whichthe Imaging Approach employs the same term.

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theories, but a relation between semantic theories (i.e. sets of models) that satisfies special condi-tions. Notwithstanding these differences, reduction is still analyzed by Bickle as a special relationbetween theories (i.e. sets of models) and not as a representational relation between models. Asstressed by Endicott (2001), Bickle shares his general view of reduction and theory structure withthe Structuralist Program (Sneed 1971; Stegmuller 1976; Mayr 1976, 1981; Balzer, Pearce, andSchmidt 1984; Balzer, Moulines, and Sneed 1987).

The general representational view of reduction that I advocate is in broad agreement withSuppes’ view of reduction.2 The so-called Suppes’ Reduction Paradigm (Schaffner 1967, 139) wasbriefly sketched in the often quoted passage below.

To show in a sharp sense that thermodynamics may be reduced to statistical mechanics, wewould need to axiomatize both disciplines by defining appropriate set-theoretical predicates,and then show that given any model T of thermodynamics we may find a model of statisticalmechanics on the basis of which we may construct a model isomorphic to T. (Suppes 1957,271)

Apart from a few other remarks, Suppes has not developed this suggestion into an explicittheory of reduction. Nevertheless, starting with Schaffner 1967, Suppes’ Reduction Paradigm hasbeen often identified with a weaker form of Nagel’s view. Schaffner based this claim on a so-called Theorem (1967, 145), according to which any Nagel-style reduction would give rise to acorresponding Suppes-style reduction. However, Schaffner’s claim is at best unwarranted for tworeasons. (i) Schaffner grounded his Theorem on a formal reconstruction of Nagel 1961 (ch. 11),but it is far from obvious that Schaffner’s formal version is a faithful rendition of Nagel’s originalview; (ii) Schaffner’s Theorem is in fact a non sequitur, for inspection of the proof reveals that,given an arbitrary Nagel-style reduction, construction of an isomorphic model is not ensured forany model of the reduced theory, but only for the particular model which is determined by itsintended interpretation.

On the other hand, it is also quite common to take Suppes’ view as the primary source ofAdams’ approach to reduction (1955, 1959) and its subsequent developments in the structuralistcamp (Sneed 1971; Stegmuller 1976; Mayr 1976, 1981; Balzer, Pearce, and Schmidt 1984; Balzer,Moulines, and Sneed 1987). However, as Day (1985) makes clear, there is no obvious conceptualrelationship between Suppes’ and Adams’ views.

The key notion in Adams’ approach is that of a reduction relation R between a reducingtheory T1 and a reduced one T2. For Adams and, more generally, for the structuralist approach,the reduction relation R “is based entirely upon certain formal correspondences linking the modelsand potential models of the two theories” (Pearce 1982, 312). By contrast, in Suppes’ ReductionParadigm (see the quotation above) such formal correspondences are not even mentioned. Rather,reduction obtains when any model MT2 of the reduced theory T2 is related to some model MT1 ofthe reducing theory T1 in such a way that, on the basis of MT1, we may construct a third modelM2(MT1) isomorphic to MT2.

Thus, according to Suppes, there is reduction when any model MT2 of the reduced theorystands in the appropriate representational relation with some model MT1 of the reducing one. But

2Section 4.2.2 (Case 2) will make clear that Schaffner’s criticism (1967, 145, par. 6) of Suppes’ ReductionParadigm does not apply to my view. Bickle refers to this criticism as the “too weak to be adequate argument”(Bickle 1998, ch. 3, sec. 3).

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this means that Suppes’ paradigm is a view of theory reduction only in a secondary and derivativesense, for the primary and basic notion is that of an appropriate representational relation betweenmodels, namely a relation that allows us to construct an isomorphic image M2(MT1) of the reducedmodel MT2 by means of the reducing one MT1. Unfortunately, Suppes has never made clear theexact meaning of “constructing an isomorphic image M2(MT1) of the reduced model MT2 bymeans of the reducing one MT1”. This crucial point is left open, but it needs to be filled in, sothat Suppes’ Reduction Paradigm may become a full fledged theory of reduction.

Compared to traditional approaches to reduction (deductivist or, more generally, theory-basedapproaches), the representational model-based one has several potential advantages, whose farreaching implications will only be apparent when the technical details of the theory are developed.Accordingly, this paper is mainly devoted to developing the theory, rather than to its criticalevaluation or defense. For the moment, it thus suffice to say that the representational view is verypromising as far as precision and depth of analysis are concerned. Also, this view is fostering aunified, conceptually crisp, and formally developed account of prima facie conflicting aspects ofreduction, on which traditional approaches hardly fare as well – conceptually homogeneous vs.heterogeneous reduction, reduction vs. multiple realization, total and exact reduction vs. partial,approximate and asymptotic one.

A further advantage of the representational model-based view is that it equally applies toeither intra-theoretic or inter -theoretic reduction, depending on whether the reducing model MT1

and the reduced one MT2 belong to the same theory (intra-theoretic reduction), or to differentones (inter-theoretic reduction). As for reduction of a theory T2 to a reducing theory T1, this isaccomplished when, for any model MT2 of theory T2 there is a model MT1 of theory T1 such thatMT2 is reduced to MT1.

I will develop this general representational approach only for the special case of dynamicalsystems. As intended here (Arnold 1977; Szlenk 1984; Giunti 1997), a dynamical system is a kind ofmathematical model that captures the intuitive idea of an arbitrary deterministic system (sec. 2).Models of this kind allow us to study in a precise way typical features of complex systems. Amongthem, in recent years, the one of emulation has gained growing attention (Wolfram 1983a, 1983b,1984a, 1984b, 2002). Intuitively, a dynamical system DS1 emulates a second dynamical systemDS2 when the first one exactly reproduces all the dynamics of the second one.

The emulation relation can be defined in a precise way for any two arbitrary dynamical systemsand it has been shown (Giunti 1997, ch.1, th. 11) that, if DS1 emulates DS2, there is a third systemDS3 such that (i) DS2 is isomorphic to DS3; (ii) all states of DS3 are states of DS1; (iii) anystate transition of DS3 is constructed out of state transitions of DS1. In this paper (sec. 3), I willfocus on a more general version of this theorem [Virtual System Theorem VST ], which is based ona weaker and simpler definition of emulation. I will then argue that this result allows us to claim:If DS1 emulates DS2, then DS2 is reduced to DS1.

The claim that emulation is sufficient for reduction (in force of [VST ]) is a precise statementof the representational model-based view of reduction for the special case of dynamical systems.Strictly speaking, this claim is intended to hold exclusively for dynamical systems as purely math-ematical models with no empirical interpretation. In a different sense, however, dynamical systemstypically function as models of real phenomena. In this second sense, a dynamical system is not apurely mathematical entity DS, but it is a pair (DS, IH), where IH is an empirical interpretationthat links the purely mathematical model DS to a phenomenon H. This paper (sec. 4) will also

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provide the main lines of an extension of the representational theory of reduction to empiricallyinterpreted dynamical systems. Sufficient conditions for reduction of empirically interpreted dy-namical systems will be given (sec. 4.2.1, Case 1) and, as a seminal example, I will show that theseconditions ensure reduction of Galileo’s falling body model to his more general projectile model.

The reduction of the falling body model to the projectile model is quite peculiar in threedifferent respects. First, it is a conceptually homogeneous reduction, for each magnitude involvedin the reduced model is also a magnitude of the reducing one. However, I will argue that the rep-resentational view does allow for heterogeneous reductions, in even a stronger sense than generallyadmitted (sec. 4.2.1, Heterogeneous reduction).

Second, it is a complete or proper reduction, for any concrete system described by the reducedmodel (i.e., any real falling body) is also a concrete system (i.e., a real projectile) described bythe reducing one – that is to say, the application domain of the reduced model is a subset of theapplication domain of the reducing one. However, in sec. 4.2.3, sufficient conditions for incompletereduction, as well as multiple reduction, will also be given. According to them, incomplete ormultiple reduction occurs when the application domains of the reduced-reducing models are not inthe subset relationship. I will then argue that multiple realization of high level functional propertiesnot only is consistent with the representational view, but is in fact a logical consequence of multiplereduction.

Third, it is a total and exact reduction, for all the dynamics of the reduced model is exactlyreproduced by the reducing one. As said, the emulation relation is the basis of the representationaltheory of reduction for dynamical systems (either empirically interpreted or not). The simplestform of such relation holds between two dynamical systems DS1 and DS2 when all the dynamicsof DS2 is exactly reproduced by DS1. This simple form is sufficient for an adequate account oftotal and exact reduction (as the reduction of the falling body model to the projectile model is),but we need a more sophisticated version of emulation for dealing with cases of asymptotic, partialand approximate reduction (Hooker 2004). Such a version will be introduced in sec. 5, where itwill then be employed for a treatment of partial and approximate reduction, as well as asymptoticone, in empirically interpreted dynamical systems. As an example, I will analyze the reductionof Galileo’s projectile model to Newton’s satellite model. Finally, I will show that a quite generalasymptotic condition ensures reduction up to an arbitrary approximation degree [Limit ReductionTheorem LRT ].

2 Dynamical systems and emulation

A dynamical system is a kind of mathematical model that formally expresses the notion of anarbitrary deterministic system, either reversible or irreversible, with discrete or continuous timeor state space. Let Z be the integers, Z + the non-negative integers, R the reals and R+ thenon-negative reals; below is the exact definition of a dynamical system.3

[1] DS is a dynamical system iff DS is a pair (M, (g t)t∈T ) such that

3Def. 1 can be made completely general by taking the time set T to be the domain of an arbitrary monoidL = (T,+) (Giunti and Mazzola 2012). For the purposes of this paper, however, it suffice to consider dynamicalsystems whose time set T is either Z, Z+, R, or R+.

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1. M is a non-empty set; M represents all the possible states of the system, and it is called thestate space;

2. T is either Z, Z +, R, or R+; T represents the time of the system, and it is called the timeset ; any t ∈ T is called a duration of the system;

3. (g t)t∈T is a family of functions from M to M ; each function g t is called a state transitionof duration t, or a t-advance, of the system;

4. for any t, v ∈ T, for any x ∈ M, g0(x ) = x and gv+t(x ) = gv (g t(x )).

[2] A discrete dynamical system is a dynamical system whose state space is finite or denumer-able, and whose time set is either Z or Z +; examples of discrete dynamical systems are Turingmachines and cellular automata.4

[3] A continuous dynamical system is a dynamical system that is not discrete; examples ofcontinuous dynamical systems are iterated mappings on R, and systems specified by ordinarydifferential equations.

[4] A possible dynamical system is a pair (M, (g t)t∈T ) that satisfies the first three conditionsof definition [1].

We can now define the concept of an isomorphism between two possible dynamical systemsas follows.

[5] r is an isomorphism of DS1 in DS2 iff DS 1 = (M, (g t)t∈T ) and DS 2 = (N, (hv )v∈V ) arepossible dynamical systems, T = V, r : M → N is a bijection and, for any t ∈ T, for any x ∈ M,r(g t(x )) = ht(r(x )).

[6] DS 1 is isomorphic to DS 2 iff there is an isomorphism of DS 1 in DS 2.

It is easy to verify that the isomorphism relation is an equivalence relation on any given set ofpossible dynamical systems. (The concept of set of all possible dynamical systems is inconsistent,5

and we must then take as the basis of the theory of dynamical systems a specific, sufficiently large,set of possible dynamical systems.)

It is also not difficult to prove that the relation of isomorphism is compatible with the propertyof being a dynamical system, that is to say: if DS 1 is isomorphic to DS 2 and DS 1 is a dynamicalsystem, then DS 2 is a dynamical system. This allows us to speak of abstract dynamical systemsin exactly the same sense we talk of abstract groups, fields, lattices, order structures, etc. We canthus define:

[7] an abstract dynamical system is any equivalence class of isomorphic dynamical systems.

It is easily shown that any two dynamical systems have exactly the same structural properties

4The term “discrete dynamical system” is often used (see, for example, Kulenovic and Merino 2002; Martelli1999; Sandefour 1990) as a synonym for “dynamical system with discrete time”, i.e., according to Szlenk 1984, acascade. My use of the term “discrete dynamical system” is in accordance with Turing 1950.

5If the set of all possible dynamical systems D existed, then the set of all sets S would exist as well, for S ={M : either M = ∅ or, for some DS ∈ D, M is the state space of DS}. But it is well known that the existence of Sentails a contradiction (Cantor’s antinomy of the set of all sets).

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iff they are isomorphic.6 Since general dynamical systems theory7 is exclusively interested in suchproperties, it regards any two isomorphic systems as identical.

Figure 1: Emulation

Dynamical systems are appropriate models to study several interesting features of complexsystems. The one of emulation is typical of computational systems (Wolfram 2002), but it can inprinciple involve any two dynamical systems. The intuitive idea is that a dynamical system DS 1

emulates a second dynamical system DS 2 when the first one exactly reproduces all the dynamicsof the second one. Here are some examples. A universal Turing machine emulates any Turingmachine; for any Turing machine TM there is a cellular automaton CA such that CA emulatesTM (Smith 1971, th. 3), and vice versa; the simple cellular automaton specified by Wolfram’s rule18 emulates the one specified by rule 90 (both CA are monodimensional, with 2 possible values forcell, and neighborhood of radius 1; see Wolfram 1983b, 20).

Giunti 1997 (ch. 1, def. 4) gave a formal definition of the emulation relation that applies toany two arbitrary dynamical systems. Here, I will employ a weaker and simpler definition (seefigure 1), which nevertheless suffices for the present purposes. Let DS 1 = (M, (g t)t∈T ) and DS 2

= (N, (hv )v∈V ) be dynamical systems;

[8] DS 1 emulates DS 2 iff there is an injective function u: N → M such that, for any v ∈ V,for any c ∈ N, there is t ∈ T such that u(hv (c)) = g t(u(c)). Any function u that satisfies theprevious condition is called an emulation of DS 2 in DS1.

6P is a structural property of a dynamical system (or a dynamical property) iff for any two mathematical modelsMS1 and MS2, (i) if MS1 has P, MS1 is a dynamical system and (ii) if MS1 has P, and MS1 is isomorphic toMS2, then MS2 has P. Thus, a dynamical property is a property specific to dynamical systems that is preserved byisomorphism. The proof that any two isomorphic dynamical systems have exactly the same dynamical properties isimmediate. Conversely, for any two non-isomorphic dynamical systems DS1 and DS2, there is a dynamical propertythey do not share; namely, the property of being isomorphic to DS1.

7By general dynamical systems theory I mean the mathematical theory whose Suppes’ style axiomatization (1957,ch. 12) is given by def. [1].

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3 Emulation is sufficient for reduction

Giunti 1997 (ch. 1, th. 11) proved that, if u is an emulation of DS 2 in DS 1, there is a third systemDS 3 such that (i) u is an isomorphism of DS 2 in DS 3; (ii) all states of DS 3 are states of DS 1;(iii) any state transition of DS 3 is constructed out of state transitions of DS 1. This result stillholds for the weaker definition of emulation [8], as the following theorem shows.

Figure 2: The u-virtual system DS 2 in DS 1

Virtual System Theorem [VST ]

• Let DS 1 = (M, (g t)t∈T ) and DS 2 = (N, (hv )v∈V ) be dynamical systems, and u be an emulationof DS 2 in DS1;

• let DS 3 = (N, (hv )v∈V ), where N = u(N ) and, for any a ∈ N, for any v ∈ V, hv (a) =u(hv (u−1(a)); the system DS 3 is called the u-virtual system DS 2 in DS1 (see figure 2);

then:

(i)u is an isomorphism of DS 2 in DS 3;

(ii) all states of DS 3 are states of DS 1;

(iii) for any state transition hv of DS 3, for any a ∈ N, there is a state transition g t of DS 1 suchthat hv (a) = g t(a).

Proof of (i)

By the definition of DS 3, for any c ∈ N, u(hv (c)) = u(hv (u−1(u(c))) = hv (u(c)). Therefore, bythe definition of isomorphism [5], u is an isomorphism of DS 2 in DS 3.

Proof of (ii)

Obvious, by the definition of DS 3.

Proof of (iii)

By the definition of DS 3, for any v ∈ V, for any a ∈ N, hv (a) = u(hv (u−1(a)). Let c = u−1(a).Since u is an emulation of DS 2 in DS 1, by definition [8], there is t ∈ T such that u(hv (c)) =g t(u(c)). Therefore, hv (a) = g t(u(c)) = g t(a). Q.E.D.

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It is my contention that, if a dynamical system DS 1 emulates a second system DS 2, [VST ]allows us to claim that DS 2 is reduced to DS 1. In other words, I maintain that, because of [VST ],emulation8 is sufficient for reduction.

Before seeing the details of the supporting argument, it is important to make clear that dynam-ical systems, as intended here, are purely mathematical entities with no empirical interpretation;that is to say, at this level of analysis, a dynamical system is just a model of the mathematicaltheory whose Suppes’ style axiomatization (1957, ch. 12) is given by def. [1]. The claim thatemulation is sufficient for reduction is thus exclusively limited to dynamical systems intended inthis sense.

As just said, when I speak of a dynamical system as a model, I mean a model of a quitegeneral mathematical theory, whose axiomatization is expressed by the definition, in set theory,of an appropriate set-theoretical predicate (def. [1]). It is important to sharply distinguish thissense of the term “model” from a different one, which also applies to dynamical systems, and isequally central to a complete understanding of their epistemological status. This second sense isthe one intended when we say that a specific dynamical system is a model of a real phenomenon;however, this sense does not refer to a dynamical system as a purely mathematical entity (i.e.,just a model of general dynamical system theory) but, rather, to such entity together with anempirical interpretation that links the mathematical model to the phenomenon which it is intendedto describe.

A simple example will make the distinction clear. Let us consider the following system oftwo ordinary differential equations 〈dy(v)/dv = y(v), dy(v)/dv = −g〉, where g is a fixed realpositive constant. The solutions of such equations uniquely determine the dynamical system DSe =(Y × Y , (hv )v∈V ), where Y = Y = V = R (the real numbers) and, for any v, y, y ∈ R, hv (y, y) =(−gv2/2 + yv + y, −gv + y). It is immediate to verify that DSe satisfies def. [1], so that it is amodel in the first sense.

On the other hand, let us consider the phenomenon of the free fall of a medium size body inthe vicinity of the earth (henceforth, Heφ),9 and let us interpret the first component Y of the statespace of DSe as the set of all possible values of the vertical position of an arbitrary free fallingbody, the second component Y as the set of all possible values of the vertical velocity of the fallingbody,10 and the time set V of DSe as the set of all possible values of physical time. Let IHeφ besuch an interpretation. Since all three of these magnitudes are measurable or detectable propertiesof the phenomenon of free fall Heφ, the given interpretation IHeφ is an empirical interpretation ofthe dynamical system DSe on Heφ, and the pair DSeφ = (DSe, IHeφ) is an empirical model ofHeφ, i.e., such a pair is a model in the second sense. DSeφ will be called the falling body model.

My claim that emulation is sufficient for reduction (in force of [VST ]) is intended to hold

8I recall that emulation, as defined here, is an exact relation between two mathematical models; this sense ofthe term “emulation” is standard in both dynamical systems theory and computation theory, and it should not beconfused with a common use of the same term, which refers to the relation involved in the simulation of a physicalsystem (e.g. a water flow) by a second one (e.g. a digital computer, which, by means of appropriate software,implements a mathematical model of the water flow).

9φ ∈ [0, ψ] is a real, non-negative, parameter; φ sets an upper bound on the maximum height (relative to theearth surface) that the falling body can reach during its motion. For further details on the meaning of φ, see sec.4.1, par. 2.

10For any falling body a, if pa is the point where a is initially released, a’s vertical position and velocity are takenwith respect to an axis with origin in the earth center that passes through pa; the positive direction of such axis isfrom the earth center to the point pa.

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exclusively for dynamical models in the first sense. This does not mean that such a claim does nothave any bearing on the further question: What are the conditions for reduction of an empiricallyinterpreted dynamical system (DS 2, I 2) to another one (DS 1, I 1)? I will return later (see sec. 4)to this question. For the moment, it suffice to say that, in my view, the conditions for reduction ofthe mathematical model DS 2 to the mathematical model DS 1 are a necessary component of themore complex conditions for reduction of (DS 2, I 2) to (DS 1, I 1).

I am now going to present a detailed argument to support the claim that emulation is sufficientfor reduction. The complete argument relies on five premises, divided into three groups. The firstpremise (A) is the most general one, for it refers to systems of any kind. Specifically, A states asufficient condition for reduction between two arbitrary systems. The premises of the second group(B1 and B2) are at an itermediate level of generality, for they refer exclusively to mathematicalsystems of any kind, that is, systems that are models of some mathematical theory. B1 explicitlystates what it is to be intended for “constitutive entity of a mathematical model”, while B2 makesclear the meaning of “whole-structure of a mathematical model”. The premises of the third group(C1 and C2) are the most specific, for they refer to dynamical systems (in the purely mathematicalsense). In particular, C1 states identity conditions for such systems, and C2 makes explicit theexact meaning of “whole-structure of a dynamical system”. Below are the five premises. Eachof them is followed by a brief elucidation, which is intended to pin point crucial features of thecorresponding premise, as well as to provide an intuitive justification for its assumption.

A For a system S 2 to be reduced to a system S 1, it is sufficient that (a) all the constitutive entitiesof S 2 are constitutive entities of S 1 and (b) the whole-structure of S 2 is a part of the whole-structure of S 1. Elucidation – In general, a system S can be thought as a whole with a definitestructure (whole-structure), which is formed by a complex of interconnected structural elements;each of these structural elements is built out of a given stock of building blocks, which we call “theconstitutive entities of S”. Thus, if two systems S 1 and S 2 satisfy conditions (a) and (b) above,the system S 2 is in fact a subsystem of S 1; this allows us to claim that S 2 is reduced to S 1.

B1 The constitutive entities of a mathematical model are the entities in its domain. Elucidation –According to standard definition, a mathematical model MS is a set D together with a family(σi)i∈I of relations on D. For any i ∈ I, there is exactly one n ≥ 0 such that σi has arity n, whererelations of arity 0 are identified with members of D, and relations of arity n > 0 are identified withsets of n-tuples of members of D ; the set D is called the domain of the model. A mathematicalmodel can thus be thought as a special kind of system, whose interconnected structural elementsare the relations in the family (σi)i∈I , and whose constitutive entities are the members of D.

B2 The whole-structure of a mathematical model MS = (D, (σi)i∈I ) is the union of all therelations in the family (σi)i∈I ;11 accordingly, if the relata of “is a part of” are whole-structuresof mathematical models, “is a part of” is to be interpreted as set-inclusion. Elucidation – Wehave just seen that a mathematical model can be thought as a special kind of system, whoseinterconnected structural elements are the relations in the family (σi)i∈I . Each of such relationsis a set of n-tuples; thus, the union of these sets is the whole-structure formed by the complex ofsuch relations. Given this interpretation of “whole-structure of a mathematical model”, it is then

11The condition in the text holds iff any relation σi has arity > 0. The general condition is as follows. Let X ={x : for some i ∈ I, x = σi and σi is a relation of arity 0}; then, the whole-structure of (D, (σi )i∈I ) is the union ofX and all relations σi of arity > 0. Obviously, this condition reduces to the one in the text when X is empty, i.e.,when any relation σi has arity > 0.

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obvious that “is a part of” should be interpreted as set-inclusion.

C1 From the point of view of general dynamical systems theory, any two isomorphic dynamicalsystems are identical. Elucidation – General dynamical systems theory studies the structuralproperties (see notes 6 and 7) of dynamical systems, and any two dynamical systems have exactlythe same structural properties iff they are isomorphic. Therefore, general dynamical systems theorydoes not distinguish between any two isomorphic dynamical systems.

C2 If a mathematical model is a dynamical system DS = (M, (g t)t∈T ), the whole-structure ofthe model is the set of all state pairs (x, y) such that, for some t ∈ T, g t(x ) = y. Elucidation –We should first of all notice that, by def. [1], a dynamical system is a mathematical model of aspecial kind, namely, such that any relation g t is in fact a function from M to M. Then, C2 is animmediate consequence of this observation and B2.12

Sufficiency of emulation for reduction1. For a mathematical model MS 2 to be reduced to a mathematical model MS 1, it is sufficientthat (a) the domain of MS 2 is included in the domain of MS 1 and (b) the whole-structure of MS 2

is included in the whole-structure of MS 1; (logically follows from A, B1 and B2;)2. hence, if u is an emulation of DS 2 in DS 1, the u-virtual system DS 2 in DS 1 is reduced to DS 1;(logically follows from 1, C2, and theses (ii) and (iii) of [VST ];)3. if u is an emulation of DS 2 in DS 1, DS 2 is isomorphic to the u-virtual system DS 2 in DS 1;(logically follows from thesis (i) of [VST ] and def. [6];)4. consequently, if u is an emulation of DS 2 in DS 1, DS 2 is reduced to DS 1. (Logically followsfrom 2, 3, C1 and the fact that dynamical systems, as intended here, are just models of generaldynamical systems theory.)

4 Models of phenomena—sufficient conditions for total andexact reduction in empirically interpreted dynamical sys-tems

Thus far, the representational theory of reduction has a precise formulation only if the modelsinvolved are dynamical systems in the purely mathematical sense. However, we have seen in sec. 3that dynamical systems can also be intended as models of real phenomena. According to this secondsense of the term “model”, a dynamical system is not a purely mathematical entity DS; rather,it is a pair (DS, IH), where IH is an empirical interpretation that links the purely mathematicalmodel DS to a phenomenon H. The representational theory should then be further developedto provide conditions for reduction of an empirically interpreted dynamical system (DS, IH2) toanother one (DS, IH1). I will briefly sketch here the main lines of such development. The followingexposition has no pretention to exhaustiveness. Its goal is just to trace a possible way along whichan adequate representational theory of reduction for empirically interpreted dynamical systemsmight be worked out.

12Thus, C2 is not an independent premise of the argument, for it is entailed by def. [1], the standard definitionof a mathematical model, and B2.

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4.1 Models of phenomena

In general, a phenomenon H can be thought as a pair (F,BF ) of two distinct elements. Its firstpart, F , is a functional description that provides a sufficiently detailed specification of (a) theconstitution and internal organization or functioning of a real system type ASF ; (b) a generalspatio-temporal causal scheme CSF of ASF ’s external interactions. In particular, the descriptionof the causal scheme CSF must include the specification of (b.1) the initial conditions that anarbitrary evolution of any real system of type ASF must satisfy; (b.2) the boundary conditionsduring the whole subsequent evolution; and, possibly, (b.3) the final conditions under which theevolution terminates. Second, the real part BF of the phenomenon H is the set of all real orconcrete systems which satisfy the functional description F or, in other words, BF is the set ofall real systems of type ASF whose temporal evolutions are all constrained by the causal schemeCSF . BF is called the application domain of H.13

For example, let Heφ = (Feφ, BFeφ) be the phenomenon of the free fall of a medium size bodyin the vicinity of the earth, where φ ∈ [0, ψ] is a real, non-negative, parameter whose meaning isexplained below (from now on, I will refer to Heφ just as the phenomenon of free fall). In this case,the functional description Feφ is as follows. The abstract type of real system ASFeφ has just onestructural element, namely, a medium size body in the vicinity of the earth. The causal interactionscheme CSFeφ consists in (i) releasing the body at an arbitrary instant, and with a purely verticalvelocity and position (relative to the earth surface) such that the body hits the earth surface atsome later instant, and the maximum height reached by the body is not higher than φ; (ii) duringthe whole motion, the only force acting on the body is its weight, and (iii) the motion terminateswhen the body hits the earth surface. Finally, BFeφ is the set of all concrete medium size bodiesin the vicinity of the earth that satisfy the given scheme of causal interactions. Any body memberof BFeφ is called a (free) falling body.

For any i (1 ≤ i ≤ n), let X i be a non-empty set, and let DS = (M, (g t)t∈T ) be a dynamicalsystem whose state space M ⊆ X 1× . . .×X n ; for any i, the set C i = {x i : for some n-tuple x ∈ M,x i is the i -th element of x} is called the i-th component of M. An interpretation IH of DS on aphenomenon H consists in stating that (i) each component C i is a subset of the set of all possiblevalues of a magnitude Mi of the phenomenon H, and (ii) the time set T is identical to the set of allpossible values of the time magnitude T of H itself. An interpretation IH of DS on H is empiricalif the time T and some of the magnitudes Mi are measurable properties of the phenomenon H.

A pair DS = (DS, IH), where DS is a dynamical system with n components and IH is aninterpretation of DS on H, is called a model of the phenomenon H. If the interpretation IH isempirical, then DS is an empirical model of H. Such a model is empirically correct if, for anymeasurable magnitude M i, all measurements of M i are consistent with the corresponding valuesxi determined by DS. An empirically correct model of H is also called a Galilean model of H(Giunti 1995; Giunti 1997, ch. 3). A Galilean model is then any empirically correct model of somephenomenon.

As an example, let us consider again the phenomenon of free fall Heφ. Let DSe be thedynamical system with two components specified in sec. 3, and IHeφ be its interpretation given insec. 3; then, according to the previous definitions, IHeφ is an empirical interpretation of DSe on

13Since the functional description F typically contains several idealizations, no concrete or real system exactlysatisfies F, but it rather fits F up to a certain degree. Thus, from a formal point of view, the application domainBF of a phenomenon H = (F,BF ) is better described as a fuzzy set.

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Heφ, and DSeφ = (DSe, IHeφ) is an empirical model of Heφ. If φ is sufficiently small, and for anappropriate value of the constant g , such a model also turns out to be empirically correct,14 andit is thus an example of a Galilean model.

4.2 Sufficient conditions for total and exact reduction in empiricallyinterpreted dynamical systems

Let us now consider two phenomena H 1 = (F 1, BF1) and H 2 = (F 2, BF2

), and two empiricallyinterpreted dynamical systems DS1 = (DS 1, IH1

) and DS2 = (DS 2, IH2) such that DS1 is an

empirical model of H 1 and DS2 is an empirical model of H 2. What are the conditions for reductionof DS2 to DS1? I will divide the discussion into three distinct cases.

4.2.1 Case 1: BF2⊆ BF1

.

Let us suppose that BF2⊆ BF1

. Under this hypothesis, it seems sensible to claim that, if DS 1

emulates DS 2, then DS2 is reduced to DS1. To see this point, let us notice, first, that thehypothesis BF2 ⊆ BF1 ensures that any concrete system b ∈ BF2 , which is described by DS2,is described by DS1 as well. Second, let u: N → M be an emulation of DS 2 = (N, (hv )v∈V )in DS 1 = (M, (g t)t∈T ), where N ⊆ Y 1× . . .×Y n and M ⊆ X 1× . . .×X m have, respectively, nand m components. Thus, by def. [8], any state transition hv : (y1, . . . , yn) → (y1’, . . . , yn ’)corresponds to a state transition g t : (x 1, . . . , xm) → (x 1’, . . . , xm ’), where u(y1, . . . , yn) =(x 1, . . . , xm) and u(y1’, . . . , yn ’) = (x 1’, . . . , xm ’). In addition, since DS2 is a model of H 2,for any j, y j and y j ’ are values of a magnitude Y j of H 2, and v is a value of the time T 2 of H 2;on the other hand, since DS1 is a model of H 1, for any i, x i and x i ’ are values of a magnitude X i

of H 1, and t is a value of the time T 1 of H 1. For any concrete system b ∈ BF2, both the DS2 and

DS1 descriptions apply to b. But then, the emulation function u tells us exactly how the DS2

description of b corresponds to the DS1 description.

As an example, let DSeφ = (DSe, IHeφ) be the falling body model, where Heφ = the phe-nomenon of free fall, and let Hpφ,θ = (Fpφ,θ, BFpφ,θ ) be the phenomenon of projectile motion, whereφ ∈ [0, ψ] and θ ∈ [0, ρ] are real, non-negative parameters whose meaning is explained below. Thefunctional description Fpφ,θ and its application domain BFpφ,θ are specified as follows. The ab-stract type of real system ASFpφ,θ is a medium size body in the vicinity of the earth, and it is thusidentical to ASFeφ . However, the causal interaction scheme CSFpφ,θ is more general than CSFeφ ,for it consists in (i) releasing the body at an arbitrary instant, and with any velocity and position(relative to the earth surface) such that the body hits the earth surface at some later instant, themaximum height reached by the body is not higher than φ, and the maximum horizontal distancecovered by the body is not greater than θ; (ii) during the whole motion the only force acting on thebody is its weight, and (iii) the motion terminates when the body hits the earth surface. BFpφ,θ is

14Quite obviously, if g = standard gravity (g = 9.80665 m/s2), the falling body model DSeφ = (DSe, IHeφ )turns out to be empirically correct within limits of precision sufficient for many practical purposes, provided thatφ is sufficiently small. The same holds for the more general model DSpφ,θ = (DSp, IHpφ,θ ) of projectile motion(provided that both φ and θ are sufficiently small; see case 1, below). Also recall that, by Newton’s law of universalgravitation, g is approximately equal to Gm/r2, where m is the mass of the earth, r its radius, and G is thegravitational constant.

12

then the set of all concrete medium size bodies in the vicinity of the earth that satisfy the givenmore general scheme of causal interactions. Any such body will be called a projectile.

Let us then consider the following system of four ordinary differential equations 〈dx (t)/dt =x(t), dy(t)/dt = y(t), d x(t)/dt = 0, d y(t)/dt = −g〉, where g is a fixed real positive constant.The solutions of such equations uniquely determine the dynamical system DSp = (X×Y×X×Y ,

(g t)t∈T ), where X = Y = X = Y = T = R (the real numbers) and, for any t, x, y, x, y ∈ R,g t(x, y, x, y) = (xt + x, −g t2/2 + yt + y, x, −g t + y).

Let IHpφ,θ be the following interpretation of DSp on the phenomenon of projectile motionHpφ,θ. In the first place, for any projectile a, let pa be the point where a is initially released; wethen consider the plane that contains a’s initial velocity vector and the earth center. On this plane,we fix both the x-axis and the y-axis of a Cartesian coordinate system, in such a way that its origincoincides with the earth center, and the y-axis passes through pa. The positive direction of they-axis is from the earth center to the point pa; accordingly, we call the y-axis the vertical axis, andthe x-axis the horizontal axis. We then interpret the first component X of the state space of DSpas the set of all possible values of the horizontal position of the projectile a, the second componentY as the set of all possible values of its vertical position, the third component X as the set of allpossible values of its horizontal velocity, the fourth component Y as the set of all possible values ofits vertical velocity, and the time set T of DSp as the set all possible values of physical time. Sinceall five of these magnitudes are measurable or detectable properties of the intended phenomenonHp, IHpφ,θ is an empirical interpretation of DSp on Hpφ,θ.

Let DSpφ,θ = (DSp, IHpφ,θ ); DSpφ,θ will be called the projectile model. By the respectivedefinitions of BFeφ and BFpφ,θ , BFeφ ⊂ BFpφ,θ . Thus, by case 1, to show that the falling bodymodel DSeφ is reduced to the projectile model DSpφ,θ, it suffice to exhibit an emulation u of

DSe in DSp. Let u: Y×Y → X×Y×X×Y and, for any y, y ∈ R, u(y, y) = (0, y, 0, y); then,quite obviously, u is an emulation of DSe in DSp.

Heterogeneous reduction. As remarked above, the emulation function u tells us how the DS2

description of an arbitrary concrete system b ∈ BF2is related to its corresponding DS1 description.

Since b is in the application domain BF2of phenomenon H2, its DS2 description is made in terms

of the time T 2 of H2 and the n magnitudes Y 1, ...,Y n of H2 which respectively interpret the ncomponents D1, ..., Dn of the state space N of DS2. On the other hand, as b also belongs to theapplication domain BF1 of phenomenon H1, its DS1 description is made in terms of the time T 1

of H1 and the m magnitudes X 1, ...,Xm of H1 which respectively interpret the m componentsC1, ..., Cm of the state space M of DS1. In addition, the emulation function u associates to anarbitrary n-tuple (y1, ..., yn) ∈ N an m-tuple u(y1, ..., yn) = (x1, ..., xm) ∈ M , in such a way thatany v-transition between two n-tuples of DS2 is transformed into a t-transition between the u-corresponding m-tuples of DS1. Therefore, the two interpretations IH2 and IH1 , together with theemulation function u, yield a correspondence between (i) the time T 2 of H2 and the time T 1 ofH1 and (ii) the n magnitudes Y 1, ...,Y n of H2 and the m magnitudes X 1, ...,Xm of H1.

This correspondence between magnitudes of the reduced-reducing models is somehow resem-blant of the correspondence between predicates in a Nagel-style reduction. In the latter kind ofreduction, the correspondence is provided by appropriate bridge-laws, while in the representationalmodel-based approach a similar correspondence is ensured by the emulation function, togetherwith the respective interpretations of the two models. In Nagel’s approach, bridge laws provide

13

connections between the vocabularies of two theories that do not share the same primitives (hetero-geneous reduction); in the representational approach, an analogous connection is ensured betweenthe different magnitudes employed by the two models.

While in simple cases, such as the reduction of the falling body model to the projectile model,the magnitudes of the reduced model may turn out to be a proper subset of those of the reducingmodel, in the general case the two sets of magnitudes are not required to be in the subset rela-tionship, and they are even allowed to be disjoint. Furthermore, while each bridge law relates apredicate of the reduced theory to a corresponding predicate of the reducing one, in the represen-tational approach the correspondence is not between pairs of magnitudes, but between an n-tupleof magnitudes on the one side and an m-tuple of different magnitudes on the other, so that thecorrespondence between the magnitudes of the two models has a global or wholistic character.The representational approach thus allows for heterogeneous reductions of a wider variety thanNagelian ones. Heterogeneous reductions of the Nagelian variety – i.e., where each magnitude ofthe reduced model corresponds but is not identical to a magnitude of the reducing one, are allowedas well, but only when n ≤ m.15

4.2.2 Case 2: BF2 ∩BF1 = ∅.

Let us suppose next that BF2 ∩ BF1 = ∅. In this case, no matter how DS1 and DS2 are related,DS2 is not reduced to DS1. For, even if DS2 is identical to DS1, any concrete system describedby DS2 (that is to say, any concrete system b ∈ BF2

) is not a system also described by DS1.

4.2.3 Case 3: BF2 ∩BF1 6= ∅ and ¬(BF2 ⊆ BF1).

The case BF2∩ BF1

6= ∅ and ¬(BF2⊆ BF1

) is still left. This case is a combination of theprevious two. In fact, for any concrete system b such that b ∈ BF2

∩ BF1, both the DS2 and the

DS1 descriptions apply to b; however, for any b such that b ∈ BF2and b /∈ BF1

, only the DS2

description applies to b. Thus, in this case, if DS 1 emulates DS 2, DS2 is incompletely reducedto DS1.

Multiple reduction of a model and multiple realization of a property. We have just seenthat case 3 only grants incomplete reduction of DS2 to DS1, provided that DS1 emulates DS2.However, DS2 may turn out to be multiply reduced to a family (DS j)j∈J = ((DSj , IHj ))j∈J ofempirically interpreted dynamical systems, each of which satisfies case 3 and emulates DS2. Thiswill be the case if the application domain BF2

is included in the union of all application domainsBFj . More precisely, for DS2 to be multiply reduced to (DS j)j∈J , it is sufficient that, for anyj ∈ J , BF2 ∩BFj 6= ∅, ¬(BF2 ⊆ BFj ), DSj emulates DS2, and BF2 ⊆

⋃j∈J BFj .

A relationship between this condition for multiple reduction and the concept of multiplerealization of a property (Fodor 1974; Kim 1998) is worth noticing. A property P is said to bemultiply realized by a family of properties (Pj)j∈J just in case (i) for any x, there is j ∈ J such

15Dizadji-Bahmani, Frigg, and Hartmann (2010, 399) deny that bridge laws always ensure that (a) every term ofthe reduced theory is connected to a term of the reducing one, and (b) a term of the reduced theory is connectedto exactly one term of the reducing one. This more liberal view of heterogeneous reduction is in better agreementwith the one provided here.

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that, if x has P , then x has Pj , and (ii) for any j ∈ J , there is y such that y has P and y has notPj . Any property Pj that satisfies both conditions (i) and (ii) is called a realizer of the property P .

Suppose now that DS2 is multiply reduced to (DS j)j∈J according to the previously stated suf-ficient condition. Let P2 be the property that corresponds to functional description F2 and, for anyj ∈ J , let Pj be the property that corresponds to functional description Fj . As BF2

⊆⋃j∈J BFj ,

it follows that, if x has P2, then x has Pj , for some j ∈ J , so that (i) is satisfied; furthermore, as¬(BF2 ⊆ BFj ), (ii) holds as well. Therefore, P2 is multiply realized by the family of properties(Pj)j∈J . Also, as BF2 ∩BFj 6= ∅, Pj is a realizer of P2, for any j ∈ J .

Since Fodor 1974, it has been customary to stress a tension between multiple realization onthe one side, and reduction on the other. According to standard anti-reductivist argument, thereis no type-type correlation between the high level properties of a special science and the low levelproperties of physics. Nevertheless, a high level property P of special science S may very well bemultiply realized by a family of physical properties (Pj)j∈J . If this is the case, P is not identical to,or coextensive with, any of the physical properties Pj . But theoretical reduction does require eitherproperty identity or property coextension. It thus follows that special science S is not reducibleto physics.

This argument relies on the key premise that, for reduction, either property identity or prop-erty coextension is needed. However, while this premise agrees with standard reading of Nagel’sparadigm of reduction (Nagel 1961), it is not obvious that a revised Nagelian paradigm needssuch a strong assumption. Marras (2002, sec. 6) sketches one such revision, where bridge-laws arenot biconditionals, and so they do not ensure property coextension or, a-fortiori, property identity.Thus, Marras’ revised paradigm, unlike Nagel’s original one, is consistent with multiple realization.

The representational, model-based, approach to reduction is not merely consistent with multi-ple realization. For, according to it, multiple realization turns out to be a necessary condition for aparticular kind of reduction, namely, multiple reduction of an interpreted model DS2 = (DS2, IH2

)to a family of interpreted models (DS j)j∈J = ((DSj , IHj ))j∈J . In fact, we have seen above thatthe sufficient condition for multiple reduction entails multiple realization of the property P2 thatcorresponds to the functional description F2 of phenomenon H2 = (F2, BF2). In addition, eachproperty Pj , which corresponds to the functional description Fj of phenomenon Hj = (Fj , BFj ),is a realizer of P2.

5 Partial and approximate emulation—sufficient conditionsfor partial and approximate reduction in empirically in-terpreted dynamical systems

From an intuitive point of view, the emulation relation holds between two dynamical systems DS1

and DS2 when all the dynamics of DS2 is exactly reproduced by DS1. I have argued so far that thisrelation might be the basis for a new approach to reduction, which I have called representational.However, it is well known that, in many cases of inter-theoretic reduction, the relation betweenthe reduced theory S 2 and the reducing one S 1 is such that S 2 is only partially and approximatelyreduced to S 1. Furthermore, such a relation typically is an asymptotic one, that is, it dependson some parameter p of either S1 or S2 in such a way that, for p tending to some fixed limiting

15

value p∗, S2 tends to be partially and approximately reduced to S1, as established according tothe limiting value p∗.16

The simple form of the emulation relation considered so far may very well be the basis fora representational account of total and exact reduction (like, for example, the reduction of thefalling body model DSeφ to the projectile model DSpφ,θ; see sec. 4.2.1, Case 1). Nevertheless, weneed a more sophisticated version of emulation for dealing with cases of asymptotic, partial andapproximate reduction. In this section, I suggest how this might be accomplished and provide (i) aformal definition of partial and approximate emulation (sec. 5.1), (ii) an example that shows howthe relation of partial and approximate emulation between the satellite model and the projectilemodel turns out to be asymptotic (sec. 5.2 and 5.3), (iii) general sufficient conditions for partialand approximate reduction in empirically interpreted dynamical systems (sec. 5.4) and, finally,(iv) a quite general asymptotic condition that ensures reduction up to an arbitrary approximationdegree (sec. 5.4, Limit Reduction Theorem LRT ).

5.1 Partial and approximate emulation

Intuitively, a dynamical system DS 1 = (M, (g t)t∈T ) partially emulates a second dynamical systemDS 2 = (N, (hv )v∈V ) if DS 1 exactly reproduces the dynamics of DS 2, limited to a fixed non-emptysubset C of DS 2’s state space N and, for any c ∈ C, to a fixed non-empty subset q(c) of DS 2’stime set V.

This concept is thus a straightforward relativization of def. [8]. Let 2V be the power-set ofV , C 6= ∅, C ⊆ N , and q : C → (2V − {∅}). The closure of C with respect to q, abbreviated ascl(C, q), is the following subset of the state space N of DS2: cl(C, q) = {a : a ∈ C or a = hv(c),for some c ∈ C and v ∈ q(c)}. Let us then define:

[9] DS 1 C-q-emulates DS 2 iff there is an injective function u : cl(C, q) → M such that forany c ∈ C, for any v ∈ q(c), there is t ∈ T such that u(hv (c)) = g t(u(c)). Any function u thatsatisfies the previous condition is called a C-q-emulation of DS 2 in DS1. The pair (C, q) is calledthe emulation space-time of DS2, and cl(C, q) its emulation domain.

Intuitively, DS 1 approximately emulates DS 2 if each state transition hv : y → y ’ of DS 2

approximately corresponds to a state transition g t : x → x ’ of DS 1. This idea can be made preciseby requiring that, for some injective function u such that u(y) = x, u(y ’) be sufficiently close tox ’, where the two states u(y ’), x ’ ∈ M are sufficiently close to each other if their distance does notexceed a fixed non-negative real δ. Thus, the concept of approximate emulation in fact presupposesthat M (i.e., the state space of DS 1) be equipped with a metric. Let d : M×M → R+ be a metricon M, let δ ∈ R+. We then define:

[10] DS 1 δ-emulates DS 2 iff there is an injective function u: N → M such that, for any v ∈ V,for any c ∈ N, there is t ∈ T such that d(u(hv (c)), g t(u(c))) ≤ δ. Any function u that satisfies

16Hooker (2004, 436) maintains that “asymptotics provides the ground on which claims about inter-theoreticexplanation, reduction and emergence must ultimately rest”. According to him, “in physics, we find that the mostfamous theory pairs are all asymptotically related” (2004, 437). Among such pairs, he explicitly mentions: (i) specialrelativity and Newtonian mechanics; (ii) optics and ray optics; (iii) quantum mechanics and Newtonian mechanics;(iv) statistical mechanics and thermodynamics (where, in each pair, the first element is the reducing theory andthe second element is the reduced theory). According to Hooker, an analogous relation may also hold between twodifferent models of the same theory; an example is the following pair of models of Newtonian mechanics: a nonlinear classic pendulum model and a harmonic oscillator model (2004, 438); in this regard, also see note 25.

16

the previous condition is called a δ-emulation of DS 2 in DS 1, and δ is called its approximationdegree. If, for some δ, u is a δ-emulation of DS2 in DS1, the minimum of all such δ must exist, forR satisfies the least upper bound property.17 Let δmin be such a minimum; δmin is then called u’sbest approximation degree. Thus, obviously, if, for some δ, u is a δ-emulation of DS2 in DS1, thenu is a δmin-emulation of DS2 in DS1 as well.

Finally, by combining definitions [9] and [10], we get a definition of the intuitive idea of partialand approximate emulation. Let 2V be the power-set of V, C 6= ∅, C ⊆ N, q : C → (2V − {∅}),cl(C, q) be the closure of C with respect to q, d : M ×M → R+ be a metric on M, and δ ∈ R+;

[11] DS 1 C-q-δ-emulates DS 2 iff there is an injective function u : cl(C, q) → M such thatfor any c ∈ C, for any v ∈ q(c), there is t ∈ T such that d(u(hv (c)), g t(u(c))) ≤ δ. Anyfunction u that satisfies the previous condition is called a C-q-δ-emulation of DS 2 in DS1, and δits approximation degree. The pair (C, q) is called the emulation space-time of DS2, and cl(C, q)its emulation domain. If, for some δ, u is a C-q-δ-emulation of DS2 in DS1, the minimum of allsuch δ, indicated by δmin, is called u’s best approximation degree.18 Thus, obviously, if, for someδ, u is a C-q-δ-emulation of DS2 in DS1, then u is also a C-q-δmin-emulation of DS2 in DS1.

5.2 The phenomenon of satellite motion and its model

Let Hs = (Fs , BFs ) be the phenomenon of satellite motion, where its functional description Fs

and its application domain BFs are specified as follows. The abstract type of real system ASFs

is any body within the gravitational field of the earth (or within the gravitational field of anyother planet, star, or similar body). The causal interaction scheme CSFs

consists in (i) the body’sinitiating its motion at an arbitrary instant and with any velocity and position (relative to theearth); (ii) during the whole motion, the only force acting on the body is the gravitational forceof the earth. BFs is then the set of all bodies within the gravitational field of the earth (or withinthe gravitational field of a similar body) that satisfy the given scheme of causal interactions. Anysuch body will be called a satellite.

Let us then consider the following system of four ordinary differential equations 〈dx (t)/dt =

x(t), dy(t)/dt = y(t), d x(t)/dt = −kmx (t)/(x (t)2

+ y(t)2)3/2, d y(t)/dt = −kmy(t)/(x (t)

2+

y(t)2)3/2〉, where k is a fixed real positive constant and m is a real positive parameter. The

solutions of such equations uniquely determine a dynamical system with four components DSs =(L, (f b)b∈B ), where B = R (the real numbers) and L = R4 −W , with W = {w : w = (0, 0, x, y),for some x, y ∈ R}.19

Let IHs be the following interpretation of DSs on the phenomenon of satellite motion Hs =(Fs , BFs

). In the first place, we think of the earth as a sphere whose mass is concentrated in itscenter, and we treat an arbitrary satellite as a particle of negligible mass; accordingly, we alsoassume that the earth position is fixed. For any satellite, we then consider the plane that containsits initial velocity vector and the earth center; on this plane, we fix both the x -axis and the y-axisof a Cartesian coordinate system, in such a way that its origin coincides with the earth center.

17According to the least upper bound property, for any non-empty subset A of R, if A has an upper bound, thenthe minimum of all upper bounds of A exists. Also recall that, for any B ⊆ R, m is the minimum of B iff m ∈ Band, for any b ∈ B, m ≤ b; u is an upper bound of B iff u ∈ R and, for any b ∈ B, b ≤ u.

18Such a minimum exists (see def. [10]).19Note that (x(t)2 + y(t)2)3/2 = 0 iff x(t) = 0 and y(t) = 0. Therefore, for any w ∈ W, w is not a possible state

of DSs, and so L = R4 −W .

17

We then interpret the first component X = R of the state space of DSs as the set of all possiblevalues of the x -axis component of the position (henceforth, x-position) of the satellite, the secondcomponent Y = R as the set of all possible values of its y-position, the third component X =R as the set of all possible values of its x-velocity, the fourth component Y = R as the set of allpossible values of its y-velocity, and the time set B of DSs as the set all possible values of physicaltime. Since all five of these magnitudes are measurable or detectable properties of the intendedphenomenon Hs, IHs is an empirical interpretation of DSs on Hs, and (DSs, IHs) = DS s is anempirical model of Hs; DS s will be called the satellite model. For appropriate values of constantk and parameter m the satellite model also turns out to be empirically correct.20

5.3 The satellite model partially and approximately emulates the pro-jectile model with respect to the latter’s empirical domain, and sucha relation turns out to be asymptotic

Let us now consider the satellite model DS s = (DSs, IHs) and the projectile model DSpφ,θ =

(DSp, IHpφ,θ ), where DSs = (L, (f b)b∈B ), DSp = (X×Y×X×Y , (g t)t∈T ), X = Y = X = Y =T = B = R (the real numbers) and L = R4−W , with W = {w : w = (0, 0, x, y), for some x, y ∈ R}.Recall that DSs and DSp are, respectively, the dynamical systems determined by the solutions of

differential equations 〈dx (t)/dt = x(t), dy(t)/dt = y(t), d x(t)/dt = −kmx (t)/(x (t)2

+ y(t)2)3/2,

d y(t)/dt = −kmy(t)/(x (t)2

+ y(t)2)3/2〉 and differential equations 〈dx (t)/dt = x(t), dy(t)/dt =

y(t), d x(t)/dt = 0, d y(t)/dt = −g〉. We take the constant k = G (the gravitational constant)and, by Newton’s law of universal gravitation, g = Gm/r2, where m is the mass of the earth andr its radius.

Let 0 ≤ φ ≤ ψ, 0 ≤ θ ≤ ρ, and Cφ,θ = {c : for some projectile a ∈ BFpφ,θ , for some y, x, y ∈ R,c = (0, y, x, y), (0, y) is the value of the initial position of a, and (x, y) is the value of the initialvelocity of a}.21 For any c ∈ Cφ,θ, let l(c) be the duration of the motion of any projectile releasedwith initial conditions c, and let the function qφ,θ : Cφ,θ → (2T − {∅}) be defined as follows: forany c ∈ Cφ,θ, qφ,θ(c) = {t : t ∈ T and 0 ≤ t ≤ l(c)}.

Then, as it can be readily verified by means of any dynamical systems software, for an appro-priately chosen δφ,θ > 0, for any c ∈ Cφ,θ, for any t ∈ qφ,θ(c), d(g t(c), f t(c)) ≤ δφ,θ, where d is the

usual Euclidean distance on X×Y×X×Y = R4. Let u be the identity function on cl(Cφ,θ, qφ,θ).

By def. [11], it thus follows that u is a Cφ,θ-qφ,θ-δφ,θ-emulation of DSp in DSs. Let δφ,θmin

be the

minimum of all such δφ,θ. Then, by def. [11], u is a Cφ,θ-qφ,θ-δφ,θmin

-emulation of DSp in DSs as

well, and so DSs Cφ,θ-qφ,θ-δφ,θmin

-emulates DSp.

It is important to keep in mind that δφ,θmin

represents the best approximation degree to whichDSs partially emulates DSp with respect to DSp’s emulation space-time (Cφ,θ, qφ,θ).

22 Besides,

20Such values are, respectively, the gravitational constant G and the mass of the gravitational source underexamination (earth, planet, star, or any other similar body).

21Values of position and velocity of an arbitrary projectile a are taken with respect to the coordinate systemspecified in sec. 4.2.1, par. 4; relative to that system, the x-coordinate of the initial position of a is always 0. Alsorecall that the possible initial values of the position and velocity of a are not completely arbitrary, but depend onboth φ and θ, for they must satisfy condition (i) of the specific causal interaction scheme CSFpφ,θ of the phenomenonof projectile motion (see sec. 4.2.1, par. 2). Cφ,θ is in fact the union, for any a, of each set of such values.

22(Cφ,θ, qφ,θ) can also be thought as the empirical space-time (see sec. 5.4, par. 2) of the projectile model DSpφ,θ

18

δφ,θmin

is a function of both φ ∈ [0, ψ] and θ ∈ [0, ρ]. Therefore, we can study the behavior of

δφ,θmin

for φ and θ simultaneously tending to 0 from the right, and it is not difficult to verify that

limφ,θ→0+ δφ,θmin

= 0 = δ0,0min

.

That is to say, for φ and θ simultaneously tending to 0 from the right, the best approximationdegree to which DSs partially emulates DSp with respect to emulation space-time (Cφ,θ, qφ,θ) tendsto the best approximation degree to which DSs partially emulates DSp with respect to emulationspace-time (C0,0, q0,0). In this precise sense, then, the relation of partial and approximate emulationof DSp by DSs (with respect to emulation space-time (Cφ,θ, qφ,θ), and to the best approximation

degree δφ,θmin

) turns out to be asymptotic (see sec. 5, first paragraph).

5.4 Sufficient conditions for partial and approximate reduction in em-pirically interpreted dynamical systems

Let us notice now that the satellite model DS s and the projectile model DSpφ,θ satisfy Case 1(sec. 4.2.1), for BFpφ,θ ⊂ BFs

(by the definitions of the respective application domains BFsand

BFpφ,θ ). Moreover, we have just seen (sec. 5.3, par. 3) that, for any φ ∈ [0, ψ] and θ ∈ [0, ρ], DSs

Cφ,θ-qφ,θ-δφ,θmin

-emulates DSp. The question then naturally arises whether this condition is sufficientfor reduction of DSpφ,θ to DS s .

On the one hand, we have seen above (sec. 5.3, par. 3 and 4) that (Cφ,θ, qφ,θ) is the emulationspace-time of DSp with respect to which dynamical system DSs partially emulates DSp to approx-

imation degree δφ,θmin

. On the other hand, we have also seen (sec. 5.3, par. 2 and note 21) that(Cφ,θ, qφ,θ) is determined, in force of interpretation IHpφ,θ , by the specific causal interaction schemeCSFpφ,θ of the phenomenon of projectile motion. As a consequence, (Cφ,θ, qφ,θ) can be thoughtas that part of DSp’s space-time that has an empirical interpretation according to IHpφ,θ . Let usthus call (Cφ,θ, qφ,θ) the empirical space-time of DSp relative to interpretation IHpφ,θ . In addition,let us call cl(Cφ,θ, qφ,θ) the empirical domain of DSp relative to IHpφ,θ , and E(Cφ,θ, qφ,θ) = {e:e = (c, g t(c)), for some c ∈ Cφ,θ and some t ∈ qφ,θ(c)} the empirical whole-substructure of DSprelative to interpretation IHpφ,θ .

23 Thus, by def. [11], all E(Cφ,θ, qφ,θ) is represented, through apartial emulation function u,24 by corresponding structure of DSs, within approximation degreeδφ,θmin

. Suppose now that ∆ > 0 is the desired approximation degree. Then, if δφ,θmin ≤ ∆, we can

safely conclude that DSpφ,θ is reduced to DS s .

In this connection, also recall that limφ,θ→0+ δφ,θmin

= 0 , where φ ∈ [0, ψ] and θ ∈ [0, ρ]

(sec. 5.3, par. 4). This means that the best approximation degree δφ,θmin

to which all the empiricalwhole-substructure E(Cφ,θ, qφ,θ) is represented by corresponding structure of DSs can be made assmall as we please, by taking sufficiently small values of both φ and θ. More precisely, for anydesired approximation degree ∆ > 0, there are sufficiently small φ∆ and θ∆ such that, for any φand θ, if 0 < φ < φ∆ and 0 < θ < θ∆, then δφ,θ

min

< ∆. In addition, recall that δ0,0min

= 0 (sec. 5.3,

par. 4); therefore, for any φ < φ∆ and θ < θ∆, δφ,θmin

< ∆. It thus follows that, for any φ < φ∆ andθ < θ∆, DSpφ,θ is reduced to DS s .

= (DSp, IHpφ,θ ) or, more precisely, of the dynamical system DSp relative to its empirical interpretation IHpφ,θ .23See van Fraassen 1980 for a general discussion of the concept of an empirical substructure.24Recall that, in this particular case, u is the identity function on cl(Cφ,θ, qφ,θ).

19

In the general case, let H = (F, BF ) be an arbitrary phenomenon, and DS = (DS, IH) beany empirically interpreted dynamical system such that DS is an empirical model of H ; let DS =(M, (gt)t∈T ). Let us assume first that, in force of interpretation IH , a one-to-one correspondencebetween the initial conditions specified by the causal interaction scheme CSF of phenomenon H(see sec. 4.1, par. 1) and a non-empty set of states of the dynamical system DS is fixed; letCF ⊆M be such a set. Second, if the causal interaction scheme CSF also specifies final conditions(see sec. 4.1, par. 1), let us also assume that, in force of interpretation IH , they uniquely determine,for any state c ∈ CF , the duration l(c) ∈ T (l(c) ≥ 0) of any evolution whose initial conditionscorrespond to state c; if CSF does not specify final conditions, let l(c) = T+, for any c ∈ CF .

We can thus define qF

: CF → (2T − {∅}) as follows. If l(c) = T+, qF

(c) = l(c); otherwise,qF

(c) = {t : t ∈ T and 0 ≤ t ≤ l(c)}. Then, (CF , qF ) is called the empirical space-time ofDS relative to interpretation IH ; the closure cl(CF , qF ) = {a : a ∈ CF or a = gt(c), for somec ∈ CF and t ∈ q

F(c)} is called the empirical domain of DS relative to IH , and E(CF , qF ) =

{e : e = (c, gt(c)), for some c ∈ CF and some t ∈ qF

(c)} is called the empirical whole-substructureof DS relative to IH .

Let H 1 = (F 1, BF1) and H 2 = (F 2, BF2) be two phenomena, and DS1 = (DS 1, IH1) andDS2 = (DS 2, IH2

) be two empirically interpreted dynamical systems such that DS1 is an empiricalmodel of H 1 and DS2 is an empirical model of H 2. Let (CF2

, qF2

) be the empirical space-time ofDS 2 relative to IH2

, and ∆ > 0 be the desired approximation degree for DS1 CF2-qF2

-δ-emulatingDS2. The previous example thus suggests that Case 1 (sec. 4.2.1) be supplemented with a weakersufficient condition for reduction, as follows.

Case 1a. Let us suppose that BF2 ⊆ BF1 . If DS 1 CF2-qF2-δ-emulates DS 2 and δ ≤ ∆, then DS2

is reduced to DS1.

A corresponding weaker condition can also be given for the case of incomplete reduction(Case 3, sec. 4.2.3), as follows.

Case 3a. Suppose that BF2∩ BF1

6= ∅ and ¬(BF2⊆ BF1

). If DS 1 CF2-qF2

-δ-emulates DS 2 andδ ≤ ∆, then DS2 is incompletely reduced to DS1.

As for multiple reduction to a family (DS j )j∈J = ((DS j , IHj ))j∈J of empirically interpreteddynamical systems, we get the following weaker condition. For DS2 to be multiply reduced to(DS j )j∈J , it is sufficient that, for any j ∈ J, BF2

∩ BFj 6= ∅, ¬(BF2⊆ BFj ), DS j CF2

-qF2

-δ-emulates DS 2, δj ≤ ∆, and BF2

⊆⋃

j∈J BFj .

A general asymptotic condition for approximate reduction with arbitrary precision.Finally, the previous example also suggests a general asymptotic condition that ensures reductionup to any desired approximation degree ∆ > 0. In fact, the following theorem25 is a straightforwardconsequence of Case 1a.

25Giunti has shown (2010, sec. 5) that the unrestricted pendulum model (a non linear model) and Galileo’s smallswing pendulum model (a harmonic oscillator) satisfy all conditions of theorem [LRT ]. It thus follows that thesecond model is reduced to the first one, whenenever pendulum swings are sufficiently small. Also see note 16.

20

Limit Reduction Theorem [LRT ]

• Let H 1 = (F 1, BF1) and H2φ1...φn = (F2φ1...φn , BF2φ1...φn) be two phenomena, where φ1 ∈ [0, ψ1],

..., φn ∈ [0, ψn] are real, non-negative, parameters;

• let DS1 = (DS 1, IH1) and DS2φ1...φn = (DS2, IH2φ1...φn) be two empirically interpreted dy-

namical systems such that DS1 is an empirical model of H 1 and DS2φ1...φn is an empiricalmodel of H2φ1...φn ;

• let (CF2φ1...φn, qF2φ1...φn

) be the empirical space-time of DS2 relative to interpretation IH2φ1...φn.

If, for any φ1, ..., φn, BF2φ1...φn⊆ BF1 , and there is u such that, for any φ1, ..., φn, for some δφ1...φn ,

(i) u is a CF2φ1...φn-qF2φ1...φn

-δφ1...φn-emulation of DS2 in DS1;

(ii) limφ1...φn→0+ δminφ1...φn = 0 = δmin0...0 ;

then, for any desired approximation degree ∆ > 0, there are φ1∆ , ..., φn∆ such that, for anyφ1 < φ1∆ , ..., φn < φn∆ , DS2φ1...φn is reduced to DS1.

Proof

a. Hypothesis: For any φ1, ..., φn, BF2φ1...φn⊆ BF1 ;

b. hypothesis: u be such that, for any φ1, ..., φn and some δφ1...φn , it satisfy (i) and (ii);

c. by b, (i) and def. [11]: For any φ1, ..., φn, u is a CF2φ1...φn-qF2φ1...φn

-δminφ1...φn-emulation of DS2 in

DS1;

d. by c and def. [11]: For any φ1, ..., φn, DS1 CF2φ1...φn-qF2φ1...φn

-δminφ1...φn-emulates DS2;

e. by b and the first equality of (ii): For any desired approximation degree ∆ > 0, there areφ1∆

, ..., φn∆such that, for any φ1, ..., φn, if 0 < φ1 < φ1∆

, ..., 0 < φn < φn∆, then δminφ1...φn < ∆;

f. by e and the second equality of (ii): For any desired approximation degree ∆ > 0, there areφ1∆

, ..., φn∆such that, for any φ1 < φ1∆

, ..., φn < φn∆, δminφ1...φn < ∆;

g. by case 1a, a, d and f : For any desired approximation degree ∆ > 0, there are φ1∆, ..., φn∆

suchthat, for any φ1 < φ1∆

, ..., φn < φn∆, DS2φ1...φn is reduced to DS1. Q.E.D.

6 Concluding remarks

This paper has developed a representational model-based approach to reduction for the specialcase of dynamical systems. Contrary to the received view, reduction has been analyzed in termsof a representational relation between models, rather than a deductive relation between theories.Namely, reduction has been construed as a manifestation of the underlying representational relationof emulation.

As said, the representational view of reduction has been developed so far only for the specialcase of dynamical systems (either empirically interpreted, or not). However, even in this specialform, the theory is far from being complete. I will mention here just two basic points that shouldbe further investigated and expanded.

The emulation relation between two dynamical systems DS 1 = (M, (g t)t∈T ) and DS 2 =(N, (hv )v∈V ) has been considered in two different forms (def. [8] and [11]), which are, respectively,based on a total and exact structure preserving mapping u, or on a partial and approximate

21

one. The crucial point is that the mapping u preserve (exactly or approximately) DS 2’s whole-structure (all of it, or just its empirically interpreted part) in DS 1’s whole-structure, and this hasbeen obtained by taking u to be an injective function from N to M. Yet, it might be possible toobtain analogous results by either dropping the injectivity requirement on u, or by taking u tobe a function from M to N. Therefore, the theory developed so far should be reevaluated, andpossibly revised or completed, in the light of more general assumptions on the structure preservingmapping u.

Nevertheless, even a complete representational theory for dynamical systems would not besufficient to account for all relevant cases of reduction, for many models in real science are not ofthis kind. What we need is a general representational theory, as precise as the one restricted todynamical systems, which apply to arbitrary models. The formulation of such a general theory,however, is not an easy matter, for it involves a preliminary investigation of fairly hard questionslike: What is, in general, a purely mathematical model?26 What is a structure preserving mappingbetween two arbitrary mathematical models? What is the relation between two arbitrary math-ematical models that generalizes the one of emulation between dynamical systems? What is, ingeneral, a phenomenon and an empirical interpretation of a mathematical model on it?

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