1 Introduction

The so-called “block view” is taken to be the metaphysical position that best accommodates the results of special relativity: the latter is generally taken to imply that no set of events displays an ontologically privileged status vis-à-vis others (Petkov 2008, 180). The block view incorporates this recommendation by portraying the universe as a “timelessly existing four-dimensional” entity (Petkov 2006, 207), in which no ontological distinction between past, present, and future events holds. All events co-exist within the block. They occupy permanent positions, as their location within the four-dimensional manifold is given permanently.Footnote 1

Precisely because the block takes events to be given once and for all, it adheres to the language of being and finds difficulties accommodating a thorough notion of becoming. To clarify the matter, a main distinction is to be drawn between occurring (or happening) and becoming (or coming-into-being). As no absolute temporal ordering between events holds, all of them are to be taken as ontologically on a par. This makes it hard to adopt a genuine notion of becoming, unless one reduces it to mere occurring. For example, while defending the block view as a suitable metaphysics for the theory of special relativity, Grünbaum (1969) is adamant that the two verbs cannot be taken as synonymous with one another. He distinguishes “to occur” as a verb that has physical meaning from “to become” that finds no room in the physical realm. In this reading, “occurring” stands for the tenseless taking place of events in their fixed position given once and for all, while events hardly happen in the tensed sense of “coming into being apart from anyone’s awareness of them” (Grünbaum 1969, 161). In short, according to Grünbaum, the lexicon of becoming is to be expelled from the metaphysics that incarnates the main lesson of relativistic physics.

This article will make the case that, if one holds onto this metaphysical view, one should coherently be prepared to strip off all elements of dynamism from one’s representation of physical reality. As the lexicon of “dynamics” is very abundant in physics, a major misunderstanding for how the term is invoked in the present article pertains the fact that physical entities are subject to forces and should thus be described in a dynamical language. This naturally applies also in the context of special relativity, whereby Minkowski spacetime allows the presence of forces encoding dynamical evolution. The theory of special relativity, so the argument goes, envisages a dynamic description of physical reality, at least within the light cone structure, and at least relatively (namely, with respect to a chosen inertial frame). While I concur with this, the point on which the present article is meant to focus is that the evolution encoded through forces within the Minkowski manifold is something different from the coming-into-being of events. One should thus not conflate genuine becoming with mere happening. Put differently, embracing a certain metaphysical view comes at a price one should be ready to face, no matter what the associated consequences. Therefore, if one sticks to the metaphysics of the block, not only should the lexicon of becoming be completely expunged; more than that, the theory has to account for every aspect of reality—including those usually defined as experiential or mind-dependent—in resolutely adynamical terms. On this “resolute view”, as Geroch (1978, 20–21) puts it, “[t]here is no dynamics within space-time itself: nothing ever moves therein; nothing happens; nothing changes”.Footnote 2 For example, Geroch continues, the collision between two particles is ultimately to be represented in a single unmoving spacetime. Accordingly, the article will mainly try to tease out the reasons for arguing that, within the block view, “dynamics” cannot be intrinsic, insofar as it does not appertain to the world of physics. Rather, it should be recovered “by comparing the situation as recorded on several horizontal 3-planes” (ibid. p. 21). Yet, I will continue to argue that, if one wants to implement a thoroughly static physics, there is a better candidate, one that leads to a metaphysics alternative to the block. This is Barbour’s (and collaborators’) early attempt to develop a relational physics program, as introduced by Barbour (1974), Barbour and Bertotti (1977, 1982) and successively dubbed “shape dynamics” by Barbour (2003). There are two main motivations for focusing on these works, rather than more recent results (such as Gomes et al. 2011, or the newly developed formulation in terms of pure shape dynamics as discussed, e.g., in Koslowski et al. 2022). First, the early attempts to formulate a neo-Machian relational program represents the foundational milestone of Barbour and collaborators’ overall enterprise and sets forth the conceptual basis for the successive development of shape dynamics. Second, the focus on this early and somewhat incomplete formulation of shape dynamics is instrumental to the overall economy of the article, namely, to compare approaches aimed at developing a fully adynamical understanding of physics—where the notion of adynamism should be read, following Geroch’s resolute view, as implying that no genuine notion of becoming (or coming-into-being) is at stake. Put differently, while I concur with Vassallo et al. (2022) that shape dynamics, especially in its most mature version, may result amenable to different metaphysical positions,Footnote 3 Barbour and collaborators’ original formulation (and associated metaphysical insights) more resolutely resembles a static understanding of physics. Within the context of this endeavor, I will argue both the relational blockworld and Barbour’s enterprise follow the so-called “Lagrangian schema” (as opposed to the “Newtonian schema”). Following Wharton (2015), these two schemata subscribe to diverging understandings of reality, whose main difference amounts to that between adynamical constrained histories and evolving dynamical states. My objective will then be to show that Barbour’s approach proves more successful in embracing adynamism, thus completely expelling the need to accommodate becoming.

The structure of the article is the following. In the first section, I will canvass some recent attempts to vindicate aspects of dynamism within the block view. Specifically, I will investigate Ismael’s and Dieks’s accounts of becoming and how they seek to make sense of the co-presence of dynamical and adynamical elements within the block. In the second section, I will explain what it means to expunge all facets of dynamism. I will dwell on the difference between the Newtonian and the Lagrangian schema, and why the latter is a promising way to capture the road to adynamism. In the third section, I will introduce Silberstein and Stuckey’s argument that the block view is to be conceived in genuinely adynamical terms. I will analyse the reasons they provide for a “relational blockworld” that pursues the end of a resolutely static metaphysics that, in their view, is not at variance with the adynamism of relativistic physics. In the fourth section, I will discuss the strengths of Barbour’s relational physics program. I will put forward the argument that his physical project as well as its metaphysical outcomes offer a more robust backdrop for adynamism than the block view.

Before concluding this introductory section, it is worth emphasizing that this article does not support an adynamical view of the universe. Rather, my argument is that, if one comes to the conclusion that our universe is a static one as block viewers concur, the most radical picture of an adynamical universe is not the block view. The theoretical program that best accommodates an adynamical image of reality is Barbour’s early attempts to promote a completely relational physics, within which the issue of becoming never comes up and the language of being survives. However, this better option for adynamism comes at a price. Barbour’s physics dispenses with the notion itself of objects to the advantage of configurations. All that is not a configuration is thus deemed to be but an illusion.

2 The Block View

Block viewers such as Price (1996), Savitt (2002; 2006), Dieks (2006), Petkov (2006), Dorato (2002; 2006), Zeh (2007), Wüthrich (2010), and Ismael (2017) argue that the metaphysical position that best fits the results of both special and general relativity is the block universe. Despite the differences in the ways it has been characterized so far, this model portrays the universe as a four-dimensional entity in which no ontological distinction between past, present, and future events obtains. Every event is located in its four-dimensional position within the Minkowski spacetime. It is only by indexically referring to a specific location within the four-dimensional manifold that a local partitioning of events in terms of temporal ordering can be defined.

A renowned argument to validate the metaphysical consequences of special relativity, with reference to the relativity of simultaneity, was put forward by Rietdijk (1966) and Putnam (1967). The Rietdijk-Putnam argument is intended to show that the relativity of simultaneity necessarily implies a four-dimensional image of the universe. In short, the argument runs as follows. Take \(R\) to be the relation of “being real with respect to”, so that \(Rxy\) means that \(x\) is real with respect to \(y\). The crucial point is then which kind of properties should be bestowed on \(R\). According to Rietdijk and Putnam, \(R\) should be reflexive (a thing is real with respect to itself), symmetrical (if \(Rxy\) then \(Ryx\)) and transitive (if \(Rxy\) and \(Ryz\) then \(Rxz\)). By combining these properties of \(R\) with the relativity of simultaneity, it follows that the universe should be conceived as a four-dimensional entity in which all events are on an equal footing: “We have learned that we live in a four-dimensional and not a three-dimensional world, and that space and time—or, better, space-like separations and time-like separations—are just two aspects of a single four-dimensional continuum with a peculiar metric” (Putnam 1967, 247).

Although some authors have tried to bring this argument into question based on different lines of thought (see e.g. Sklar 1977; Stein 1968, 1991), the block universe within a relativistic context is agreed upon by so many scholars that it has by now become common wisdom.Footnote 4 Petkov (2006) is crystal-clear about this. On the one hand, he contends that “the block universe view, regarding the universe as a timelessly existing four-dimensional world, is the only one that is consistent with special relativity” (Petkov 2006, 207). On the other hand, he is also convinced that the theory of special relativity successfully settles the controversy between three-dimensionalism and four-dimensionalism.Footnote 5

Once the block universe is admitted as the only metaphysical complement to relativity, we are left with one major problem, which in the last decades has attracted abundant attention. Since all aspects of the world are to be included within the set of events making up the four-dimensional manifold, where no distinction of past, present and future applies, how should one account for (at least apparently) dynamical aspects of reality? In other words, what are the types of constraints imposed by the metaphysical view married to relativity? While many scholars have dealt with this issue from a variety of angles (see e.g., Dainton 2010; Huggett 2010; Dorato 2015; Dorato and Wittmann 2019; Callender 2017), it is worth emphasizing that this article does not discuss the ways to reconcile experiential time with physical time. My point is both more general and more specific, which is to say that the block view should strip off any element of dynamism, including experiential time when it is accounted for in dynamical terms. Hence, experiential time is just one further case of dynamical features within the theory. As I noted above, if one advocates the block universe, then one has to rule out notions that contradict it in all fields of reality, at all levels of analysis. However, some authors aspire to accommodate aspects of dynamism in a way that (allegedly) does not contradict the static nature of the block.

Ismael advances a fundamental distinction between the all-encompassing view of the block and the situated viewpoint of human beings. She dubs the former “History sub specie aeternitatis” (Ismael 2016, 110) and the latter the “embedded, embodied” (ibid., p. 107) position of a participant. History corresponds to the total sum of events that are not relativized to any temporal frame of reference, whereas the participant enjoys the limited point of view inescapably affixed to a temporal frame of reference. On Ismael’s account, the experience of becoming is the outcome of a transformation that turns the History view of the four-dimensional manifold (which is invariant under temporal transformations) into the embodied view producing an evolving image of the universe. From this latter vantage point, what we get is a representation of the world through a series of events ordered in a temporal sequence with a fixed past and an open future. However, sub specie aeternitatis, everything is fixed, including human actions and the seeming evolution of historical events—even though from the particular point of view of an embodied participant nothing looks fixed.

In sum, her defense of the block view leads Ismael to adopt a twofold type of analysis, in which the invariant History of the universe gets represented time by time by the embodied participant. The term she refers to as a situated participant in History is “information-gathering and information-utilizing system (IGUS).”Footnote 6 In doing so, the dynamical aspect of becoming is reduced to the way in which local systems represent the invariant structure of the static relations captured by the block view. Notwithstanding Ismael’s insistence on the difference between levels of analysis, her understanding still begs the question: If the block universe contains the set of all past, present, and future events, should it then not also encode all local representations within it? Put otherwise, does the type of becoming she advocates possess the genuine characteristic of becoming, or should it rather be treated as an illusion? Ismael opts again for a dual approach. Becoming is real but is not actual. It is real, in the sense that is not a pure illusion. It is the way in which human beings experience the world. However, physics does not have to level with such a “folk ontology” (Ismael 2016, 122). Actual becoming is nothing that the universe undergoes.

Therefore, although Ismael distinguishes between illusionary and perspectival experience, actual becoming is not an inherent feature of physical reality. This dual level of analysis that leads to differentiated perspectives is rejected by Dieks (2006; 2016). Nor does he accept the idea of different levels of reality, especially if one level entails static events and the other a dynamical reconstruction of them. All the events of the universe, together with their characteristics and mutual relations, figure in the block universe and thus are on an equal footing. Therefore, events from a local perspective and the structure of physical events must coincide in their corresponding spacetime collocation. Despite this, Dieks does insist that such a static image of the world and the existence of change are not at variance. For science still makes room for change as a variation of properties: “Basic science has no problem dealing with change: change is variation in the values of quantities during time” (Dieks 2016, 19). This is how Dieks comes to promote the idea of “local becoming”, that is, the local succession of the coming-into-being of events: “Events come into being by occurring, by happening; what other coming into being could there be?” (Dieks 2006, 170).

A major issue with Dieks’ analysis is that it equates being and becoming. And this comes down to a rather thin notion of becoming as the mere occurrence of events. Therefore, the identity of being and becoming simply reduces the semantic scope of the latter. Indeed, the notion of coming-into-being continues not to be compatible with the stillness of events within the block. Events that simply are do not come into being in the sense of something that passes from a state of not being to a state of being. While for Ismael this passage is a matter that does not relate to the field of physics, but that of human experience, Dieks rejects this dichotomy and yet treats events as though they could become from a physical standpoint (on the same wavelength, Dorato 2002; 2006). But, in the end, this turns out to be a denial of genuine becoming in the sense of coming-into-being: “[B]ecoming is nothing but the happening of events, in their temporal order” (Dieks 2006, 171). The meaning of “happening” here is exactly the one that Grünbaum attributed to it – one that according to Grünbaum has nothing to do with “coming into being”. As a result, the dynamism of physical reality is effaced to the advantage of the metaphysics of the block.

It should come as no surprise that some critics of this conception found it “yawn-inducing” (Earman 2008, 159)Footnote 7 in the sense that the problem of becoming is simply swept under the carpet. In the case of Ismael, becoming is not a physical issue, but an experiential one. In the case of Dieks, becoming is the happening of events in their assigned position. But none of them faces the question of what becoming is in ontological and physical terms. Again, physics only makes room for the lexicon of being. But if one intends to take becoming seriously, the coming into being of events is to be given an explanation able to reconcile physics and metaphysics. Put it differently, the double standard view which tries to reconcile a static, fixed, all-encompassing universe with seemingly genuine dynamical aspects at the experiential level is unsatisfying and ultimately flawed. To this end, there are various theoretical perspectives aimed at bridging the gap between physical time and experiential time. The one advocated by Earman (2008) builds on a thicker notion of events as discrete elements that do become. He insists that, to do so, the metaphysics of the block should be abandoned, while, drawing from works such as Rideout and Sorkin (1999) and Sorkin (2007), the physics of special relativity should be regarded as a continuous approximation of a more fundamental, discrete underlying theory of causal sets.

It is not for this article to go down this road. My objective in this section was to cast doubt on the possibility of reconciling dynamical aspects of reality—a genuine becoming—with the metaphysics of the block. As I will insist in the subsequent sections, a consistent block view is forced to subsume all aspects of reality (including the dynamical aspects of human experience) under a totally static, adynamical physical conception, and to privilege being over becoming.

3 The Lagrangian Schema and the Road to Adynamism

Critics of dynamism think that residues of dynamical laws within contemporary physics are due to a fundamental bias that exists within Newtonian physics and that persists in its effects even within successive physical paradigms, such as relativistic physics and quantum mechanics. It has to do with the understanding itself of physical phenomena, whose evolution through time is what physical laws are alleged to map and predict. Critics of dynamism contend that this very understanding of physics is inevitably burdened by the notion of something that evolves through time, no matter how time is conceived. This is why even theories that espouse timelessness cannot completely purge themselves of this bias. The only way to implement an effectively adynamical physics is to give up such a Newtonian legacy. To understand what a resolutely adynamical block world implies, as well as how one can achieve it, it is worth expanding on this insight.

In his treatise on the variational principles of mechanics, Lanczos (1952) points out that, ever since Newtonian foundations of mechanics, the science of mechanics has followed two main paths. Vectorial mechanics determines the vector sum of the forces acting on a given point-particle, its motion being uniquely governed by the totality of the forces acting on it at every instant. Newton’s mechanics relies on the determination of the momentum of a force to calculate its action. Contrary to Newton’s vectorial mechanics, Leibniz took the kinetic energy of the system as a proper gauge for the dynamical action of a force. Therefore, he replaced Newton’s force with the work function. In this regard, Leibniz can be considered as the initiator of a second branch of mechanics, namely analytical mechanics. It defines states of equilibrium and/or motion based on two scalar properties, or rather, the kinetic energy and the work function (the latter being frequently replaced by the potential energy of the system under consideration). Since motion is a directed phenomenon, it might at first look unclear how two scalar quantities can instantiate the corresponding vectorial properties. Indeed, the energy theorem, according to which the sum of the kinetic and the potential energy of an isolated system is constant, only results in a single equation, while the motion of a particle in the three-dimensional space requires three independent equations.

However, as Euler and Lagrange first discovered, the energy theorem is to be used as the basis of a principle rather than an equation. To understand how this principle operates, suppose that one has to study the motion of a particle from an initial spacetime point, having a certain initial velocity, to another spacetime point. Even if the actual path followed by the particle is unknown to us, we can mathematically establish all its virtual paths (all the curves that connect the initial and the final position of the particle), provided that the energy theorem holds. These virtual paths are trial curves we can allow the particle to ideally move along in accordance with the energy principle. Based on this principle, one can determine the velocity at each point in all these virtual paths, and thus the motion of the particle in that specific path. We can arbitrarily choose one of these virtual paths, but once the path is chosen, the motion follows uniquely. In particular, we can determine the time at which the particle reaches an arbitrarily chosen point in a specific virtual path and thus to calculate the time integral of a specific quantity between the initial and the final position of the particle. This time integral is called “action”. It has a definite value for all virtual paths. Euler and Lagrange found out that the actual path of motion is the one that minimizes the action,Footnote 8 this condition being called the principle of least action.

The conceptual differences between vectorial mechanics and analytical mechanics have been recently addressed by Wharton (2015). He argues that, ever since Newton, we have been accustomed to the idea that the universe is a system that takes certain initial conditions, applies a set of dynamical equations, and then traces the time-evolved results in terms of physical predictions. This understanding of the universe is so widespread that Smolin (2009) has dubbed it the “Newtonian schema”. Despite some divergences on how to best understand the Newtonian schema, both authors claim that this anthropocentric view of physical reality introduces a conceptual bias that may ultimately impede a more comprehensive account of the universe. It might well be the case, they contend, that the fundamental laws governing the universe cannot be squared with such a conceptual framework.

According to Wharton, a viable alternative—one that makes sense of the fundamental and yet too often neglected conceptual difference between the vectorial and analytical formulations of mechanics—is the so-called “Lagrangian Schema.” These two schemata operate in utterly diverging ways, as the Lagrangian one requires logical rather than initial inputs. On this respect, one starts by parametrizing physical events, then constrains these parameters, and finally applies the principle of least action to obtain the unconstrained parameters. In this frame, there are no initial values that yield the need for dynamical laws that allow the prediction of temporally evolved trajectories. Rather, initial and final states are logically constrained according to a principle of minimization.

There are two main attitudes towards the Lagrangian schema. The first is to regard it as a mathematical methodology that bears no genuine physical meaning. The second is that action minimization is physically equivalent to the Newtonian formulation of mechanics.Footnote 9 He decidedly rejects both these interpretations and advances the idea that the Newtonian and the Lagrangian schema are opposing perspectives on physical reality. The Newtonian schema is hospitable to elements of dynamism, in that, by devising a predictable trajectory evolving in a 3 + 1 framework, it embodies the idea of a temporal evolution. Quite the reverse, the Lagrangian schema is utterly adynamical, in that there are no initial and final conditions; and, more importantly, there is no evolution which takes from the former to the latter. Nor is there a trajectory that is to be predicted based on dynamical laws. There is no evolution whatsoever and a fortiori no need for the dimension of time: “There’s no longer any mathematical difference between the past and the future” (Wharton, p. 184).

Ultimately, then, the difference lying between the Newtonian and the Lagrangian schema amounts to the difference between evolving dynamical states and adynamical constrained histories. For the limited purposes of this article, Wharton’s reasoning helps introduce a theme that will be key to the following sections, that is, a more resolutely adynamical understanding of physical reality in terms of the Lagrangian schema. The two theoretical frameworks I will discuss, namely, the relational block world and Barbour’s relational program, are both couched in the spirit of such schema.

4 The Relational Block World

Silberstein et al. (2012; 2018) pivot their critique of dynamism within theories of the block on the Lagrangian schema. They state that it “is supremely ironic that the dynamism and unificationism historically driving physics led us directly to blockworld and frozen time” (ibid., p. 504).Footnote 10 Although dynamical physics has led to adynamical paradigms, the co-presence of dynamical and adynamical aspects within contemporary paradigms generates a conflict that needs settling, and the conflict is to be settled in favour of adynamism. As their argument goes, dynamical explanations are still the leading assumption in physics because of the Newtonian bias that I focused on in the previous section.

Silberstein, Stuckey and McDevitt get to an adynamical view of the block within the scope of a broader research line towards unification. To this end, they build on the notion of “spacetimematter” co-construction per a global constraint equation. In this view, the fundamental elements of physical reality are tetrahedra, namely 4d-simplices, whereby both quantum and classical physics can be recovered only statistically. It is a form of monism in which space, time, and matter still figure at the most fundamental level but are characterized neither in causal nor in dynamical terms (unlike other approaches such as loop quantum gravity or causal set theory). This research line takes the lead from Feynman’s rationale for the introduction of the path integral approach to quantum mechanics, based on a generalized version of the principle of least action. In this regard, Silberstein, Stuckey and McDevitt’s approach is genuinely Lagrangian. As I hinted above, they argue that their discrete graph theory can be used as a starting point to recover both quantum theory and general relativity. Let me emphasize that the “Relational Block Universe” is a rather speculative attempt to come up with a unificationist framework in which the theoretical articulation directly percolates from the metaphysical desideratum of providing, in the spirit of the Lagrangian schema, a fully adynamical understanding of physical reality, in all its facets.

As far as quantum theory is concerned, they maintain that it can be reconstructed through boundary operators on the graph satisfying an adynamical constraint equation they dub the “self-consistency criterion”. More specifically, they assume as a starting point that a discrete path integral over graphs is more fundamental than the conventional continuum path integral which represents quantum field theory within Feynman’s interpretation. As I understand it here, the self-consistency criterion is analogous to Einstein’s field equation (which demands consistency between the stress-energy tensor, on the one hand, and the spacetime metric, on the other) in which fundamentally there is not spacetime, but spacetimematter.

As far as general relativity is concerned, they regard it as an approximate continuous theory of an underlying, fundamental discrete spacetimematter structure. Since their underlying approach is graphical, they start with a graphical version of GR, that is, Regge calculus (Regge 1961). Regge calculus is typically understood as a discrete approximation to GR, where the discrete counterpart of Einstein’s equation is calculated via a discrete version of the minimization procedure. This enables one to come up with a methodology whereby the 4d simplices can recover the continuous structure of GR. Naturally, the smaller the dimension of the simplices, the better the continuous structure of GR can be ensured. Regge calculus also comes equipped with a consistency criterion, since the stress-energy tensor is associated with the discrete graph, so much so that any particular choice of the metric tensor has to be consistent with the matter-energy distribution. Silberstein, Stuckey and McDevitt propose a modified version of Regge calculus in which all simplices give a non-zero stress-energy contribution. This means, as I anticipated above, that space, time and matter are co-constructed per an adynamical constraint equation.

Based on this theoretical model, Silberstein, Stuckey and McDevitt set out to expel all facets of dynamism from their conception of the block universe. For they argue that the truly static block comes from the implementation of the Lagrangian schema. It is the recurring presence of the Newtonian bias that is held responsible for the inconsistencies relating to the block universe and its proclivity for dynamical explanations that both physicists and philosophers of physics share–and in particular those block viewers who try to accommodate becoming within the realm of being. Based on such an unreserved exclusion of dynamical aspects, their theoretical enterprise sets itself up to demonstrate that the universe does not comprise any dynamical entity. Every accepted statement about the (claimed) dynamical universe is ultimately flawed.

With their reconstruction of GR through the Lagrangian formalism, Silberstein, Stuckey and McDevitt offer a strong version of the block universe view dubbed “Relational Block World” (RBW). In their view, even that which Ismael treats as the experiential level that needs an alternative type of analysis should be explained with no reference whatsoever to dynamism. The conflict between time as experienced and time as characterized within foundational physics can be settled by reconceiving even the experiential realm within the RBW. The physical and the experiential are non-dual features of a more fundamental, adynamical reality.

All in all, what these authors refute is the idea that one can separate the genuine physical content of the universe, on the one hand, and human experiential representation, on the other. As Silberstein and Stuckey (2020, 6) comment, “conscious experience of an external world is not some virtual model or construction of the world trapped in the mind and generated by the brain, while the actual external physical world lies outside us forever as I-know-not-what noumena.” Rather, the physical and the experiential realms are to be regarded as coherent aspects of a self-consistently constrained adynamical 4d universe. While all seemingly dynamical features of the universe are to be accounted for adynamically through the Lagrangian formalism (temporal experience being one of them), it cannot be described in terms of representation or re-representation (as Ismael does). All that exists, whether physical or experiential, obeys the Lagrangian schema, whereby they arrive at the admittedly radical conclusion that even conscious experience gets constrained per a self-consistency criterion. Put differently, as radical (and speculative) as it may sound, their point is to ground both mental and physical phenomena in what they dub “neutral monism” which is something co-extensional to spacetime. On this view, physical reality and the experience of situated selves as humans are not to be portrayed as disjunct. Rather, “there is only the spatiotemporally extended world of experience” (Silberstein and Stuckey 2020, 6). Let me emphasize, though, that how neutral monism is meant to help us solve this and the more specific issue of temporal experience remains rather unspecified (more on this point in the next section).

Silberstein, Stuckey and McDevitt’s effort to strike off dynamism from physics is impressive. They offer compelling reasons against all theoretical accounts that, whether knowingly or not, reinstate a dual type of inquiry. Yet, they admit that there is an attempt at adynamism that is even more resolute. They write “Julian Barbour […] advocates what is arguably an even more radically static conception of the universe than our own RBW […].”Footnote 11 This is particularly interesting, because this model, which they claim to be more radical in terms of adynamism, has nothing to do with a block world; for Barbour implements a physical paradigm that claims special relativity and the block view as its metaphysical counterpart to be only partially correct. Barbour moves away from the four-dimensionalism of Newtonian physics (and relativity) in a way it will be worth analysing in some detail in the subsequent section. However, for what concerns Silberstein and co-authors’ theoretical enterprise, the juxtaposition with Barbour’s alternative paradigm casts an interesting light on the RBW. While providing the ground for a genuinely adynamical block world, Silberstein, Stuckey and McDevitt make room for trans-temporal objects, that is, objects that persist.

This is evidence that their four-dimensional account of the universe conjures up a notion of persistence of physical objects, whether it is cast in terms of perdurance or transdurance. So, this begs the question if four-dimensionalism can really do away with the need to justify the persistence of objects and therefore of something that, whether constrained or not, can be said to keep its properties through time. This makes the comparison with Barbour’s enterprise even more interesting. For the latter completely dismisses the notion of persisting objects, in that the only thing that exists, within his neo-Leibnizian and neo-Machian theoretical frame, is a maximum variety of points that should be conceived not as traditional points on the Minkowski spacetime, but as “different (coherently related) viewpoints of the universe as a whole” (Barbour and Bertotti 1982, 265). So, it is up to the following section to discuss if Barbour’s reformulation of mechanics in a Lagrangian fashion and the related three-dimensional picture of the universe successfully discards the notion of objects, and thus the problem of their persistence. My claim will be that it is precisely because no account of persistence needs to figure in Barbour’s theoretical framework that shape dynamics makes a stronger case for adynamism.

5 Adynamism in Shape Dynamics

As I anticipated, Barbour’s implementation of a static physics leads to a metaphysical apparatus alternative to the block. His enterprise pivots on three-dimensional geometries that do without relativity spacetime. His thought-provoking relational program, initially introduced by Barbour (1974), Barbour and Bertotti (1977, 1982) and successively refined by Barbour (2003), describes the evolution of a system of a finite number of point-like interacting particles in a purely relational fashion. Although Barbour develops a Machian theory of point-like particles, he is obviously conscious of the modern conception of matter in terms of matter fields. In this respect, he considers his relational account of object-like particles as preparatory or heuristic.Footnote 12 Barbour’s (1974, 1989, 1994, 1999, 2010, 2012, 2003, 2009) contribution to theoretical physics is fairly broad, as it covers three main areas, namely classical physics, quantum physics and quantum gravity. However, I will only consider his reformulation of Newtonian classical physics. As already explained in the introduction, this is because it is the foundational stone of the successive development of shape dynamics and it is particularly compliant with the adynamism here envisioned.Footnote 13 I think that in the present context this is more relevant than his reformulation of relativity, in that it is by revising classical physics that Barbour firstly and more clearly implements the Lagrangian schema. Let me add that, as the paper explores viable routes to adynamism in the light of the Lagrangian schema, this is another reason why I decided not to focus on the more recent approaches to shape dynamics (such as the replacement, developed in Koslowski et al. 2022, of least-action principles on shape space with a Hamiltonian generated dynamics). As already mentioned, whether the more recent formulation of Barbour and collaborators’ ideas in terms of pure shape dynamics can be squared with adynamism remains an open question that is not tackled in the present context.

In stark contrast to Newton’s reliance on non-directly observable structures, the relational program (as developed in the three seminal works Barbour 1974; Barbour and Bertotti 1977, 1982) gets underway with the application of what is currently dubbed the Mach-Poincaré principle, whereby the dynamical evolution of closed systems is defined solely in terms of the relational starting configurations and associated (intrinsic) first derivatives (Vassallo et al. 2022, 109).

It follows that both (absolute) space and (absolute) time are banned. Drawing from the insights of both Poincaré (1902) and Mach (1911; 1919), Barbour (2010, 1273) commences with a distinction between three kinds of spaces. The first is the Newtonian configuration space, which, for N non-interacting particles, is composed of 3N Cartesian coordinates. This kind of space can also be referred to as the Extended Configuration Space. Starting from this set, all the configurations that are congruent by transformations of the Euclidean group (translations and rotations) form the Relative Configuration Space. If one, then, considers all the similar configurations by transformations of the similarity group (dilations), or rather, if one also adds scale invariance considering all shape-similar configurations, these latter form the Shape Space. This means that, if we have a configuration of N point-like particles in the Euclidean space, we can apply translations, rotations or dilations and get respectively to a congruent or a similar configuration.Footnote 14 In shape space there are only sets that constitute “snapshots” of the instantaneous shapes of the system.

In this regard, there are only particles “moving” relative to each other and no external temporal parameter is needed. A shape configuration is a specification of all the inter-particle distances at some instant, without considering the position of the system in absolute terms, its orientation, and scale metric. The shape-space is thus a 3N-7-dimensional space. It is worth noting that this type of configuration space implies that, starting with a specification of relative distances between point-like particles and their rate of change, no unique future evolution is established. However, if a small amount of further information is provided, this issue can be resolved. From a Machian perspective, the procedure whereby this is achievable is of central interest. However, the technical details of this implementation are not relevant for present purposes. Suffice it to say that the Newtonian equations are a subset of all the possible configurations in shape space, which are defined either by the specification of an initial point along with a direction, or by a point and a tangent vector, respectively termed the stronger or the weaker form of Mach’s principle.

While Newtonian equations represent specific solutions within the shape space, Barbour christens Platonia the sum of all the possible configurations. The universe is thus the arena of all the possible instantaneous configurations – where the term “possible” is to be conceived not in a logical sense, but as referring to all the physically possible configurations. In other words, Platonia is the timeless landscape of all possible arrangements of particles whose position is given once and for all. Nothing changes in Platonia. All configurations simply occupy permanent, fixed collocations. It is the arena that, for Barbour (1999, 45), should replace the notions of space and time in the Newtonian frame. On this view, he writes that “we must think of Newtonian-type dynamics as something that ‘paints a path’ onto the timeless landscape of Platonia” (ibid., p. 45). On the one hand, there is a timeless structure of point-like configurations. On the other hand, there is a subgroup of solutions satisfying the timeless principle of least action, which corresponds to Newton’s laws. While sets of configurations can be mathematically defined and while continuous curves obey some variational principle, no actual physical history holds (Butterfield 2002, 317). This is why Butterfield (2002) argues Barbour’s theoretical framework somewhat recalls a type of modal realism à la Lewis (1986). The idea is that all possible paths or courses of history are equally real, none of them being favoured (in terms of actuality) vis-à-vis others: all relative configurations are accorded equal ontological status, being thus equally actual.

Again, a doubt naturally springs to mind: From “a scheme without the vestige of time” (Barbour 1999, 46), what to make of the impressions of movement and change as well as the temporal evolution of systems? In the previous sections, I made the point that the block view has difficulties explaining the coexistence of dynamism and adynamism because of its four-dimensional conceptualization of physical reality which—also in the Lagrangian formalism—allows accounting for the persistence of objects. Not surprisingly, then, Barbour’s strategy to overcome such unpleasant coexistence of dynamical and adynamical features relies on the dismissal of the very notion of objects, let alone their persistence. Let me pinpoint how he reasons.

Barbour recognizes that shape dynamics needs to deal with the problem of temporal evolution and that an explanation is needed for the impression of motion and change in physics. Importantly, one should carefully distinguish between the impression of time (and change) and the issue of time in physics. Barbour seems to identify these phenomena, as he regards time and change as related illusions originating from the viewpoint of a human being. In reality, he insists, nothing moves, nothing changes, time does not pass. But Barbour’s assumption is correct only insofar as time and becoming are conflated. However, according to a few scholars, he successfully dismisses becoming, while a notion of time—emergent time (see also Smolin and Unger 2015)—remains in his physical theory. For example, Baron et al. (2018) state that, at least in Barbour’s reformulation of Newtonian mechanics, time figures as an ordering principle among three-dimensional spatial configurations (remarkably, this flaw of Barbour and collaborators’ earlier relational program has prompted the successive refinement of the approach in terms of “pure” shape dynamics). On a similar vein, Butterfield’s (2002) early assessment of shape dynamics has it that time is not expunged completely. This is why I argue one should distinguish timelessness and adynamism, the former having to do with the impossibility of defining a unique ordering parameter, the latter concerning the absence of a genuine notion of becoming or coming-into-being of physical events.

While talking about the illusory nature of motion, Barbour makes the example of a cat, Lucy, catching swifts in flight. He questions the common view that the cat on the ground and the cat taking the leap are one and the same cat. He writes that in Platonia.

Lucy never did leap to catch the swifts. The fact is, there never was one cat Lucy—there were (or rather are, since Lucy is in Platonia for eternity, as we all are) billions upon billions upon billions of Lucys. This is already true for the Lucys in one leap and descent. […] Because we do not and cannot look closely at these Lucys, we think they are one. And all these Lucys are themselves embedded in the vast individual Nows of the universe. Uncountable Nows in Platonia contain something we should call Lucy, all in perfect Platonic stillness. It is because we abstract and “detach” one Lucy from her Nows that we think a cat leapt. Cats don’t leap in Platonia. They just are. […] We think things persist in time because structures persist, and we mistake the structure for substance. But looking for enduring substance is like looking for time (Barbour 1999, 48-49).

This is accordant with a becomingless, adynamical world: as dynamics vanishes, so does the idea of change over time in the sense of the differences between successive temporal parts of objects. Based on this, Barbour’s rejection of dynamics comes down to a denial of persistence. Ultimately, the pillar of his adynamism is the denial that objects persist over time. As Rickles (2008, 159) comments, Barbour’s three-dimensional items “do not change or endure and they cannot perdure […]”. The mutual consistency among paths is an illusion produced by the “juxtaposition of several subpatterns within one pattern” (Barbour 1999, 30), whereby becoming and change are somehow encoded in each of these subpatterns.

Barbour calls the temporal stratifications within patterns from which an impression of dynamism arises “time capsules”. This term refers to any fixed arrangement encoding the impression of motion, change or history. Barbour (1999, 31) writes that “any static configuration that appears to contain mutually consistent records of processes that took place in a past in accordance with certain laws may be called a time capsule”. A time capsule is conceivable as a point of view which displays stratifications, records, traces. It is intended to enable a perspective shift in which to see “perfect stillness as the reality behind the turbulence we experience” (Barbour 1999, 32). In other words, the universe is a network of time capsules within time capsules. A human body represents a time capsule, as well as its several components. Broader formations to which human bodies belong form other time capsules. In a few words, time capsules are distributed along a variety of scales.Footnote 15

To summarize, Barbour views the universe as composed of static configurations that do not unfold over time and, more specifically, do not become. The ultimate arena of reality is the set of all these possible configurations, each of which is a possible, unchanging “now”. If there is a scale where temporality comes into play, that is the encoding of time each instantaneous configuration entails. The illusion of change and motion is thus contingent upon local stratifications of a static set of non-uniquely ordered sequence of states. The impression of a coherent and unfolding sequence of events emerges within specific subsets of the universe having a “special structure”. In particular, these specific configurations contain sub-arrangements which seem to suggest the existence of a common, fixed, unique past.

Clearly, one of the most controversial and hardly explainable point of Barbour’s shape dynamics is how to deal with the mutual consistency of time capsules. One aspect that seems difficult to justify is how a time capsule, which ultimately belongs to an alleged timeless structure, should display (at least partially) traceable stratifications. The alternative, promoted by Silberstein et al. (2018) is to embrace what they dub “neutral monism”, which is meant to adynamically ground temporal experience through something neutral termed “Presence”. The reason why Presence is supposed to provide a more coherent account of temporal experience with respect to, e.g. Ismael’s and Barbour’s accounts, is that while a spatiotemporal event being present is relative to a system’s perspective, Presence is something universal. In their own words, “Presence is neither physical nor mental as these are typically characterized. It isn’t a thing, substance, entity, property, law, qualia, event, or representation. Presence doesn’t exist in space and time, it is the adynamical basis for them. Everything is grounded in Presence” (p. 392). This neutral Presence is meant to provide the basis to a form of contextualism for both spacetime and matter. I find at least two problems with this perspective. The first is that their definition of “Presence” remains somewhat elusive. The second is that they do not fully spell out why it should guarantee greater coherence between the static structure of physical reality and temporal experience as compared to Ismael and Barbour’s accounts.

At the same time, if Barbour’s theory of time capsules resembles a form of modal realism, the configuration space of actual paths would become so huge that the probability of mutually traceable strata should approximate very low values (especially if mediated over a rather vast configuration space). Barbour seeks to tackle this issue by stating that mutually consistent paths get higher probability amplitudes than others. While this is undoubtedly one of the most critical points within his early attempts at the development of a fully relational theory, what interests me in the present context is the fact that Barbour’s theoretical framework is able to provide a viable explanation for (apparent) dynamical features of physical reality, devoid of an upsetting double-standard view. Coherently, he discards the notion of spacetime points as appertaining to primitive ontology and is prepared to accept a metaphysics where no persisting object figures.

6 Conclusions

The reason lying behind my analysis of Barbour’s neo-Machian relational strategy was the need to find a resolutely adynamical approach. As I argued in the previous sections, a limit that affects most block universe views is that they endorse the outcomes of relativistic physics and yet aim to rescue a dynamical explanation of physical reality. I went on by showing that, as supporters of the static block like Silberstein and co-authors outline, it is theoretically wrong to accept the adynamism of fundamental physics and yet to make room for dynamism at higher levels of reality. However, the more consistent attempt of Silberstein and co-authors still can accommodate residues of change as long as they identify trans-temporal objects as objects that persist through time.

Barbour (1999, 143) was alert to this theoretical inconvenience. While he sees the block view as close to his own, he replaces four-dimensional spacetime events with three-dimensional spatial configurations. Put otherwise, the Minkowski space-time is not made of points of four-dimensional space–time, but three-dimensional configurations of extended matter (Barbour 1999, 143). In doing so, Barbour overcomes the limits of the block view by elaborating on a physical theory that truly expels motion and change, and by purging physics from the ambiguity inhering in the concept of four-dimensional objects. The disappearance of four-dimensional objects makes becoming irrelevant in both physical and metaphysical terms. Experientially, it is nothing other than an illusion. This is why Anderson (2009; 2017) comments that this approach, and the more refined framework of shape dynamics, entirely replaces the “becoming” lexicon with the “being” lexicon. All that we have is a semblance of dynamics due to the role of time capsules: some configurations play out as stratifications or traces in other configurations. Stratifications or traces are the only correlation between configurations. The universe Barbour has in mind is not hospitable to four dimensional objects that persist. They are the effect of an illusion that leads human beings to merge experience within time capsules. So, the rejection of four-dimensionalism comes at a price—no entities other than three-dimensional configurations survive within Platonia. Therefore, no notion of persistence needs to figure in the universe envisaged by Barbour.