The Empirical Under-determination Argument Against Scientific Realism for Dual Theories
aa r X i v : . [ phy s i c s . h i s t - ph ] J a n The Empirical Under-determination ArgumentAgainst Scientific Realism for Dual Theories
Sebastian De Haro
Institute for Logic, Language of Computation, University of Amsterdam Institute of Physics, University of AmsterdamVossius Center for History of Humanities and Sciences, University of Amsterdam [email protected]
January 5, 2021
Abstract
This paper explores the options available to the anti-realist to defend a Quinean em-pirical under-determination thesis using examples of dualities. I first explicate a version ofthe empirical under-determination thesis that can be brought to bear on theories of con-temporary physics. Then I identify a class of examples of dualities that lead to empiricalunder-determination. But I argue that the resulting under-determination is benign, andis not a threat to a cautious scientific realism. Thus dualities are not new ammunition forthe anti-realist. The paper also shows how the number of possible interpretative optionsabout dualities that have been considered in the literature can be reduced, and suggestsa general approach to scientific realism that one may take dualities to favour. Forthcoming in
Erkenntnis. ontents Introduction
Over the last twenty to thirty years, dualities have been central tools in theory construc-tion in many areas of physics: from statistical mechanics to quantum field theory toquantum gravity. A duality is, roughly speaking, a symmetry between two (possibly verydifferent-looking) theories. So in physics: while a symmetry typically maps a state of thesystem into another appropriately related state (and likewise for quantities); in a duality,an entire theory is mapped into another appropriately related theory. And like for sym-metries, there is a question of under what conditions dual theories represent empiricallyequivalent situations. Indeed, in some cases, physicists claim that dual pairs of theoriesdescribe the very same physical, not just the very same empirical, facts (more on this inSection 3).Thus there is a natural question of whether dualities can generate interesting examplesof empirical under-determination, or under-determination of theory by empirical data.The empirical under-determination thesis says that, roughly speaking, ‘physical theoryis underdetermined even by all possible observations... Physical theories... can be logicallyincompatible and empirically equivalent’ (Quine, 1970: p. 179). Despite the wide philosophical interest of the empirical under-determination thesis,it is controversial whether there are any genuine examples of it (setting aside under-determination ‘by the evidence so far’). Quine himself regarded this as an ‘open question’.One well-known example is the various versions of non-relativistic quantum mechanics,including its different interpretations; however, it is controversial whether these are casesof under-determination by all the possible evidence, or by all the evidence so far . Likewise,Laudan and Leplin (1991: p. 459) say that most examples of under-determination arecontrived and limited in scope:
It is noteworthy that contrived examples alleging empirical equivalence always in-volve the relativity of motion; it is the impossibility of distinguishing apparent fromabsolute motion to which they owe their plausibility. This is also the problem in thepre-eminent historical examples, the competition between Ptolemy and Copernicus,which created the idea of empirical equivalence in the first place, and that betweenEinstein and H. A. Lorentz.
But dualities certainly go well beyond relative motion and the interpretation of non-relativistic quantum mechanics. For they are at the centre of theory construction intheoretical physics: and so, if they turned out to give cases of under-determination, thiswould show that the problem of under-determination is at the heart of current scientificresearch. And since philosophers have thought extensively about under-determination,this old philosophical discussion could contribute to the understanding of current devel-opments in theoretical physics. The qualification ‘all possible’ distinguishes this type of under-determination thesis from the under-determination of theories by the available evidence so far, also called ‘transient under-determination’(Sklar, 1975: p. 380), which is of course a very common phenomenon. Transient under-determinationtends to blur the distinction between the limits of our current state of knowledge and understanding ofa theory vs. the theory’s intrinsic limitations. Furthermore, under-determination by all possible evidenceis closer to discussions of dualities. For these reasons, in this paper I restrict attention to the originalQuinean under-determination thesis. not illustrate under-determination, then this particular argument against scientific realism would be under-mined. Either way, the question of whether dualities give cases of under-determinationdeserves scrutiny.So far as I know, and apart from an occasional mention, the import of dualities forthe under-determination debate has so far only been studied in a handful of previouspapers: see early papers by Dawid (2006, 2017a), Rickles (2011, 2017), Matsubara (2013)and recent discussions by Read (2016) and Le Bihan and Read (2018). Given that arich Schema for dualities now exists equipped with the necessary notions for analysingunder-determination, and several cases of rigorously proven dualities also exist, the timeis ripe for a study of the question of empirical under-determination vis-`a-vis dualities.In this paper, I aim to undertake a study of the main options available to produceexamples of empirical under-determination using dualities. Thus, rather than giving aresponse to the various under-determination theses or defending scientific realism, myaim is to analyse whether dualities offer genuine examples of under-determination—as itis reasonable to expect that they do. We will find that, although the examples of dualitiesin physics do illustrate the under-determination thesis, they do so in a benign way, i.e. theyare not a threat to—cautious forms of—scientific realism. In particular, I will argue that—leaving aside cases of transient under-determination —not all of the interpretative optionsthat have been considered by Matsubara (2013) and Le Bihan and Read (2018) are distinctor relevant options for the problem of emprical under-determination, and that ultimatelythere is a single justified option for the cautious scientific realist. Thus dualities may betaken to favour a cautious approach to scientific realism.In Section 2, I introduce the Quinean empirical under-determination thesis. Section3 introduces the Schema for duality. Section 4 then explores whether dualities give casesof under-determination, and whether scientific realism is in trouble. Section 5 concludes. This Section introduces the Quinean empirical under-determination thesis, and the alliedconcepts of empirical and theoretical equivalence.
The word ‘under-determination’ is notoriously over-used in philosophy of science. Theunder-determination thesis that I am concerned with here is what Quine (1975: p. 314)called ‘empirical under-determination’, which is not to be confused with the Duhem-Quine For a discussion of the import of dualities for the question of scientific realism, see Dawid (2017a). See De Haro and Butterfield (2017), De Haro (2019, 2020), and Butterfield (2020). Transient under-determination for dualities is discussed in Dawid (2006). Quine gives several different formu-lations of his own thesis, of which the simplest version is:
Quinean under-determination : two theory formulations are under-determined if theyare empirically equivalent but logically incompatible.Under-determination theses are prominent as arguments against scientific realism. For ifthe same empirical facts can be described by two theories that contradict one another,why should we believe what one of the theories says? There are, roughly, two strategies for realists to try to respond to the challenge ofempirical under-determination, as follows:(i) One can try to undercut the under-determination threat through appropriate mod-ifications of one’s notions of equivalence, so that on the correct notions there is nounder-determination after all, i.e. what one thought were examples of empirical under-determination turn out not to be. Since under-determination involves two notions ofequivalence in tension (under-determination “lives in the space between empirical andlogical equivalence”), this can be done in two ways. First, one can try to argue thatan appropriate notion of empirical equivalence is sufficiently strong, so that the class ofpotential examples of under-determination is reduced (i.e. because the two theory for-mulations in the putative example are not empirically equivalent). Or one can try toargue that the appropriate notion of logical equivalence (or, more generally, theoreticalequivalence: see Section 2.3) is sufficiently weak, so that the class of potential examples ofunder-determination is reduced (i.e. because the criterion of logical equivalence adoptedis liberal, and makes the pairs of theory formulations in the putative examples logicallyequivalent).(ii) One can try to use alternative assessment criteria that do not involve modifica-tions of the notions of equivalence. These criteria may be empirical, super-empirical, oreven non-empirical. They are often forms of ampliative inference which offer more “em-pirical support” to one theory than the other; or one theory may be “better confirmed bythe evidence than the other” (Laudan, 1990: p. 271).I take Quine’s challenge of under-determination to be primarily about (i) not (ii). Thisfollows from the formulation of the challenge itself. Thus my position is that in so far asQuinean under-determination poses a challenge to scientific realism, one must aspire tomeet it in its own terms, i.e. by adopting the first strategy. Lyre (2011: p. 236) gives an interesting discussion of the relation and contrast between under-determination, on the one hand, and Duhem-Quine holism and Humean under-determination, on theother. I concur with Stanford (2006: p. 17) in rejecting the use of fictitious examples to put forward under-determination theses: ‘[T]he critics of underdetermination have been well within their rights to demandthat serious, nonskeptical, and genuinely distinct empirical equivalents to a theory actually be producedbefore they withhold belief in it and refusing to presume that such equivalents exist when none can beidentified.’
5f (i) turns out to fail, so that we need to adopt strategy (ii) instead, this might still notbe a blow to scientific realism, since (ii) can indeed solve the empirical under-determinationproblems that the practicing scientist might be concerned with. In particular, if we adoptthe strategy (i), we may still be left with some cases of under-determination, and then wemay, and should, still ask: which of these two theory formulations is better confirmed,and which one should we further develop or accept? That is, the strategy (ii) remainsrelevant.Also, one should note that, although the strategy (i) is semantic and (ii) is epistemic,epistemic considerations are relevant to both approaches, since we are interested in the-ories that are proposed as candidate descriptions of the actual world. Thus in variousparts of the paper, when I talk about holding a certain thesis (e.g. giving a verdict oftheoretical equivalence), I will also consider the justification that we have for this thesis.Fortunately so far, for realists, cases of under-determination seem hard to come by.Quine (1975: p. 327) could find no good examples, and concluded that it is an ‘open ques-tion’ whether such alternatives to our best ‘system of the world’ exist. (See also Earman(1993: pp. 30-31)). As late as 2011, Lyre (2011: p. 236) could write that empirical under-determination ‘suffers from a severe but rarely considered problem of missing examples’. In the rest of the paper, I will discuss strategy (i), and how dualities address this prob-lem of missing examples. As we will see, my analysis will in effect be as follows. (A) I willonly minimally strengthen the notion of empirical equivalence (in Section 2.2), althoughI will also use this notion in a liberal way (in Section 4.3). (B) I will also strengthen thenotion of equivalence that is to replace Quine’s condition of logical equivalence. Thusthe effects of my two modifications on the empirical under-determination thesis are intension with each other. (A) strengthens empirical equivalence, which should help strat-egy (i) (cf. the beginning of this Section), but I will also allow a liberal use of it, whichagain runs against (i). (B) is a substantial strengthening of the logical criteria of equiva-lence. So the overall effect of my analysis will not necessarily be good news for scientificrealists—and this is of course the case we should consider, if we are to make the challengeto scientific realism as strong as possible. My liberal use of (A) will be motivated by howphysicists use dualities, and by a discussion of the semantic and syntactic conceptionsof empirical equivalence. The strengthening (B) will also be based on work on dualities,and how dualities give us a plausible criterion of theory individuation, that was developedindependently of the question of empirical under-determination. Laudan and Leplin’s (1991: p. 458) example, the “bason”, illustrates how hard-pressed the groupof philosophers involved in the empirical under-determination debate have sometimes been to find whatremotely look like genuine physics examples. The “bason” is a mythical particle invented to detectabsolute motion: it fortuitously arises as a result of absolute motion, and ‘the positive absolute velocityof the universe represents energy available for bason creation’. Unfortunately, this cunning philosophicalthought experiment hardly deserves the name ‘scientific theory’. See my endorsement of Stanford’scritique of the use of fictitious theories, in footnote 7. Lyre (2011: pp. 237-241) contains a useful classification of some available examples. I agree withLyre’s verdict about most of these examples being cases of transient under-determination, so that ‘inretrospect such historic cases appear mainly as artefacts of incomplete scientific knowledge—and do assuch not provide really worrisome... cases’ of empirical under-determination (p. 240). .2 Empirical equivalence Quine’s (1975: p. 319) criterion of empirical equivalence is syntactic: two theories are empirically equivalent if they imply the same observational sentences, also called obser-vational conditionals, for all possible observations—present, past, future or ‘pegged toinaccessible place-times’ (p. 234)Another influential and, as we will see, complementary account of the meaning of‘empirical’ is by van Fraassen (1980: p. 64), who puts it thus:
To present a theory is... to present certain parts of those models (the empiricalsubstructures ) as candidates for the direct representation of observable phenomena.The structures which can be described in experimental and measurement reportswe call appearances : the theory is empirically adequate if it has some model suchthat all appearances are isomorphic to empirical substructures of that model.
Van Fraassen famously restricts the scope of ‘observable phenomena’ to observation bythe unaided human senses. Accordingly, his mention of ‘experimental and measurementreports’ is restricted to certain kinds of experiments and measurements. Thus I will setvan Fraassen’s notion of observability aside but keep his notion of empirical adequacy asa useful semantic alternative to Quine’s syntactic construal of the empirical—and in thissense the two views are complementary to each other.In Section 4.3, I will give a judicious reading of these two criteria of empirical equiva-lence, which will give us a verdict that duals, i.e. dual theories, are empirically equivalent,as a surprising but straightforward application of van Fraassen’s and Quine’s proposals.
The second notion entering the definition of under-determination —namely, logical equiv-alence—was replaced by Quine, and later by others, by other (weaker) notions, oftenunder the heading of ‘theoretical equivalence’. In this Section, I will introduce some ofthese notions, and then present my own account, following De Haro (2019).Note that ‘logical equivalence’ is defined in logic books as relative to a vocabulary orsignature, and so it is obviously too strict. For example, one would not wish to countFrench and English formulations of the theory of electrodynamics as different theories,while they would count as logically inequivalent by the criterion in logic books, since theirvocabularies are different.Quine also argues that logical equivalence is too strong a criterion. He proposes thefollowing criterion of equivalence between theory formulations: ‘I propose that we counttwo formulations as formulations of the same theory if, besides being empirically equiv-alent, the two formulations can be rendered identical by switching predicates in one of For example, van Fraassen’s conception of observability rules out collider experiments and astro-nomical observations, where the reports are based on computer-generated data that encode observationsby artificial devices. A conception of observability that is more straightforwardly applicable to modernphysics is in Lenzen (1955). See, for example, Hodges (1997: pp. 37-38). n -placepredicates to n -variable formulas). For a formalization of this, see Barrett and Halvorson(2016: pp. 4-6). I will follow these authors in calling this new kind of theoretical equiva-lence Quine equivalence . Thus we arrive at:
Quinean under-determination: two theory formulations are under-determined if theyare empirically equivalent but there is no reconstrual of predicates that renders themlogically equivalent (i.e. they are Quine-inequivalent).Barrett and Halvorson (2016: pp. 6-8) argue that Quine equivalence is too liberal acriterion for theoretical equivalence (which, in turn, means that one ends up with a strictnotion of empirical under-determination, which will have few instantiations). Indeed, theygive several examples of theories that are equivalent according to Quine’s criterion, butthat one has good reason to consider inequivalent. They then introduce another criterion,due to Glymour (1970), that better captures what one means by intertranslatability, i.e. the existence of a suitable translation between two theory formulations. The criterionis, roughly speaking, the existence of reconstrual maps “both ways”, i.e. from T to T ′ andback. They show that, in first order logic, intertranslatability is equivalent to the notionof definitional equivalence : which had already been defined in logic and advocated forphilosophy of science by Glymour (1970: p. 279). Since the criterion of intertranslatabilityis Quine’s criterion taken “both ways”, it can be seen as an improvement of it.In De Haro (2019), I argued that there is an interesting project of finding criteria ofequivalence that are mostly formal, while the full project, of formal plus interpretativeequivalence that I am interested in here, requires the consideration of ontological matters.In particular, theoretical equivalence requires that the interpretations are the same. Thuswe can give the following definition: Theoretical equivalence: two theory formulations are theoretically equivalent if theyare formally equivalent and, in addition, they have the same interpretations.The following Sections will further articulate the above definition. Thus pace
Quine, thecriterion of individuation of theories that is relevant to scientific theories is not merelyformal, but is a criterion of theoretical equivalence. Interpretation matters in science, and two theories can only be said to be equivalent if they have the same interpreta-tions, i.e. they have the same ontology. Thus taking theoretical equivalence as our notionof equivalence, we arrive at the following conception of under-determination, based onQuine’s original notion, but now with the correct criterion of individuation of physicaltheories: Empirical under-determination: two theory formulations are under-determined ifthey are empirically equivalent but theoretically inequivalent. See e.g. Coffey (2014).
8s we will see in the next Section, the most straightforward way to look for examples of empirical under-determination is if the theories have different interpretations . Thisis because, as I argued above, empirical under-determination is primarily a matter ofmeaning, interpretation, and ontology.In what follows, we will always deal with theory formulations, and so will not need todistinguish between ‘theory’ and ‘theory formulation’. So for brevity, I will from now onoften talk of ‘theories’ instead of ‘theory formulations’.
This Section introduces the notion of duality, along with our Schema (with Butterfield)for understanding it, and a few examples. The Schema encompasses our overall treat-ment of dualities: which comprises the notions of bare theory, interpretation, duality,and theoretical equivalence and its conditions (Section 3.1); and two different kinds ofinterpretations, dubbed ‘internal’ and ‘external’ (Section 3.3). Section 3.2 discusses twoexamples of dualities.
In this Section, I illustrate the Schema for dualities with an example from elementaryquantum mechanics.Consider position-momentum duality in one-dimensional quantum mechanics, repre-sented on wave-functions by the Fourier transformation. For every position wave-functionwith value ψ ( x ), there is an associated momentum wave-function whose value is denotedby ˜ ψ ( p ), and the two are related by the Fourier transformation as follows: ψ ( x ) = 1 √ π ~ Z ∞−∞ d p ˜ ψ ( p ) e i ~ x p (1)˜ ψ ( p ) = 1 √ π ~ Z ∞−∞ d x ψ ( x ) e − i ~ x p . Likewise for operators: any operator can be written down in a position representation, A , or in a momentum representation, ˜ A . Because of the linearity of the Fourier transfor-mation, all the transition amplitudes are invariant under it, so that the following holds(using standard textbook bra-ket notation for the inner product): h ψ | A | ψ ′ i = h ˜ ψ | ˜ A | ˜ ψ ′ i . (2)Since the Schr¨odinger equation is linear in the Hamiltonian and in the wave-function, theequation can itself be written down and solved in either representation. The full Schema is presented in De Haro (2020), De Haro and Butterfield (2017), Butterfield (2020),and De Haro (2019). Further philosophical work on dualities is in Rickles (2011, 2017), Dieks et al. (2015),Read (2016), De Haro (2017), Huggett (2017), Read and Møller-Nielsen (2020). See also the special issueCastellani and Rickles (2017). duality is an isomorphism between theories, which I will denote by d : T → T ′ . States, quantities,and dynamics are mapped onto each other one-to-one by the duality map, while the valuesof the quantities, Eq. (2), which determine what will be the physical predictions of thetheory, are invariant under the map.Notice the dependent clause ‘which determine what will be the physical predictionsof the theory’. The reason for the insertion of the italicised phrase is, of course, thatmy talk of quantum mechanics has so far been mostly formal. Unless one specifies aphysical interpretation for the operators and wave-functions in Eq. (2), the formalism ofquantum mechanics does not make any physical predictions at all, i.e. only interpreted theories make physical predictions. Without an interpretation, the formalism of quantummechanics could equally well be describing some probabilistic system that happens toobey a law whose resemblance with Schr¨odinger’s equation is only formal.This prompts the notions of bare theory and of interpretation which, together, formwhat we call a ‘theory’. A bare theory is a theory (formulation) before it is given aninterpretation—like the formal quantum mechanics above: it was defined in terms of itsstates (the set of wave-functions), quantities (operators) and dynamics (the Schr¨odingerequation). Thus it is useful to think of a bare theory in general as a triple of states,quantities, and dynamics. I will usually denote bare theories by T , and a duality will bean isomorphism between bare theories . Interpretations can be modelled using the idea of interpretation maps . Such a mapis a structure-preserving partial function mapping a bare theory (paradigmatically: thestates and the quantities) into the theory’s domain of application, i.e. appropriate objects,properties, and relations in the physical world. I will denote such a (set of) map(s) by: i : T → D . The above notions of bare theory, of interpretation as a map, and of duality as anisomorphism between bare theories, allow us to make more precise the notion of theoreticalequivalence, from Section 2.3. First, suppose that we have two bare theories, T and T ,with their respective interpretation maps, i and i , which map the two theories intotheir respective domains of application, D and D . Further, assume that there is a One may object that the position and momentum representations of quantum mechanics are not twodifferent bare theories, but two different formulations of the same theory. But recall that I have adopted‘theory’ instead of ‘theory formulation’ for simplicity, so that it is correct to consider a duality relatingtwo formulations of the same theory. If a bare theory is presented as a triple of states, quantities, and dynamics, then the interpretationis a triple of maps, on each of the factors. However, I will here gloss over these details: for a detailedexposition of interpretations as maps, the conditions they satisfy, and how this formulation uses referentialsemantics and intensional semantics, see De Haro (2019, 2020) and De Haro and Butterfield (2017). d ←→ T i ց ւ i D Figure 1:
Theoretical equivalence. The two interpretations describe “the same sector of reality”,so that the ranges of the interpretations coincide. T d ←→ T y i y i D = D Figure 2:
Theoretical inequivalence. The two interpretations describe “different sectors ofreality”, so that the ranges of the interpretations differ. duality map between the two bare theories, d : T → T , i.e. an isomorphism as justdefined. Theoretical equivalence can then be defined as the condition that the domainsof application of the two theories are the same, i.e. D := D = D , as in Figure 1, sothat the two interpretation maps have the same range. Note that, while the domains ofapplication of theoretically equivalent theories are the same, the bare theories T and T are different, and so the theories are equivalent but not identical .As I mentioned in the definition of empirical under-determination in Section 2.3, inthis paper we are mostly interested in situations in which the two theories are theoretically in equivalent, i.e. the ontologies of two dual theories are different. This means that theranges of the interpretation maps, i.e. the domains of application of the two theories,are distinct, so that: D = D . Thus the diagram for the three maps, d, i , i , does notclose: the diagram for theoretical inequivalence is a square, as in Figure 2. This notionof theoretical inequivalence makes precise the first condition for under-determination, atthe end of Section 2.3.This will allow us, in Section 4, to give precise verdicts of under-determination. Inorder to do that, we also need to make more precise the second condition for under-determination in Section 2.3, i.e. the notion of empirical equivalence. We turn to thisnext.To illustrate the notion of empirical equivalence, let me first briefly discuss the stan-dard textbook interpretation of quantum mechanics, in the language of the Schema—theother interpretations are obtained by making the appropriate modifications. Since myaim here is only to illustrate the Schema, rather than to try to shed light on quantummechanics itself, I will set aside the measurement problem, and simply adopt the stan-dard Born rule for making probabilistic predictions about the outcomes of measurements.On this understanding, the interpretation map(s) i map the bare theory to its domain of11pplication as follows: i ( x ) = ‘the position, with value x , of the particle upon measurement’ i (cid:0) | ψ ( x ) | (cid:1) = ‘the probability density of finding the particle at position x , upon measurement’ i ( ψ ) = ‘the physical state of the system (the particle)’ , etc., where x is an eigenvalue of the position operator. Although all of the above notions (outcomes of measurements, Born probabilities,physical states of a system, etc.) are in the domain D , and they all enter in the assess-ment of theoretical equivalence, not all of them are relevant for empirical equivalence,since not all of them qualify as ‘observable’, in the sense of Section 2.2. For while theposition of a particle can be known by a measurement of the particle’s position on adetection screen or in a bubble chamber, the state of a particle cannot be so known:thus it is a theoretical concept. On the other hand, a probability (on the standard—andadmittedly simplistic!—frequentist interpretation of probabilities in quantum mechanics,where probabilities are relative frequencies of events for an ensemble of identically pre-pared systems) is an observational concept, since it can be linked to relations betweenmeasurements (i.e. frequencies), even if it does not itself correspond to an outcome of ameasurement or physical interaction.Thus in the case of quantum mechanics, the empirical substructures in van Fraassen’sdefinition in Section 2.2 are the subsets of structures of the theory that correspond tooutcomes of measurements, in a broad sense, and relations between measurements. Thesestructures are, quite generally, the set of (absolute values of) transition amplitudes of self-adjoint operators, Eq. (2), and expressions constructed from them, including powers oftransition amplitudes. These are the empirical substructres (on van Fraassen’s semanticconception), which make true or false the observational conditionals of the theory (Quine’sphrase). The remaining terms of the theory may have a physical significance, but theyare theoretical. Thus this discussion distinguishes, for elementary textbook quantummechanics, between what is theoretical and what is empirical. I will now give two other important examples of dualities: T-duality in string theory, andblack hole-quark-gluon plasma duality.To this end, I will first say a few words about string theory. String theory is ageneralisation of the theory of relativistic point particles to relativistic, one-dimensionalextended objects moving in time, i.e. strings. On analogy with the world-line swept outby a point particle, the two-dimensional (one space plus one time dimension!) surface See for example van Fraassen (1970: pp. 329, 334-335). Note that there are different kinds of elements that are here being mapped to in the domain. Thefirst example is a standard extensional map, which gives a truth value to an observational conditional ina straightforward way. The second maps to the probability of an outcome over measurements of manyidentically prepared systems, rather than measurement of a single system. The third map describes thephysical situation of a particle with given properties: thus the map’s range is a concrete rather than anabstract object. R .T-duality first appears when one of these ten dimensions is compact, for example acircle of radius R . Thus the spacetime is R × S , where S is the circle. The periodicityof the circle then entails that the centre-of-mass momentum of a string along this directionis quantised in units of the radius R ; so p = n/R , where n is an integer. Furthermore,closed strings can wind around the circle (this is only possible if the spacetime is R × S not R ): so they have, in addition to momentum, an additional winding quantum number, m , which counts the number of times that a string wraps around the circle. Now thecontribution of the centre-of-mass momentum, and of the winding of the string aroundthe circle, to the square of the mass M of the string, is quadratic in the quantum numbers,as follows: M nm = n R + m R ℓ , (3)where the subscript nm indicates that this is the contribution of the momentum and thewinding around the circle—there are other contributions to the mass that are independentof the momentum, winding, and radius, which I suppress—and ℓ s is the fundamentalstring length, i.e. the length scale with respect to which one measures distances on theworld-sheet. We see that the contribution to the mass, Eq. (3), is invariant under thesimultaneous exchange: ( n, m ) ↔ ( m, n ) (4) R ↔ ℓ /R . In fact, one can show that the entire spectrum, and not only the centre-of-mass contri-bution Eq. (3), is invariant under this map (cf. Zwiebach (2009: pp. 392-397), Polchinski(2005: pp. 235-247)). This is the basic statement of T-duality: namely, that the theoryis invariant under Eq. (4), i.e. the exchange of the momentum and winding quantumnumbers, in addition to the inversion of the radius. T-duality is a duality in the sense of the previous Section, in that it maps states oftype IIA string theory to states of type IIB, and likewise for quantities, while leaving all For an early review, see Schwarz (1992). See also Huggett (2017), Read (2016), Butterfield (2020). Ifollow the physics convention of setting ~ = 1. Taking into account how T-duality acts on the fermionic string excitations, one can show that itexchanges the type IIA and type IIB string theories compactified on a circle. show that nothing that I will have to say dependson the conjectural status of the string theory dualities. Indeed, my analysis here will beindependent of those details. In the previous two Sections, I defined dualities formally (Section 3.1) and then I gavetwo examples (Section 3.2). The examples already came with an interpretation, i.e. Idiscussed not just the bare theories but the interpreted theories, since that is the bestway to convey the relevant physics. In this Section, I will be more explicit about the kindsof interpretations that can lead to theoretically equivalent theories.Not all interpretations lead to theoretically equivalent theories: and this fact—thattheoretical equivalence is not automatic—creates space for empirical under-determination.For, as I will argue in the next Section, the interpretations that fail to lead to theoreticalequivalence introduce the possibility of empirical under-determination.Which interpretations lead to theoretically equivalent theories, as in Figure 1, or totheoretically inequivalent ones, as in Figure 2? Note that theoretical equivalence requiresthat the domains of the two theories (the ranges of the interpretation maps) are thesame, i.e. the two theories have the same ontology. This imposes a strong conditionon the interpretations of two dual theories: whereas theoretical inequivalence comes ata low cost. Thus one expects that, generally speaking, an interpretation of two dual Another example of a mathematically proven duality is bosonization, or the duality between bosonsand fermions in two dimensions. See De Haro and Butterfield (2017: Sections 4 and 5). Cf. also Butterfield(2020). As I stress in this Section, duality does not automatically give theoretical equivalence, because dualtheories can have different ontologies. external .An external interpretation is best defined in contrast with an internal interpretation,which maps all of and only what is common to the two theory formulations, i.e. an in-ternal interpretation interprets only the invariant content under the duality. An externalinterpretation, by contrast, also interprets the content that is not invariant under theduality—thereby typically rendering two duals theoretically, and potentially also empiri-cally, inequivalent.I will dub this additional structure, which is not part of what I have called the ‘in-variant content’, the specific structure.
An isomorphism of theories maps the states,quantities, and dynamics, but not the specific structure, which is specified additionallyfor each specific theory formulation. Indeed, theory formulations often contain struc-ture (e.g. gauge-dependent structure) beyond the bare theory’s empirical and theoreticalstructure (which is gauge-independent).In the example of T-duality, the interpretation of string quantum numbers as ‘mo-mentum’ or as ‘winding’ is external and makes two T-dual theories inequivalent, since theinterpretation maps to distinct elements of the domain under the exchange in Eq. (4).If an interpretation does not map the specific structure of a theory but only com-mon structure, I will call it an internal interpretation . This means that it only maps thestructure that is common to the duals. Such an interpretation gives rise to two maps, i and i , one for each theory: their domains differ, but their range can be taken to bethe same, since for each interpretation map of one theory there is always a correspondinginterpretation map of the other theory with the same domain of application. Thus weget the situation of theoretical equivalence, in Figure 1. Unextendability justifies the use of an internal interpretation.
There is an im-portant epistemic question, that will play a role in Section 4.2’s analysis, about whetherwe are justified in adopting an internal interpretation, according to which the duals aretheoretically equivalent.The question is, roughly speaking, whether the interpretation i of a bare theory T ,and T itself, are “detailed enough” and “general enough” that they cannot be expectedto change, for a given domain of application D . I call this condition unextendability. This conception gives a formal generalisation of an earlier characterisation, in Dieks etal. (2015: pp. 209-210) and De Haro (2020: Section 1.3), of an internal interpretation as one wherethe meaning of the symbols is derived from their place and relation with other symbols in the theoreticalstructure, i.e. not determined from the outside. Those papers were concerned with theory construction,the idea being that when, in order to interpret T ’s symbols, we e.g. couple a theory T to a theory ofmeasurement, we do this either through T ’s specific structure, or by changing T in some other way. Ei-ther way, we have an external interpretation, because the specific structure makes an empirical difference.Thus an internal interpretation, which does not introduce an external context or couplings, only concernsfacts internal to the triples and our use of them. The phrase ‘the internal interpretation does not map the specific structure’ can be weakened: onecan allow that the interpretation map maps the specific structure to the domain of application, as long asthe duality is respected (i.e. the domains of two duals are still the same), and the empirical substructuresof the domain do not change.
15e Haro (2020: Section 1.3.3) gives a technical definition of the two conditions for unex-tendability, the ‘detail’ and ‘generality’ conditions. Roughly speaking, the condition thatthe interpreted theory is detailed enough means that it describes the entire domain ofapplication D , i.e. it does not leave out any details. And the generality condition meansthat both the bare theory T and the domain of application D are general enough thatthe theory “cannot be extended” to a theory covering a larger set of phenomena. Thus T is as general as it can/should be, and D is an entire “possible world”, and cannot beextended beyond it.To give an example involving symmetries: imagine an effective quantum field theorywith a classical symmetry, valid up to some cutoff, and imagine an interpretation thatmaps symmetry-related states to the very same elements in the domain of application.Now imagine that extending the theory beyond the cutoff reveals that the symmetry isbroken (for example, by a higher-loop effect), so that it is anomalous. The possibility ofthis extension, with the corresponding breaking of the symmetry, will lead us to questionthat our interpretation is correct: maybe we should not identify symmetry-related statesby assigning them the same interpretation, especially if it turns out that the states aredifferent after the extension (for example, if higher-loop effects correct the members of apair of symmetry-related states in different ways). And so, it will probably prompt us todevelop an interpretation that is consistent with the theory’s extension to high energies,so that symmetry-related states are now mapped to distinct elements in the domain. Inconclusion, we are not epistemically justified in interpreting symmetry-related states asequivalent, because we should take into account the possibility that an extension of thetheory to higher energies might compel us to change our interpretation.External interpretations of two dual theories do not in general classify them as havingthe same interpretation. And so, although these interpretations could be subject tochange, this does not affect the verdict that the theories are theoretically inequivalent.There is a weaker sense of unextendability, which allows that a theory might be ex-tended in some cases, but only in such a way that the interpretation in the original domainof application does not change, and the theories continue to be theoretically equivalentafter the extension. In what follows, I will often use ‘unextendability’ in this weaker sense.The example of position-momentum duality in elementary quantum mechanics canbe interpreted internally once we have developed quantum mechanics on a Hilbert space.The two theories are then representations of a single Hilbert space, and we can describethe very same phenomena, regardless of whether we use the momentum or the positionrepresentation. And quantum mechanics is an unextendable theory in the weaker sense(namely, with respect to this aspect with which we are now concerned, of position vs. mo-mentum interpretations) because, even if we do not have a “final” Hamiltonian (we canoften add new terms to it), the position and momentum representations keep their powerof describing all possible phenomena equally well. Namely, because of quantum mechan-ics’ linear and adjoint structure, unitary transformations remain symmetries of the theoryregardless of which terms we may add to the Hamiltonian: so that our interpretation interms of position or momentum will not change.Dualities in string and M-theory are also expected to be dualities between unextend-able theories, at least in the weak sense. For example, the type II theories discussedin Section 3.2 have the maximal amount of supersymmetry possible in ten dimensions;16nd the number of spacetime dimensions is determined by requiring that the quantumtheories be consistent. Also, their interactions are fixed by the field content and the sym-metries. If we imagine, for a moment, that these theories are exactly well-defined (sincethe expectation is that M-theory gives an exact definition of these theories, and that T-duality is a manifestation of some symmetry of M-theory), then they are in some sense“unique”, constrained by symmetries, and thus unextendable: they are picked out by thefield content, their set of symmetries, and the number of spacetime dimensions. Thus,if these conjectures can be fleshed out, taking type IIA and type IIB on a circle to betheoretically equivalent will be justified, because the theories describe an entire possibleworld, and there is no other theory “in their vicinity”.
In this Section, I first make some remarks about aspects of scientific realism relevant fordualities (Section 4.1). Then I will illustrate the notions of theoretical inequivalence ofdual theory formulations (Section 4.2) and empirical equivalence (Section 4.3). This willlead to a surprising but straightforward conclusion, which will give us our final verdictabout whether dualities admit empirical under-determination. Section 4.4 discusses howto obtain internal interpretations. Section 4.5 will then ask whether scientific realism isin trouble.
My discussion of scientific realism aims to be as general as possible, i.e. as independentas possible of a specific scientific realist position. The notion of under-determinationthat we are considering has both semantic and epistemic aspects (see Section 2.1), andalso the interesting scientific realism to consider has both semantic and epistemic aspects.Roughly, the relevant scientific realism is the belief in the approximate truth of scientifictheories that are well-confirmed in a given domain of application.The semantic aspect does not involve a na¨ıve “direct reading” of a scientific theory (assome formulations, like van Fraassen’s (1980), could lead us to think). Rather, it involvesa literal, but nevertheless cautious, reading, informed by current scientific practice and byhistory. This is essential to secure that the belief in the theory’s statements is epistemicallyjustified. Indeed, one does not simply take the nearest scientific textbook and quantifyexistentially over the entities that are defined on the page. There are many cases whereone is not justified in believing in the entities that are introduced by even our best scientifictheories, and so one proceeds tentatively. In such cases, a cautious scientific realist oughtto suspend judgment about the existence of the posited entities, until further interpretativework has been carried out that justifies the corresponding belief. For example, considerthe cases of local gauge symmetries, and of a complex phase in the overall wave-functionof a system: in both cases, belief in the corresponding entities ought to be postponeduntil the further analysis is done. This contrasts with, say, the particle content of our See also Dawid (2006: pp. 310-311), who discusses a related phenomenon of ‘structural uniqueness’. My own position is in De Haro (2020a: pp. 27-59). While I do not myself endorse the details of these proposals, I am sympathetic to them:and they do illustrate the general idea of ‘caution’ about realist commitments endorsedabove. Namely, that determining the realist’s commitments is not a matter of “readingoff”, but sometimes involves judgment.Let me now say more about how this applies to dualities (more details in Section 4.5),without aiming to develop a new scientific realist view here: since, as I said, my argumentsshould apply to different versions of scientific realism. Rather, I wish to express a generalattitude—which I denote with the word ‘caution’—towards the role of inter-theoreticrelations in the constitution of scientific theories.In so far as a duality can have semantic implications—namely, in so far as dualitiescontribute to the criteria of theory individuation—the cautious scientific realist shouldtake notice of those implications, and suspend judgment about whether two dual theory-formulations are distinct theories, until those criteria have been clarified. Indeed, theaccount of dualities that I favour proposes that, under sufficient conditions of:(1) internal interpretation,(2) unextendability,(3) having a philosophical conception of ‘interpretation’, one is justified (but not obliged) to view duals as notational variants of a single theory.This view of theory individuation takes a middle way—“the best of both worlds”—between two positions that, in the recent literature, have been presented as antagonistic.Read and Møller-Nielsen (2020: p. 276) defend what they call a ‘cautious motivationalism’:duality-related models may only be regarded as being theoretically equivalent once aninterpretation affording a coherent explication of their common ontology is provided, andthere is no guarantee that such an interpretation exists.Huggett and W¨uthrich (2020: p. 19) dub this position an ‘agnosticism about equiv-alence’ and object that, though it is cautious, it is less epistemically virtuous than theirown position: namely, to assert that ‘string theory is promising as a complete unifiedtheory in its domain, and so it is reasonably thought to be unextendable. And from thatwe do think physical equivalence is the reasonable conclusion’.The position that I defend takes a middle way between these apparently contrastingpositions: one is justified, but not obliged, (i) to take duals to be theoretically equivalent These accounts have been criticised on various points, most notably because the defence of selectiveconfirmation appears to involve a selective reading of the historical record (Stanford, 2006: p. 174). This is my own version of what Read and Møller-Nielsen (2020: p. 266) call ‘an explication of theshared ontology of two duals’. See De Haro (2019: Section 2.3.1) and footnote 36. But even if the three conditions above, (1)-(3), are not fully met, or are not explicitlyformulated, it may still be legitimate, lacking full epistemic justification, to take the dualsas equivalent and to be a realist about their common entities: as a working assumption,a methodological heuristic, or a starting point for the formulation of a new theory.
In Section 3.1, I defined theoretically equivalent theories in terms of the triangular diagramin Figure 1, and theoretically inequivalent theories in terms of a diagram that does notclose, i.e. a square diagram as in Figure 2.In Section 3.3, we made a distinction between internal and external interpretations.Thus the Schema gives us two cases in which dual theories can prima facie be theoreticallyinequivalent. The first is through the adoption of external interpretations. These inter-pretations lead to the square diagram in Figure 2, with different domains of application.I will discuss this case in Section 4.5.The second case is that of two theoretically equivalent but extendable dual theories.In this case, although the judgment, based on a given pair of internal interpretations, isthat the theories are theoretically equivalent, this judgment is epistemically unjustifiedbecause the theory is extendable (as I discussed in Section 3.3). And since the judgmentof theoretical equivalence is unjustified, this could prima facie give a new way to gettheoretically inequivalent theories.However, this second case is not a genuine new possibility. For the lack of justificationfor adopting the internal interpretations prompts us to either: (i) Justify the use ofthese, or some other, internal interpretations, thus getting a justified verdict of theoreticalequivalence; or (ii) conclude that the internal interpretations under consideration areindeed not adequate interpretations, and adopt external interpretations instead, whichthen judge the two theories to be inequivalent.
In case (i), where one finds internal interpretations whose use is justified, we end upwith a case of theoretical equivalence and not inequivalence, i.e. a case of the trianglediagram Figure 1; and so this case is irrelevant to the under-determination thesis. In case(ii), we are back to the situation of external interpretations. Thus external interpretationsexhaust the interesting options for empirical under-determination. Recall the examples of external interpretations from Section 3.3. External interpreta-tions of T dual pairs of strings winding on a circle interpret the momentum and windingquantum numbers, n and m respectively, as corresponding to different elements in thedomain (namely, to momentum and to winding, respectively), and in addition they would This notion appears to be close to Ney’s (2012: p. 61) ‘core metaphysics’, which retains those common‘entities, structures, and principles in which we come to believe as a result of what is found to beindispensable to the formulation of our physical theories’. However, one should keep in mind that I amhere concerned with semantics and epistemology, and not chiefly with metaphysics. I thank an anonymousreviewer for pointing out this similarity. My reasoning here bears a formal similarity with Norton’s (2006) analysis of Goodman’s new riddleof induction. d ←→ T y i y i D d ←→ D Figure 3:
Empirical equivalence. There is an induced duality map, ˜ d , between the domains. interpret the circle as having a definite radius of either R or R ′ := ℓ /R : so that the dual-ity transformation Eq. (4) leads to physically inequivalent theories, despite the existenceof a formal duality that pairs up the states and quantities. Such an external interpretationcan for example include ways to measure the quantities of interest—momentum, winding,and the radius of the circle—so that they indeed come to have definite values, accordingto the external interpretation. An indirect way to measure the radius is by measuring thetime that a massless string takes to travel around the circle. In this Section, I will use Section’s 2.2 account of empirical equivalence, which will enableour final verdict about empirical under-determination in cases of duality. The Sectionsummarises De Haro (2020b) (see also Weatherall, 2020).According to the syntactic conception of theories, two theories are empirically equiva-lent if they imply the same observational sentences. As I argued in Section 3.3, externallyinterpreted theories are in general not empirically equivalent, in this sense. The domainsare distinct, as Figure 2 illustrates.On the semantic conception, two theories are empirically equivalent if the empiricalsubstructures of their models are isomorphic to each other (cf. Section 2.2).For dualities, the duality map d gives us a natural—even if surprising—new candidatefor an isomorphism between the empirical substructures of the models: I will dub it the‘induced duality map’, ˜ d : D → D . It is an isomorphism between the domains ofapplication, subject to the condition that the resulting (four-map) diagram, in Figure 3,commutes. This commutation condition is the natural condition for the induced dualitymap to mesh with the interpretation (the condition for its commutation is that i ◦ d =˜ d ◦ i ). If such a map exists, then the two theories are clearly empirically equivalent onvan Fraassen’s conception, even though they are theoretically inequivalent, because theinduced duality map is not the identity, and the domains differ: D = D .Thus, if such an induced duality map exists, on this literal account the dualities thatwe have discussed do in fact (and surprisingly!) relate empirically equivalent theories. See Quine (1970, 1975) and Glymour (1970, 1977). Since we are discussing empirical equivalence, the domains of application can here be restricted tothe observable phenomena. However, I will not indicate this explicitly in my notation. De Haro (2020b) argues that duality is indeed the correct type of isomorphism to be considered, onthe semantic criterion of empirical equivalence. n, m ) and radius R to the quantum numbers( m, n ) and radius R ′ = ℓ /R , and likewise in the domain, by swapping measurements ofpositions of momenta, and inverting the physical radii.The semantic notion is prima facie more liberal than Quine’s syntactic notion (in thesense that it is less fine-grained, because it gives a verdict of empirical equivalence moreeasily), and more in consonance with the scientific practice of dualities—thus it promptsa judicious reading of the syntactic notion of empirical equivalence. To this end, it is notnecessary to change Quine’s criterion of empirical equivalence from Section 2.2; all we needto do is to change one of the theories, generating a new theory by giving a non-standardinterpretation to the bare theory.Since we have a non-standard and innovative interpretation, we have abandoned whatmay have been the theory’s intended meaning, i.e. we have changed the theory, stipulatinga reinterpretation of its terms, thus producing the desired observational sentences. Thuswe have been faithless to the meanings of words. But this is allowed, since we are dealingwith external interpretations anyway: and external interpretations can be changed if one’saims change. Indeed, nobody said that we had to stick to a single interpretation of a baretheory in order for it to make empirical predictions: for although theories may haveintended interpretations, assigned to them by history and convenience, nothing—moreprecisely, none of the Quinean notions of empirical under-determination and empiricalequivalence—prevents us from generating new theories by reinterpreting the old ones,thus extending the predictive power of a bare theory: but also creating cases of under-determination!In Section 4.4, I first sketch how internal interpretations are usually obtained. For thiswill be important for the resolution of the problem of under-determination, in Section 4.5.
Internal interpretations of dual theory formulations are often obtained by a process of abstraction from existing (often, historically given) interpretations of the two formulations.Let me denote the common core theory obtained by abstraction, by ˆ t , and let ˆ T and ˆ T betwo empirically equivalent dual theories from which ˆ t is obtained. Here, the hats indicatethat these are interpreted, rather than bare, theories.We can view the formulation of the common core theory ˆ t as a two-step procedure.We first develop the bare i.e. uninterpreted theory, and then its interpretation:(1) A common core bare theory, t , is obtained by a process of abstraction from the twobare theories, T and T , so that these theory formulations are usually representations (inthe mathematical sense) of this common core bare theory. This is discussed in detail inDe Haro and Butterfield (2017).(2) The internal interpretation of the common core theory, ˆ t , is similarly obtainedthrough abstraction: namely, by abstracting from the commonalities shared by the inter- For unextendable theories, the theory’s natural interpretation is surely an internal, not an external,interpretation. T and ˆ T . In this way, one obtains an internal interpretation that mapsonly the common core of the two dual theory formulations. The incompatibility of the interpretations of two theory formulations is thus resolvedthrough an internal interpretation that captures their common aspects, and assigns notheoretical or physical significance to the rest.Let me spell out the consequences of this a bit more. The result of the processof abstraction, i.e. points (1) and (2), is that the interpreted theories ˆ T and ˆ T arerepresentations or instantiations of the common core theory, ˆ t . Thus in particular, wehave: ˆ T ⇒ ˆ t and ˆ T ⇒ ˆ t , (5)where I temporarily adopt a syntactic construal of theories (but the same idea can also beexpressed for semantically construed theories). That is, because ˆ T and ˆ T are representa-tions or instantiations of ˆ t , and in particular because ˆ t has a common core interpretation(more on this below): whenever either of ˆ T and ˆ T is true, then ˆ t is also true. (Forexample, think of how a representation of a group, being defined as a homomorphismfrom the group to some structure, satisfies the abstract axioms that define the group, andthus makes those axioms true for that structure).Further, for empirically equivalent but theoretically in equivalent theories, we have:ˆ T ⇒ ¬ ˆ T . (6)Let me first give a simple example, not of a full common core theory, but of a singlequantity that is represented by different theory formulations. Consider the quantity thatis interpreted as the ‘energy’ of a system: while different theory formulations may describethis quantity using different specific structure, and so their external interpretations differ(even greatly) in their details, the basic quantity represented—the energy of the system—is the same, as described by the internal interpretation. Indeed, in all of the followingexamples, the energy is indeed represented, on both sides of the duality, by quantities thatmatch, i.e. map to one another under the duality: position-momentum duality (Section3.1), T-duality and gauge-gravity duality (Section 3.2; Huggett 2017; De Haro, Mayer-son, Butterfield, 2016), bosonization (De Haro and Butterfield, 2017), electric-magneticduality. ‘Abstraction’ does not necessarily mean ‘crossing out the elements of the ontology that are notcommon to the two theory formulations’. Sometimes it also means ‘erasing some of the characterisations given to the entities, while retaining the number of entities’. For example, when we go from quantummechanics described in terms of position, or of momentum, space (Eq. (1)), to a formulation in terms ofan abstract Hilbert space, we do not simply cross out positions and momenta from the theory, but rathermake our formulation independent of that choice of basis. In this sense, the formalism of the commoncore theory is not always a “reduced” or “quotiented” version of the formalisms the two theories. Since the sentences of a scientific theory, observable or not, are given by the theory’s interpretation,the above entailments should be read as entailments between statements about the world. As I mentionedin (2), this requires that we adopt internal interpretations: so that ˆ t ’s interpretation is obtained from theinterpretations of ˆ T and ˆ T by abstraction, and the entailments are indeed preserved. t always exists,since they require an explication of the shared ontology of two duals, before a verdict oftheoretical equivalence is justified. Since points (1) and (2) sketch a procedure for obtaining ˆ t by abstraction, the questionof “whether ˆ t exists” is effectively the question of whether the common core theory ˆ t thusobtained has a well-defined ontology. This can be divided into two further subquestions:(1’) Is the thus obtained structure t , of which T and T are representations, itself abare theory, i.e. a triple of states, quantities, and dynamics?(2’) Does t also have a well-defined ontology?If these two questions are answered affirmatively, then ˆ t is a common core theory inthe appropriate domain of application.While examples of dualities have been given in De Haro (2019) for extendable theories,where the answer to at least one of these questions is negative (so that a common coretheory does not exist): I am not aware of any examples of unextendable theories for whichit is known that a common core does not exist.Let me briefly sketch whether and how some of the familiar examples of dualitiessatisfy the requirements (1’) and (2’) (this paragraph is slightly more technical and canbe skipped). Consider bosonization (De Haro and Butterfield, 2017). (1’): The set of quantities is constructed from an infinite set of currents that are the generators of theenveloping algebra of the affine Lie algebra of SU(2) (or other gauge group). The states are the irreducible unitary representations of this algebra. Finally, the dynamics is simplygiven by a specific Hamiltonian operator. (2’): The theory has an appropriate interpre-tation: the states can be interpreted as usual field theory states describing fermions andbosons. The operators are likewise interpreted in terms of energy, momentum, etc. Andthe dynamics is ordinary Hamiltonian dynamics in two dimensions. Thus bosonizationanswers both (1’) and (2’) affirmatively.Other string theory dualities are less well-established, and so the results here arerestricted to special cases. In the case of gauge-gravity dualities, under appropriate con-ditions, the states include a conformal manifold with a conformal class of metrics on it(De Haro, 2020: p. 278). The quantities are specific sets of operators (which again can beinterpreted in terms of energy and momentum), and the dynamics is again a choice of aHamiltonian operator. Thus, at least within the given idealisations and approximations,and admitting that gauge-gravity dualities are not yet fully understood; they also seemto answer (1’) and (2’) affirmatively. There are two reasons to set aside the worry that ˆ t does not exist, together with itsthreatened problem of under-determination: so that in Section 4.5 I can safely assume I endorse their requirement, which is more specific than my own earlier requirement of a ‘deeperanalysis of the notion of reference itself’ (2020: Section 1.3.2) and ‘an agreed philosophical conception ofthe interpretation’ (2019: Section 2.3.1). Note that this answer to (2’) about the domain of application does not just mean getting correctpredictions. For example, common core theories are also being used to give explanations and answerquestions that are traditionally regarded as theoretical, e.g. about locality and causality (Balasubrama-nian and Kraus, 1999) and about black hole singularities (Festuccia and Liu, 2006). For a variety of otherphysical questions addressed using the AdS/CFT common core, see Part III of Ammon and Erdmenger(2015). t exists. (I restrict the discussion to unextendable theories).First, as just discussed, there is evidence that such common core theories exist inmany (not to say all!) examples of such dualities. Indeed, dualities would be much lessinteresting for physicists if the common core was some arbitrary structure that does notqualify as a physical theory in the appropriate domain of application. Thus it is safeto conjecture, for unextendable theories, that of all the putative dualities, those that aregenuine dualities and are of scientific importance (and that would potentially give rise toa more serious threat of under-determination, because the theory formulations differ fromeach other more) do have common cores.The second, and main, reason is that there are, to my knowledge, no examples ofdualities for unextendable theories where a common core is known to not exist. For somedualities, it is not known whether a common core exists. For example, Huggett andW¨uthrich (2020) mention T-duality as an example where a common core has not beenformulated explicitly: working it out explicitly would, in their view, require a formulationof M-theory. It might also be possible to work out a common core theory perturbatively.But so long as it is not known whether a common core exists, rather than having positiveknowledge that it does not exist, this just means that there is work to be done: thelack of an appropriate common core theory ˆ t can, at best, give a case of transient under-determination, which, as I argued in Section 2, reflects our current state of knowledge,and is not really worrisome.This agrees with Stanford’s requirement that, before such putative cases of under-determination are accepted, actual examples should be produced (cf. footnotes 7 and 8).And so, this worry can be set aside. In this Section, I address the question announced at the beginning of Section 4.2: namely,whether, for dual bare theories, the under-determination of external interpretations givesa problem of empirical under-determination. Let us return to the distinction betweenextendable and unextendable theories:(A)
Extendable theories:
I will argue that here we have a case of transient under-determi-nation (see Section 1 and footnotes 2 and 9), i.e. under-determination by the evidence sofar. And so, I will argue that there is no problem of empirical under-determination here,but only a limitation of our current state of knowledge.The reason is that extensions of the theory formulations that break the duality areallowed by the external interpretations, i.e. such that two duals map to a different domainof application with different empirical substructures (or different observation sentences).Interpreting the specific structure, as external interpretations do, introduces elements intothe domains that, in general, render the two theory formulations empirically in equivalent.This is in the ordinary business of theory construction. For, although in our currentstate of knowledge it may appear that the theory formulations are empirically equivalent,the fact that the theory is extendable means that future theory development could wellmake an empirical difference. Thus one should interpret such theories tentatively. This isthe ordinary business of transient under-determination: and one should here look for the24rdinary responses of scientific realist positions.(B) Unextendable theories:
In this case, the theories are somehow “unique”, perhaps‘isolated in the space of related theories’ (cf. Section 3.3).I will argue that a cautious scientific realism (see Section 4.1) does not require beliefin either of the (incompatible) interpretations of the dual theory formulations. Rather,belief in the internal interpretation, obtained by a process of abstraction from the externalinterpretations, is justified. And so, there is under-determination, but of a kind that isbenign. Thus dualities may be taken to favour a cautious approach to scientific realism.In more detail: as I discussed in Section 4.1, in the presence of inter-theoretic relations(and given the three conditions discussed at the end of Section 4.1), the cautious scientificrealist will take the inter-theoretic relations into account when she determines her realistcommitments. Although the under-determination prevents her from being able to choosebetween ˆ T and ˆ T , she has an important reason to favour ˆ t . Namely, she will prefer ˆ t over ˆ T and ˆ T on logical grounds. For, on the basis of the implications Eqs. (5) and (6),and regardless of which of ˆ T and ˆ T is true, she can consistently (and at this point, alsoshould) accept the truth of ˆ t . Thus, since her scientific realism commits her to at leastone of these three theories (notably, because they are empirically adequate theories, andthey otherwise satisfy the requirements of her scientific realism), she will in any case becommmitted to ˆ t . Namely, a commitment to either ˆ T or ˆ T commits her, in virtue ofEq. (5), ipso facto to ˆ t , while she cannot know which of ˆ T or ˆ T is true.This does not prevent the scientific realist from, in addition, being committed toone of the two external interpretations, i.e. committing herself to one of the two theoryformulations (perhaps using alternative assessment criteria, i.e. point (ii) in Section 2.1):ˆ T , say. Indeed, ˆ t and ˆ T are of course compatible, by Eq. (5): and so, there is no under-determination here either.In any case, the scientific realist does not make a mistake if (perhaps in the absence ofadditional assessment criteria) she commits only to ˆ t . Thus the cautious scientific realistis, in the cases under consideration, always justified in believing the common core theory. This is a conclusion that the multiple interpretative options considered in Le Bihanand Read (2018: Figure 1) obscures. They also consider a “common core” theory, buttheirs is not constructed by abstraction: in any case, they do not consider the possibilityof a constraint, Eqs. (5) and (6), that follows from the process of abstraction. Thisconstraint eliminates four out of six of the options in their taxonomy. The remainingtwo are the two cases just discussed, i.e. commitment only to ˆ t , or possible commitmentto both ˆ t and ˆ T . Thus, as just argued, the problem of under-determination is therebyresolved. In sum, a scientific realist may lack the resources to know which of two dual theory This is clear from e.g. the following quote (p. 5): ‘[T]here is a sense in which, absent further philo-sophical details, such a move [i.e. identifying the common core] has made the situation worse : we have,in effect, identified a further world which is empirically adequate to the actual world’. Together with the assumption that at least one of ˆ T , ˆ T , ˆ t is worth scientific realist commitment—elsethere is no question of under-determination either! It is of course not guaranteed that a common core theory ˆ t always exists. However, as I argued inSection 4.4, this does not lead to cases of empirical under-determination. T or ˆ T , describes the world, but is not required to believe either of them.Rather, the cautious scientific realist is justified in adopting an internal interpretation,which abstracts from the external interpretations, and thereby resolves their incompati-bilities. For unextendable theories, a scientific realist may take theoretical equivalence tobe a criterion of theory individuation. Thus a catious scientific realism recommends onlybelief in the internal interpretation of unextendable theories.My overall view can be well summarised in terms of how it clarifies previous discussionsof under-determination for dualities: by Matsubara (2013) and Rickles (2017). Matsub-ara presents two approaches to dualities: his first approach (‘Accept the different dualdescriptions as describing two different situations’) corresponds to the Schema’s case oftheoretically inequivalent theories, and he says that these theoretically inequivalent the-ories are nevertheless empirically equivalent (‘the world may in reality be more like onedual description than the other but we have no empirical way of knowing this’).While I broadly agree with Matsubara, my analysis makes several clarifications. Forexample, my characterisation of the under-determination is slightly different: there is anunder-determination specific to dualities only in the case of unextendable theories, but oneis always justified in adopting internal interpretations. (For extendable theories, furthertheory development can distinguish between two theory formulations, so this is ordinarytransient under-determination.) Since, in the relevant examples, these are obtained byabstraction from the external interpretations of the various theory formulations, they donot contradict them (Eq. (5)). Thus I disagree with Matsubara’s conclusion that ‘Thismeans that we must accept epistemic anti-realism since in this situation it is hard to findany reason for preferring one alternative before another.’ Indeed, as I have argued, acautious scientific realist does not need to choose one dual over the other: such a realistis justified in adopting an internal interpretation of the kind discussed, which agrees—in everything it says—with the external interpretations, and is silent about their otherspecific aspects.Matsubara’s second approach (‘We do not accept that [the duals] describe differentsituations; instead they are descriptions of the same underlying reality’) corresponds tomy case of theoretically equivalent duals. I agree with Matsubara that there is no under-determination, but I think this position is less tangled with ontic structural realism thanhe seems to think (although it is of course compatible with it). As the notions of aninternal interpretation and its justification through unextendability should make clear,judging two duals as being theoretically equivalent is a matter of setting both formal and interpretative criteria for individuating theories.In the case of unextendable theories, a methodological preference for internal inter-pretations surfaces—because justified by unextendability and abstraction. This feeds intophysicists’ main interest in dualities, which comes from their heuristic power in formulat-ing new theories. On an internal interpretation, a duality is thus taken to be a startingpoint of theory individuation, with an interpretation of the theory’s common core, ofwhich the various theory formulations are specific instantiations. This further motivatesDe Haro and Butterfield’s (2017) proposal to lift the usage of ‘theory’ and ‘model’ “onelevel up”, so that the various theory formulations are ‘models’ (i.e. instantiations or rep-26esentations) of a single theory. And this now holds not only for bare theories and models,but also—on internal interpretations—for interpreted ones. The discussion of under-determination begins slightly differently, and is complementary, tousual discussions of duality. While usual discussions of duality aim to establish conditionsfor when duals are theoretically equivalent, when analysing under-determination we beginwith theoretically inequivalent theories, whose inequivalence—as my analysis in Section4.2 shows—ends up depending on external interpretations. Thus our discussion has puttwo under-emphasised aspects of duality to good use: namely, theoretical inequivalenceand external interpretations.The Schema’s construal of bare theories, interpretation, and duality prompts a notionof theoretical equivalence that, although strict, turns out to generate cases of under-determination: but I have also argued that these do not present a new problem for acatious scientific realism. For theory formulations that can be extended beyond theirdomain of application, we have the familiar situation of transient under-determination,where further theory development may be expected to break the duality. For theoryformulations that cannot be so extended (or only in a way that does not change theirinterpretations in the old domain), the under-determination is benign, because—for du-alities in the literature that are sufficiently well-undestood—a common core theory canbe obtained by abstraction from the external interpretations. In this case, a cautious sci-entific realism does not commit to the external interpretations, but belief in the internalinterpretation is justified. Thus the under-determination is here benign, and dualities donot provide the anti-realist with new ammunition, although they may be taken to favoura cautious approach to scientific realism.Just as dualities bear on the problem of theory individuation that is central to thediscussion of theoretical equivalence, they also bear on the question of scientific realism.Namely, they suggest a cautious approach, according to which inter-theoretic relationsshould be taken into account when determining one’s realist commitments.
Acknowledgements
I thank Jeremy Butterfield, James Read, and two anonymous reviewers for comments onthis paper. I also thank John Norton for a discussion of duality and under-determination.This work was supported by the Tarner scholarship in Philosophy of Science and Historyof Ideas, held at Trinity College, Cambridge. My conclusions are similar to some of Dawid’s (2017b). Where he says that there is a shift in the roleplayed by empirical equivalence in theory construction (and I agree with this, see De Haro (2018)), I haveargued that the traditional semantic and syntactic construals of empirical equivalence are in themselvessufficient to analyse dualities, and that the main difference is in the theories whose empirical equivalenceis being assessed. eferences Ammon, M. and Erdmenger, J. (2015).
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