The interpretation and justification of Earman’s symmetry principles (stating that any spacetime symmetry should be a dynamical symmetry and vice-versa) are controversial. This is directly connected to the question of how certain structures in physical theories acquire a spatiotemporal character. In this paper I address these issues from a perspective (arguably functionalist) that relates the classical discussion about the measurement and geometrical determination of space with a characterization of the notion of dynamical symmetry in which its application to subsystems that act as measuring devices plays an essential role. I argue that in order to reformulate and justify Earman’s principles, and to provide a general account of the chronogeometrical character of some structures, the existence of a coordination between two notions of congruence, one mathematical and one dynamical, must be assumed for the interpretation of physical theories. This coordination provides the basis on which we can understand spacetime in physical theories as the codification (representation) of certain features of the access ideal observers have to experience.

La interpretación y justificación de los principios de simetría de Earman (que establecen que toda simetría espaciotemporal debería ser una simetría dinámica y viceversa) son controvertidas. Esto está directamente conectado con la cuestión de cómo ciertas estructuras en la teorías físicas adquieren su carácter espacio-temporal. En este artículo abordo estos problemas desde una perspectiva que relaciona la discusión clásica sobre la determinación geométrica del espacio con una caracterización de la noción de simetría dinámica en la que juega un papel central su aplicación a subsistemas que actúan como aparatos de medida. Defiendo que para reformular y justificar los principios de Earman, y para proporcionar una caracterización general del carácter crono-geométrico de algunas estructuras, debe asumirse, en la interpretación de las teorías físicas, la existencia de una coordinación entre dos nociones de congruencia, una matemática y otra física. Dicha coordinación proporciona el marco en el que podemos entender el spaciotiempo en las teorías físicas como la codificación (representación) de ciertos rasgos del acceso a la experiencia de observadores ideales.

Many recent discussions of spatio-temporality in physical theories consider the idea of spacetime not being ontologically fundamental. Moreover, some recent proposals take a functionalist perspective and regard structures as spatio-temporal in virtue of them playing certain roles in our physical theories. These are often appealed to when considering questions about the emergence of spacetime and are also referred to in discussions on the interpretation of symmetries.^{[1]} If spacetime is as spacetime does, as the functionalist mantra is sometimes put (Lam and Würthrich , 2018), the question is: What does spacetime do? Or to be more explicit: What roles do spatio-temporal notions play in our physical theorizing and what consequences can we extract from those roles for specific questions regarding the status of spacetime symmetries, in particular, and the interpretation of spacetime theories in general? Diﬀerent answers to this central question are possible, with Knox’ inertial frame functionalism probably being the most commonly discussed, and there seems to be a general feeling that there is no single function that is apt to be used to identify space-time generally.^{[2]} In this paper, I defend a particular answer to this central question which is linked to a general strategy that allows us to dispel some problems that have worried philosophers of physics for the last few decades: those relate d to the origin of the relation b etween spacetime and dynamical s ymmetries.

Functionalism can be seen as a way of framing one of the most important questions in the interpretation of physical theories: the question about the conditions/criteria for certain (mathematical) structures to be considered spatio-temporal. This, ultimately, in a more ontological fashion is the question about how space and time are represented in physics. A general (natural) scheme adopted to tackle the problem consists of thinking that the relation must come from some common element present both in how the metric of spacetime (and any other spatio-temporal structures) is determined and in some general conditions for the formulation of the laws that describe the dynamics. A traditional answer has to do with noting the fact that the chrono-geometry of the metric of spacetime is determined through the operations of measuring physical/dynamical systems like rods and clocks. This hint (as Weatherall (2021) notes) is also the original inspiration for the so-called dynamical approach to relativity.

The general perspective I have just alluded to, which embraces an interpretive core according to which determination of the metric (the chronogeometrical significance of the metric) is dynamical, seems to have become obscured at times in recent debates. Nonetheless, it is always there, lurking in the wings. Take, for instance, the recent debate concerning the two primary perspectives on the relation between spacetime and dynamics in relativity theory: the geometrical approach (GA) and the dynamical approach (DA).^{[3]} Although some eﬀorts have been made to play down the diﬀerences in this dispute (Weatherall, 2021; Read, 2020), it is often understood in an extremely stylized and highly formalistic fashion.

In such interpretations of the dispute, the GA is seen as assuming that some structures are primitively spatio-temporal and that they somehow constrain the dynamics, while the DA takes certain features of the dynamical laws to be primitive and it is these that eventually define some structures as being spatio-temporal. Undoubtedly, in any reasonable understanding of the two perspectives, in the characterization of their diﬀerent starting positions, a common reference to the role of rods and clocks must be acknowledged. But despite the fact that this common ground can be seen as containing the seeds of their mutual relation, the discussion tends to forget this dimension. An example of this, diﬀerent from the aprioristic version of the GA I have just given, is the defence of the DA that takes the coincidence of the symmetries of all the matter laws as a brute fact (a ‘miracle’) and considers their relation to spacetime symmetries to be analytical or definitional.^{[4]} What happens in both cases, it can be argued, is that certain relevant features of spacetime and dynamics are first separated from their physical origin and then a question about how one of them explains or can be reduced to the other is posed. We encounter this together with a tendency to frame the discussion only in terms of the formal structures of theories without explicitly co nsidering how such str uctures, according to some assumed int erpretation of the theor y, are supposed to come into con tact with actual expe rience.

Let me focus on why I think that the DA, even if correctly embracing the dynamical origin of spacetime structures in physical theories, falls short of providing a fully satisfactory account of the relation between spacetime chronogeometry and dynamics. Put simply, I maintain that the declared aim of the DA, “to account for the chronogeometry of metric structure” (Brown and Read, Forthcoming, p. 9), cannot be achieved within a version of the approach in which the coincidence of the space-time symmetries and the dynamical symmetries of a theory is taken to be analytical or definitional. This ana lytical version of the DA ^{[5]} m ay be a simplification that does not do justice to a more sophisticated version, but it does indicate that we need some account of how, starting from the assumption of certain symmetries of the dynamical laws, we arrive at spacetime symmetries. This is usually completed in the DA by appealing to the strong equivalence principle (SEP), which imposes the condition that the local symmetries of matter laws must be such and such, together with a functionalist perspective (in particular, inertial frame functionalism) that would identify some features of certain structures—those that determine the local inertial frames—with spacetime.^{[6]}

This is problematic for diﬀerent reasons. Weatherall (2021) mentions the diﬃculty of arriving at a formulation of the SEP that would allow us to identify which are the relevant symmetries of the equations, and also the question of whether this is suﬃcient for us to recover spacetime as we understand it. Without going into too much detail, the problem with this approach can be simply put like this: if the formulation of the SEP is equivalent to the claim that the symmetries of the dynamical laws coincide (locally, if you want) with the symmetries of a metric field, then you have not gained much at all by introducing the SEP; you are still determining the spacetime symmetries of some structures by definition and then justifying their spatio-temporal character by projecting a functional perspective onto this state of aﬀairs. It might be true that some progress has been made with respect to the crude analytical DA, but the work is done by hiding the definition under the label of SEP (reference to which awakens some epistemic intuitions) and by bringing in some form of functionalism (which justifies why we can call those structures spacetime). So, the progress achieved depends, to a high degree, on how much you depart from the miracles formulation of SEP. If the formulation adopted was seen as diﬀerent from the start, then a full specification of it would be required. In any case, what the DA seems to be lacking is an elucidation—not some postulation—of the connection between the dynamical symmetries (again, which symmetries?) and what we call spacetime symmetries.

This problem also aﬀects Knox’ functionalist extension of the DA: if you define spacetime by the role that structures play in determining local inertial frames, and assume a version of SEP that declares that locally the symmetries of the laws of matter coincide with those of the metric thus determining local inertial frames, then you entrust all the functionalist work to SEP. But then you certainly have a problem: you are assuming that the notion of dynamical symmetry can be defined uncontroversially without any previous determination of spatio-temporal structures and that you can then define spacetime symmetries from those dynamical symmetries. This might be all right if an independent account of dynamical symmetry is provided. But what may well actually be going on in such approaches is that the notion of spacetime in sneaked in through the back door via some implicit reference to rods and clocks. What is needed is an explicit connection between the notion of rods and clocks and dynamical symmetry. The use of such ellipsis must stop at some point!

We need then criteria to identify which dynamical symmetries define spacetime ones. My general perspective is based on the following: it is precisely because some structures play the role of codifying the ways in which we (ideal observers) gain access to empirical content, which is implicit in using certain systems to probe spacetime, that we identify some dynamical symmetries as spacetime symmetries and therefore some structures as spatio-temporal.^{[7]}

What this initial take assumes is that the spatio-temporal character of some structures in a physical theory is derived from the fact that we can interpret them as encoding the structural (formal) characteristics of the way observers gain access to the empirical world. However, the approach that I adopt in this paper can be read in a more down-to-earth way. I will demonstrate that some features in the characterization of (systems that act as) measuring devices are such that they can be (and have been) naturally interpreted as being spatio-temporal. This being the case, it is not too far removed to take a general characterization of measuring devices as a putative abstract representation of an ideal observer, and then to see the spacetime role as the codification of some general features of idealized observers. Wherever one starts, the key point of my analysis is the connection between certain general features of the behaviour of measuring devices and some structures that can be taken to be part of the formal determination of spacetime.

So, the plan for the rest of this paper is as follows. In Section 2, I present a general overview of the framework of my proposal. Then I introduce the so-called problem of space (Section 3) and the discussion of the interpretation of dynamical symmetries in which the treatment of subsystems plays a central role (Section 4). After that I bring the two discussions together to oﬀer a justification of Earman’s principles (Section 5). I finish with some conclusions (Section 6).

What is the origin of the relation between spacetime symmetries and the symmetries of dynamical laws? If one rejects the possibility of it being a simple definitional relation, and in the previous section I have provided reasons to do so, then this question is in need of an answer. In this paper I propose one. A simple way to state the underlying general motivation for my response is the following: the justification for such a relation is connected to how, according to a given interpretation, a physical theory is taken to represent ideal observers. However, an explicit representation of observers is nowhere to be found in spacetime theories, so it might initially seem to be a dubious strategy to refer to one in order to justify the relation between two, in principle, uncontroversial features of the theory: its spacetime and dynamical symmetries. Perhaps the claim I support can be understood in these less contentious terms: certain elements in the formulation of spacetime theories, in particular those that allow us to interpret some structures as spatio-temporal and some symmetries as dynamical in a physically relevant way, can be understood as traces of the implicit representation of ideal observers. Spacetime, then, from this perspective, will be identified with certain structures that can be interpreted as playing the role of codifying formal features of the access observers have to experience.^{[8]} The expected benefits of this approach are that, understood in this manner, we have a natural justification for the relation between spacetime and dynamical symmetries. Precisely the extent to which this is a faithful presentation of things can only be decided after an explication of such a relation has been given in detail.

Let me advance the general features to be developed in the rest of the paper. As mentioned above, there is a venerable approach to the nature of spacetime in physical theories that links its determination to the standard operation of rods and clocks. In general, one can say that the empirical determination of physical chronogeometry will always involve some procedures governed by certain dynamics and therefore constrained by some principles. On the other hand, we also have physical principles that are implicit in the codification of some specific processes through which it is assumed that empirical content is acquired or, in other words, in the descriptions of measuring apparatus. And finally, we may consider that this characterization also imposes constraints on the dynamics of matter, as described by the theory. So, the basic assumption here is that the dynamics of these two processes (determination of physical geometry and the empirical content that is evidence for a theory) can be taken to be the same if we interpret certain features of the theories as somehow codifying the role of idealized observers. From here, I will argue, we can derive a relatio n between spacetime symmetries and dynamical symmet ries. It must be clear that t his is not a version of the GA view in which geometry is ta ken to explain dynamics, but neither does it involve a reduct ion of spacetime symme tries to dynamical symmetries. Geometry, in this approach, is dynamically constructed, but at the same time it is recognized that t his construction invol ves some principles which contain or imply general restrictions on matter dynamics. The existence of these cons traints on the formulat ion of dynamics is a consequence of inter preting the constructions as being derived from the physical desc ription of measuring devices (which might be interpreted as part of the codification of the empirical receptivity of idealized observ ers).

The key notion that technically bridges the two types of symmetries is that of congruence, which originated in geometry and has been everpresent in the debates about the true geometry of physical space motivated initially by the discovery of non-Euclidean geometries and then later by the eruption of relativity theory. I will argue that the same transformations (motions) that are part of the definition of the notion of congruence, and therefore can be interpreted as spacetime symmetries, from the point of view of the description of the dynamics of subsystems are symmetry transformations with features that make them ideal for the formulation of a dynamical notion of congruence. In particular, these transformations are unobservable from the interior of the subsystem but detectable because they change some quantities that encode relations between subsystems. Through the use of some technical machinery introduced by David Wallace, ^{[9]} this will become the basis for esta blishing the connection between spacetime symmetries and symmetries of the dynamics in physical theories (my main claim). It will also allow us to make the limits and conditions of such a relation explicit, and to tackle such a relation in the context of particular theories (a claim that would involve correcting some ideas about how to interpret the situation in some paradigmatic cases).

Through developing the details of this schematic presentation, I will also bring together two much discussed themes in spacetime theories. One is the determination of physical geometry that I have already mentioned, the so-called problem of space (PoS). The other is the observability of dynamical symmetries.

The question concerning which geometrical structures are suitable to be used to describe physical space and what justification can be given for this is generally referred in the literature as the problem of space (PoS). Even if it is obvious that formulated in this way the question only fully makes sense after the discovery of non-Euclidean geometries, much of the reflection that has occurred during the search for responses has its roots in the Kanti an analysis of space and time as forms of intuition. Irrespecti ve of whether Kant was aw are of the challenge that the new geometr ies posed, his analysis of the notio ns of space and time has been highly influential in the diﬀ erent formulations of the PoS due to the fact that he placed the question of how to give an account of the ph ysical/empirical valid ity of geometry centre s tage. Furthermore, w e must distinguish t wo stages in the histor y of the discussion: the classical pre-relati vistic era, mainly carried out by mathematicians like Riemann, Helmholtz, Lie and Poincaré ; and the relativistic s tage, formulated mainly by Hermann Weyl.

There are a fair number of presentations of the history of the PoS in the literature.^{[10]} My intention is not to repeat the story; although we will need a brief account to be able to focus on some aspects of the problem that I think are essential for my discussion and that perhaps have not been suﬃciently stressed to date.

Even if Riemann can be considered the initiator of the classical formulation of the PoS, I will take some features of Helmholtz’ approach as a reference to understand the dynamical dimension of the problem. The basic question that Helmholtz was trying to answer is: How can the geometry of physical space be determined? His answer is based on the idea that the measure of spatial geometry requires a notion of congruence for physical bodies and this, in turn, is made possible by the condition of free mobility of bodies. The notion of free mobility, as it is generally recognized, plays a central role in Helmholtz’ conceptualization of the PoS. From this condition, Helmholtz claimed to derive that the geometries that are able to represent physical space are those of constant curvature (although he originally excluded the Lobachevskian geometry). This result was rigorously derived later, through applying group theory, by Sophus Lie.^{[11]}

The mathematical derivation of the conditions that the geometries (the metric) of physical spaces must satisfy if one assumes free mobility is one side of the problem. In fact, this comprises the purely mathematical part of the question: starting from a notion of congruence, which must be specified through the formulation of a number of axioms, one extracts the consequences for the geometries that are compatible with it. This part is what Lie perfects. But, one can argue, this makes up only half of the problem, at least as it seems to be understood by Helmholtz and, more importantly, if one wants to fully answer the question of the physical validity of geometry. So, in this case, one must also ask about the consequences of the attribution of a certain spatial (and temporal) geometry for the formulation of dynamical laws. To tackle this, it is necessary to reflect on the status of free mobility as a physical condition, in addition to the derivation of the formal restrictions on the metric.

We find the first seeds of this kind of reflection in Helmholtz (1876). There, Helmholtz’ discussion about how the axioms of physical geometry are based on the notion of congruence, presupposes the possibility of moving solids without deformation. At the end of these considerations, he explicitly refers to the question of the mechanical principles that must be conjoined to the geometrical propositions in order for them to be more than mere definitions without empirical validity. He eloquently adds that without presuming such mechanical principles, the answer to the question regarding the geometry of physical space hides the presumption of a pre-established harmony between form and reality (Helmholtz, 1876, p. 17).

Let me reformulate the core of Helmholtz’ position. This can be done in the following way: in order to claim that the geometry of space is such and such, some mechanical principles are necessarily involved and these are involved in the functioning of the systems through which we gain empirical access to the geometry. In Helmholtz’ case, the relevant physical systems are rigid bodies and the principles concern the independence of the mechanical properties of bodies and their interactions under certain physical operations (translations, rotations and so forth). The reason for this choice is that these are the systems that are involved in the empirical determination of spatial geometry. To go beyond these specific systems, we need to deepen and generalize the principle.

It is evident that Helmholtz’ particular formulation of the PoS, linked to the notion of the free mobility of rigid bodies conceived as a procedure that measures spatial geometry, cannot withstand the progression to a relativistic context. To have a general scheme that is applicable to physical theories in this new scenario, two generalizations would be needed: the problem would need to be formulated in a way that can be interpreted as referring to measurements of spacetime metric; and it would need to be detached from the narrow, finite notion of a rigid body in a way that extends its validity to the infinitesimal domain. Weyl addresses this task in his development of a purely infinitesimal geometry around 1920. Alt hough his reformulation of the PoS passes through di ﬀerent stages, ^{[12]} it seems clear that he understands that his approac h is partly a generalization of the Helmholtz–Lie strat egy that is now compatible with the theory of general relati vity. In a stylized manner, w e can present its main poi nts as follows. The fundamental question that guides the enquiry is how to justify the not ion that the metric which describes spacetime has a certain general form; in particular, the P ythagorean form. The strateg y adopted to arrive at an ans wer consists of starting from a notion analogous to the congruence by free mobility in the Helmholtz–L ie problem, which is given by the definition of infinitesi mal congruences at each point and f or displacements betw een infinitesimally close points. ^{[13]} Weyl realizes that it is necessary to define the congruence of displ acements by introducing a metric connection that s ets the standard of comparison between close points. The condition s that define such an infinite simal notion of congruence are expressed in two postulate s, named by Weyl the Principle of Freedom and the Principle of Coherence. The former can be understood as a principle of free mobility at each poin t, while the latter expresses the condition of compatibility betwee n the metric connection and the aﬃne connection. Finally, W eyl is able to prove a result which constrains the form of the metric. Glossing over ma ny diﬃculties and subtlet ies, we can say that he arriv es at the result (Scholz, 2016) that a metric satisfyin g the conditions of infinites imal congruence, for which the metric connection uniquely determines the aﬃne connection, has the form of a Weylian metric (a Riemmannian metr ic of fixed signature plus a metric connection) with P ythagorean line element.

We have here a general formal scheme that connects a mathematical notion of congruence with certain restrictions on the metric, which furthermore can be formulated in terms of a group of symmetry transformations. In a sense, these symmetry transformations can be interpreted as providing the definition of a notion of congruence through the specification of a mathematical group.^{[14]} Now, in order for this metric to be considered a property of physical space(time), we should be able to interpret the congruence transformations as motions of physical systems which—despite the fact that in idealized form they are defined merely by the mathematical notion of congruence—insofar as they are taken as valid surveyors of the spacetime metric, must be governed by dynamical laws that satisfy certain constraints. This perspective thus has two questions at its core which must be answered in order to say something specific about spacetime and its relation to dynamics: Which chronogeometrical structure is determined by the assumptions of the idealized systems; and what constraints does such an idealization impose on the dynamics of the systems?

The first question, the mathematical part, is answered in the classical problem of space by Helmholtz and Lie through the proofs that free mobility, mathematically defined in a certain way, constrains the metric in such a way that it has to be of constant curvature. And for the infinitesimal case, it is answered by Weyl’s generalization.

The second, the dynamical part, is more conspicuous for the problem of the physical validity of geometry. That it must always be taken into account is revealed by this simple fact: without it, we only have the definition of a mathematical structure with no claim concerning its physical relevance. Only by assuming that there are physical systems that fit the mathematical axioms, is this applicability endorsed. But the question that is rarely brought to the fore concerns the consequences that this has for the formulation of dynamics. Helmholtz suggests, rightly I think, that these consequences can be formulated in terms of some symmetry principles that the dynamics must satisfy. Nonetheless, this demands a precise formulation. My intention is to provide this through the ensuing discussion of the notion of dynamical symmetry as applied to subsystems.

A central aspect of the present approach to the issue of the relation between spacetime symmetries and dynamical symmetries is how, in a given theory, the procedures through which we acquire empirical content (that confirms/refutes the theory) are reflected. I assume that every physical theory, even if it does not have the resources to model measuring devices explicitly, must at least include some features whose interpretation can be linked to measuring procedures performed by ideal observers. This seems unavoidable when the models of a physical theory are taken to represent parts of the world that we experience. So, I must now turn to the question of how these measuring procedures are encoded in features of the formalism of the theory and what consequences this has for its symmetries.

As my starting point, I take a basic, minimal characterization of measuring as a physical process in which two diﬀerent subsystems interact, with the result that the final state of one of them—the measuring device— can be taken as providing information on the state of the other—the target system—just before the measuring took place.^{[15]} As I hope to show, from this extremely schematic characterization it is already possible to extract some general consequences for the definition of dynamical symmetries and their relation to spacetime symmetries for a theory whose interpretation incorporates such minimal modelling of measuring devices.

In order to do this, I must delve into the discussion about whether quantities that are variant under symmetry transformations are observable. A perspective on this issue developed by David Wallace, that takes the role played by subsystems as central, will prove essential. In a series of works Wallace (2022a, 2022b, 2022c) emphasizes that the answer to questions concerning the observability of symmetries are always linked to how the symmetry transformations behave when interpreted as being applied to subsystems. He develops a powerful framework to tackle the main problems in the interpretation of symmetries. I fully agree with this perspective. Wallace argues that the preponderance given to the behaviour of subsystems for the interpretation of symmetries stems from the usual treatment that physicist aﬀord them. Moreover, I would add that the special role that subsystems play in the characterization of measuring devices explains why the notions of symmetries that matter most in physics are connected to their interpretation in terms of the behaviour of subsystems.

So we have a general strategy to tackle some of the main issues concerning the interpretation of symmetries which, starting from a general formal characterization of the notion of dynamical symmetry (basically, a transformation that takes solutions to solutions), complements this sparse definition, which by itself seems unable to provide answers to questions about the representational capacity or the observability of symmetries, with the idea that such issues must be interpreted in the context of the application of symmetries to subsystems. In particular, in order to decide whether certain quantities that are variant under symmetry transformations (and therefore usually considered to be unobservable) are obser vable, one must look at how the symmetry extends from i ts application to a given s ubsystem to the interact ion between that subsystem and its environment. Only in cases in which a symmetry transformation of a giv en subsystem is also a symme try of the composition sub system-plus-environme nt (and it is, using Wallace’ termi nology, extendible), can some variant quantities be observed despite the ‘common wisdom’ that only i nvariant quantities are observable. ^{[16]} Let me sketch Wallace’ argument, as it intro duces some elements that are e xtremely fruitful wi th regard to the relation be tween the characterization of measuring devices and judgements ab out the dynamical symmetries of a theory.

Wallace starts from the aforementioned notion of dynamical symmetry and assumes a physical description of a measuring device: a system that has a ready state that is independent of the target system and which, after interacting with it, ends up in a state that is a function of the pre-measurement state of that target system (Wallace, 2022a, pp. 8-9). From this, it follows that a measurement that is internal to the system cannot detect whether a dynamical symmetry transformation has been performed. This is proof of what Wallace calls the Unobservability Thesis. Now, the interesting question is what happens to the measurements of quantities for systems that can be considered as external to the subsystem in which the measuring device is placed (that is, measurements external to the subsystem). This involves considering the device itself as a subsystem interacting with a target system that can vary independently of it. Wallace introduces the following notation to express the combined state of the two subsystems: (^{[17]} If the symmetry is extendible and global, it is possible to define the invariant quantity: ^{−1} g. Now, assuming that the primed system is a measuring device as characterized above, and in particular that it meets the condition of having a dynamics that is independent of the target system, then we can fix the quantity ^{−1 }g covaries with g, it is possible to measure g. So, this amounts to an account of how a quantity that is variant under a dynamical symmetry transformation can be measured, if we are ready to interpret it as a relational quantity expressing some kind of target–device relation. The synopsis of this argumentation is that, for subsystem-global symmetries,^{[18]} globally variant quantities are observable via measuring devices outside the system, but such observations can always be reinterpreted as o bservations of an invari ant relation between sy stem and measuring device (at least, this is so in the context of the theories that Wallace considers).

We need now to reverse Wallace’ argument: instead of starting by assuming a given dynamics with a certain type of symmetry, as Wallace’ does for the case of Newtonian particle mechanics, I will explore what can be inferred about the relation between the internal dynamics of a subsystem that acts as a measuring device and the dynamics of target systems measured by it, if one starts from just the general characterization of a device that measures some quantities of external target systems.

Famously, Earman (1989) explicitly expresses two heuristic principles for the formulation of theories of motion declaring the equality of spacetime symmetries and dynamical symmetries. My aim in this section, through making use of the analysis in the two previous sections, is to address the question of the foundation of Earman’s principles and, in general, to discuss the possibility of formulating principles that relate spacetime symmetries and dynamical symmetries. This must necessarily involve a discussion of the motivation behind the definitions of the symmetries that the principles interrelate. In particular, because it is usually taken for granted, I am especially interested in discussing the notion of spacetime symmetry.

First, we need to consider what kind of principles Earman’s principles are. For this we must make explicit what definitions of symmetry they presuppose. Let me start with the notion of spacetime symmetry. Earman’s discussion assumes that a formulation of a physical theory (a theory of motion) incorporates the identification of certain structures as spatio-temporal. If this is the case, then we can define spacetime symmetries as transformations that leave these structures invariant. From this posit, Earman’s principles are understood as providing criteria to establish which formulations of a given theory are preferable in virtue of their not containing spacetime structures whose symmetries do not coincide with the dynamical symmetries. Dynamical symmetries, on the other hand, are defined in the standard way (see the previous section). This is consistent with Earman’s understanding of the principles as heuristic: one begins with some posit on what the spacetime symmetries—implicitly encoded in a given interpretation of a theory—are and attempts to refine it by recourse to the principles. To avoid circularity, though, the justification of the principles must be independent of the definition of what a spacetime symmetry is. This is why Earman stresses that these are not principles of meaning (they are not analytical) and he invokes some epistemic considerations, allegedly related to the general notion of spacetime, to try to provide a justification for the principles. I think that the two central ideas in Earman’s discussion of the principles are right: that the principles should not be taken as analytical and that their force derives from the connection between the notion of spacetime and its epistemic role in physical theories. Nonetheless, keeping the nominal definition of the notion of spacetime symmetries (i.e., as symmetries of spacetime structures), it is easy to fall into one of two interpretive traps (that excessively burden the discussion). The first consists of taking the nominal definition as substantive and thinking that what the justification of the principles would determine is that the dynamics is adapted to spacetime structures that are not dynamically determined. The second, partly motivated by dissatisfaction with the first, is to think that dynamics, transparently and without presupposing any further epistemic input, dictates what the spacetime symmetries are. To avoid these extremes, it is advisable to note from the beginning that in the determination of which structures are spatio-temporal, and therefore what spacetime symmetries are, epistemic considerations of a dynamical character must be taken into account. This is why I propose to make it explicit from the start that in the determination/definition of spacetime symmetries, general conditions that can be interpreted as proceeding from the characterization of measuring devices are essential, and the precise sense in which they are. These are the epistemic considerations of a dynamical character that might also be taken as providing content for a definition of the notion of spacetime symmetry that goes beyond the nominal definition.

Let me try to make all of this more precise. The connection between the determination of geometry and dynamical conditions was at the centre of the responses to the PoS. The link, in those frameworks, was provided by coordinating a notion of congruence (finite, in the Helmholtz–Lie classical response; infinitesimal, in Weyl’s version) with some transformations that are taken to be the correlate of the motions of physical systems that would measure physical geometry, expressed as the condition of free mobility and its translation to the relativistic context. The general assumption here is that a determination of physical geometry is always going to be through the identification of certain transformations that can be interpreted as defining a notion of congruence (some kind of relation of equivalence for physical systems that meet certain criteria that permits us to interpret them as congruences). These transformations of congruence are the natural candidates for providing the definition of spacetime symmetries. So far, they are the transformations that can be used to define a geometry, in line with Klein’s Erlangen programme, from the structures which are invariant under them. The connection to (physical) spacetime, in the PoS approaches, comes from assuming that such transformations can represent physical motions of systems that measure the geometry of space. In the case of the classical solution to the PoS, the physical interpretation of the notion of congruence is given by the notion of free mobility of rigid bodies, which is then associated with the group of transformations that are permissible according to the mathematical notion of congruence. In other words, we have a mathematical notion, congruence, that provides a sense of correspondence for mathematical objects, and its physical counterpart given by the idea of rigid body. This allows us to determine a group of transformations as those respecting certain internal relations, and from then to determine, a least partially, the geometry of space.

In any case, I want to stress that these approaches provide a framework within which to formulate the connection between geometry and dynamics. Note that the general strategy can be taken, independently of the specific results that it renders, to cons ist of providing the mathemat ical characterization of a certain notion of congruence t hrough a group of transfor mations, which will be int erpreted as defining a geometr y. If one starts with a prior notion of congruence (equality of lengths for vectors, for instance), this det ermines the group of transf ormations. But we could also think, inversely, of the group as defining congruence. This perspective m ight be especially relevant when we leave the context of pure mathematics. In this case, we might think that a no tion of mathematical cong ruence will be justified insofar as it represents a certain concept of equ ivalence for physical systems that is relevant in some specific wa y. Whatever that notio n of equivalence may be, by generalizing the lessons from the PoS we can see that it is its eventual associ ation with a group of trans formations (if they can be interpreted—using Weyl’s terminology—as allowing congruence transfers) that will provide the connection with the geometr y of spacetime. This point s to the desired link between spa cetime symmetries and dynamical sy mmetries, as I will next el aborate, but coming from the opp osite direction.

Let us now consider the treatment of measuring devices as subsystems interacting with other subsystems. Generally, we can take them to be measuring empirical quantities that can be used to confirm/refute a given dynamical theory. The quantities themselves need not be spatio-temporal, but the assumption is that, in order for them to have empirical relevance (some authors call this empirical salience), the measuring must provide some kind of parametric marking of the events in such a way that the measured relations can be taken as data to test the theory. It seems diﬃcult to see how this assumption could be avoided (which does not mean that it is being assumed that the relations are determined). Perhaps a less loaded assumption about the functioning of certain subsystems as devices would just be that the measurement contents must be coordinated in such a way that relations between the events can be expressed, and some of them sanctioned, as being derived from the dynamics.

In any case, initially bracketing the question about the degree of commitment that one is ready to make to the minimal structure of events needed to formulate a dynamical theory, measuring devices can be taken to be physical systems that can be characterized as subsystems that interact with other subsystems. Since we will be interested in the description of dynamical symmetries as they apply to diﬀerent subsystems, we can use the framework discussed in the previous section. The dynamics of subsystems acting as measuring devices can be represented in a configuration space with coordinates capable of encoding the dynamics of target systems (whether this is the same subsystem device or other subsystems). Borrowing Wallace’ notation that I previously introduced, we can represent the combined target system–device state as (^{−1} g would be invariant under symmetry transformations (if they are symmetries of the combined system) but could be interpreted, assuming that the change in the device is undetectable (therefore taking ^{[19]}

This is the conceptual basis that connects certain dynamical symmetries with the notion of congruence: they share some formal characteristics (being defined by a group of transformations that are not observable via the change in quantities measured internally, but nonetheless making sense of the claim that the transformation has taken place because some quantities that encode external relations have changed). From establishing this connection, the next step is to say something about the relation between the symmetries of the dynamics of the device and the dynamics of target systems. For this, we just need to recover the following result: in order for these quantities to be observable externally, the symmetries of the subsystem device must be extendible and, using Wallace’ terminology, subsystem-global: the same group must be a symmetry group of the diﬀerent interacting subsystems. This means that the dynamics of target systems must have the same symmetries as the dynamics of the subsystem device.

Formally, the rationale for such a connection is given by the equivalence of some structure in both cases: the existence of transformations with the structure of a group (which is given as part of the definition of congruence, and eventually of geometry) and the identification of relevant invariances of the dynamics. This is one way of expressing what has been done here: a mathematical notion of congruence (which can be taken as defined through a group of tra nsformations) has been c oordinated with a dynamical n otion of congruence. This latter is based on the idea of subsystems that can detec t some symmetry transfor mations that are internall y unobservable but observable by measuring qua ntities whose variation wit h the transformation is interpreted as detecti ng change in some relation b etween the subsystems. The motivation for the coordi nation between these no tions comes from the idea that such t ransformations for measuring devices share essential features with mathematical cong ruence transformations; they provide a criterion of equivalence which is somehow inter nal, together with a di stinction between init ial and final state which is ext ernal. There is a class of dynamical symmetries that can accomplish t his: those that can be i nterpreted as dynamical congruences and d efine a subclass of dynamical symmetr ies.

We can define

The general motivation behind this definition is that ^{[20]} one can say that the geometrical notion of congruence is defined by some procedure for determining the equivalence between bodies at the same place and some rules for comparing distant bodies, all of which determine certain transformations. Mathematically such transformations form a group. Alternatively, one can think of this characterization as providing a procedure to determine certain intrinsic properties of the figures (length, angles...) and a group of transformations that keeps the intrinsic properties invariant while changing the extrinsic relations to other figures. Naturally, depending on what the procedure to determine the intrinsic property is, the group of transformations found is going to be diﬀerent and, one might think, the fact that a certain group of transformations define a geometry is dependent on having originally chosen properties that are, let us say, spatial or geometrical.

The bold step taken here, inspired by Helmholtz’ treatment, consists on abstractly focusing on the properties of the dynamics that the physical systems that implement the notion of congruence must meet and, together with this, generalizing by abstracting the initial geometrical features. The main leading question can be posed in the following terms: What general conditions must the symmetries of the dynamics meet in order to be at the base of a definition of congruence? The leap is taken by thinking that any dynamical symmetry that meets such conditions could be considered as able to support a definition of congruence. To put it diﬀerently, if we blindly started by looking at the dynamics without a previous geometrical ba ckground, we could use those pro perties to define a subset of dynamical symmetries that, eventually, might be taken to define a cong ruence. The answer to the ques tion, I claim, is found in the feat ures that certain dynamical symmet ries when applied to subsystems have. They define a group of trans formations that are not detect able by measuring devices detecti ng intrinsic quanti ties but can be detected by variations in som e relational quantities between subsystems. F ormally, they will be congrue nces. ^{[21]}

At this point, it is important to stress a couple of things. The first is that there might be further requirements that are needed to coordinate the dynamical notion of congruence with a geometrical one, but it seems unlikely that we could formulate them in general without using concepts that are already chronogeometrical. Here I have specified the minimal structure that is behind the connection between spacetime and dynamics in a theory, intentionally eluding any mention of plainly phenomenological notions.

The second remark has to do explicitly with the nature of the convention involved here. This can be seen from two complementary perspectives. From the geometrical point of view, it involves taking some physical systems as suitable for the implementation of a notion of geometrical congruence; from the perspective of the dynamics, it means assuming, in cases in which a full dynamical description of the measuring devices is inviable, that the laws governing the dynamics of the devices have certain symmetries. These two aspects are derived from the original conventional dimension involved in the coordination of a notion of dynamical symmetry and a geometrical congruence. This could also be expressed in a slightly diﬀerent way: through its coordination to a notion of mathematical congruence, we are using a notion of dynamical congruence to define spacetime symmetries. In this sense, this approach is at the base of an eventual dynamical definition of spacetime symmetry, which is the principal motivation of the DA.

The relation between spacetime and dynamical symmetries can be traced back to the interpretation of some features of physical theories as potentially codifying the notion of ideal observers. This interpretive framework provides a justification for the connection between a notion of congruence, from geometry and essential for the determination of physical space(time), and certain features that characterize the measuring devices that can be used to empirically test the theory. So, such a connection can be understood as a way of giving content to an epistemological framework that assumes that the same procedures that are used to measure the geometry of spacetime are also part of the means through which we arrive at the empirical content that suppor ts our physical theories. In both characterizations (congruence and measuring devices) certain tran sformations play an esse ntial role; the present prop osal attempts to state under what condi tions those transform ations can be equated. This is what provides the qualified rel ation between spacetime and dynamical symmetries express ed by the symmetry princi ples.

These are the terms involved in this formulation of the principles. Spacetime symmetries, nominally invariances of spacetime structures, must be understood as being determined by the congruences associated with certain (idealized) systems that are taken to probe spacetime. Dynamical symmetries are transformations that take solutions to solutions; and are such that when applied to subsystems that act as measuring devices, they are internally unobservable but detectable as changes in quantities that are relational between subsystems. With these definitions, and the discussion in this paper, we have a justification for Earman’s type of symmetry principles.

Such principles are restricted in two senses. First, this formulation assumes that every dynamical symmetry so defined is going to be a space-time symmetry but, as I have suggested, this assumes that all such dynamical symmetries can be interpreted as congruences. This might not always be the case (think of the controversial case of global “internal/phase” symmetries). Second, it depends on assuming a certain degree of idealization for the physical systems that determine spacetime structures. Such an idealization might be based on physical considerations or, as in the case of Weyl, on phenomenological ones.

The principles so explicated can be understood as heuristic principles for the interpretation of spacetime theories. They recommend an interpretation of the theory in which certain dynamical symmetries (those meeting the conditions referred to above) are interpreted as a group of congruence transformations that is at the base of the definition of spacetime structures. Such an interpretation might have to be accompanied by diﬀerent kinds of formal modifications; sometimes, but not always, these might recommend considering some structures as surplus.

We can distinguish diﬀerent interpretive levels in this proposal. At the base, we have the claim that the relation between spacetime symmetries and dynamical symmetries arises from the coordination between some dynamical symmetries and a mathematical notion of congruence. This might be inserted into an interpretation of the formalism of a theory in which certain features are taken as codifying the work of measuring devices. Finally, the connection can be justified in a general framework in which this is related to the representation of ideal observers. The full package is what I believe motivates us to regard the proposal as a version of a functionalist approach to spacetime.

Acknowledgements

I am grateful to diﬀerent anonymous referees for their very helpful comments and suggestions. Research for this article has been supported by the following project: Reassessing Scientific Objectivity (PID2020-115114GB-I00), Ministerio de Ciencia e Innovación (Spain).

Adán Sus is an Associate Professor at the Department of Philosophy of the University of Valladolid. His research is focused mainly on the philosophy of physics, in particular the philosophy of spacetime theories and symmetries.

Address: Department of Philosophy. Universidad de Valladolid. Pza del Campus, s/n. 47011. Valladolid, Spain. E-mail:

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