An Investigation Into Temporal Becoming Within Timeless Strategies in Physics

An investigation into temporal becoming within timeless strategies in physics

Facoltà di Lettere e Filosofia Dipartimento di Filosofia Corso di laurea in Filosofia

Emilia Margoni Matricola 1538340

Relatore Relatore esterno Prof. Emiliano Ippoliti Prof. Mauro Dorato

A.A. 2019-2020

Contents

Introduction ...... 2 Chapter 1. Ordinary experience time and physical time ...... 11 1.1. Introduction ...... 11 1.2. The challenge of dynamism in accounting for experienced time ...... 12 1.3. The adynamism of the block universe ...... 23 1.4. Concluding remarks ...... 30 Chapter 2. On processual becoming ...... 31 2.1. Introduction ...... 31 2.2. Point-like occurrences and the dismissal of becoming ...... 32 2.3. Processual becoming ...... 37 2.4. Concluding remarks ...... 48 Chapter 3. Shape dynamics: A review ...... 49 3.1. Introduction ...... 49 3.2. Relational dynamics ...... 50 3.3. Shape dynamics ...... 56 3.4. What is left of time? ...... 62 3.5. Barbour’s relationalism: becomingless but not timeless ...... 68 3.6. Concluding remarks ...... 72 Chapter 4. General covariance and Quantum Gravity: a discussion ...... 74 4.1. Introduction ...... 74 4.2. The principle of general covariance and gauge theories ...... 76 4.3. The gauge-theoretic formulation of GR and its conceptual interpretation ...... 81 4.4. Gauge invariance and Quantum Gravity ...... 84 4.5. Rovelli’s Evolving Constant approach ...... 86 4.6. Concluding remarks ...... 93 Conclusions ...... 94 Bibliography ...... 97

1 Introduction

In the last two centuries, human beings’ understanding of time has undergone momentous changes. The advent of relativistic physics along with the advances in quantum mechanics have overridden previous conceptualizations of physical time and also profoundly ques- tioned the time of ordinary experience. It comes as no surprise that one of the persisting conundrums in foundational physics and the philosophy of physics is the gap that this epochal shift has introduced between physics and human experience. This work does not certainly dare to resolve such a basic problem. Rather, it intends to analyze how the conceptual toolkit elaborated within the frame of those contemporary physical theories espousing timelessness – in the sense that physical time is deemed not to be part of the fundamental ontology – can contribute to a better understanding of one of the key aspects of temporal experience, that is, (temporal) becoming. Indeed, not all theoretical strategies to implement a timeless physics do away with becoming, although they all treat time as non-fundamental. To bring out this difference, I will mainly look into Julian Barbour’s shape dynamics and Carlo Rovelli’s evolving constant approach. “In a timeless world, verbs of becoming like ‘happen’ have no place”, says Barbour (1999, p. 45). “This impossibility [of a single simple succession of global instants] is not absence of becoming. It is the fact that becoming is more complex than a naive non-relativistic extrapolation assumed”, writes Rovelli (2019, 1332). Evidently, addressing these theories requires treating time and becoming as distinct phenomena. Based on a rich wealth of literature, my analysis will come to the conclusion that Barbour’s denial of time is denial of temporal becoming, while Rovelli’s more nuanced notion of becoming successfully rules out time. Contrary to Barbour’s straightforward rejection of becoming, for Rovelli, the naïve notion of becoming is to be described in terms of a one- parameter of evolving constants of motion. Obviously, this work cannot accommodate a full-fledged analysis and a subsequent juxtaposition of such complex theories. So, I will scrutinize them with the aid of a particular conceptual lens, that is, the notion of process. I will take Barbour’s view to be the most radical and coherent refutation of process as a physical notion and Rovelli as the advocate of processes (not states or objects or other entities) being at the heart of physics. “There is a long tradition, going back at least to Hamilton, that seeks to make process the

2 most basic thing in the world. Roughly, the idea is that physics should be built up using verbs, not nouns […]. It all sounds very exciting, but I just do not think it can be done” says Barbour (1999, p. 329). “The universe is an ensemble of processes that happen”, writes Rovelli (2020, p. 124). It will be my claim that one of the crucial differences in Barbour’s and Rovelli’s timeless theories is that theirs are differing types of relationalism that diverge on what is implicated in relation. Barbour’s shape dynamics is an adynamical, configurational relationalism, in the sense that configurations, as I will concisely indicate below, are non-dynamical entities. Relationalism is but of locations and scale (see Gomes, 2020). On the contrary, Rovelli makes room for processual dynamics, to the extent that speaking of objects or states is but a way to offer a partial understanding of something that happens within the process. Yet, to make this latter claim clearer, it is worth briefly expounding its meaning at this initial stage. As I will argue more extensively in Chap. 2 and Chap. 4, the processual approach entails a four-dimensional theory of spacetime which admits no “natural breakup” of spacetime into spaces and times. Any such breakup comes from the introduction of an arbitrary frame of reference (a time-like fibration or a space-like foliation of the region of spacetime) whereby a given state of (what hence becomes) an observed system arises. In this sense, the state of a system is but a partition of a continuous process by means of a frame of reference which does not belong to the system1. According to John Stachel (2006; 2014), in physics processes precede states. For him, a processual view takes seriously the distinction between physical processes and physical events (Stachel, 2006, p. 56). The former has an extension in the sense that it occupies a finite region of spacetime, while a physical event occupies a point of spacetime. But insofar as spacetime is represented by a continuum, an event is nothing but “the limit of a portion of some physical process as all the dimensions of the region of spacetime occupied by this portion are shrunk to zero” (ibid.). To put it otherwise, an

1 I would like to stress again that I will address processual views within the set of theories that dismiss time as fundamental. This is the reason why I will not take into consideration other significant approaches that regard processes as “the stuff” of physics but treat time as fundamental. For example, Lee Smolin (2001, p. 53) avers that the world is a “history of processes”, where dynamical change is primary. But Smolin’s processual view pivots on time’s being fundamental, not emergent. Accordingly, his understanding of process implies time in the sense that the process is an activity generating a thick present, that is to say, two events that can be causally related to each other in a present that is continually growing by addition of new events. As to events in the thick present lose their ability to influence future events, they move into the always growing past (see e.g. Smolin 2001, 2020; Cortês, Gomes & Smolin, 2015).

3 extensionless event is an ideal limit2. However, it is of primary importance to specify what extension amounts to in this context. While it is wrong to think of processual extension as something unfolding through time (as this would beg the question of what time is), a process, as Stachel recommends, should rather be conceived of as related to the issue of the identity that a given entity takes up within a set of relations. To simplify his reasoning, mathematical points are invariant under permutation group actions, meaning that they do not possess intrinsic qualities that allow distinguishing one from another. He draws an analogy between electrons, which have nothing intrinsic whereby one can be distinguished from another, and points of spacetime in general relativity. What makes the difference – or, better, what allows distinguishing one entity from another, whether an electron or a point in spacetime – is its position within a relational structure. They acquire a degree of individuality by becoming the bearers of relations3. To sidestep a whole swath of debates that, though rich, is not directly relevant to the present work (see e.g. Redhead, 1982; Castellani, 1998; French and Krause, 2006; Ladyman and Ross, 2007; Dorato & Morganti, 2013; Morganti, 2013), the meaning of the term process which I am working with has to do with the acquisition of individuality of a given entity within a set of relations, short of which that entity would not be identifiable. Therefore, the issue of process is tied up with the issue of whether or not entities can be extrapolated from broader set of connections among more entities that confer identity on each of them. Though various details of various processual perspectives may not accord with one another, processualism is a view that deems the individuality of an entity as deriving from a more fundamental interaction with other entities – none of which is distinguishable based on intrinsic properties. The distinction of these entities is possible only through an activity of individuation that conceptually (i.e., abstractly, for certain

2 Iftime and Stachel (2005) construct a robust case that the points of the manifold can be identified as individualized points of that spacetime only once a particular metric field has been specified. 3 Stachel’s (2005, p. 205) example is that of an electron that is characterized by a set of quantum numbers that describe the atomic shell in which it is and the component of its spin with reference to a fixed axis in the context of atomic spectroscopy. His position has been widely commented in recent contributions, such as Adam and Butterlfield, 2012; Pooley, 2013; Glick, 2015. More in general, the debate on whether quantum particles could/should be treated as individuals is obviously broader than what I briefly summarized for my limited purposes in this Introduction. Three texts that make this point very clearly are Ladyman and Ross, 2007, Dorato and Morganti 2013, and Morganti 2013 (especially Chap. 2). Mauro Dorato and Mas- simo Pauri (2006) advance a notion of the identity of point-particles which resonates with Stachel’s but emphasizes the theoretical distance with traditional space-time relationalism. They argue for a mitigated relationalism, whereby “the point-events (the relata) exist (entity realism), but their nature (their magni- tudes, or properties they instantiate) are extrinsic or relational and not intrinsic or monadic” (pp. 147-148).

4 descriptive purposes and at a certain energy scale) isolates an entity from its nested set of relations. The metaphysics ensuing from processual physics is nicely encapsulated by Ladyman and Ross (2007, p. 263): “[t]he entire universe is a graph of real processes, where the edges are uninterrupted processes, and the vertices the interactions between them”. Based on this understanding, the idea of process being the central feature of physical theories can be squared with timeless physical paradigms and also allows distinguishing time from becoming. It would be pointless to expand on this topic here. Suffice it to say that, as Rovelli puts it, a processual view dispenses with a notion of time as absolute and yet holds onto becoming, and more precisely processual becoming. While becoming concerns individual time-like worldlines describing sequence of events, this becoming is processual insofar as each of these events is in its turn comprised in a set of various other relations that connect them to other events in other worldlines in a non-linear way. Such an idea of processualism makes it very unlikely that anybody would ever be able to describe the universe as a whole, if only because “in a process, variables change value” (Rovelli, 2020, p. 124). On this view of what physical theories do, as both Stachel and Rovelli insist, physics has little to do with states or things. These latter terms give the idea of a fixed, frozen reality that cannot be but the result of a breakup of something that can be isolated from the rest only conceptually and approximately. This is what Rovelli (2020, p. 128) means when he writes that quantum theory describes events “that happen when systems interact”– he goes so far as to speak of a “mental” separation of a quantum system for a certain time interval from the rest of the world with a view to describing how it interacts with its surroundings. In short, a processual view implies there being a whole set of interacting processes that cannot be individuated in isolation – each system of relation is isolable only insofar it is regarded as interacting with other systems; within each system of relations, no individual entity can be individuated as an individual entity, since it can only be individuated as an entity in relation to the other entities of a system. It is evident that a processual view gestures to a kind of relationalism that seeks to obtain a background independent account of physical processes. As I insisted, though, it makes room for “real becoming in the universe” (Rovelli, 2019, p. 1331). This allows differentiating it from Barbour’s type of relationalism, where what gets out of the picture is becoming – to such a degree that, paradoxically, time sneaks in through the backdoor

5 (at least as far as the classical reformulation of Newtonian mechanics and general relativity is concerned). In his background independent formulation of general relativity in a Machian style, the metric based on best matching between configurations grounds dynamics in relative three-dimensional configurations. The idea of becoming gives way to a becomingless universe. Accordingly, there is no sequence of events, let alone any process – which is a (physically) meaningless term in Barbour’s view. All we have is paths in a timeless, static configuration space – where space is truly fundamental. This type of configurational relationalism pivots on the notion of instantaneous relative configurations. This is a radically adynamical relationalism which, as I will explain in Chap. 3, is truly becomingless, but not timeless. To briefly anticipate the point that I will make later in this work, all that exists in the universe, at any particular instant, are patterns of configurations relative to each other. These configurations replace instants of time of classical Newtonian mechanics. Within the distinct configurations there is no evolution or change. The classical New- tonian trajectories correspond to a subset of the total possible relative configurations satisfying a timeless implementation of the principle of least action. Barbour’s configurational relationalism addresses dynamics in terms of the definition of the action between any two neighbouring points of the universe with respect to the best matching of the intrinsic differences between the two points in question. Clearly, while becoming is not present in the three-dimensional relative configurations, time can be accounted for as an ordering of the instants along a geodesic. In short, Barbour’s take on timelessness boils down to the claim that the only elements of reality are frozen instants (three-dimensional relative configurations) lacking temporal extension. To put it crudely, there are many nows that exist timelessly. These nows exist simultaneously within the instant. A non-linear temporal evolution reappears in a probabilistic fashion, since there are jumps from now to now in keeping with the Hamiltonian constraint. Thus, a notion of time reappears, whereas becoming is banned. This type of becomingless, non-linear temporality debases the idea that systems persist through time and this is a serious sticking point with the processual view that I sketched above. In Barbour’s theoretical frame, there are discontinuous sequences of configurations where the sequence obeys a probability distribution. This fundamental

6 discontinuity, entrenched in the relational theory as formulated by Barbour, presupposes a basic discretization of the universe, which most processualists refute4.

Outline of the work

To carry out my analysis of physical theories espousing timelessness through the prism of processualism, this work is divided into four Chapters. I first clarify why and how the issue of temporal becoming arises within timeless physical paradigms (Chap. 1). Subse- quently, I discuss the notion of process in physics and its metaphysical counterpart (Chap. 2). By employing this conceptual toolkit, I provide an assessment of Barbour’s shape dynamics to show how his rejection of becoming cannot be qualified as a wholesale dismissal of time (Chap. 3). Finally, I focus on Rovelli’s evolving constant approach to make the point that the denial of time as a basic feature of physical reality does not imply the erasure of processual becoming (Chap. 4). A quick chapter-by-chapter illustration will unveil the common thread of my overall work. In the first Chapter, I make my way into how the contemporary debate in physics gets to grips with humans’ temporal experience by drawing a basic distinction between experience-based and ontology-based accounts of time. This analysis is preliminary to the assessment of timeless strategies because, as I stressed above, not all of them reduce temporal becoming to an illusion. So, some of them might still be used to reconcile the ordinary experience of time with physical time, although this attempt at a reconciliation is not something that I embark on in this work. Rather, my interest in ontology-based accounts relates to those physical theories claiming that human temporal experience, in a world of processes, cannot be treated as something other than a process. Human cognition is made up of the same “stuff” as the physical world5. In short, while experience-based approaches, as I will try to clarify in Chapter 1, pursue a dual type of analysis, whereby physical time and ordinary experience time are to be treated as distinct phenomena,

4 The idea that the world does not have basic, ultimately measurable, constituents, identifiable as minimal quanta, is a main tenet of processualism. Of course, Rovelli is committed to the opposite view stating that ultimately reality displays a granular structure formed by individual quanta of spacetime. This, however, does not strip Rovelli’s approach of his processual character for the reasons that I will provide in Chap. 4. 5 This applies to a variety of physical and metaphysical projects, even when they have altogether different ontological commitments. Two remarkable examples in this regard are Bohm and Hiley 1993; Silberstein and Stuckey 2019; 2020.

7 ontology-based ones deal with humans’ perception of temporal reality in a way that brings out its relationship with physical phenomena. As I stressed, the objective of this work is not to expand on how to recover the gap between ordinary experience and physics; rather, I am interested in ontology-based accounts insofar as they try to reconcile both natural and mental features with the basic laws of physics. The key to my investigation in Chap. 1 is that some of the strongest ontology-based accounts that reject processualism embrace an adynamical 4D block universe married to some form of monism (this is the case with, e.g., physicists and philosophers of physics such as Jenann Ismael or Michael Silberstein and Mark Stuckey). However, this particular blend of physics and metaphysics is beset by a problem relating to the nature of point- like events. As Barbour (1999, pp. 142-146) comments with regard to the block universe, this widespread metaphysical view posits the existence of point-like events that are not given in the physical realm. A truly adynamical, non-processual view, Barbour insists, should get rid of this notion to implement a notion of configuration that is not affected by the problem of the temporal extension of events. On the contrary, on the opposite side of the camp, processualism and its inclination to (admittedly differing versions of) relationalism are hospitable to the intercourse of mental and physical processes, as scholars try to justify a notion of extended processes that neither implies absolute time nor needs the conceptual presupposition of unextended, frozen point-likeness. This conclusion leads to the subsequent discussion, in Chap. 2, of how the notion of point-likeness emerges and to what extent block-viewers can successfully claim that (point-like) events occur but do not become. By drawing on processualist views, I argue that, first, events are not point-like, and, second, the extendedness of events does not invoke any external, privileged temporal parameter. As a consequence, the notion itself of instantaneous becomingless events turns out to be an unnecessary idealization. Making this point is key to identifying the consequences of repudiating the possibility itself of becoming as an inner feature of events and provides the kernel of an idea of timeless local events that do become. Timeless physical theories that neglect the role of processes and privilege instantaneous events are forced to accept the idea that the universe is fundamentally adynamical (Silberstein and Stuckey, 2018). However, if adynamism is the road that physicists and philosophers of physics go down, then, as I will go on to argue in Chap. 3, the most consistent defence of an adynamical world is Barbour’s. For it has the virtue of

8 not having to commit to the ontology of the block universe and thus to the correlated idea of point-like events. The third Chapter puts forward a critical assessment of Barbour’s physical theory, which unfolds as a configurational relationalism that firmly replaces the notion of point- like events for that of point-like particles giving rise to configurations. Points are given once and for all, but, unlike the block universe model, they are not arranged into events. The latter are not physical entities, as the answer to “the question ‘what is real?’” is “configurations” (Barbour, 1999, p. 228). The universe is nothing other than the collection of all possible configurations. In doing so, Barbour (see e.g. 1989; 1994) reclaims a truly Leibnizian relationalism. In summary, as I will explain in some detail, spacetime is no longer conceived as a collection of events, but as a set of configurations assembled by the principle of best matching. Barbour’s theoretical approach, whose main objective is to develop a fully background-independent account of dynamics, is analysed from a variety of standpoints relevant to the overall economy of my work, that is, its premises, its inner consistency, its implications for temporal becoming. The main purpose is to bring to the surface the bearings of the relational model which shape dynamics is pivoted on the understanding of becoming. If, as I briefly summarized above, the universe is a set of relative, static, unchanging configurations, temporal becoming only arises within the context of human cognitive experience, not in the physical realm. Based on this critical reconstruction, I will explain why shape dynamics is the most consistent physical paradigm among those physical and metaphysical frameworks that refuse to accommodate the notion of processual becoming. The fourth Chapter stars off by explaining the rationale for one of the most challenging projects within current theoretical physics, namely quantum gravity. The quest for a micro theory of gravity arises from the inconsistencies of both quantum field theory and general relativity. Many and at time incompatible approaches have been developed for this purpose. This chapter concentrates on canonical quantum gravity, a theoretical perspective which starts with general relativity and operates via quantization procedures à la Dirac. Within the various timeless physical paradigms, canonical quantum gravity is particularly congenial to the removal of time from the fundamental structure of reality.

9 Classical observables are replaced by corresponding quantum operators in the Hil- bert space. What proves crucial is the interpretation of the principle of general covariance required in order that a proper quantization may be implemented. In this respect, different approaches towards the ultimate significance of general covariance lead to different types of quantization in quantum gravity. Within them, I draw attention to Rovelli’s relational account of dynamics to demonstrate that it is consistent with processual becoming while successfully eliminating time from the basic equations describing reality. On his view, physical processes are mapped in terms of interaction/correlation between physical systems. In line with what I argued at the very outset, a process is not something that evolves through time, but is what happens to a given system while it interacts with other physical systems – and this calls for quantization, as a process can be qualified as such only when it can be characterized by its “boundary quantities”. Contrary to timeless theories endorsing the adynamical block view, Rovelli recognizes the physical pre-eminence of dynamics, “which are expressed in terms of relations between the values of physical variables at the boundaries of a process” (Rovelli and Vidotto, 2014, p. 52). Even more importantly from a genuinely processual view, the boundaries of a process are not pre-established, as they can be any physical interaction of the system with another physical system. The same applies to spacetime, where boundaries are arbitrarily drawn in spacetime. Rovelli’s processualism depicts the universe as an unbroken process, which cannot be broken but conceptually and for specific theoretical purposes. Accordingly, a physical theory is nothing other than “a description of how arbitrary partitions of nature affect one another”, while these partitions have no stable identity of their own, because they “are at the same time subdivided subsystems” (ibid., p. 55).

10 Chapter 1. Ordinary experience time and physical time

1.1. Introduction

Time and the experience of time are related phenomena. While this work will concentrate on physical theories that do not treat time as fundamental, it is imperative, as a preliminary step, to explore how the experience of time has been discussed in the recent literature on human conscious experience of time and physics. The main objective of this preliminary analysis is to understand how physical theories that hold onto the basic assumptions of relativist physics make sense of something that appears to be dynamical in nature, that is, temporal experience. This is what section 2 will revolve around. While many scholars have dealt with this issue from a variety of angles, I will make my way into the contemporary debate by drawing a basic distinction between experience-based and ontology- based accounts. Whereas the former treat experienced time and physical time as distinct phenomena, the latter try to reconcile the human experience of time with the laws of physics. This will be key to highlighting that dualistic types of inquiry could be a nail in the coffin for physical theories that accept the metaphysical perspective which best fits relativity, that is, the so-called block universe view. In other words, my primary concern in this chapter is the extreme consequences of physical theories that start from the outcomes of relativistic physics. As I will explain in section 3, the block universe view calls for an utterly static explanation of physical phenomena, including human cognition. This requires rejecting either the dynamism of experienced time or the image of the universe captured by the block view. As staunch de- fenders of adynamism Michael Silberstein, W.M. Stuckey and Timothy McDevitt (2012, p. 530) remark, one cannot, for example, endorse the image of a frozen universe and at the same time allow for the dynamical character of human conscious experience: “The other alternative, that conscious experience emerges from or is realized in neuro-dynamical activity, is problematic in a block universe in which everything, past, present and future is just there ‘at once’ (including conscious experiences throughout the block) and brains are just worldtubes like everything else”. In this argumentative frame, the challenging contradiction between the dynamism of experienced time and the adynamism of physical laws will be instrumental in

11 illustrating the constrains that are imposed on truly consistent ontology-based accounts that do not want to undermine the basic assumptions of relativist physics. This analysis will make sense of the topics of the subsequent chapters. For they will juxtapose timeless processual theories that consistently rescue dynamism in physics with a most robust timeless paradigm that compellingly takes dynamics out of the picture.

1.2. The challenge of dynamism in accounting for experienced time

Despite geo-historical differences that cannot be addressed here1, experienced time as a particular dimension of human life is hallmarked by three features. The first is the distinction between that which is no longer present, that which is present, and that which is not yet present – usually referred to as “past”, “present”, and “future”. For the time being, we can say that all that exists in a given moment is the dimension that, within that very moment, one can call “present”. It can be distinguished by the other temporal dimensions in that the latter do not comprise any existing things. The present as the privileged moment where things exist can be named “now”. We will later have to complicate these definitions and see that they do not withstand deep scrutiny; but let us use them for the sake of the argument in this section. The second feature is the idea that time passes, in the sense that certain things cease to exist and thus produce a different “now”, which becomes the present (now) vis-à-vis that which has become the past (now). The third feature relates to the fact that the passage from one now to another one is an irreversible process – what is generally referred to as “temporal anisotropy”. While my conceptualization so far insists on what exists to determine what is experienceable in temporal terms, a predominant way to thematize time is to start with humans’ perception of reality – what we may term experience-based accounts2. Ontology- based and experience-based accounts are not necessarily incompatible, given that, as I will discuss below, the analysis of temporal perception often calls into question either

1 See e.g. Gell, 1992 and Greenhouse, 1996. 2 Michael Silberstein and W.M. Stuckey (2018) stress that the number of scholars in foundational physics and the philosophy of physics interested in time as experienced is raising mostly because they are concerned about the gap between physical theories and the advances in cognitive science and neuroscience: “many hope to reduce the experiential arrow of time to some physical arrow of time” (63). More worryingly, as Silberstein and Stuckey explain and as I will elaborate on below, these scholars fear that the gap increases as some physical theories espouse an adynamical view of the universe where events have no temporal extension and fundamental laws are symmetric.

12 physical events and the world structure around which cognitive processes revolve or the actual functioning of these cognitive processes and their neurophysiological basis. Both physical events and the neurophysiological substrate of human information processing capacities exceed the study of the phenomenal contents of human perception – this means that most of the time experience-based accounts are not only concerned with how things appear to the experiencing subject. However, there are experience-based accounts that are purely “phenomenological”, in the sense that they deliberately only home in on the phenomenon of time as it is captured by our consciousness3. This is the starting point of Barry Dainton’s (2010, 103-120) influential discussion of human temporal experience, as he seeks to provide a reliable description of the conscious experience of time without dealing with the neurophysiology of the mind. It is worth commencing my analysis with Dainton’s phenomenological experience-based account, as it lies at the opposite extreme of the continuum that has ontology-based accounts at one end and experience-based ones at the other end. Dainton focuses on the experience of variation as well as its opposite, which he calls “immanent phenomenal flow”. This is described as “a dynamic feature of experience that is independent of changes of the ordinary qualitative sort” (p. 104). What is to be noted, then, is that, while his subject-matter is human experience (rather than time as an empirical study topic), not only is time equated to change, but humans are said to be able to directly experience time in the form of passage. In other words, Dainton tries to make the case that the experience of change can be described with reference to the phenomenal flow and the apprehension of one experience flowing into another as the source of time awareness. As for many other authors, the mainstay of Dainton’s argument is the difference between “a succession of experiences” and “an experience of succession” – the latter being the conscious experience of time passage. He initially draws from a two-dimensional explanation that presents momentary awareness as containing a complex content

3 As I will explain shortly, I will now dwell on Barry Dainton’s exemplar analysis of time awareness. As to the meaning of the label “phenomenological”, he specifies that his task is to describe humans’ short-term experience of time. In holding this methodological tenet, he adds: “Other interesting problems, such as how our brains manage to perform the impressive feat of integrating the inputs from our different senses to produce a real- time representation of our usually changing surroundings (the typical time-lag between initial stimulation and experience is under half a second) will be ignored” (Dainton, 2010, p. 103). In the glossary, he insists that phenomenology “ventures no claims about the causes of experience, or what lies outside or beyond experience” (p. 436).

13 that is temporally extended. This hinges on the notion of presentedness, positing that contents of experience are present in varying degrees in a given act of experience. The complex contents of different acts are interrelated. The example provided by Dainton is a sequence of notes in which the first note in the musical line is less and less present as the other notes are apprehended in acts of consciousness: “This slippage of contents into the phenomenal past creates the impression that our immediate experience is advancing into the future” (p. 110). Yet, Dainton points out that acts of consciousness are not only interrelated, as they also overlap. While experiencing a sequence of three notes, we experience the movement from the first to the second, and from the second to the third, but it is not the case that we experience the second note twice. For Dainton claims that these two acts of consciousness overlap partially. He reasons in this way to avoid acts of consciousness being atomistic, that is utterly separated from one another. By considering the phenomenon of overlapping, he argues that “an awareness of change or succession is itself temporally extended” (p. 113). At this stage, it is crucial to note that Dainton recognizes no need to distinguish between acts of consciousness and their contents. He insists that phenomenal contents directly present themselves to human consciousness: contents are experiences in their own right. This is why he says that contents are “co-conscious”, and then goes on by claiming that co-consciousness “not only connects simultaneous experience, it also ranges a very short way over time” (p. 114). Interestingly, he attaches to this the experience of a particular direction, or, as in his words, “an inherent directional dynamism” (p. 115). As far as a movement is concerned, we do not perceive a body being located in different points, but we see a body moving. This is the reason why Dainton claims that movement is itself an intrinsic feature of one’s visual experience. Just in the same way, he claims, phenomenal temporality is contingent upon one’s being aware of change by dint of co-consciousness over time. Di- rect temporal awareness (what is also called the “specious present”4) is very short, because co-consciousness extends only very short over time and in accordance with its inherent dynamic5. In short, the overlap theory Dainton advocates has it that one experience

4 This expression was used, though not coined, by William James. For a contemporary understanding of it, besides Dainton, see Kelly, 2005. 5 Dainton recognizes that postulating the immanent flow as an intrinsic characteristic of experience could be charged with expediency – though he replies that this feature unquestionably exists.

14 flowing into another is possible because they are co-conscious, that is, they overlap: “When two experiences are (diachronically) co-conscious, they are experienced together, in a unified temporally extended episode of experiencing” (p. 118). The overlap theory implies that co-consciousness is that which creates “phenomenal presents” insofar as it extends over short periods and successive phenomenal presents overlap – although in a dynamic, directional way. Paavo Pylkkanen (2007) chides Dainton’s (2000; 2001) limiting himself to phenomenology, as he thinks that leaving aside the empirical level to consider only (human) experience of time from a conscious point of view engenders a series of conceptual diffi- culties – in particular, a confusion between description and explanation along with the unwarranted substitution of the explanans for the explanandum. Pylkkanen builds on Antti Revonsuo’s (2003) critique of “co-consciousness” as a fundamental experiential relation that is claimed to make sense of the unity and continuity of consciousness. How- ever, according to these Dainton’s critics, as an effective explanatory mechanism, it itself is not supported by any justification. It is a main assumption for Dainton’s overall argument to get off the ground. Presumably, Revonsuo comments, the way to rescue it would be to ground it on some lower-level features that neuroscientists might be able to pinpoint. Pylkkanen’s (2007, p. 217) criticism is even blunter, since he lays emphasis on the matter- of-factness that Dainton seems to attribute to co-consciousness. It is nothing other than a basic relationship of being experienced together. But, as such, the impression it gives is that the explanandum becomes party to the explanans (on this point, see also Arstila, 2016, p. 172). A few other elements, according to Pylkkanen, are left unquestioned by Dainton. For one thing, the duration of direct awareness; for another, the idea that phenomenal contents possess an inherent direction. In sum, the phenomenological approach Dainton espouses builds on premises that seem apposite to make his own case. A problem that is even more relevant to my argument is brought out by Mauro Dorato (2015a) as he questions Dainton’s (2010, p. 114) “simple view”, which I hinted at above, that “phenomenal contents (such as sounds, colours, pains) are intrinsically conscious items: they do not need to be apprehended by a separate awareness to be experienced, the contents themselves are experiences in their own right”. Here the limits of purely phenomenological perception-based accounts of temporal experience come into light even more blatantly. Indeed, the simple view reduces phenomena to the mental

15 contents and thus fails to address the question of if “the content of perception involving time” could not be merely mental but “a physical or a mind-independent entity” (Dorato, 2015a, p. 269). Pace Dainton, then, it still makes sense to wonder “whether the content of our acts of temporal perceptions is itself something mental (and therefore refers to the external world only indirectly) or whether we perceive temporal aspects of a mind-independent world directly” (pp. 269-270). By adopting an experience-based account that includes both lower-level neurophysiological processes and the mind-independent, physical world, Dorato and Wittmann (2015) overcome such a thorny issue by investigating the relation between the experience of succession and the unfolding of physical events. They concur that a succession of experiences does not lead to the experience of succession but insist that the grounds for the latter cannot be confined to one’s mental experience. If one is able to capture events in succession within the same extended simultaneity, the physical basis of such a mental experience of succession is “the local, irreversible succession of events relative to a particular worldline” (p. 208). What looks particularly relevant here is that the experience of time needs not only the mental component but also a physical grounding, although physical events are not arranged along the temporal line that humans experience. Based on this view, humans’ ability to experience succession relies on successive occurrences of events that happen with intervals longer than the temporal-order threshold in perception6. Physically successive events are captured within “the same window of ‘extended simultaneity’”: this “is the necessary condition for perceiving a melody, for understanding a sentence, and for coordinating an action” (p. 208). In Dorato and Wittmann’s view, the experience of time is founded on humans’ conscious integration of local events that are ordered in a specific frame that is dependent on humans’ neurophysiological apparatus – a relation that makes the now a mind-dependent phenomenon that is not merely mental. On this wavelength, Robin Le Poidevin (2015) also interrogates the relation between the perception of the external world and the passage of time from an experience-

6 Dorato and Wittmann (2015) argue for a threefold integration comprising three elements. First, the experienced moment, which has to do with the feeling of phenomenal unity within the conscious stream of perception. Second, the functional moment allowing the perception of complex sequences running from 300 ms (that is indispensable for humans to perceive individuated events) to 2-3 s length. Third, the mental presence, or rather, the feeling of oneself being part of the world at this present moment, that integrates all conscious experience, including the possibility for personal identity and its continuity over time. The mental presence is that which merges the experience of the immediately past with the experience of the immediately anticipated future, and thus produces “the now”.

16 based account that integrates temporal experience, cognitive processes and the physical structure of the world. He makes two main contentions. On the one hand, the act of perception always involves a temporal dimension, so much so that time can offer fruitful insights into the mechanisms underlying information-processing. On the other hand, human temporal account of reality is strictly connected to human perception, which is regarded as a temporal process. To this end, Le Poidevin argues against a cinematic model of temporal experience7. This model entails that human perception corresponds to a series of perceptual states (or snapshots of reality), each of which depicts an instantaneous configuration of the world. According to this view, while certain physical configurations display persistence, human beings perceive their states at a specific instant in time, and these states are not themselves persistent. Le Poidevin criticizes this view because it makes the implausible claim that we are perceptually sensitive to durationless states. Humans are wrongly believed to be able to obtain still snapshots of a “metaphysical present” corresponding to an indivisible moment. Rather, Le Poidevin remarks that there is a lag between the human elaboration of the perception of things and the things perceived. While this lag characterizes human cognition, no perception can be instantaneous: perception is temporally structured, while the temporal structure reveals itself to human perception. Le Poidevin in this text is particularly interested in debunking presentism – which is a topic that I have to leave aside in this work8. Still, what deserves mention for my purposes is the notion of temporal perception as a process, or better, a construction based on the perceptual ordering of physical events. With reference to Rick Grush (2007), Le Poidevin espouses a notion of the experienced temporal content as a not merely passive registering of the temporal features. Rather, the temporal content of experience is a construction whereby our perceptual system uses information about regularities it expects in

7 There are various rival models that claim to describe the experience of time. The cinematic model relies on passage, motion and change being enabled by the succession of static scenes, each of which represents one barely different phase of the moving object. This yields the experience of continuous passage. The retentional model has it that passage is enabled by the “retention” of a collection of representations of the recent past, while direct awareness has not extension in time. The act of consciousness owns a compound structure also involving retention of the past. The extensional model looks at time awareness as extending through time – which permits experiencing events that have temporal structure. Therefore, this model posits a genuine experience of succession which is not based on a simple succession of experiences. So far, see Dainton, 2018 and Mölder, 2014. Dorato (2015a) enlarges this taxonomy by introducing two further models. The anti-retentional model implies temporally extended time awareness whose content is not temporally extended. The retentional+protentional model characterized by acts of extended-in-time consciousness that not only contain retentions of the past but also anticipations of the future. For further complications of this taxonomy see Grush, 2016 and Arstila, 2016. 8 For robust and detailed analyses and defences of presentism, see Ingram and Tallant, 2018; Tallant, 2019a.

17 the perceived environment to interpret what incoming sensory information suggests. In other words, the perceptual system, based on current observations and past regularities, constructs representations whose content anticipates the perceived situation to come. A very similar conclusion is also offered in Dennis Dieks’s (2016) analysis of the relationship between physical time and experienced time. While his main focus is to defend a tenseless theory of time – an issue I will address later in Chap. 2 –, it is worth discussing his characterization of temporal experience because it can be regarded as a first important step towards an ontology-based account. Indeed, a key feature of Dieks’ proposal is the link he establishes between the biological background and the physical structure of reality. By taking stock of the outcomes of special relativity, he comes at two main conclusions. First, a naturalistic explanation of temporal experience does not benefit from the notion of an “extended now”. Second, although it would be wrong to qualify the temporal flow as an illusion, temporal change is always local – viz., if there is a viable way to account for it, it is always in accordance with a particular position within the four- dimensional spacetime manifold. In other words, an analysis of temporal experience that does not conflict with a physical account of time does not need either the postulation of the simultaneity of events or of a flow that occurs outside human experience – while this does not belittle human beings’ awareness of change. Dieks argues that events can be defined simultaneous if and only if they are causally independent of each other. A first relevant consequence is that these events cannot display a physically coherent behaviour, though this does not exclude the possibility of a relation between them in accordance to a common cause in the past. Another noticeable consequence bears upon the alleged existence of a global, extended now, as no simultaneous events can be experienced from within a local position – for all physical interactions in relativistic physics are local: “Time has become a local and path-dependent quantity, so that any flow that could be made compatible with special relativity necessarily must become local and path-dependent itself” (Dieks, 2016, p. 9). This means that there exists an infinite set of “preferred” nows, each depending on the position within the four-dimensional manifold. Therefore, a time lapse between events makes sense only in a worldline that represents a material process linking these events – and that rules out the plausibility of time lapses being inherent in the spatiotemporal framework as such, because the occurrence of a physical process is needed. To complicate this scenario even further, one

18 might consider general relativity, according to which it is impossible to fix a special temporal background geometry. Geometrical properties arise out of the interaction between the gravitational field and the material content of the universe. More clearly than special relativity, in the light of general relativity, time in itself and for itself finds no room, as it is always affected by the material content of the universe that produces deformations in the space-time (dynamical) structure. Despite the analysis above, Dieks continues, we do not need an alternative theory that accounts for time experience. A scientific analysis of time should be able to explain time awareness. The main point of Dieks’ reasoning is that science makes room for change as a variation of properties. In short, the disposal of an “extended present” and the confinement of time intervals to the properties that are instantiated at each stage of a given process introduce a notion of change in which there is no “moving now”, but still explains change within a finite region of spacetime. To this end, Dieks, too, rejects a cinematic view of time awareness: humans do not perform instantaneous experiences. Were our temporal awareness confined to a durationless instant, it would not be possible to grasp change since “there could be no comparison, in direct experience, of states of objects at different physical instants” (Dieks, 2016, p. 15). The flow is due to sensory inputs that are not uniform during the physical temporal extension of the specious present, and this is why he defines it as a secondary quality – what we could describe as the encounter between specific physical characteristics and the human cognitive apparatus. Awareness of time does indicate the presence of variation: “It is this joint presence of different inputs that is responsible for our being aware of a flow” (p. 17). Whether or not Dieks’ argumentative strategy succeeds (see contra Dolev, 2016), the crucial passage I want to emphasize is one that views the explanation of time awareness as (also) based on the physical structure of the world. Another remarkable step towards an ontology-based account of time is the one performed by Jenann Ismael (2016a). She intends to bridge the gap between experienced and physical time in a way that treats the former as a view of the latter from an “embedded, embodied” position (p. 107). This reconciliation usually follows the broader path of a reconciliation of the so-called “manifest image of the world”9 with the image provided by physics through a logical

9 Wilfrid Sellars (1963) first introduced the distinction between “manifest image” and “scientific image” of the world. They relate to distinct conceptual frameworks whereby human beings conceive and interpret the world they inhabit. The manifest image is the framework whereby one comes to be aware of herself as a

19 relationship between micro and macro descriptions of reality – thus, treating high-level structures as collections of low-level structures. Ismael gets rid of this methodology because recent developments both in physics and metaphysics have clearly brought to light the limits of such a reductionist view of ontological dependence10: “High level objects are, rather, configurations of low-level objects that gain and lose parts but maintain enough internal integrity to be tracked through change and reidentified across contexts” (p. 119). This entails that each context instantiates its own functional patterns as suitable “objects”. Put otherwise, it is not sufficient to know the microscopic constituents of reality to account for higher-level organization systems. For one also needs to display the emergent patterns of the collective behaviour of those elementary building blocks. This is a viable entry point to the notion of emergence, or rather, “behaviour that is novel and robust relative to some comparison class” (Butterfield, 2011, p. 920). As it is particularly useful to present an ontology-based view, let us see in more detail how Ismael’s argument unfurls. As for Le Poidevin and Dieks, so is for Ismael motion perceived directly and the idea of static snapshots is to be rejected. Perceptual states capture variations over very small temporal intervals – while, Ismael continues, for longer intervals memory plays a role in various ways. In a way that recalls Spinoza more than she is prepared to acknowledge, Ismael draws a distinction between two points of view. On the one hand, there is the view from what she calls “History sub specie aeternitatis” (Ismael, 2016a, p. 10), that is all that happens in the history of the universe, from the beginning to the end – or, better, the total sum of events that are not relativized to a temporal frame of reference.

being-in-the-world and that she uses to account for what she comes across in everyday experience. This image is by no means stable and unchanging, as it can be amended empirically, based on better observation- level generalizations about the world, and categorially, based on a conceptual revision of the basic objects comprised in the image. So, while the manifest image cannot be charged with being unscientific, it is methodologically less reliable than scientific methods. According to Sellars, methodologically the scientific image is a development within the manifest image. However, it postulates new kinds of basic entities (scientific notions such as subatomic particles and fields) that contribute to producing a new conceptual framework that claims to be an accurate and complete explanation of the world. Following in Sellars’ footsteps, Craig Callender (2007, p. 5) comments that, like the manifest image, manifest time is more complex than a simple map of temporal order and does not coincide with experience time. On the manifest image of time see also Dorato 2015a, p. 15 and Esfeld, 2020, pp. 111-161. 10 This is a topic that I will debate in another chapter. For now, it is worth referring to Matteo Morganti’s (2009) reconstruction of ontological dependence, as well as emergence asymmetry. He asks whether or not we have to assume a fundamental level to reality, and whether or not ontological dependence needs an ultimate ground. In particular, he argues that the possibility of no ultimate level of basic constituents (what he calls “metaphysical infinitism”) can be defended from both metaphysical as well as methodological reasons.

20 On the other hand, the point of view through the eyes of a participant, who introduces a temporal frame of reference. This latter point of view is something arising within a situated, embodied position11. Passage and flow are 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. A representation of the world from this latter point of view is the description of the world as events ordered in a temporal sequence with a fixed past and open future. This representation is intertwined with perception and memory. Building on the identification of these points of view, Ismael sets herself up to a very demanding task, that is, reconcil- ing everyday experience and physics-informed ontology. In other words, in her ontology- based account, ontology is a description of the world that accommodates physics without demoting the human intuitions about time. More than this, it is an ontology that grasps the intrinsic relation of the embodied view with the science-informed view coming from physics. The lynchpin of Ismael’s argument is a fairly explicit distinction between intrinsic relations among events and a perspectival representation of them. While History events in the four-dimensional manifold display fixed invariant relations, it is the introduction of an embodied participant that produces a temporal ordering with divisible sections like past, present and future. Thus, it is the lived experience of the participant in History that orders events according to extrinsic relations, that is, ones that do not actually pertain to events in History. The perception of History through an embedded, embodied participant results in an ordered sequence which itself changes during the course of that participant’s life. In other words, events are intrinsically located in a stable concatenation, whereas it is the human being’s ever-changing representation of them that puts events in ever-shifting sequences. The link between temporal sequences and the embodied perspective is contingent upon Ismael’s distinction between representations and re-representations. While representations capture the encounter between events and the participant, re-representations are the ever-changing product of a complex and continuous re-configuration of that encounter from the participant perspective – namely, a representation of the

11 Ismael points out that this analysis of temporal experience relates to human beings only, as temporal experience of animals is still under question (p. 114).

21 participant’s own representations. Ismael expands on this point in a former book, The Situated Self, where she presents ideas12 as

connected to the world by a long chain of representation and re-representation, all kept in coordination by a combination of physical law and (more or less) loose and local regularities. Some of the links are ordinary physical interactions among naturally occurring systems, some are interactions between artificially designed systems, some involve interactions with other people or the interaction of people with physical systems, and some involve interactions with other ideas. Your body is just a physical system in interaction with all these others, a kind of mobile measuring device, with your phenomenal states playing the role of pointer observables (Ismael, 2007, p. 71).

Such an underpinning epistemology, combined with a view of human beings as information-gathering and information-utilizing systems, explains the role of re-represen- tations13. Human beings loosen the internal ties between ideas and sensory representations as they loosen the ties between things and their sensory representations. This ensures a relation between ideas and things. They establish and maintain informational links with parts of the external world even though those parts are not directly experienced. The pro- duction of internal image of the world occurs within a bounded region of space and in a small number of circumstances. It is the supplement with language – as a set of “plastic media” – recording information that allows this information to circulate widely through socially and linguistically mediated channels. This is the genesis of re-representations: within an enormous representational environment, human beings develop the capacity to self-reflect and thus to re-present their representations. In this way, human beings continuously reconfigure previously acquired phenomenal contents by organizing them into morphing patterns of ideas. It is humans’ situated partial representation of events that allows for the emergence of a perspectival temporal sequence – one that is not an illusion, but a situated organization of experience along the line of temporal frames. Although this account certainly presupposes a particular theory of knowledge, Ismael’s strikes me as a clear example of ontology-based accounts. The crux of her analysis is the interaction between the external

12 Ideas are mental “particulars located in a nexus of informational connections with the world” (Ismael, 2007, p. 19) that coordinates the behaviour of the embodied agent. 13 This notion was originally developed by James Hartle (2005). Also see comments on the IGUS model by Callender (2017, pp. 231-247).

22 environment and human beings as complex systems who navigate a complex and moving environment. It is how the world is structured that prompts the capacity for thought. Cognition is a “matter of nonconceptual interaction among the brain, the body, and the environment” (Ismael, 2007, p. 19). Temporal dynamics arise from the interaction between a universe characterized by intrinsic relations among events and the (human) system producing distinctions that have a merely perspectival connotation. The perspectival structure of human experience “is recovered in the view of time sub specie aeternitatis as explicitly relational” (p. 118). A representation is not an illusion; it is just a perspective that springs from the encounter between two systems. For my purposes in the subsequent section, it is important to remark that Ismael seizes on the so-called “block universe” view as a metaphysical background. She deems it to be an accurate image of the universe that includes a representation of time sub specie aeternitatis. I have already illustrated Ismael’s point that time sub specie aeternitatis does not rule out either time passage or the openness of future. In what follows, I want to address these claims from a perspective that radicalizes them to advance the bold view that the world of the block universe is utterly adynamical.

1.3. The adynamism of the block universe

A good deal of scholars, such as Steven Savitt (2002; 2006), Dieks (2006), Vesselin Petkov (2006), Dorato (2002; 2006; 2013), Christian 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 hitherto been characterized14, this model portrays the universe as a four-dimensional entity in which no ontological distinction between past, present and future events holds. Every event is located in its four-dimensional position within the Minkowski spacetime. It is only by indexically picking out a specific location within the four-dimensional manifold that a local partitioning of events in terms of temporal ordering can be defined.

14 Richard Arthur (2019, pp. 53-62) avers that there are at least three versions of the block universe view. First, a manifold composed of static events which are given once and for all. Second, a “growing block” conception in which the totality of existing things gets continuously updated. Third, a four-dimensional entity which includes all events that have been, are, or will be, together with a genuine notion of becoming as the succession of events. I will return to this point later on.

23 Historically, the first argument proposed to validate the metaphysical consequences of the theory of special relativity, with reference to the relativity of simultaneity, was put forward by Willeke Rietdijk (1966) and Hilary Putnam (1967). The Rietdijk-Putnam argument is intended to show that the relativity of simultaneity necessarily implies a four- dimensional image of the universe. Therefore, only the block universe view proves compatible with the outcomes of the theory of special relativity. In short, the argument runs as follows. Take � to be the relation of “being real with respect to”, so that �� means that � is real with respect to �. The crucial point is then which kind of properties should be bestowed on �. According to Rietdijk and Putnam, � should be reflexive (a thing is real with respect to itself), symmetrical (if �� then ��) and transitive (if �� and �� then ��)15. By combining these properties of � with the relativity of simultaneity, what follows is 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, p. 247). Though some authors have tried to bring into question the Rietdijk-Putnam argument by rejecting either the symmetrical or the transitive property of � (see respectively e.g. Stein 1968, 1991; Sklar, 1977), the block universe within a relativistic context is agreed upon by such a great number of scholars that it has by now become common wis- dom. Petkov (2006) is quite resolute in asserting this conclusion. 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, p. 207). On the other hand, he is also convinced that the theory of special relativity is able to settle on its own the controversy between three-dimensionalism and four-dimensionalism. To corroborate his reasoning, he provides two empirical examples (the phenomena of length contraction and time dilation16) to make the point that, as a consequence of the relativity of simultaneity, no three-dimensional account of the world is

15 For an exhaustive reconstruction of the argument see e.g. Petvok (2006, pp. 211-225), Dainton (2010, pp. 328-331) and Dorato (2013, pp. 31-34). 16 Famously, one of the most important consequences of the theory of special relativity is the fact that time- duration and metric-extension are no absolute terms. Rather, their value is contingent upon the mutual relation between the observed system and the reference system. Galilean transformations have thus to be replaced by the Lorentz transformations so to accommodate the principle of the relativity of simultaneity.

24 tenable. He concludes that “special relativity alone appears to provide a definite proof of the block universe view” (Petkov, 2006, p. 227). What concerns me in the context of this chapter is that, once the block universe is admitted as the only metaphysical complement of relativity, we are left with one major problem. 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 can one account for human temporal experience as dynamical and processual? While I am not so much concerned with human temporal experience as such, what interests me is the type of constraints imposed by the metaphysical view married to relativity. In a few words, if one advocates the block universe, then she has to rule out notions that contradict it in all fields of reality. As I will say below, this is what is required by a truly consistent espousal of the block view. Ismael is well aware of the problem. As she puts it, we human beings “occupy the space depicted in the view of our History sub specie aeternitatis” (Ismael, 2016, p. 186), although no bird’s-eye view is ever available to us. Now, under the aspect of eternity (the block view), everything is fixed, including our actions and the history of the world. On the contrary, from the particular point of view of one human being, nothing is fixed. What makes this issue relevant to my argument is that, according to some resolute supporters of the block view, this predicament is unavoidable and must be resolved to the advantage of the underlying static structure of the universe. But before reaching this extreme conclusion, let us see how other, less resolute thinkers have tried to get out of the quagmire. As I illustrated in the previous section, Ismael’s (2017) solution pivots on the notion of temporal perspective. Her intention is to offer a “schematic formal understanding of the logical contours” (ibid., p. 24) of human experience, where the logic of temporal perspective should be conceived as the formal connection between the contents of representations from diverse temporal perspectives. She contends that the human brain processes information over a finite temporal interval, thus producing a representational content with a temporal extension. Ismael reasons as follows. One starts with a system that is able to process input information from the external environment, allows a temporal dimension to the representational contents of the system, grants memory to run on these representational contents and let the memory operate recursively on them. This allows to organize the representational contents of the system in terms of an “explicit autobiographical

25 history whose contents are always being reconsidered and retrospectively modified” (Ismael, 2017, p. 27). If one considers a temporal cross-section of the resulting stream one gets to a specific temporal perspective. Clearly, this temporal perspective displays an evolution – what Ismael calls the temporally evolving point of view. Evidently, this explanation of how human temporal experience emerges forces her to adopt a double-standard view, in which the impersonal, invariant History of the universe gets represented time by time by the re-representations of the embedded, embodied participant. It is the complex, ever-changing process of “summarizing, constructing, interpreting and condensating life experiences” that enables the IGUS to “produce a more or less coherent narrative sense of a personal past” (ibid., p. 26). What is not entirely clear is whether the process of knowledge is a truly dynamical process – which in principle should be ruled out within a block-universe frame. Ismael keeps on contrasting the invariant structure of the four-dimensional manifold, representing the unfolding of History (capital letter), with the “frame-dependent representation of time associated with every point along a psychological history of a situated being” (Ismael, 2017, p. 30). According to her, human experience of time over History depends on the perspectival access that the embedded, embodied IGUS exhibits with respect to the invariant, underlying structure of the universe. Still Ismael (2017, p. 33) claims that there is no conflict between time as captured in a frame-independent way and time as represented relative to a specific reference frame. While physics deals with increasingly finer level of resolution, human cognition produces a coarse-grained representation of the external environment. She is right when she says that, within her account, the experience of time is not illusory. While the invariant structures correspond to static relations captured by the block view, the perspectival representations of human experience correspond to the evolution of the point of view of the IGUS. But the non-illusory character of human temporal experience is exactly what ignites the problem that I am trying to foreground: if the process that the perspectival account refers to is real, then the physical world (of which human cognition is part) entails dynamism. Therefore, despite Ismael’s insistence on the difference between levels of analysis, her understanding begs the crucial question: If the block universe contains the set of all past, present and future events, should it then not also encode all humans’ representations and re-representations within it? This is, for example, Dieks’s (2006) consistent position.

26 He recognizes that the block view needs to accommodate not only the totality of events of the four-dimensional manifold, but also their properties and mutual relations. In this respect, it should also contain individual humans’ events, experiences and impressions: “All actual events, experiences and intuitions must be there in the block representation, exactly at the spacetime position where they actually occur” (ibid., p. 169). This means that there cannot be any separation between levels of analysis, let alone levels of reality, especially if one level entails static events and the other a dynamical account of other events. All the events of the universe, together with their characteristics and mutual relations, figure in the block universe and are thus on an equal footing. As a consequence, human temporal representation and the structure of physical events must coincide in their corresponding spacetime collocation. Oliver Pooley (2013a) casts compelling doubts on how Ismael and Dieks address the issue of the dynamical nature of time as he insists that the block view makes no room for events that happen, rather than simply occur. That between happening and occurring is a distinction which will be central to my argument in the subsequent chapter, but at the moment it is sufficient to state that, according to Pooley, the block view fails to account for the dynamics of time other than a static arrangement of (still) events. Pooley agrees that Ismael’s view does not reduce temporal experience to an illusion, and, as I noted, this is the interesting aspect for my purposes. Indeed, Pooley points out that it is exactly the account of the dynamical nature of experienced time that is flawed within the block view. If Pooley is right, then the problem I detected in Ismael’s double-standard view is something that she fails to resolve. Ultimately, as Pooley asserts, the block view implies a thoroughly adynamical image of the universe where the stillness of events applies to all levels of reality. Let me briefly go into the detail of his reasoning. Pooley defines a block viewer as someone who thinks that a complete account of temporal reality amounts to an exhaustive catalogue of which events occur and the way they are related, where relatedness is completely “tenseless, holding once and for all” (ibid., p. 324). Again, this will be explained in more detail in the following chapter. The way Pooley puts it is a spatial analogy, that is, a description of reality that accounts for the spatial dispositions of all events relative to one another. In this frame, “one does not need to further specify that object a is here, or twenty meters to the left. Such information serves to locate objects spatially relative to ourselves, […]: if the view-from-nowhere

27 characterization of reality is really complete, then one can in principle locate oneself spatially, and determine one’s orientation in the world so described” (ibid., p. 324). In this block view, there is no need to specify the events that occur now. This view puts all nows on a par, so much so that no privileged now can in any way be identified. Pooley is also dubious of Dieks’s characterization of temporal becoming in a non- perspectival way as the successive coming-into-being of events. He argues that either Dieks’s position is incompatible with the block universe, where no change whatsoever is admitted, or he is using a very flimsy notion of coming-into-being, whereby coming-into- being, occurring, and happening can be used interchangeably. If the latter is the case, as in fact is, the kind of dynamical nature that Dieks is invoking, according to Pooley, is quite shallow. With reference to John Earman (2008), Pooley pushes to the extreme the adynamical aspect of the block universe, as both authors claim that conceiving time as the simple ordering of events effaces all aspects of actual change; and this explains why the verbs “to occur” and “to happen” are equated17. I do not need to explore Pooley’s or Earman’s alternative proposals here. As I noted, their contribution is relevant to my argument insofar as they persuasively emphasize the utterly static nature of all events, re- gardless of the fact that they are ordered in local sequences within the Minkoswki spacetime. This idea of there being a conflict between dynamism and the block view has recently been advocated even more radically by Silberstein, Stuckey and McDevitt (2012; 2018). They straightforwardly “reject dynamism at its foundation” (2012, p. 529) and argue that all puzzlements related to the block universe are due to a lack of resoluteness in accepting a thoroughly adynamical physical theory. Coherently, their understanding of the block world dismisses all facets of dynamism. Although I cannot properly expand on their instructive analysis, suffice it to say that, according to them, if one takes the consequences of the block universe seriously, then one cannot venture into a dynamic explanation of phenomena, including temporal experience. What is to be rejected is exactly the bipartition perspectival vs. aperspectival, which sneakily reintroduces the need for a dynamical explanation of the universe. In this sense, theirs is one of the strongest ontology- based accounts of time rooted in a block universe view. The authors recognize that human beings are prone to think dynamically, but at the same time contend that dynamics does

17 This will be one of the main argumentative thread in the next chapter.

28 not explain anything and has to be discarded as a non-physical notion. They treat the idea that in physics there is dynamics as a bias which is of no aid either in explaining physical time and in describing experienced time. Within the Minkowski spacetime nothing ever moves, nothing happens, nothing changes. Events are just in spacetime once and for all. Accordingly, the bold argument they develop is that the incompleteness and inconsistencies relating to the block universe arise out of the penchant for dynamism that both physicists and philosophers of physics share. They go so far as to ask the reader to “make every effort to suspend this dynamical bias” (ibid., p. 17) while reading their book: this, they write, is the only way to get a fully accountable grasp of the static universe we inhabit along with all the events occurring therein. Based on such an unreserved exclusion of dynamical explanations, their theoretical enterprise sets itself up to demonstrate that the universe does not comprise any time-evolved entity and that also experienced time is a feature that must be addressed with recourse to an adynamical fundamental physical theory. Unfortunately, here I cannot explore such a fascinating hypothesis. For the path that I decided to take in the next chapters is one that leaves aside the block view on purpose. Silberstein and co-authors’ effort to strike off dynamism from physics is impressive. They also allow bringing to the surface the potential contradictions of various ontology-based accounts that, whether knowingly or not, reinstate a dual type of inquiry. However, they do not fully escape the predicament they pinpoint so vividly. They give away a major inconsistency besetting their conceptualization of events as they paradoxically represent them as both point-like and extended. While events are modelized as occurring points in the spatialized model, as the block view recommends, they are nonetheless forced to admit that events cannot be completely extensionless. Silberstein et al. (2018, pp. 65-66) write that they consider an event as possessing an “infinitely small extension and duration”. The unresolved problem is that, no matter how small, it is still extension and duration that qualify events – which is to say that events are not extensionless. While it is interesting to note that they do so to duck a debate that I will heavily draw from in the subsequent chapter to defend a processual view and to offer an alternative account of events, this inconsistency is enough for me to decide to concentrate on an alternative adynamical paradigm that is not affected by it.

29 While talking of a fully adynamical view, Silberstein and co-authors refer to Julian Barbour’s similar enterprise, as he tries to explain the key elements of the universe in a totally adynamical way. Barbour’s physical theory does not need to hypothesize point- like events that are also extended. Consistently, he maintains that events are neither point- like nor durational. Properly speaking, there are no events. This is what explains me fo- cusing on Barbour’s ontology-based and its extreme consequences, as I will briefly say in the next final lines.

1.4. Concluding remarks

The main points of my theoretical journey so far are two. First, ontology-based accounts try to steer clear of dualistic explanations of physical time and experienced time. How- ever, when it comes to the basic models for these phenomena, many scholars unwarrant- edly misread key aspects of them either to include a dynamism that physics excludes or to make ordinary experience of time utterly adynamical. While the focus of my analysis here was not experienced time per se, I took it to be an illustrative case for bringing out potential inconsistencies of physical theories that recognize the relativity of simultaneity. Yet – and this is the second point I wanted to make – a consistent physical theory that recognizes the relativity of simultaneity and accepts the metaphysics of the block view must provide a fully adynamical description of the universe. The vindication of these two points will be key to the subsequent chapters. In Chap- ter 2 I will expand on the nature of physical events to explain what it means to say that they have (or lack) extension – and thus to rescue the basic difference between occurring and happening. In this frame, I will advocate a processual view of events that does not need to accept the resoluteness of the adynamical view of the universe. Chapter 3 will take the opposite track to bring to light the extreme consequences of what I see as the most consistent and compelling adynamical theory of the universe, that is, shape dynamics. Although I am inclined to endorse a processual view, as will become apparent, the virtues of both processualist and Barbour’s adynamical ontology-based accounts is that they circumvent the flaws of all physical theories that postulate adynamism at the bottom and tend to neglect its extreme consequences at less fundamental levels of reality.

30 Chapter 2. On processual becoming

2.1. Introduction

The first Chapter centred on the predicament of the dynamical vs. adynamical nature of both human temporal experience and physical events. The conclusion was that theories that embrace the outcomes of relativistic physics and accept the block view are forced to adopt the image of a static universe where becoming finds no place. This Chapter picks up where the previous one left off. It intends to expand on the presuppositions and the consequences of adopting a becomingless physical approach to vindicate the theoretical virtues of a notion of events that is hospitable to a genuine notion of becoming. Interrogating the nature of events will be instrumental in determining what is at stake in the age-old dispute between so-called A-views and B-views of time. I will show that the matter in question is not (only) the idea of there being a moving now, but the very possibility of two events being contiguous. If duration is enabled by the continuity of non- contiguous events, no notion of becoming is needed – whereas becoming is introduced by the indexical expressions of the human tensed discourse. Becoming is, and cannot be but, perspectival (Sec. 2.2). I will continue by saying that, even if the notion of contiguous events is dismissed, there is a way to conceptualize the processual nature of events, and thus to justify a notion of becoming which is not at odds with the laws of physics. This rests on a conceptualization of processes as the discretization of continuous networks of relations where entities acquire their identity as they are the bearers of these relations. On this view, becoming is nothing but the unfolding of relations among them, while the various networks of events and their relations to each other give a system its specific configuration (Sec. 2.3). To this end, I will follow three paths. First, I will investigate the insights of one of the forerunners of processualism, Alfred N. Whitehead, to show how his theorizing paves the way for contemporary interpretations of the relational nature of individuality within the foundations of physics (Sec. 2.3.1). I will then go on by introducing John Stachel’s processual model, which opens up to an idea of becoming as the set of processes that are needed in order for physical individual entities to be identifiable (Sec. 2.3.2). Finally, I

31 say a few words on why a processual view is relevant to a viable concept of dynamics that can be employed by physical paradigms taking time not to be fundamental.

2.2. Point-like occurrences and the dismissal of becoming

Within an idealist framework, J.M.E. McTaggart (1908) introduced a distinction that hallmarked subsequent debates on time. While the numerous discussions about his proposal have taken many routes, for present purposes the appropriate way to represent McTag- gart’s argument is to envisage a set of events alongside one another23. One way to understand this arrangement of events accords privilege to a position (in McTaggart’s terminology a moment, each position of these events being a specific moment in time) with respect to which some events are said to be past and some others are said to be future. This is what he calls the A-series, that is, the continuous shift of this moment in time running from the past to the present, and from the present to the future. That moment can be called the “now”. Importantly, on this view, becoming is qualified as the movement of this ever-shifting now. However, McTaggart questions the tenability of the idea that there is such thing as a moving now and consequently dismisses the implicated notion of becoming (insofar as he qualifies becoming in that way). This leads him to espouse a becomingless notion of the relation between events which he dubbed B-series. The A- series and the B-series are characterized by the same items (Markosian, 2016), whereas the only difference between the series is the absence of a now that changes its position with respects to the items. In the B-series events are positioned to one another according to relations of antecedence, simultaneity and subsequence. It deserves mention that, as the set of items in the two series is equal, all that changes is its (ontological) ordering principle. Given that in the B-series no shifting now turns a

23 Although this bespeaks my theoretical inclination towards a dynamic notion of becoming, events should be couched in terms of an isolated multiplicity of non-isolable relationships. Along these lines, Alfred N. Whitehead (1925, p. 126) insists that events are, to some extent, abstractions, in the sense that no event is a truly independent entity. The “event” as a unified entity is hardly conceivable as secluded, because it originates from a unification of processes within a processual network. The strong conclusion Whitehead reaches is that “an isolated event which has lost its status in any complex of events is equally excluded by the very nature of an event” (ibid.). However, this does not introduce, as I will argue more extensively below, a notion of asymmetric dependence between the (alleged) totality of events and the single event. For, within a processual view, there would be no totality of events without there being each single event. In conclusion, each single event is the localized actualization – describable as a discretization – of a broader series of related events, which in their turn are the localized actualization of broader series of related events.

32 present event into a past event, or a future event into a present one, events are alleged to occupy permanent positions. The only thing that can be said about them is that they are positioned to one another according to relations of antecedence, simultaneity and subsequence. It is exactly the (supposed) equivalence between becoming as a shifting now and time that shored up McTaggart’s conclusion that the B-series is a timeless one24. If it is the case that time amounts to becoming in the sense of a moment changing position throughout the series of events, then, no time is admitted in the B-series, as events occupy a fixed position. In this context I do not want as much to put McTaggart’s overall argument to the test, as to determine more clearly what the notion of becoming is that he discredits. For, as I just noted, becoming is understood as ensuing from a privileged moment that moves directionally from the past to the future (while – it is to be noted – these pasts and futures are not positioned in themselves, but their being in the past or in the future is determined by the movement of the shifting now). So, in brief, the becomingless B-series events are already-and-always ordered with respect to one another because their position is fixed. It is just that the order of events does not rest on anything that becomes. In a few words, some events are earlier than others, while the latter are later than the former. The now is reduced to simultaneity, meaning that it is only by indexically pointing at a moment in time that events can be said to occur “now”. This entails that the ordering of events according to a now is not an inherent feature of events. Rather, it is introduced by an observer or a speaker whose act or utterance, as it were, “freezes” an instantaneous moment by means of her very act or utterance. In this sense, Adolph Grünbaum (1969) distinguishes the (actual) order of events, which does not depend on any observer or utterer, from the (experiential) order of events as it is reconstructed by the observer or utterer. The actual order is mind-independent and amounts to the network of events occurring in such a way that some are earlier, and some are later, than others. The experiential order is exactly the consequence of there being a (non-inherent, mind- dependent) flowing now, a preferred time-frame, which moves along a specific line and thus splits up the present (which corresponds to the observer’s or utterer’s specific position), the past and the future. Grünbaum’s insight is instrumental in my argument here insofar as it allows detailing a distinction between occurring and becoming, which has a

24 According to McTaggart, the B-series is not sufficient to guarantee the existence of time, as the latter requires the ontological distinction between past, present and future events that only the A-series would imply. This is why, as McTaggart denies the reality of the A-series, he concludes that time is unreal.

33 remarkable impact on how the latter is conceived. According to Grünbaum, occurring is the mind-independent, tenseless occurrence of events in their fixed position given once and for all. These events do not happen in the tensed sense of “coming into being apart from anyone’s awareness of them” (ivi, p. 161)25. Of course, events do happen, but independently of the observer/utterer. Put otherwise, they occur “at certain clock times in the context of objective relations of earlier and later” (ibid.)26. Yet, such a type of occurring disqualifies a coming-into-being in terms of actual becoming inhering in events. Becom- ing is, and can only be, experiential, or rather, produced by the representation and reconstruction of events from one’s standpoint. In this wholesale dismissal of becoming, a distinction looks to me as particularly salient, that is to say, the problem of the moving now as distinct from the problem of the positions that the now is alleged to occupy. McTaggart’s critique of the A-view of time stems from the untenability of the moving now, which in its turn is alleged to be bound up with someone’s doings or sayings that put events in a mind-dependent order. On this account, becoming is discredited as not appertaining to the structure of the world because it is coalesced with the idea of there being a moving now. Since there is no actual point sliding over a line that determines the now as it changes position, this now is introduced by the indexicality of a cognitive experience that does not belong to actual events. This is why becoming cannot be but experiential: it is to be confined to how humans arrange the order of events from their own vantage point. Of course, such an understanding is limited because it misreads, or at least curbs significantly, the notion of becoming. It is precisely the more or less implicit assumption of becoming being conditional on the moving now that leads to the double standard whereby physics contains no time whereas time originates from the organization of human experience. The occurrence of physical events and how they are arranged within human cognition turn out to be utterly separate questions. In summary, actual becoming

25 In slightly different terms, while “there cannot be becoming where there are no events” (Dorato, 2006, p. 564), Grünbaum turns the table and argues that no physical event can be conceived unless we introduce becomingless occurrence. This is due, as I will argue, to a flawed conception of becoming. 26 At the same time, this does not mean that events are all given in their totality as in a sort of simultaneity, because relations of antecedence and subsequence obtain between them. Although the question exceeds the limits of my discussion in this section, it is to be mentioned that the relations of earlier and later are articu- lated within the scope of a causal theory of time that takes causal terms as primitives. A causal definition of temporal order entails temporal relations being reduced to causal relations by which saying that an event occurs before another is reducible to saying that the former is the cause of the latter. For a thorough critique of Grünbaum’s own overhaul of the causal theory of time, see Hugh M. Lacey, 1968.

34 inexorably implies a moving now and, since no moving now obtains, events do not come into being from the future: they just occur. However, it is one thing to pin the problem of A-theories on the moving now, quite another to pin it on the position that this now is supposed to occupy. Indeed, the genuine sticking point between B-theorists à la Grünbaum and those whom he criticizes has to do with the more substantial issue of how to characterize the continuum of change and whether this continuum can be divided at all. This is a crucial philosophical question that between the 19th and the 20th century was revived in a variety of disciplines beyond philosophy, including the foundations of mathematics. The main problem is how to square the notion of continuity with the idea of a state of change. Put otherwise, what the idea of variation is that proves compatible with the property of denseness of the continuum (i.e., the fact that, for any couple of points, infinite ones exist between them). In this light, the matter of the question changes. It is not the fact that a now moves that is problematic in the first place; rather, it is the very possibility of identifying the various positions that this now is alleged to take as it moves. Therefore, as I will briefly comment, for Grünbaum the trouble with the moving now has first and foremost to do with the fact that it is supposed to move along points next to each other. Based on Cantor’s set theory, he denies points being contiguous, and therefore disposes of the idea that becoming derives from the contiguity of events. According to Grünbaum (1950), scholars who admit actual becoming are bound to accept a “pulsational” theory of time whereby occurrences are successive and contiguous parts giving life to a divisible continuum. In other words, he goes on to say, events must be held to occur in a discrete sequence. Nevertheless, Grünbaum (ivi, p. 171) continues with reference to Cantor’s set theory, the continuum is not divisible in ultimate lengths or parts. Between events only “earlier than” and “later than” relations exist within a network of dependence on a non-atomizable continuum27. If that is the case, it is mistaken to assume that becoming can be qualified as a sequence of extended events next to each other. Actual becoming is banned from the realm of physics inasmuch as events are not extended, or rather, they do not endure. This does not prevent thinking of duration as a

27 The reasoning that backs up Grünbaum’s position relates to his take on the Zeno paradox (see 1952). In the very same way Leibniz criticized Bernoulli for treating the actually infinite set of natural numbers as having a last term which can be “reached”, he criticizes scholars who reach “final elements” generated by a completed progression of division operations. A “last” division which ensures the completability of the process of “infinite division” is not available.

35 “continuous set of instants” (ivi, p. 171). In sum, the notion of unextended occurrence goes hand in hand with the denseness of the continuum, which rules out the idea of events being contiguous. The only admitted coordinates organizing actual events are “later- than”, and relations between events are given “in a network of dependence such that the events constitute a linear Cantorean continuum […]” (ivi, pp. 168-169). It follows that the concept of “later-than” proves the key to a temporal order that does not require the nextness property. The problem of the moving now is evidently less stringent than the problem of a set of states of change. An argument that aspires to solve it is called upon to prove that denseness poses no threat to a notion of actual becoming through an interval. Cantor’s set theory conceives a line as composed by a non-denumerably infinitely ensemble of points. Its application to measure theory entails that even though each of these points (or any count- able number of them) is unextended, still a non-denumerably set of them displays a finite measure. As a consequence, while each of them is unable to produce it, their being col- lectively taken into consideration results in a finite measure – this theory also implying that the measure of a segment is not an intrinsic property of the segment itself (viz., it does not depend on the number of points of the segment). This means that the lack of passage of singular instants does not at all entail the lack of passage across a finite interval. In brief, while a singular point-event has no next-one point-event (denseness or no- contiguity between point-like events), each of these events can be taken as the terminus of a continuous process in which variation and passage obtain (non-denumerably set of point-events or continuity). As Robert Arthur (2019, p. 32) argues, “from the fact that there is only point-event or instantaneous state of change (and no actual becoming) at each instant of the interval, it does not follow that there is no passage in that interval”. While this argument seems to stifle the opposition between occurrence and becoming, it does not suffice to justify a more detailed notion of the latter. In other words, a further central question that needs addressing is if events can really be thought of in terms of instantaneous occurrences, or rather, if this latter notion is eventually tenable.

36 2.3. Processual becoming

2.3.1 The pioneering contribution of Alfred N. Whitehead’s process philosophy One of the polemical targets of Grünbaum’s critique is Alfred N. Whitehead’s (1925) conception of time. Whitehead, too, holds the conviction that the denseness of moments in the continuum makes it impossible that they be consecutive. Yet, contrary to Grünbaum, he thought the only way to accommodate the notion of change is to invoke an atomistic account: becoming can only be ensured by discrete variations where every point has an immediate successor. Temporalisation, says Whitehead (ivi, p. 129), is not a continuous process, but a succession of contiguous points. As such, it appears to entail a form of divisibility that reaches a minimum length: “Thus time is the succession of elements in themselves divisible and contiguous” (ibid.). All in all, it is apparent that Whitehead fails to take into account the application of Cantor’s set theory to events, based on which point-like events are not contiguous, and hence his view of temporality turns out to be more than wanting. Contrary to this picture, however, Whitehead’s understanding of atomized temporality is much more refined than it might prima facie appear. As a preliminary step to our foray into his complex philosophy, it is important to notice that he distinguishes between the potential (continuum) and the actual (atomistic). This suggests that, as I hinted at the beginning of this section, Whitehead is conscious of the fact that the property of the continuum of being dense does force us not to conceive points or instants as possible, rather than actual divisions. These instants do not need to be extensive to give rise to a definite duration. However, Whitehead believes that this difficulty cannot be ducked by simply assuming that something becomes at each non-extensive step of the interval. His (dis)solution of the problem is that it is not necessary to hold that an infinite series of acts of becoming, with a first act, and each act with an immediate successor, is inexhaustible in the process of becoming. While every act of becoming involves the becoming of something with temporal extension, the act itself is not extensive. More precisely, it is not divisible into earlier and later acts of becoming. This solution cannot be properly understood unless we introduce a key element of Whitehead’s philosophy, that is, the intercourse between the continuum, made up of potentialities, and its divisions, namely, actual entities. The extensive continuum – divisible

37 but not divided – is a relational complex of “potential objectifications”, connected by various relations and overlapping parts. It is both indefinitely divisible and unbounded in extension: “This extensive continuum expresses the solidarity of all possible standpoints throughout the whole process of the world” (Whitehead, 1925, p. 66)28. The continuum satisfies the property of denseness, because the potentiality of division is indefinite, and thus no notion of contiguity obtains. It is merely composed of a huge number of events – none of which has any proper parts – contained in multifarious relations to one another. The continuum compenetrates, and at the same time is comprised by, all these potential configurations29. Actual entities actualize the continuum, or rather, atomize it. They evolve through discrete steps and display coherent actual systems of divisions. A given configuration of the universe, viewed as a multiplicity of relations among actual entities, is definable as an actual occasion, that is, the whole universe in process of attainment of a particular satisfaction (Whitehead, 1930, p. 200) – where this latter term refers to a completely formed actual entity and how it relates to the other contiguous actual entities within its neighbourhood. Put otherwise, every actual entity is thus atomistic and holds a relationship with all the other actual entities to the extent that they form a local network which comes to occupy a specific position in the continuum and thus to atomize it30. Still, we should not conceive of the continuum as a container and the entities as mere subsections in spatialized terms. Holding a processual view implies rejecting the idea of processes as unfolding

28 To make my case clearer, I think this is the appropriate place to anticipate a definition of process that will have to be expanded later. First, process is prior to substance, in the sense that objects are the expression of the interaction among covariant quantum fields (see Rovelli, Vidotto, 2014, p. 20). Second, no being is conceivable without becoming, since entities are the continually evolving outcome of the interactions between quantum fields. Third, experiential units of reality are not severable objects, as they co-produce each other by means of their mutual relations (therefore, as stated above, Whitehead’s view is an atomism without isolation: nothing exists in isolation, as all entities, in their coming into being, are linked to all other events, and each in turn becomes the objective data for future events). Fourth, as a result, complexes of relations are primary objects, which entails the primacy of networks of relations over discrete objects. 29 A few remarkable emendations and integrations characterize the evolution of Whitehead’s thinking in between Science and the Modern World (1925) and Process and Reality (1930). This theme cannot of course be broached here. For a perceptive analysis, see Stengers, 2011. 30 Other process philosophers have found the interplay of the continuum and its atomization an unnecessary complication. For example, Nicholar Rescher (2000, p. 31) cannot make sense of Whitehead’s commitment to process atomism and his idea of “ultimately undissolvable processual units”. He goes on to say that this atomistic doctrine is nothing but a conceptual burden: “Nothing is more natural than that miniprocesses should join and combine into macroprocesses, and a process metaphysic that does not commit itself to a Whiteheadean atomism needs no special machinery to accommodate this fact because it allows reality to be seen as processual ‘all the way down’” (ibid.). My reading, however, align with other interpreters who underscore the dynamic nature of actual entities, which are not ultimately undissolvable as they morph based on their relations as the process evolves.

38 through time and occupying specific regions of space. A process is rather understandable in the sense of a network of interacting components, where none of them can be isolated from the others. An actual configuration of the physical world – an actual occasion, in Whitehead’s parlance – is a contingent state of affairs that originates from, and gives a particular shape to, the continuum, in the sense that the continuum is fully expressed in and by the relations between actual entities. The continuum can therefore be identified with this network of actual entities insofar as all possible relations to each other are considered. Complexes of relations are primary, not discrete objects. The actual world is a particular configuration determining one coherent system of real divisions. Potentiality – that is, all of the relations, including those that are not actualized in specific configurations – is therefore not a metaphysical notion, but the full spectrum of possible relations between entities. On this account, actuality is the fact that entities are so disposed as to act in a way that favours some relations to some entities vis-à-vis other entities. So, in this regard, a dispositional reading of actual entities seems to make sense of why the continuum acquires a certain configuration – as I will also point out later. With no need to expand fully on Whitehead’s somehow quixotic terminology, it is worth quoting him at length to give a sense of the relationship between potentiality and actuality. This is key to understanding why instants can at best be models, while their genuine existence is to be excluded. To this end, I can quickly define “prehensions” as the way in which an actual entity relates to each other – in a way, it is how they become experiential subjects31 that modify each other. A “negative prehension” is that which the actualization of certain relations among entities leaves unexpressed. With this short ter- minological proviso, we can get to Whitehead’s quote:

A negative prehension is the definite exclusion of that item from positive contribution to the subject’s own real internal constitution. This doctrine involves the position that a negative prehension expresses a bond. A positive prehension is the definite inclusion of that item into

31 While saying that an actual entity becomes a subject, I should like to note that Whitehead neither holds a simplistic view implying that only sentient beings have experience nor a surreptitious pan-psychism implying that all entities have a psyche. Rather, he endorses a form of pan-experientialism, namely, the idea that there is experience at all levels of reality (see e.g. Eastman and Keeton, 2003). Without delving further into this issue, a prehension is the way in which actual entities take into account and thus experience each other. If no solitary entities but only solidary entities exist, experience can be rethought in the sense of how the relations of each entity to one another shapes the actual configuration of the world. This is why White- head can claim that his philosophy aims to clarify the notion of “being present in another entity” (White- head, 1930, p. 50).

39 positive contribution to the subject’s own real internal constitution. This positive inclusion is called its “feeling” of that item. Other entities are required to express how anyone item is felt. All actual entities in the actual world, relatively to a given actual entity as “subject”, are necessarily “felt” by that subject, though in general vaguely. An actual entity as felt is said to be “objectified” for that subject. Only a selection of eternal objects are “felt” by a given subject, and these eternal objects are then said to have “ingression” in that subject. But those eternal objects which are not felt are not therefore negligible. For each negative prehension has its own subjective form, however trivial and faint. It adds to the emotional complex, though not to the objective data. The emotional complex is the subjective form of the final “satisfaction” (Whitehead, 1930, p. 41).

What should strike the reader is that, on this view, subjects and objects are by- products of the process of actualization. From the bundle of all interconnected potential relations, certain entities get actualized that relate to each other. They do not pre-exist the continuum, while the continuum cannot be said to pre-exist them. Leaving aside the question of whether the terminology of subject and object is appropriate (which I think is not), it is worth mentioning in passing that this account implies that it is wrong to characterize the relation between the continuum and the actual entities as one between the whole and its parts. A symmetric relation holds between continuum and actual entities: every single actual entity is inseparable from the complete network of potentialities, and the continuum is inseparable from every single actual entity which forms it32.

32 In this light, Whitehead’s characterization is one way to unravel the thorny issue of existence vs. priority monism. I shall have to return to this issue in other places of this work, so for the time being it is enough to say that existence monism is the view that there is only one concrete existing object, while priority monism is the view that there is one basic concrete object on which parts are ultimately grounded, and hence dependent. In Whitehead’s view it makes no sense to say that one single object exists, as the only existing thing is process; while it is just as incorrect to say that actual entities precede the process that actualizes them. The fact that the event has an unextended nature, as I will shortly underline, means that no actual entity can be conceived in isolation; but, by the same token, the continuum cannot be conceived short of its actual parts. We can say that there is a conceptual containment relation between them whereby the continuum can be conceived only starting from the actual entities, while the entities can be conceived only as forming a network. This also makes sense of Whitehead’s reference to Spinoza: “In the analogy with Spinoza, his one substance is for me the one underlying activity of realisation individualising itself in an interlocked plurality of modes. Thus, concrete fact is process. Its primary analysis is into underlying activity of prehension, and into realised prehensive events. Each event is an individual matter of fact issuing from an individualisation of the substrate activity. But individualisation does not mean substantial independence” (Whitehead, 1925, p. 71). Interestingly enough, this allusion to Spinoza jibes with Samuel Newland’s (2018) recent reinterpretation of Spinoza’s monism as a conceptual dependence one which evades the Scylla and Charybdis of existence vs. priority monism. Finally, further evidence for the connection between Whitehead’s idea of process and Spinoza’s overall metaphysics is provided by Francesca di Poppa (2010) as she avers that the notion of essence as power suggests that “substance” and “modes” should be considered as processes.

40 It is in the light of the complex interplay of continuum and actual entities that Whitehead can reject the notion of instantaneity. More precisely, he thinks it is meaningless to speak of instants of time as real physical entities. Rather, they are practical tools for the modellization of time which are nowhere to be found in the real unfolding of processes. Instants do not belong to the structure of the physical world. While the instant as an intellectual construal gestures towards isolable events that just do happen in their fixed position, the idea of process emphasizes the non-separability of actual entities in terms of the relations that obtain between them – and that make them what they are. Therefore, an event is naturally extended because it comprises a network of actual entities with differing temporalities (i.e., the processes whereby they get satisfied). Events are describable as a nexus of interrelated actual entities. As a consequence, a state of change in a durationless instant is hard to conceive. This characterization of events as extended in time has been recently defended by Richard Arthur (2019), who considers the point-events of physics as nothing other than useful abstractions obtained by narrowing physical events to the limit. This extendedness, which allows conveying events as processes, results from the partitioning or atomization of dispositional potentialities. And this partitioning introduces a notion of becoming as the order of discrete variation. On this view, becoming cannot arise from non-extensive instants of time, but “there is no need to assume that an infinite series of acts of becoming, with a first act, and each act with an immediate successor, is inexhaustible in the process of becoming” (Whitehead, 1925, p. 69). Rather, becoming is a process with a temporal extension: bits of becoming are atomization of a process as a continuum. This partition of the continuum reflects the way in which entities get actualized into bundles of experienceable qualities. Actual entities, therefore, split the continuum into subdivisions. However, as I have already shown, actual entities are themselves processual, in the sense that they form a network with the other actual entities. This processual becoming forestalls the idea of true atoms occupying fixed space. Atomization, as it were, is the dynamic discretization of a continuous indivisible process – indivisible in the sense that the actual entities belonging to these processes are not severable from the complex network of actual entities. In the light of this, becoming should not be measured against a temporal metric. Quite the contrary, the temporal metric arises as the unfolding ordering of such a process of discretization, thus as the order of the

41 relations among actualized entities. Time is the organization on an actual scale of the continuum. It expresses the unfolding of events, which together form the world as a structure of evolving processes.

2.3.2. Processual becoming as the discretization of relational networks In the last twenty years, processualism has been invoked in a variety of contexts and reinterpreted in accordance with the peculiarities of these contexts. Amongst the most relevant contributions are, in theoretical physics, Bohm, 1965, 1980; Eastman & Keeton, 2003; Epperson, 2004; De Saint-Ours, 2008; in metaphysics and the history of philosophy, Rescher, 1996, 2000; Di Poppa, 2010; in biology Nicholson & Dupré, 2018; in theoretical chemistry, Earley, 2008; Sukumar, 2013; in social sciences Stengers, 2011; Helin et al., 2014. Though the meaning of the term “process” inevitably undergoes changes depending on its field of application, there is one central feature that all versions of processualism agree upon. I will try to unpack it with reference to the physicist and philoso- pher of science John Stachel, as his elaborations are particularly appropriate to the other arguments of the present work. Indeed, Stachel (1993; 2005; 2006; 2014) has provided a clear and robust processual interpretation of both general relativity and quantum field theory and also made the claim that the notion of process should be taken seriously for a proper theory of quantum gravity to be ever developed. Stachel’s analysis gets underway with the discussion of the so-called hole argument, a version of which dates back to Einstein himself. The general theory of relativity (GTR) is generally covariant, which, in a nutshell, means that the significant laws of the theory do not depend on the adopted system of coordinates. As a consequence, if one starts with one model of GTR and applies what is technically called a diffeomorphism, namely a permutation of the elements of the model, then a certain physical process can be described in two different manners, respectively corresponding to one or the other model. The hole argument comes down to the fact that, while the two models describe observationally equivalent state of affairs, the applied diffeomorphism produces a different correspondence between the physical process and spacetime points33.

33 As I will return in more detail on the principle of general covariance and the hole argument in Chap. 4, my discussion here is merely instrumental in reconstructing the basis of the notion of process in the way it was developed by Stachel.

42 In the first decades of the 20th century this issue sparked various interpretative con- flicts among scholars (see e.g. Kretschmann, 1915; 1915 a; Hilbert, 1917; Kretschmann, 1917; Einstein, 1918; Darmois, 1927; Einstein, 1956). After a period in which the issue was put in abeyance, the argument encountered a modern revival starting with the 1980s, with John Earman and John D. Norton (1987) and John Stachel (1993). In a recent survey of the hole argument, Stachel (2014, p. 6) observes that it is of particular theoretical import, as it “shows that, for any theory defined by a set of generally-covariant field equations, the only way to make physical sense of the theory is to assume that the entire equivalence class of diffeomorphically-related solutions to the field equations represent a single physical solution”. Because of this, he concludes that physical theories should be written in a fully background independent fashion so as to satisfy the principle of general covariance. While, as I noted, I will get back to this, what is worth noticing in this juncture is that Stachel invokes the hole argument to defend a processual account of physics. To this end, he starts off by differentiating algebraic and geometric structures. If one considers a set of elements � = {1, 2 … �} together with a set of relations between its elements, call it �, there is a significant distinction between a geometry and an algebra. In a geometry, the elements of � = {1, 2 … �} display the same quiddity (i.e., the same nature) but no haecceity (i.e., non-inherently individuated properties)34. The differentia- tion between the elements originate from the set of internal relations � between them. If one omits these relations, then the set � = {1, 2 … �} is invariant under the permutation group. The set of relations specifying a geometric structure on � is termed �. The maximal subgroup of � for which all the relations between the elements of � is preserved is called the symmetry or automorphism group of this geometry. In the case of geometry, the identity of a certain element can only be defined in terms of the relations it bears. Quite the opposite, in an algebra, each element of a set S= {1, 2 … �} displays both

34 Stachel adopts here Teller’s (1998) terminology. For the sake of clarity, I quote Stachel (2006, p. 56) at some length: “‘Quiddity’ refers to the essential nature of an entity, its natural kind; and – at least at the deepest level which we have reached so far – entities of different natural kinds exist, e.g. electrons, quarks, gluons, photons, etc. What distinguishes entities of the same natural kind (quiddity) from each other, their unique individuality or ‘primitive thisness’, is called their ‘haecceity’. Traditionally, it was always assumed that every entity has such a unique individuality: a haecceity as well as a quiddity. However, modern physics has reached a point at which we are led to postulate entities that have quiddity but no haecceity that is inherent, i.e. independent of the relational structures in which they may occur.”.

43 quiddity and intrinsic haecceity, as it is put in a one-to-one correspondence with a num- bered coordinate system35. As Stachel (2014, p. 20) goes on to explain, since Descartes’s introduction of analytic geometry, it is generally convenient, or even necessary, to employ algebraic methods in the solution of geometrical problems. This is performed via a coordinatization procedure by which each geometrical element is assigned an algebraic coordinate. In this way, the homogeneity of the geometrical structure is lost. If one wants to restore it, the class of all admissible coordinatizations of the geometry, based on the given algebra, must be considered. In this way, each geometric point is assigned every admissible element of the algebra as a possible coordinate in at least one admissible coordinate system. A transformation between two admissible coordinate systems is called an admissible coordinate transformation. This can be achieved in two different manners, each corresponding to a different transformation. In an active point transformation, one keeps the coordinate system unchanged and permute the points of the geometry. In a passive coordinate transformation, one vice versa keeps the geometric points fixed, and performs an admissible coordinate transformation of the elements of the algebra. Each set � = {1, 2 … �} is then characterized by equivalence relations. An equivalence relation is a reflexive, symmetric and transitive two-place relation. An equivalence relation splits the set � = {1, 2 … �} into equivalence classes, either termed orbits. A certain theory is permutable if, given a certain model for the theory, the overall equivalent class of that model is still a model for the theory. A certain theory is generally permutable if the entire equivalence class is taken as a single model of the theory. This reasoning has bearings on the hole argument in as long as GTR is a generally covariant theory. In this respect, one may wonder which conceptual interpretation should be accorded to the ontological status of spacetime. Typically, the two main contending positions are substantivalism and relationalism. I will expand on the historical background of this dispute in Chap. 3 while introducing Barbour’s shape dynamics. Substan- tivalism is the metaphysical position which accords an independent existence to spacetime, while relationalism claims that spacetime displays a derivative existence with respect to matter. In Belot and Earman’s words (2004, p. 227):

35 For a punctilious critique of Stachel’s reasoning, see Pooley (2006).

44 Substantivalists understand the existence of spacetime in terms of the existence of its point- like parts, and gloss spatiotemporal relations between material events in terms of the spatiotemporal relations between points at which they occur. Relationists will deny that spacetime points enjoy this robust sort of existence, and will accept spatiotemporal relations between events as primitive.

In the light of the hole argument, Stachel believes there is a third viable option, termed dynamic structural realism, which, according to him, resonates with Dorato’s (2000) structural spacetime realism or the sophisticated substantivalism advanced by Pooley (2000). Dynamic structural realism entails that the points of the spacetime manifold do have physical character even before a certain metric field is chosen. However, spacetime points lack individuality, or haecceity. Stachel leans towards a form of structuralism, different from the ontic structural realism à la Ladyman or Steven French (see e.g. Ladyman, 1998; French and Ladyman, 2003; French, 2017). Ontic structural realism is a metaphysical conception whose main tenet is that “what there is in the world at the fundamental level […] are physical structures, in the sense of networks of concrete physical relations among concrete physical objects (relata)” (Lam, 2014, p. 1157). Differently, Stachel endorses a form of philosophical realism which “stresses the priority of processes over states” (Stachel, 2014, p. 38). If we follow Stachel, the connection between this type of structuralism and a process view in physics proves particularly crucial to a covariant approach to quantum gravity – which will be key in the fourth chapter of this work to accounting for the processual nature of becoming36. A covariant approach to quantum gravity develops a four-dimen- sionally invariant theory from the very beginning. On the contrary, a canonical approach to quantum gravity puts stress on the three-dimensional state of things, in the light of which a process can be described as a succession of different states of those things37.

36 On the backdrop of the canonical vs. covariant split, see Rovelli, 2004, Appendix B and Rickles, 2020, pp. 124-127. Two alternative strategies to derive a covariant form of canonical quantum mechanics are Reisenberger and Rovelli, 2002 and Arnsdorf 2002. Importantly for the notion of process itself, Reisen- berger and Rovelli argue that a covariant formulation of canonical quantum theory can be obtained by getting rid of the “idealization” of measurements as instantaneous. They conclude that giving up such an “the unrealistic idealization” opens up to a formulation of quantum theory sufficiently general to deal with covariant theories. 37 In this context Stachel refers to a debate that I cannot address in this work, that is, perdurantism vs. endurantism. For a clear discussion of the topic, see Barons and Miller, 2018, pp. 171-191; Gilmore et al., 2016. On the dependence of this debate on the controversy between presentists and eternalists, see Dorato, 2010.

45 According to Stachel, what is at stake in this alternative is the physical and metaphysical difference between four-dimensional processes and things as three-dimensional states of these processes. A thing in the way Stachel describes it, viz., as a state, is “just a particular spatial cross-section of a process and of secondary importance: all such cross-sections are equal, and each sequence of states represents a different ‘perspective’ on the same process” (pp. 65-66). As far as special and general relativity are concerned, four-dimensionality is such an inherent feature that spacetime can admit no “natural” breakup into spaces and times. Any such breakup comes from the introduction of an arbitrary frame of reference (a time- like fibration or a space-like foliation of the region of spacetime) which leads to a given state of (what thus becomes) an observed system arises. This all means that the state of a system – a thing – is nothing other than a partition of a continuous process obtained through the introduction of a frame of reference that does not belong to the system.

2.3.3. Processual becoming: relations and relata on a par I began with the controversy between Grünbaum and Whitehead to make one point clear: there is an uninterrupted intercourse between the continuum and what the latter calls actual entities. If we combine Whitehead’s and Stachel’s insights, we can say that actual entities are individuals with haecceities. As Stachel notes with reference to GTR, where the continuum is the basis, any breakup is to some extent arbitrary – a conceptual abstraction for specific purposes. With reference to the hole argument, Stachel concludes that the points of spacetime are not self-standing individuals but inherit haecceity from the metrical field or other physical relations imposed on them. It is here that the notion of process proves crucial to the theoretical project of both authors: it is only when something is taken to be part of a set of relations that one can speak of entities with haecceities, as long as these entities are not abstractively isolated from the rest but are taken to be bearers of relations. A process, therefore, is the moment at which an entity is taken to be an individual within a set of relations, while the basic structure of reality is a continuum networking. A processual view is one that never neglects the intrinsic nexus between the identity of entities and the relations they are in. So, processualists break with the metaphysics of things – where things are understood à la Stachel as spatial cross-sections of a process – which

46 deems individuals in spacetime to be endowed with an existence that is independent of the existence of other individuals. The processual view as advocated by its pioneer, Whitehead, and subsequently by other scholars does not see the world as composed of building-blocks that are naturally separate from each other. On the contrary, facts about quantum particles and spacetime points gesture to the fact that “their identity and diversity are not intrinsic to them but rather are determined by the relational structures into which they enter” (Ladyman and Ross, p. 251). A concluding point needs to be stressed, though. The type of relationalism that both Whitehead and Stachel hold does not demote relata to pure effects of their relations. In Whitehead’s view, the continuum and its atomistic division presuppose each other. Real- ity only expresses itself through actual entities that are so-called insofar as they actualize the continuum. Similarly, Stachel, unlike Ladyman and Ross, rejects the idea that there are no relata but only relations. However, the physical and metaphysical impossibility that relations exist prior to relata does not lapse back into commitments to any notion of a fundamental ontological level comprising individuated entities. The anti-foundational- ist inspiration is preserved as one embraces two tenets. First, no entity exists out of the network of relations in which it acquires identity. Second, networks of relations are nested into other networks of relations. These relations, as it were, express themselves through the individuated entities on which they confer identity. No relationship of priority can thus be established between relata and relations – as they are not conceivable without one another. This view bears resemblance to the Spinoza-inspired revision of moderate structural realism proposed by Vincent Lam and Michael Esfeld (2013, p. 255) stating that “relations can be, like intrinsic properties, the ways in which objects are”. The idea that relations require objects as that which is given within the relations means that certain “relations are the ways of being of the objects in question” (ibid.). Objects are tied together by certain relations and do not own any identity independently of them. Accordingly, a structure is nothing but a set of objects whose essential ways of being (Spinozian modes) are or at least include certain relations that they bear to each other. Entities do not exist apart

47 from the structure they are part of, while the structure would not be without the entities it puts in relation38. Structures are networks of concrete physical relations. To conclude, this is the bottom line of a processual view: identifying things or states requires seeing them as non-isolable, individuated sections of concrete physical relations. To make sense of their behaviour as things or states requires studying how these individuated sections relate to the other sections of the chain of relations in which they stand.

2.4. Concluding remarks

As I discussed in the previous chapter, relativity and the block view seem to rule out becoming. Despite this, this chapter mounted the argument that there is a way to justify a dynamical explanation of physical processes that is consistent with relativity and also shines a light on the nature of physical events. It is a type of relationalism that puts relations on a par with relata and understands the identity of the latter in terms of interactions between systems. As I will contend in Chapter 4, this will be key to timeless paradigms that aim to rescue a robust notion of process and, based on this, distinguish time from temporal becoming. Before doing so, however, it will be instructive to explore the opposite path – one that radically rejects becoming and arrives at a type of relationalism that admits no motion at all. Time and change are considered as illusions. This is Julian Bar- bour’s thought-provoking shape dynamics, which hinges exactly on the exclusion of events as primitives. The radicality of this view, which goes so far as to refute not only time but also the self-identify of individual things, lies in the contention that only becomingless configurations among particles obtain. The juxtaposition between one of the most compelling becomingless theories with processual ones will allow realizing what is gained and what gets lost in the erasure of becoming within physics.

38 As Di Poppa (2010) compellingly argues, Spinoza’s metaphysics is indeed a processual one, where Na- ture as expressed by its modes should be conceived as an infinite chain of interacting processes rather than in terms of subject-properties or substance-predicates. While modes are certainly not “things” or “parts” of Nature, no absolute individuation of them is feasible for Spinoza. The only way to individuate modes is to identify the power they exert in the set of dynamical interactions in which they take place. It belongs to the nature of modes not to be independent individuals, both ontologically and conceptually. Importantly, each mode is dependent not only on Nature but also on infinite other modes. However, Lam is spot on when he insists that in Spinoza’s metaphysics modes are “concrete, particular ways in which objects are”. This means that, while there is no ontological distinction between the objects’ properties and relations in which they stand in the sense of modes, the latter are the only way in which Nature expresses itself. Therefore, the distinction between relations and relata is ultimately conceptual.

48 Chapter 3. Shape dynamics: A review

3.1. Introduction

This Chapter is centred on a critical evaluation of Julian Barbour’s thought-provoking theoretical framework, known as shape dynamics, whose main intent is to implement a purely relational description of dynamics. Famously, Newton’s laws rest on the alleged existence of a privileged frame of reference, compared to which “real” motion is ascer- tainable. In stark contrast to Newton’s reliance on non-directly observable structures, Barbour develops an entirely relational account of the evolution of a system of point-like interacting particles39. Barbour’s contribution to theoretical physics is fairly broad, as it covers three main areas, namely classical physics, quantum physics and quantum gravity. However, this chapter will only inspect Barbour’s reformulation of Newtonian classical physics, because it is the foundational stone of his overall approach. My discussion will therefore address the essential traits of shape dynamics in rigorous but inevitably limited terms. Moreover, within the boundaries of this research, I will have to skip a thorough discussion of technical details to foreground the tie between the relational description of dynamics and the issue of temporal becoming.

39 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. I will get back to this point later on. For a thorough discussion of this topic, see Esfeld, 2020, pp. 1-62. See also Vassallo, Deckert and Eslfed, 2016, which exposes two alternative strategies to combine a minimalist ontology of the physical world (matter points and their mutual distance relations) with physics. One main assumption in their argument, though, is ques- tionable. They write: “assume, as is well supported by all the available physical evidence, that the configuration of matter consists in finitely many discrete objects, such as point particles”. Yet, it is worth noting at least in passing that the problem of the existence of such things as point particles is far from settled. To provide a telling example, one of the fiercest critics of this notion, David Bohm (1984, p. 83), writes: “When one applies the existing quantum theory to the electrodynamics of ‘elementary’ particles (such as electrons, protons, etc.), internal inconsistencies seem to arise in the theory. These inconsistencies are connected with the prediction of infinite values for various physical properties, such as the mass and the charge of the electron. All these infinities arise from the extrapolation of the current theory to distances that are unlimit- edly small. Among the things that make such an extrapolation necessary, one of the most important is the assumption, which seems to be an intrinsic part of current theories that ‘elementary’ particles, such as electrons, are mathematical points in the sense that they occupy no space at all. On the other hand, in spite of many years of active searching on the part of theoretical physicists throughout the whole world, no way has yet been found to incorporate consistently into the current quantum theory the assumption that the electron occupies a finite region of space. While it has been suggested that perhaps the infinities come from an inadequate technique of solving the equations (i.e. perturbation theory), persistent efforts to improve this technique have not yet produced any favourable results, and indeed those results that have been obtained favour the conclusion that basically it is not the mathematical technique that is at fault, but rather the theory itself is not logically consistent”.

49 Section 2 will explore the theoretical background that lies beneath Barbour’s approach. Attention will be drawn to a few key features of the relational perspective so as to pave the way for the discussion of his physical paradigm. Commencing with the theoretical commitments of the relational model, this section surveys Mach’s contribution to the conceptual toolkit of relationalism and the development of analytical mechanics as an alternative path to Newtonian vectorial mechanics. Section 3 gets a grip on Barbour’s reformulation of Newtonian mechanics from a purely Machian point of view. Section 4 outlines a more critical perspective on shape dynamics. In particular, three aspects will be put under scrutiny. First, the tenability of Barbour’s theoretical premises. Second, the internal consistency of his relational account of dynamics. Third, the question of whether or not shape dynamics is able to rule out, as Barbour hopes, a genuine notion of time and temporal becoming. Section 5 will offer a brief but much needed discussion of the radical consequences of shape dynamics. I will construct the case that Barbour’s relational paradigm successfully eliminates becoming but reintroduces time. These final notes will show the reasons for the subsequent comparison with a type of relationalism that rescues becoming but is truly timeless.

3.2. Relational dynamics

One of the most interesting, and yet challenging, projects within the philosophy of physics is the quest for the most fundamental constituents (if any)40 of reality. Scholars who work in this research area generally refer to the notion of primitive ontology to identify the basic building-blocks (if any) of the universe. Although the link between fundamentality and the ontological status of spacetime is still underexplored, the latter is certainly relevant in this context. For, the two main contending views, namely substantivalism and relationalism, come to a head about what is fundamental. On the one hand, substantivalists argue that both material entities and spacetime are to be conceived as fundamental. In this view, which is generally traced back to Newton, spacetime is a substance in its own right. A particular version of this perspective, which

40 There is a variety of topics concerning fundamentality, which cannot be properly analysed in the present work. For what concerns here, suffice it to say that there are scholars, such as Matteo Morganti (2009, 2019), Gian Francesco Giudice (2019), Daniele Oriti (forthcoming), who have been questioning the idea of an ultimate, fundamental level of reality.

50 is called super-substantivalism, goes as far as to recognize only spacetime as fundamental, claiming that material objects are identical to spacetime regions (see e.g. Lehmkul, 2018). On the other hand, relationalists believe that only material entities are fundamental, while spacetime exhibits a derivative existence. In Oliver Pooley’s41 words (2013b, p. 523) “claims apparently about spacetime itself are ultimately to be understood as claims about material entities and the possible patterns of spatiotemporal relations that they can instantiate”. In its latest developments, this second approach, which was famously elaborated by Leibniz42, takes particles as the building-blocks of reality. The distance of particles from each other varies and this results in the evolution of the system under scrutiny. While it would be pointless for the purposes of this work to provide a full reconstruction of all the various positions on relationalism, it is imperative to note that the

41 Oliver Pooley (2013b) offers a robust, clear-cut account of the main tenets of the relationalist perspective. As a preliminary step, he considers, first, the ontology of a theory as the objects to whose existence the theory is committed, second, the theory’s ideology as the set of (primitive) distinctions the theory is able to provide in terms of its (primitive) predicates and terms, namely its (primitive) properties and relations. In this respect, he argues that different strategies are available for relationalists. A first solution consists in allowing the theory’s ideology to expand so as to recover the same physical distinctions the substantivalist inertial structure is able to determine in a way that is compatible with a relationalist ontology. In order to perform this, new relational quantities have to replace the standard Newtonian theory. A second strategy is to implement an alternative theory to Newtonian mechanics that engages only with relational quantities but is empirically adequate, the most successful model being – according to Pooley – Julian Barbour’s shape dynamics. A third option starts with Nick Huggett’s (1999) remark that no theory displays all of the following features, namely 1) spatiotemporal ideology restricted to Leibnizian relations, 2) Newtonian dynamically allowed histories, 3) supervenience of inertial effects on the specified spatiotemporal relations between objects. According to Pooley, while the first two strategies respectively waive 1 and 2, the third alternative consists in relinquishing 3. In this latter case, neither variation in the theory nor additional ideology are required: a “have-it-all” relationalism is implemented. 42 In his correspondence with Samuel Clarke, Leibniz (1717) critically addressed Newton’s approach from a variety of standpoints. First, he identified two implications of Newton’s mechanics, each corresponding to a different collocation the actual world could exhibit in respect to the alleged absolute substantival space, namely the static shift (different position of the universe in absolute space) and the kinematic shift (different state of motion of the universe in absolute space). Leibniz argued that, if in these two configurations, the spatiotemporal relations among particles are preserved, then these diverse situations are indistinguishable from any observative point of view. This is indeed an entry point to the principle of identity of indiscerni- bles, according to which two objects displaying the same set of properties cannot be distinguished, and thus must be identified. Another aspect encoded in the static shift is Leibniz’s formulation of the principle of sufficient reason, which entails a strict connection between a certain state of affairs of the universe and the causes that produced it. In this respect, Leibniz contends, there is no sufficient reason to believe that a certain initial configuration of the universe should be favoured vis-à-vis other possible configurations. Again, the idea of an absolute space according privilege to a specific configuration of the universe seems untenable. Finally, Leibniz condemned Newton’s position also from a methodological point of view, as no credit should be given to non-empirically verifiable claims. While all of these criticisms have been properly addressed, analysed and countered by the proponents of substantivalism (see e.g. Sklar, 1977), this last point is generally regarded as the least defensible. In fact, many research programmes within contemporary theorical physics postulate the existence of non-directly observables quantities, provided that they result in directly observable effects. For a clear discussion about the advantages/disadvantages of both substantivalism and relationalism see also Barry Dainton (2010, chap. 10), Oliver Pooley (2013b) and Michael Esfeld (2018, chap. 11; 2020, chap. 1).

51 conceptual commitments of the relational perspective owe very much to Ernst Mach’s research, though his contribution came in terms of hints rather than a systematic conceptualization. Contrary to Newton’s reliance on non-directly observable structures, Mach (1919) believed the dynamics of the universe should be traced in terms of variations among observable entities. Newtonian time is an independent, exogenous parameter against whose background motion is evaluated. However, as Mach (1919, p. 223) noted, “we must not forget that all things in the world are connected with one another and depend on one another”. As a consequence, time cannot be conceived as an absolute, external entity whose existence is independent of directly observable entities. Quite the contrary, any viable measure of time is contingent upon variations among those entities, while (1919, p. 224) “time is an abstraction, at which we arrive by means of the changes of things”. Absolute time, obtained by comparison with no motion, has neither practical nor scientific value. It is an “idle metaphysical conception”. According to Mach, the universe is an all-encom- passing system in which all individual parts mutually determine one another. Physics generally operates by subdividing the universe into ever smaller components, whose dynamics can be described almost neglecting the influence of the rest of the universe. At the same time, inertial phenomena “cannot be understood and explained within the context of the subsystems themselves, but at best by taking into account the rest of the universe” (Barbour and Pfister, 1995). This consideration directly leads to Mach’s idea that the inertial motion could arise in accordance with the causal action of all the masses of the universe. All things considered, Mach’s intuition to dismiss redundant parameters opens up to an account of dynamics in terms of the mutual positions of the subsystems of the universe in its configuration space. In order to fully grasp the import of this perspective, let me expand on the notion of configuration space. In his treatise on the variational principles of mechanics, Cornelius Lanczos (1952) points out that, ever since Newtonian foundation of dynamics, the science of mechanics has followed two main paths43. 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

43 Lanczos’s is a text that Barbour (2010, p. 1269) considers as a most instructive reconstruction of the developments of variational mechanics.

52 a force to calculate its action. Contrary to Newton’s vectorial mechanics, Leibniz took as the proper gauge for the dynamical action of a force the kinetic energy of the system. As a consequence, he replaced Newton’s force with the work function. In this regard, Leibniz was 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 its 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 we want 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 completely 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. This principle enables us to 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 action44, this condition being called

44 Although it is not for this work to further expand on the insights of analytic mechanics, it is important to notice that the Lagrangian formulation of mechanics works for isolated systems, viz. systems for which the sum of the kinetic and the potential energy is constant over time. However, there are systems for which the work function does not only depend on the position of the particle but also on time. For these systems the principle of Euler and Lagrange is not applicable. In these cases, it is the principle of Hamilton that needs to be applied. The characteristic quantity that we use to define the action is now the difference between the kinetic and the potential energy. As Lanczos (1949, p. XIX) explains, “the Hamiltonian formulation of the

53 the principle of least action. As we will see, in his Lagrangian formulation of shape dynamics, Barbour adopts a timeless version of this principle. To sum up what I have so far written, while vectorial mechanics is based on the determination of forces and momenta in the vectorial space, analytic mechanics is based on a principle of minimization of scalar quantities. This means that analytical problems of motion need a generalization of the Euclidean coordinates in terms of parameters encoding the position of a mechanical system. These parameters are called the “generalized coordinates” of the mechanical system, the vector space defined by these coordinates being termed the configuration space. For a single particle in Euclidean space its configuration space is � = ℝ. For N disconnected, non-interacting point-particles, the configuration space is � = ℝ. In this formalism, the points in Q that express an instantaneous configuration of the system rest on a curve in Q denominated the dynamical orbit of the system. At this stage, it is important to stress two aspects. First, this representation is based on positions in an inertial frame of reference, and thus it does not meet Mach’s requirements. Famously, Newton himself acknowledged that only relative positions and times are observable. He then added, though, that absolute space and time could be inferred from directly observed data. This issue, to which he never returned in the Principia, has been neglected until then, and is generally referred to as the scholium problem. Barbour (2010, p. 1266) summarizes it as follows:

Given only the successive separations � of a system of particles that form a closed dynamical system in Euclidean space and told that there does exist an inertial frame of reference in which the particles obey Newton’s laws and are interacting in accordance with his law of universal gravitation, how can one confirm this fact and find the motions in, for definiteness, the system’s center-of-mass inertial system? For simplicity and without loss of insight, it may be assumed that the particle masses are given.

The question, then, is how to define an inertial frame of reference starting from purely relative motion. According to Barbour, this has been successfully performed by Peter Guthrie Tait (1883-1884), of whom he reports a generalization that takes into consideration not only the relativity of position and time but also that of scale. Suppose there is a

principle of least action asserts that the actual motion realized in nature is that particular motion for which this action assumes its smallest value”.

54 system of N point-like particles which are said to be moving inertially and whose instantaneous configurations are defined. In order to implement a purely relational account of this motion, only dimensionless quantities should be accorded physical meaning. This entails that standard Cartesian distances must be replaced by the dimensionless separations �:

� = , where � = ∑ �

where � is the separation between particles “i” and “j” in some arbitrary reference scale45. As discussed above, in a Cartesian representation, a system of N particles is associated with a configuration space of � = ℝ. However, if one cannot describe the system in terms of a privileged frame of reference, then only 3N-7 objective (observable) data remain: three of them get lost because the centre of mass is unknown, three because the spatial orientation is unknown, and finally, one because the scale is unknown. This means that the positions are described only up to Euclidean translations, rotations and dilations46. Starting from these considerations, Tait demonstrated that, in order to confirm whether or not particles are moving inertially, three snapshots of the system are required. This result, which I will not examine technically, is key to furthering a relational model. For it cast light on the amount of additional information that is necessary for a purely relational account of dynamics to be implemented. We will come back to this point while discussing shape dynamics. The second relevant aspect for the analysis of the foundations of Barbour’s theoretical framework derives from the definition of the configuration space. The standard account of dynamics refers to instantaneous configurations of the system moving along the orbit as time passes. This time is generally referred to as an external parameter. If no access to anything outside the system is provided, then a measure of time should be

45 In this representation, rij is the mutual distance between any couple of the N particles in an arbitrary system of reference. The parameter r expresses the sum of all these distances, and, for N particles, the sum is composed of N(N-1)/2 terms. The index of the sum “i

55 somehow extrapolated from within the system itself. However, all information is encoded in the configuration space, and this leads to a vicious circle: it is not possible to define the evolution parameter, which is to be extrapolated from differences along the curve, before the curve itself has been defined. The solution to this problem, which Barbour himself takes up, will be discussed in the subsequent section. For now, I just want to remark that, in order for a purely Machian account of dynamics to be implemented, two steps are to be taken. First, dynamics should be expressed only in terms of relative quantities. Second, the inertial motion needs to arise not as the effect of an absolute background space, but from the dynamical effect of the universe as a whole. To apply this procedure, which Einstein (1918) named Mach’s principle, Barbour’s shape dynamics intends to develop a universal framework for the direct and explicit implementation of a completely background-independent theory.

3.3. Shape dynamics

Barbour’s technical contribution to classical physics is twofold. First, along with Bruno Bertotti, he has elaborated a Machian account of classical dynamics. Second, he has developed a Machian analysis of the structure of GR. For present purposes, I will concentrate on Barbour’s reformulation of Newtonian mechanics in terms of a system of non- relativistic point-like interacting particles. This is instrumental to the overall economy of my work, whose intent is to analyse the consequences of Julian Barbour’s theoretical framework on the notion of temporal becoming. To this end, a preliminary, concise account of shape dynamics is to be provided. Let us start with a system of N classical, individually identifiable, particles, which evolves through time. Then assume these point-like particles move in a spacetime equipped with an absolute simultaneity structure, a Euclidean geometry for each simultaneity slice, and an absolute time metric. As both (absolute) space and (absolute) time are non-directly observable, one may wonder whether there is a way to reformulate mechanics with less kinematic structure and more predictive power. The idea is to eliminate potentially redundant structure (Barbour, 2012, p. 259). In other words, one would like to single out the minimum set of possible observable initial data that governs the evolution of the system, thereby eliminating disposable information. In this view, only

56 relational ingredients à la Mach should obtain. Whereas Mach’s ideas come in terms of intuitions rather than technical requirements, Barbour (1999, p. 71) reminds us that it was Henri Poincaré (1902) who offered a precise solution to this issue. He suggested that a relational theory of mechanics should consider, as initial data, the instantaneous relative distances between the particles and the distances’ change rate47. Though problematic from a computational standpoint, Poincaré’s proposal can be vindicated from a variety of other standpoints. First, the homogeneity and isotropy of Euclidean geometry seem to entail that the instantaneous configurations of the point-like particles relative to any alleged absolute space and the orientation of the system relative to such a space should be unobservable. Second, Poincaré’s aim was to find an alternative solution to Newtonian mechanics, whose merit was to uniquely predict the evolution of the system, once the initial position and velocity of the particles under scrutiny are given. This means that there is a price to be paid for the dismissal of illegitimate metaphysical commitments48. The objective of a follower of Mach’s who embraces Poincaré’s proposal is thus to find a theory for which the initial data give a unique future evolution, and which is empirically equivalent (if not superior) to Newtonian theory. In order to implement this account, Barbour (2010, p. 1273) starts with a distinction between three kinds of spaces. The first is the aforementioned Newtonian configuration space, which, for N non-interacting particles, is composed of 3N Cartesian coordinates. This space can also be called the Extended Configuration Space (ECS). Starting from this ensemble, all the configurations which are congruent by transformations of the Euclidean group (translations and rotations) form the Relative Configuration Space49 (RCS). If one, then, considers all the similar configurations by transformations of the similarity group (dilations), or rather, if one adds also scale invariance considering all shape-similar configurations (Lie group50), these latter form the Shape Space (SS). This means that, if we

47 As Jeremy Butterfield outlines in a long discussion of Barbour’s book The End of Time, Poincaré’s initial data may result as an odd request for an empiricist. Indeed, though instantaneous inter-particle distances are relational, it might well be the case that they are not observable. Making measurements takes time. This begs a question I will be addressing later on, namely the computational limit of Barbour’s relational dynamics. 48 The three snapshots required for Tait’s account of the evolution of the system is an example of the consequences of the implementation of a purely relational description of dynamics. 49 A relative configuration space is a specification of all the inter-particle distances at some instant. 50 Lie groups are “groups that are simultaneous manifolds, i.e., their elements are parametrized by continuous parameters” (Barbour, 2012, p. 260). In the context of shape dynamics, the Lie group constitutes the fundamental structure group. It plays a dual role. First, it enables to eliminate redundant configurations. Second, it provides the right tools for a purely relational theory of dynamics.

57 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 configura- tion51. In shape space there are only sets which 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 a 3N-7-dimensional space. As already discussed, 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 by which this is achievable is of central interest. However, the technical details52 of this implementation are not relevant for our purposes. Suffice it to say that the Newtonian equations are a subset of all the possible configurations in SS, which are defined either by the specification of an initial point together with a direction or by a point and a tangent vector53, respectively being termed the stronger or the weaker form of Mach’s principle. While Newtonian equations represent specific solutions within SS, the sum of all the possible configurations is called by

51 The congruence is obtained when two configurations, using translations or rotations, can be brought to exactly overlap, and they are similar if we can exactly overlap them, given also a scale (dilation or contraction) transformation. 52 The point is that the notion of initial velocity, which is a condition for the application of Laplacian determinism in Newtonian dynamics, can be generated by purely group actions. To be more precise, different Newtonian velocities can arise from identical data in SS. This is because configurations in shape space are invariant under translations, rotations and dilatations. These actions are “invisible” in SS, which define only the shape and the way it is changing. According to Galilean relativity, translations of the system have no effect in SS. While we can ignore them, we cannot do the same with rotations and dilatations. These imply four dimensionless dynamically effective quantities. First, two angles which determine the direction in space of a rotation axis. Second, from the kinetic energies associated with rotation, �, dilatation, �, and change of shape, �, we can form two dimensionless ratios, namely and . This means that the kinematic action of the Lie group generates four parameters that affect the histories in SS without changing the initial shape and dilatation. If there is also an acting force, then there is a fifth parameter to be considered, namely T/V (T=kinetic energy of the system, V=its potential energy). As a consequence of this, Newtonian dynamics are not deterministic in SS. 53 While it is not for the present analysis to enter into the details of shape dynamics, it is worth noting that the difference between these two cases is the required amount of information to specify a unique evolution. In the first case, there is a unique curve that passes through the initial point � ∈ � (where q is the initial configuration state of the system under scrutiny) in a given direction d at that point. In the second case, there is a one-parameter family of curves passing through q, all being tangent to each other at q.

58 Barbour Platonia. 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 rather as referring to all the physically possible configurations. In sum, gluing together Mach’s intuitions and Poincaré’s technical analysis, Barbour aims to establish a purely relational description of dynamics that disposes of the notion of time on a fundamental level. While in standard dynamical theory, time is an independent variable supplied by an external clock, he lays out an alternative path in which the relation between time and dynamics of the system is reversed, the former being contingent upon the latter. Indeed, the clock itself is a mechanical system. So, if one wishes to treat the universe as a single system, the issue of what clock, if any, should be used becomes critical. I would now like to expand on the main tenets of the theory. Consider the relative configuration space of a system of N point-like particles and define an intrinsic difference between two configurations. This is called the best-matching procedure. In the book The End of Time, Barbour (1999, pp. 116-117) begins by discussing the intrinsic difference and the best matching procedure for a system of N=3 point-like particles. This system forms the SS of all possible triangles. As he explains, the distance between two configurations can be defined in several ways, but one of the sim- plest is the following. Take one of the two configurations as fixed and let the second configuration move freely in any position relative to the first configuration. In this way one can match, respectively, the three vertices of the first triangle (corresponding to the fixed configuration) with the three vertices of the second triangle (corresponding to the movable configuration). At this point, form a “trial distance” by calculating a weighted average on the masses54. This quantity, which is arbitrary, is taken in accordance with the relative positioning of the two triangles. However, Barbour goes on to say, it is possible to consider all the relative positionings of the aforementioned triangles and thus to find the one for which the distance is minimized. This “measures the intrinsic difference between the two matter distributions represented by the triangles. It is completely determined by them and does not rely on any external structure like absolute space” (Barbour,

54 Formally, define the first triangle as ABC and the second triangle as A*B*C*. Imagine that ABC is fixed, and that A*B*C* moves freely with respect to ABC. This forms “distances” AA*, B B*, C C* between the corresponding vertices, at which their mass (respectively a, b, c) is positioned. In this way, “we form a ‘trial distance’ d by taking each mass and multiplying it by the square of the corresponding distance, adding the results and taking the square root of the sum” (Barbour, 1999, p. 116). We finally obtain the following formula: � = �(��∗) + �(��∗) + �(��∗)

59 1999, pp. 116-117). The intrinsic distance between two arbitrary matter distributions can be calculated in a similar fashion. Consider two configurations and form a “trial distance” by matching each point-like particle position in one configuration with its corresponding position in the other configuration. One distribution is kept fixed, and the other is allowed to be moving freely with respect to the other. For any trial position, the analogue of the above expression is evaluated, and the positioning in which the trial distance is minimized is searched. This intrinsic difference, which minimizes the difference between two matter distributions, is called the best-matching position. The resulting metric is called the “intrinsic difference” between two matter configurations and the resulting theory is called “intrinsic dynamics”55. The configuration space of the universe is the fundamental concept of Barbour’s shape dynamics56. In a universe composed of N point-like particles the configuration space has 3N coordinates, each particle being identified by a triplet of them. As the universe evolves, the system traces a curve in the configuration space, where the latter is composed of all the positionings of the particles relative to each other. Among the relative matter distributions in the configuration space, there is a subset of special curves for which Newton’s laws hold. Here comes the most crucial and maybe controversial point of Barbour’s theoretical framework. In the first section, I sketched the differences between Newtonian vectorial mechanics and Leibnizian analytic mechanics. In that context, I commented that, whilst vectorial mechanics calculates forces and momenta, analytic mechanics considers only scalar quantities, namely the kinetic and the potential energy of a system. The advantage of the principle of least action is that it is

55 According to Butterfield, this procedure allows to provide a solution to a problem that emerges when we consider each relative configuration as in its own instantaneous space. This is what Barbour calls the problem of equilocality, namely (Butterfield, 2002, p.302) “the problem of identifying spatial points (not point- particles) between two such spaces”. The solution states that two spatial points of two diverse matter distributions are equilocal if they occupy the same position in those Cartesian coordinate systems which min- imize the trial distance. Says Butterfield (ivi, pp. 302-303), “two relative configurations with their spatial points thus identified are called “horizontally stacked”. The idea is that, as usual in spacetime diagrams, the vertical dimension […] represents time, so that “horizontal stacking” refers to placing relative configurations relative to one another in the horizontal dimensions while stacking them in a vertical pile (i.e. in the time dimension)”. 56 I should like to recall that Barbour has identified three types of configuration space, namely ECS, RCS and SS, the latter encoding all the shapes of the system. As I have already discussed, ECS is larger than SS because it contains information about the position, orientation and size of the shapes. Even if ECS has unphysical information, it is convenient to work in it as its math is simpler. As Sean Gryb and Flavio Mercati (2015) explain “this is called a gauge theory. We can work with gauge theories provided that we remove, or quotient, out the unphysical information”. This is done by arrowing a 1-to-1 correspondence between the ECS and SS. This procedure is called picking a gauge. Mathematically, this is obtained by applying constraints on the ECS (such as fixing the centre of mass, orientation and size). This is why Bar- bour speaks of the configuration space, while maintaining that only SS has physical validity.

60 diffeomorphism invariant, viz., it does not change value in different system of coordinates. This means that the methods of the calculus of variations automatically satisfy the condition of background independence. Thus, it comes as no surprise that Barbour’s intrinsic dynamics advocates analytic mechanics and applies a version of the principle of least action to calculate the evolution of the system. I will now present it in its essential traits. Let us start with two configurations of the universe and consider, as in classical analytic mechanics, all the trial curves which continuously join these two configurations. In the standard formulation, all trial curves are considered with respect to a fixed pre- existing temporal parameter. Barbour’s version does not make this assumption and is thus a timeless principle of least action. Each trial curve is then divided into very short segments, for each of which the action is calculated57. Clearly, the action along a certain trial curve is given by the sum of the action of all the singular segments. As in the usual formulation, for one of the trial curves joining the extremities the action is minimized, and this extremal curve defines the subset of solutions for which Newton’s laws hold. Im- portantly, Barbour’s version of the principle of least action is timeless, meaning that the action is calculated for an arbitrarily chosen parametrization along the curve, so that there is no need for a time metric. The main idea of intrinsic dynamics is thus to replace classical quantities with intrinsic differentials. We can then recover the familiar Newtonian time metric if the arbitrarily chosen parameter along the curves in the configuration space is fixed so that the energy theorem obtains. In this respect, one is able to determine a time interval as the result of the dynamics of the system. I will conclude this section with this calculation. Recall that the energy theorem holds that, for an isolated system, � = � + �, where T is the kinetic energy and V is the potential energy of the system. The kinetic energy of a system of N point-like particles is given by:

1 1 �� = �� = � 2 2 ��

57 The action Barbour defines is a timeless application of the principles of least action and is calculated as: �� = √2(� − ∑ ) ∑ �� (��) where E is a fundamental constant, ∑ corresponds to the potential energy of the system, and (��) is the distance the particle “i” has moved.

61 This means, considering the energy theorem, that:

1 �� + � = � 2 ��

Finally getting to:

∑ (�� ) �� = 2(� − �)

This is what Barbour (2012, p. 265) calls the emergent time. He refers to this timeless version of the principle of least action as the first example of the holism of relational dynamics. In Barbour’s (2012, p. 265) own words, “the time that we take to flow locally everywhere is a distillation of all the changes everywhere in the universe. Since everything in the universe interacts with everything else, every difference must be taken into account to obtain the exact measure of time. The universe is its own clock”.

3.4. What is left of time?

Shape dynamics is fascinating from a variety of standpoints. The main virtue and origi- nality of this theoretical framework arguably consists in the development of intrinsic dynamics. The idea of time being dependent upon the dynamics of a system, rather than the evolution of a system being punctuated by an external temporal parameter, appears as a potential strength of the theory, at least from a metaphysical point of view. However, this comes at a high cost. Unfortunately, to fully grasp the tenability and consistency of shape dynamics, one would require an in-depth analysis of its application to the fields of quantum mechanics and quantum gravity. This exceeds the scope of the present discussion. Nonetheless, a few remarks can be made. First, intrinsic dynamics rests upon the idea of three-dimensional instantaneous configurations. One may wonder whether this accords with Mach’s desideratum of the dynamics being contingent upon observable entities. A coherent relational approach

62 should advocate observable and relational quantities. However, observability and relationality are two distinct concepts, whose co-presence is not warranted. There are cases in which observable quantities (such as the absolute temperature of a state of a matter field at a certain spacetime region) are non-relational, and cases in which relational quantities (such as the distance between simultaneous spacetime points) are unobservable58. In this regard, it is unclear how Barbour’s instantaneous configurations would meet both the requirements of observability and relationality. Second, Barbour’s intrinsic dynamics of a system of N point-like particles needs to confront the emergence of the modern conception of matter in terms of matter fields59. Apparently, Barbour aims to tackle the outcomes of field theory. In this regard, he argues that the description of Machian theories of point-like particles is intended as a preliminary, simplified version of a more complete account of relational dynamics. Barbour deals with classical matter fields. However, in its latest developments, the formalism of matter fields treats particles as local excitations of the corresponding quantum fields (see Zee, 2010). This aspect, which is not present in Barbour’s treatment, signals a serious flaw of the model. Indeed, there is a specific aspect in quantum field theory which would hardly coexist with shape dynamics, namely the fact that the field operator does not commute with the particle-number. Barbour’s shape dynamics takes as a starting point a system of N, identifiable point-particles. This means that there are two conditions which undermine the tenability of this premise. The first is the just mentioned non-conservativeness of the number of particles within the context of quantum field theory. The second is the possible existence of an infinite universe. As Barbour admits (2010), “the Machian view point is only possible if the universe is a closed dynamical system”.

58 These two examples are taken from Butterfield (2002, p. 309). 59 This is connected with the broader introduction of the formalism of fields. While in the Newtonian model it is assumed that the gravitational force is directly acting between distant objects, late in the 19th century studies on electromagnetism eventually undermined this assumption. Faradays’ hypothesis, later formal- ized in Maxwell’s equations, is that of introducing a real entity spread in space, which modifies and at the same time is modified by the presence of electromagnetic objects. This entity, which defines local properties, is currently described as the electromagnetic field. Faraday conceived of it as a sort of meshwork formed by thin line bundles, which he called “lines of force”. The impact of such a change of perspective is so remarkable that it plays a big role in the formulation of both Quantum Field Theory (QFT) and GR. The idea is that of exploiting the concept of field and to apply it to all known forces. QFT is a theoretical framework which combines quantum mechanics, special relativity and classical field theory to describe the properties of particles and quasiparticles in terms of excitations of underlying fields. It is a physical model which is perfectly suitable to describe atomic and subatomic structures. On the other hand, GR main achievement is that of applying the concept of field to the gravitational force.

63 Finally, I want to tackle Barbour’s thought-provoking dismissal of the notion of time as such to critically assess how his approach deals with temporal becoming. In order to address this aspect, let me recall a few key concepts of Barbour’s proposal. Shape dynamics identifies as the ultimate arena of reality the totality of all the relative configurations of a system of N, finite and identifiable, point-like particles. This arena is what Barbour calls Platonia. It is the timeless landscape of all the possible arrangements of particles whose position is given once and for all. Nothing changes in Platonia. All the configurations simply occupy permanent, fixed collocations. It is the arena that, according to Barbour (1999, p. 45), should replace Newtonian space and time. 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. Agreed, sets of configurations can be mathematically defined. Agreed, there are continuous curves obeying some variational principle. However, no actual physical history holds (Butterfield, 2002, p. 317). According to Butterfield (2002), this aspect of Barbour’s theoretical framework somewhat reminds a type of modal realism à la David Lewis (1986). The idea is that all possible paths or courses of history are all equally real, none of them being favoured (in terms of actuality) vis-à-vis others. To be more precise, Butterfield explains, Barbour’s position is a combination of realism and (what Butterfield dubs) Spontaneity. The latter term, whose most prominent formulation is attributable to John S. Bell (1981), arises in the context of Everettian interpretations of quantum theory. While examining Spontaneity in its fullest form is unnecessary, what counts here is the key concepts it implies. Its main tenet is that, “unbeknownst to us, the actual history jumps between disparate instantaneous states” (Butterfield, 2002, p. 313). In order to understand Spontaneity, suppose we are given a set of possible paths. We tend to conceive them as real, actual and continuous. This means that we “think of a possible history as a sequence of instantaneous states of the world (in metaphysics) or of the system (in physics)” (ibid., p. 313). The impression we get from our direct experience of the world is that of a continuous unfolding of events, not of a sequence of discrete jumps. Spontaneity directly attacks this point and argues vice versa that all possible paths (including the actual one) are by no means continuous.

64 Rather, the unfolding of events proceeds in terms of sudden, arbitrary jumps in the space of instantaneous states. This entails that human impression of continuity is just the effect of correlations between memories and records defined at an instant, which topologically may be tremendously distant60. To sum up, shape dynamics implies that all relative configurations are actual, and that specific curves or paths unfold via arbitrarily chosen instantaneous jumps. At this stage, one question naturally springs to mind: How is it possible, from “a scheme without the vestige of time” (Barbour, 1999, p. 46), to recover the temporal evolution of systems and eventually human awareness of time? In the previous chapter, I defended the idea that temporal becoming is not just the successive occurrence of events but rather the coming-into-being of extended events, which I called processual becoming. Based on my reconstruction above, it is evident that Barbour goes down the opposite road. He defends a particularly strong version of the block-universe, according to which every configuration is frozen in its own permanent position. Coherently, he discards the notion of events as appertaining to primitive ontology. I shall now continue my analysis with Barbour’s own account. He recognizes that shape dynamics needs to deal with the problem of temporal experience and that an explanation is needed for the impression that time passes. Barbour (1999, pp. 27-28) cursorily refers to Boltzmann’s suggestion that conscious beings could emerge on either side of a point of lowest entropy. Time does not possess a direction. It can at best be conceived as a line, of which the conscious mind only explores partial segments. While Newtonian time is an ideal line with a fixed direction, Boltzmann disposes of the direction but keeps the line. Here comes the most crucial point. Barbour suggests that it might well be the case that we also need to discard the line. He advocates a retentional model of temporal experience whereby the brain collects several snapshots within its specious present and thus produces the impression of motion. Two phases cause temporal experience. First, several successive positions of the external environment are stored in each instantaneous brain pattern. Second, the impression of motion is generated by the co-presence of the stored information within the same specious present. It is an illusion produced by the

60 One of the things that make Barbour’s record theory less palatable is that it lacks a “concrete mathematical model evidence for there being any correlation between […] time capsules and their being near a dis- tinctive feature of configuration space such as a change of stratum or a point of great uniformity” (Ander- son, 2009, p. 662).

65 “juxtaposition of several subpatterns within one pattern” (Barbour, 1999, p. 30). Time is somehow encoded in each of these subpatterns. Barbour calls the temporal stratifications within patterns time capsules. By this term, he refers to any fixed arrangement encoding the impression of motion, change or history. Barbour (1999, p. 31) writes that “any static configuration that appears to contain mutually consistent records of processes61 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, p. 32). In other words, the universe is a network of time capsules within time capsules. A human body represents a time capsule. Its sub-components form a different time capsule. A broader group to which that human body belongs displays other time capsules. Time capsules are thus distributed along a variety of scales. To summarize, Barbour views the universe as composed of static configurations that do not unfold over time, hence 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 time is thus contingent upon human representation of a static set of non-uniquely ordered sequence of states. Human impression of a coherent and unfolding sequence of events emerges within specific subset 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. More precisely, if Barbour combines a form of modal realism with Spontaneity, then 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

61 Oddly enough, Barbour employs the term “process” which, strictly speaking, should not figure in his conceptual and lexical framework.

66 paths get higher probability amplitudes than others. Despite this, I do not see any plausible reason for such an ad hoc condition62. In a recent paper, Sam Baron, Peter Evans and Kristie Miller (2018) tackle the issue of how and to what extent Barbour’s relational formulation of GR and QG can be conceived as genuinely timeless. For that to be the case, it is vital to define which notion of time Barbour is suggesting not to appertain to the fundamental structure of the universe. For while he claims that human experience of motion and change are but illusionary phenomena, he never directly addresses which features of time are currently lacking in his theoretical framework. As we have discussed in the previous Chapters, the notion of time can be defined from a variety of standpoints. According to Rovelli (1995; 2004) various attributes can be contextually accorded to this term. In GR, the two properties which can be referred to time are linearity (a one-dimensional ordered sequence of events) and met- ricity (a proper measure of distance between events). While Barbour avers that the ultimate constituents of the universe are three-dimensional relative configurations, he nevertheless provides a meaningful temporal distance between any two points of the configuration space, and thus defines an ordering sequence of events along a geodesic. In this regard, Barbour’s Machian reformulation of GR is consistent with temporality: “The relevant features exist, it is just that they emerge out of the three dimensional points in the relative configuration space via this specific best-matching algorithm” (Baron, Evans and Miller, 2018, pp. 44-45). Time turns out to be an ordering principle among three-dimensional spatial configurations. Their conclusion is that Barbour’s classical account of dynamics cannot eventually be regarded as a timeless one63. Quite the contrary, though not figuring in the fundamental ingredients of reality, time can still be accounted for via the implementation of the best-matching procedure. Finally, there is at least one last detail of Barbour’s proposal that deserves mention in the present section, namely whether or not his account of temporal experience, on the one hand, and the underlying static structure of reality, on the other, are mutually consistent with one another. In the first Chapter I explored the idea of temporal perception

62 For a clear discussion on analogous ambiguities of shape dynamics in the field of quantum gravity see Butterfield (2002, pp. 325-328). 63 I cannot further expand on Barbour’s Machian formulation of QG in the present context. Suffice it to say that, in that respect, his account seems to hold up to scrutiny, so much so that “if there is any decent chance that Barbour’s interpretation of quantum gravity is correct, it is startling indeed to realise that this would entail that ours is a timeless world” (Baron, Evans, Miller, 2018, p. 58).

67 being conceivable as a process, rather than a series of perceptual states (or snapshots of reality), each of which depicting an instantaneous configuration of the world64. As I tried to argue, processing cannot be squared with adynamism. Indeed, Barbour (1999, pp. 28- 29) states that “the brain cannot process data instantaneously, and it is known that the processing involves transmission of data backward and forwards in the brain”. So, I wonder how this account of temporal experience can be realigned with the physical structure advocated by Barbour. It is true that human consciousness is a particularly insidious subject of inquiry, so maybe this is not anything that a physical theory can provide on its own. Still, one would expect a more stringent consistency of such a radical theoretical model. Not only does Barbour’s shape dynamics repudiate the existence of a uniquely ordered sequence of events65; it also disposes of their processual nature. It is then unclear where, provided this background, the processual nature of temporal experience should arise from. In other words, if the ultimate arena of reality is the ensemble of all the relative, static configurations of point-like particles, why should conscious beings, in their embodied instantaneous collocation, elaborate such a nuanced though deceptive temporal experience?

3.5. Barbour’s relationalism: becomingless but not timeless

The reason lying behind my analysis of Barbour’s shape dynamics was the need to find a timeless approach that would be prepared to face the consequences of a universe without becoming. As I argued in Chapter 1, a limit that affects most block universe views is that they endorse the principle of the relativity of simultaneity and yet aim to defend local becoming. As pointed out by critics like Oliver Pooley and John Earman or supporters of the static block like Michael Silberstein, the adynamism of fundamental physics acquires dynamism at lower levels of reality without there being any sound justification for the leap from one level to the other. This unresolved tension between adynamism and dynamism eventuates in an ambiguous conceptualization of events and a thin notion of becoming.

64 For a recent discussion on the preferability of a version of the extensional model for the experienced present, see e.g. Dorato and Wittmann (2019). 65 Where events are conceived as instantaneous, not discrete.

68 As I noted there, Barbour (1999, p. 143) himself gnaws at this flaw: “The block universe picture is in fact close to my own, but the idea that Nows have no role at all to play in physics, and must be replaced by point-like events, would destroy my programme”. He clearly identifies the modellization itself of point-like events as the source of problems in the block view and proposes to replace it with that of configurations:

Now, what is Minkowski space-time made of? The standard answer is events, the points of four-dimensional space-time. But there is an alternative possibility in which three-dimensional configurations of extended matter are identified as the building blocks of space-time. The point is that the three-dimensional hyperplanes of relative simultaneity are vitally important structural features of Minkowski space-time. It is an important truth that special relativity is about the existence of distinguished frames of reference. And an essential fact about them is that they are “painted” onto simultaneity hyperplanes. As a consequence, simultaneity hyperplanes, which are Nows as I define them, are the very basis of the theory (ibid., p. 143).

Barbour’s peculiar type of relationalism overcomes the limits of the block view by fully embracing the idea of a world without motion, change, passage, and the related concepts of events and becoming. As both Edward Anderson (2017) and Flavio Mercati (2018, p. 5) explain in their sympathetic accounts, shape dynamics is pivoted on two basic kine- matical principles. First, spatial relationalism does not ascribe any absolute properties to space as it posits that the positions and sizes of objects are defined relative to each other. Second, temporal relationalism implies that the universe as a whole does not feature any absolute time. The static nature of shape dynamics derives from the combination of these two principles and gets glued to the absence of becoming in a congruent manner: time is implemented by actions which are free of extraneous time-like quantities and are calculated in terms of an arbitrarily chosen parametrization, thereby resulting reparametriza- tion invariant. This combination, according to Anderson, lays the foundations for what can sensi- bly be called “configurational relationalism” (see Anderson, 2017, pp. 125-129) – one that considers a continuous group of transformations (the Lie group) acting on the configuration of the system as physically irrelevant. The best matching is what implements this relationalism, as it is a minimization procedure establishing the least incongruence between adjacent physical configurations. In other words, it excludes any extraneous

69 configurational structures. Time-parametrization makes sure that there is no fixed background. Based on the above, as I anticipated in the Introduction, Barbour’s configurational relationalism successfully erases all references to events and becoming. More correctly, the disappearance of becoming makes the notion of events physically irrelevant. In this regard, Anderson (2009; 2017) comments that shape dynamics entirely replaces the “becoming” lexicon with the “being” lexicon. The record theory enshrined into shape dynamics qualifies the passage from one instant to another as illusory66. Past instants are in no way connected to present ones. All that happens is that past instants play out as memories or traces in the present instant. Memories are the only correlation between instants, so much so that it is meaningless to speak of a sequence of instants. All that can be said is that present instants contain memories or traces of other instants. As I illustrated above, Barbour’s term of art to render this totally adynamical and becomingless scenario is time capsules, understood as configurations from which a semblance of dynamics and a semblance of becoming can arise (Anderson 2009; 2017). But what are the consequences of a fully becomingless world? A twofold striking outcome is that time reappears under the (dis)guise of a radical theory of objects and a global preferred parametrization. Above I mentioned Butterfield’s (2002) and Baron, Evans and Miller’s (2018) cri- tiques stating that, at least in Barbour’s classical reformulation of Newtonian mechanics, time does not really vanish. For Butterfield, Barbour’s is a blend of Spontaneity and modal realism. For Baron, Evans and Miller, time figures as an ordering principle among three-dimensional spatial configurations. However, there is a more compelling critique that spots the presence of time in the way Barbour is forced to conceptualize the existence of objects through time once becoming has been dismissed. Dean Rickles (2008, pp. 157- 160) rightly points his finger at the endurance of objects in the physical world as it is portrayed by Barbour. Let us look at the latter’s own reasoning about this. While talking about the illusory nature of motion, he 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

66 Anderson notes that Barbour’s is just one of the various record theories, which he classifies and discusses. See Anderson, 2017, pp. 343-346.

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.

This is quite consistent with a becomingless world: as becoming vanishes, so does the idea of change over time in the sense of the differences between successive temporal parts of an object. Based on this, Barbour’s rejection of becoming boils down to a denial of persistence. As Rickles (2008, p. 159) has it, Barbour’s view

is also perfectly compatible with a kind of temporal parts type theory. However, rather than the structure of time being linear, it is non-linear (as encoded in the relative configuration space) and the “temporal evolution” is probabilistic (governed by a solution to the Hamilto- nian constraint). We see that the parts themselves do not change or endure and they cannot perdure since they are three-dimensional items and the parts occupying distinct 3-spaces (and, indeed, the 3-spaces themselves) are not genidentical; rather, the quantum state “jumps” around from Now to Now in accordance with the Hamiltonian constraint in such a way that the parts contain records that ‘appear’ to tell a story of linear evolution and persistence.

A becomingless denial of persistence, however, should not be confused with the absence of time. On the contrary, as I demonstrated at the end of sec. 3.3, shape dynamics goes so far as to reintroduce a global temporal parameter as a distillation of all the changes in the universe. It is based on this consideration that Lee Smolin takes shape dynamics to be the most convincing path to abandoning the principle of relativity of simultaneity. Smolin (2013, p. 170) says that the universe can be described with the language of general relativity, where the definition of time is arbitrary and time in se is relative. An alternative that he considers more valuable is to describe the universe in the language of shape dynamics, where a universal notion of time comes to the surface. Along with Roberto Unger

71 (Smolin and Unger, 2015), Smolin explains that Barbour’s model permits a preferred choice of time coordinate which comes with the admissibility of the simultaneity of distant events. As I have tried to demonstrate between Chapter 1 and this chapter, it is precisely here that shape dynamics breaks with the metaphysics of the block universe – which I showed to derive from a modellization of the relativity of simultaneity. Clearly, the global temporal parametrization that Barbour derives from the configuration space is relational through and through, and is abstracted from the dynamics of the universe as a whole. This undermines the physical relevance of locality, as time does not need a local observer to be determined. Shape dynamics allows reformulating GR as a theory with a preferred dynamically determined global slicing. For GR is “defined on a fixed three-surface which evolves in a global time coordinate. This formulation […] shares with general relativity diffeomorphism invariance on the three-dimensional spacelike surfaces but replaces the many-fingered time invariance with a new local gauge invariance which is invariance under local three-dimensional conformal transformations” (Smolin and Unger, 2015, p. 420). Of course, Barbour is far away from the conclusions reached by Smolin, as the latter claims shape dynamics lays the foundation for a processual view of the physics that puts at its heart becoming rather than being. Nonetheless, Smolin makes a nice point when he foregrounds the notion of time that re-emerges in shape dynamics, one that yields a relational theory of spacetime pivoted on a global time measurable by taking into account the whole universe.

3.6. Concluding remarks

This chapter has centred on a critical assessment of Barbour’s reformulation of Newto- nian mechanics. This theoretical approach, whose main objective is to develop a fully background-independent account of dynamics, has been analysed from a variety of standpoints relevant to the present work, that is, its premises, its inner consistency, its implications for temporal becoming. Needless to say, a proper appraisal of the whole philosophical relevance of shape dynamics and its more general aspiration to break a path in contemporary physics could not be undertaken within the limited scope of my

72 investigation, as it would require an exhaustive analysis of its application in the areas of quantum theory and quantum gravity. What I tried to do was to bring to the surface the theoretical import of a type of relationalism that promotes configurations as that which is in relation and sets aside other notions that create frictions within the block view (such as point-like events and local becoming). I sought to foreground the merits as well as the less palatable features of such a radical theory. My review was key to showing that the exclusion of becoming does not in itself entail the exclusion of time and that therefore time and becoming are to be treated as non-interchangeable physical concepts. Pace Barbour, shape dynamics, I maintained, is becomingless but not timeless, and this is exactly due to the relationalism of configurations. The task of the subsequent Chapter will be to justify a type of relationalism that comes about from a proper interpretation of the principle of general covariance and that, despite this, is fertile soil for a defensible notion of becoming.

73 Chapter 4. General covariance and Quantum Gravity: a discussion

4.1. Introduction

In the previous chapter, I offered a concise account of Julian Barbour’s shape dynamics with a view to exploring its consequences on the physical notions of time and temporal becoming. The main conclusion was that shape dynamics profitably disposes of temporal becoming: the ultimate arena of reality is the configuration space of all the possible arrangements, whose collocation is given once and for all. Nothing changes, nothing becomes. And yet, Barbour’s theoretical model makes room for time as a global parameter for slicing the configuration space, so much so that time and temporal becoming are to be conceived as distinct phenomena. Barbour’s is certainly not the only physical theory that undermines or at least deeply challenges these physical notions. In this Chapter I want to concentrate on a context that, though endorsing relationalism and timelessness, offers an utterly opposite view of becoming. Current research within theoretical physics shows that different energy regimes result in significant behavioural discrepancies. Correlations between distinct domains are thus under exploration whenever regime transitions occur (see e.g. Crowther, 2018; Oriti, forthcoming). Quantum Field Theory (QFT) and General Relativity (GR), which are the most fundamental theories we currently have, turn out to be effective field theories (EFTs)67. They present restricted domains of prediction, out of which divergence arises. As a consequence, EFTs have led to a remarkable reconfiguration of the relationship between theories. Some authors, such as Cao and Schweber (1993, p. 69), claim that the EFT program supports “a pluralism in theoretical ontology, antifoundationalism in epistemology and an antireductionism in methodology”. Others, such as Hartmann (2001) and Castellani (2002), while recognizing the relevance of the EFT program, do not believe that this theoretical approach definitely rules out the possibility of an underlying, unifying theory. Following this line of thought, while a map of theories in terms of

67 For a proper characterization of the notion of EFT see e.g. Castellani (2002). As she explains, an effective theory (ET) is a theory which has a certain domain of validity. Drawing from Georgi’s (1997) analysis, Castellani conceives an ET as “an appropriate description of the important (relevant) physics in a given region of the parameter space of the physical world” (Castellani, 2002, p. 260). As physics displays highly differentiated behaviour at different energy scales, ETs are intrinsically approximate. In this article, Cas- tellani explores the relevance of EFTs in the context of particle physics.

74 applicability domains can be traced, the quest for a more fundamental theory of reality is still desirable among various scholars. One of the most promising and yet challenging branches of theoretical physics, currently working on possible intersections between QFT and GR, is Quantum Gravity (QG). This theory is intended to overcome both QFT and GR’s incompleteness68. Though several approaches have been developed within QG, there is general agreement about its general definition: it is a theory about the microscopic structure of spacetime. The main disagreement lies in the methodology from which such a theory can be developed (Hu, 2009). For the sake of simplicity, three main approaches can be singled out. The first holds that QG results from a quantization of GR, a highly prosperous theory for the macroscopic, classical structure of spacetime – the most promising representative being Loop Quantum Gravity (LPQ). The second draws from quantum theories of matter and aims at extending them to make sense of the gravitational field and their mutual interactions – the most important representative being String Theory (ST). Finally, the third one claims that classical spacetime is emergent from a more complex, behaviourally dissimilar, background – causal dynamical triangulations (CDT) being one of the most successful repre- sentatives. In this Chapter I will focus on the first approach, which is generally referred to as canonical QG. While it is not for the present work to offer a full reconstruction of the various and at time incompatible approaches in canonical QG, my purpose here is to concentrate on a specific theoretical issue, which has proven crucial in the road towards QG. This concern, which arises in the context of GR, aims at clarifying, from an interpretative point of view, the role and significance of the principle of general covariance. While this principle is strictly connected with Barbour’s requirement of a purely relational account of dynamics, I will show that general covariance can lead to an interesting interpretation of dynamics in terms of processual becoming. Section 2 explores the key elements of the principle of general covariance and the connection of this principle with gauge theories. Section 3 is devoted to an interpretation of GR in terms of a gauge theory. Section 4 concentrates on the role of gauge theories

68 QFT offers an account of all known fundamental forces, except gravity. The latter is described via the classical theory of GR. This entails that both theories discard certain aspects of physical reality. According to GR, spacetime is not a fixed background. Instead, it needs to be conceived as a dynamical entity which interacts with matter fields. On the other hand, QFT describes the intertwining of fundamental fields on a static background spacetime.

75 and the concept of gauge invariance in the context of QG. In this respect, three main alternatives will be outlined as a consequence of different interpretations of the principle of general covariance, namely the Evolving Constants approach, the Constant Mean Cur- vature approach and the Internal Time approach. Section 5 will expand on Carlo Rovelli’s Evolving Constant approach with a specific focus on the conceptual consequences of this theoretical model. In particular, I will mainly address whether or not this approach opens up to a viable notion of processual becoming.

4.2. The principle of general covariance and gauge theories

The main tenet of the principle of general covariance, which arises in the context of GR, is that the physical laws need to be written in a covariant form under arbitrary coordinate transformations. A theory is expressed in a covariant form if there is a function that connects different descriptions of the same physical system in terms of the relevant parameters of the theory (see e.g. Stachel, 2014). In this respect, John Earman (2006) distinguishes two notions of general covariance, namely formal and substantive. A spacetime theory fulfills formal general covariance (FGC) if its physical laws are valid in every coordinate system, provided that they are valid in one of them. Drawing from Kretschmann’s (1917) analysis on general covariance, the formal version of general covariance bears no real physical content. Rather, it is a certain type of formulation which can be applied to any physical spacetime theory (such as Newtonian spacetime theory or STR): “This is a condition of the well-formedness of a theory, not on its content” (Ear- man, 2006, p. 4). On the other hand, a spacetime theory satisfies the substantive requirement of general covariance if its laws are diffeomorphism invariant. A theory is diffeomorphism invariant if its physical laws are invariant under arbitrary coordinate transformations. In this regard, Karel Kuchar (1988) states that the content of the general covariance of GR does not consist in the fact that it can be expressed in a generally covariant formulation (like every other physical spacetime theory). Rather, the point is that GR ought to be written in a generally covariant fashion. For the very content of the theory resides precisely in that it does not depend on the existence of a preferred coordinate system or geometrical structure.

76 This aspect is crucial insofar as it shows that misunderstanding the physical content of a certain theory could lead to incorrect formulations of new theories. Empirically equivalent formulations of a certain theory can produce novel inequivalent theories, if one does not adopt the correct formalism: “This link between content and method is the source of the sentiment – which is widespread among physicists working on canonical quantum gravity – that there is a tight connection between the interpretative problems of general relativity and the technical and conceptual problems of quantum gravity” (Belot and Earman, 2004, p. 214). In this respect, Belot and Earman (2004) offer a robust discussion of the connection between the general covariance of GR and gauge theories. This connection is valuable insofar as different interpretations of this relation result in diverse quantization in canonical QG. In addition, it has a remarkable impact on the nature of physical observables, as well as the existence of a temporal parameter in quantum theory (QT). This reconstruction calls for a preliminary inspection into Hamiltonian and gauge systems. As discussed in the previous Chapter, there are several ways to account for the dynamical evolution of classical systems. One of the most widespread procedures is to represent the configuration space, encoding all the physically possible states of the system under scrutiny, and to single out a set of curves in this space which corresponds to dynamically possible trajectories of the system. In the previous Chapter I sketched the La- grangian formalism within the configuration space. Belot and Earman (2004, pp. 215- 220) explore two other implementations, namely the Hamiltonian formalism and the gauge-theoretic formalism – where the latter consists but in a modest generalization of the former. The Hamiltonian formalism corresponds to a variety of classical physical systems. Hamiltonian systems consist of a triplet of mathematical objects (M, �, H), where M is a manifold69, � is a tensor, called a symplectic form, which endows the manifold M with a geometric structure, and H is the Hamiltonian70 of the system, which defines the

69 In mathematics, a manifold is a topological space which corresponds to a collection of points composing a certain kind of set. 70 In analytical mechanics, the Hamiltonian corresponds to the total energy of the system under scrutiny, and is given by � = � + �. In the previous chapter, we have seen that, in the Lagrangian formalism, the Lagrangian � = � − � can be used to define the dynamical trajectories of an isolated system (one for which the total energy is conserved). This is performed via the application of the principle of least action. For generalized systems, viz. for non-isolated systems, the Lagrangian formalism needs to be replaced by the Hamiltonian formalism.

77 dynamical trajectories over the manifold. Typically, in the context of classical mechanics, one starts with the configuration space and defines the phase space. The latter is the space of all the possible positions and momenta (respectively termed as “q” and “p”) of a system of N point-like particles. The Hamiltonian H of a point in the phase space (q, p) defines the energy of a system with that position and momentum. The dynamical trajectory of the system under scrutiny is defined via a variational principle, analogous to the one I mentioned in the previous Chapter for the Lagrangian formalism. Given two positions in the phase space, the principle of least action71 selects the unique trajectory for which the action is minimized, this trajectory corresponding to the dynamical physical one. The gauge-theoretic formalism represents an interesting generalization of the Ham- iltonian formalism. A gauge theory is a theory that has mathematical parameters or degrees of freedom that can be arbitrarily changed without affecting the predictions of the theory, viz. a theory with redundancies (Mercati, 2018, p. 6). A gauge freedom or a gauge symmetry can vary arbitrarily without affecting the relevant aspects of the theory72. In order to implement it, one starts by relaxing one of the conditions imposed upon symplectic forms. This procedure brings to a more general class of geometries, namely the presymplectic geometries, which define the phase space of gauge theories. In this case, the manifold is divided into submanifolds, each corresponding to a foliation with some fixed dimension. Each point of the manifold denotes a submanifold. These submanifolds are called the gauge orbits. We define as a diffeomorphism a gauge transformation which preserves the gauge orbit. A function on the manifold is gauge-invariant if it is constant on each gauge orbit. The Hamiltonian, which is a gauge-invariant function on the manifold, defines, together with the manifold and the presymplectic form, the gauge system. Again, the dynamical trajectories generated by the Hamiltonian can be investigated. However, while in the case of Hamiltonian systems the dynamical trajectory follows uniquely, in the case of gauge-theoretic systems there are infinitely many trajectories through each point of the manifold. The only thing that the infinite class of dynamically possible trajectories through a given point share is the gauge orbit in which the common

71 In the case of the Hamiltonian formalism, the principle of least action is defined as � = ∫ � ��, where t1 and t2 corresponds to the initial and final temporal coordinates of the system, and L is the Lagrangian. 72A gauge symmetry is present in those theories whose equations of motion are derivable from an action principle. An action principle comes equipped with a variational symmetry group, namely a group which leaves the action invariant up to a divergence term.

78 point lies. Thus, although no dynamical trajectory is defined in a unique or ordered manner, if a gauge-invariant function, call it f, on the manifold is given in terms of an initial value, this function is constant all over the manifold. This means that, for any two dynamical trajectories �(�) and �′(�), which share the same origin �(0) = � (0) = �, it follows that ��(�) = ��(�), ∀ �: “Thus specifying the initial state of the system completely determines the past and future values of any gauge-invariant quantity” (Belot and Earman, 2004, p. 218). The most interesting types of gauge systems arise as constrained Hamiltonian systems, which are obtained through a set of real-valued vanishing functions on M. Such functions are called constraints73. What is important to notice in the present section is that gauge-invariant functions commute with all the constraints, meaning that they are constant over gauge transformations. Gauge-invariant functions are thus background independent. After this concise introduction to both Hamiltonian and gauge formalisms, it is now necessary to expand on their relative interpretations. As will become clear in the following sections, this preliminary analysis is a crucial step towards a proper evaluation of the correct quantization in canonical QG. Given a certain Hamiltonian system, defined by the triplet (M, �, H), the physical interpretation reads that this formalism fixes a one-to-one correspondence between dynamically possible states with the points of the manifold M. Typically, in the classical context, M corresponds to the phase space, viz. the space of possible configurations (both in terms of position and momentum) of a set of particles or fields relative to some inertial frame. As a consequence, Hamiltonian systems lead to a uniquely determined dynamical evolution. Quite the contrary, the interpretation of gauge systems is not trivial at all. As already discussed in the previous Chapter, the introduction of a background independent account of dynamics has to pay a cost for the dismissal of redundant metaphysical structures. In particular, no dynamical trajectory follows uniquely from an initial state in the configuration space. As a consequence, there are two viable interpretations of the gauge-theoretic formalism. The first (the literal interpretation) argues that the gauge-theoretic approach, though essentially correct, needs to be “supplemented with an account of measurement

73 As Belot and Earman (2004, p. 219) explain, there are two types of constraints, namely first-class and second-class constraints. The first-class constraints are the one which commute with all of the constraints and that generate gauge transformations.

79 which will ensure that the predictions derivable from our gauge theory are perfectly determinate” (Belot and Earman, 2004, pp. 220-221). In this case, one needs to explain how this approach, despite its indeterminism, can make determinate predictions. The idea is to state that some physically real quantities are not measurable. To make a proper prediction, physical quantities with well-posed initial values are needed. A function in the phase-space displays a well-posed initial value if it is gauge-invariant. In this respect, only gauge-invariant quantities should be accorded measurability. This grants prediction. The second (the gauge-invariant interpretation) states that the formalism of this type of gauge theory contains surplus structure, which must be eliminated for a physically viable understanding of the theory. On this view a formal move, namely a reduction procedure, is applied to the manifold. While it is not possible to further expand on the technical aspects of this second interpretation, suffice it to say that in this case attention is devoted to gauge orbits, rather than points. We thus build a subset of the original manifold, the so-called reduced phase space, whose points are the gauge orbits of the original manifold: “Giving a gauge-invariant interpretation of the original gauge theory is the same thing as giving a literal interpretation of the reduced phase space” (Belot and Ear- man, 2004, p. 222). The difference between the Hamiltonian and the gauge-theoretic formalism lies in the fact that, while the former establishes a uniquely determined dynamical trajectory, the latter is not sufficient to determine the evolution of its variables. In classical physics, one typically regards this aspect as a consequence of the inclusion of redundant variables. On this view, it is recommended to favour interpretations which only include the variables whose evolution is provided by the equations of motion as physically real quantities. The point is thus to find a suitable set of physically sound gauge-invariant quantities for GR. Before I get on to the discussion of the quantization technique of gauge theories, let me emphasize the reasons lying behind the present analysis. The conceptual incompatibility between GR and QFT is generally regarded as one of the most compelling reasons for the development of a theory of QG (see e.g. Hedrich, 2010). On the one hand, QFT combines quantum mechanics, special relativity and classical field theory. It provides a theoretical framework for the description of the properties of particles and quasiparticles in terms of excitation of the underlying fields. It is a physical model which is perfectly suitable to describe atomic and subatomic structures. However, the gravitational force is

80 not included in the model. On the other hand, GR is a very successful classical theory for the macroscopic description of the gravitational force. It depicts a continuous, rather than discrete spacetime structure. As the other fundamental forces (electromagnetic, strong interaction, weak interaction) are defined as discrete quantum fields, one would expect that, at the micro level, this should apply also to the gravitational force. This is why the type of quantization that is to be applied is crucial to a proper characterization of QG. In the case of canonical QG, one starts with GR and looks for the most adequate quantization technique. In order to perform a quantization of Hamiltonian systems, classical observables have to be replaced by corresponding operators in the Hilbert space. The question is thus how this is applied in the case of the gauge-theoretic formalism. There are two viable routes. The first starts with the reduced phase space and apply canonical technique to the resulting Hamiltonian system. However, this is generally unattainable, as the structure of the reduced phase space is very difficult to define. The second strategy is to apply a quantization of the gauge systems implementing a technique first explored by Dirac (1964). Suppose we have a gauge system defined by a certain class of constraints. Then one defines a set of coordinates on M and looks for corresponding operators satisfying Heisen- berg’s principle of indetermination. Next, one looks for a quantum analogue (viz. an operator) of the classical constraint and imposes the quantum constraint to construct the space of physical states. This procedure warrants that the quantum states are gauge invariant.

4.3. The gauge-theoretic formulation of GR and its conceptual interpretation

One of the thorniest aspects of GR, at least among philosophers of physics, is whether or not its core commitments favour substantivalism or relationalism. This theory starts with the definition of a manifold M as a collection of points possessing certain topological properties (i.e. four-dimensionality and continuity) and considers some additional structures, namely a metric field (the metric tensor “g”) and a matter field (the stress-energy tensor “T”). GR then establishes a correlation between the metric structure and the matter distribution, and this seems to imply that GR renders substantivalism hardly avoidable.

81 However, starting by the end of the 20th century, some problematic aspects of a substantivalist account of GR started to emerge (see e.g. Earman and Norton, 1987; Stachel, 1993; Norton, 2008). The so-called hole argument shows the ultimate consequences of a substantivalist interpretation of GR. In its standard formulation (Earman and Norton, 1987) this argument is structured in the following way. Take a model (�, g) of GR, and let �: � → � be a diffeomorphism. The general covariance of the theory implies that � = (�, �∗�) is also a model for GR. This simply means that, provided a diffeomorphism on the manifold M, there must be an equivalent formulation of the theory under coordinate transformations, viz. the theory must be diffeomorphism invariant. If one views � and � as corresponding to distinct physically possible worlds, she is also committed to regard GR as an indeterministic theory. Indeed, substantivalists claim that � and � refer to distinct states of affairs74, so they are committed to the claim that GR does not provide a deterministic evolution of physical systems. The hole argument can thus be viewed as an interpretative issue analogue to the one arising in the context of a literal vs gauge-invariant interpretation of gauge theories. According to Stachel (2014), the ultimate significance of the hole argument is that any viable fundamental theory should be background independent. In order to outline the strict connection between a literal interpretation of GR and its subsequent indeterministic nature, one starts by defining GR in terms of a gauge theory. As Belot and Earman (2004, p. 223) expound, one defines a gauge-theoretic formulation of GR by “considering how to represent an instantaneous state of a general relativistic world”. To this purpose, one takes a manifold, provided with a diffeomorphism and considers a globally hyperbolic vacuum solution of the Einstein field equations75. A procedure, whose technical content cannot be further expanded in this context, can be implemented to describe GR as a constrained Hamiltonian system. This means that one requires a set of constraints, in this case a scalar and a vectorial constraint. Each constraint corresponds to an equation which apply to all the points of the manifold76. Dynamical

74 The fact that, in a substantivalist perspective, � and � correspond to different physical states of affairs is due to the fact that these two models relate differently with the (alleged) substantival space. 75 This procedure is performed in order to calculate the geometry that the vacuum (� = 0) solutions induce on the manifold. 76 The notion of constraint encodes the dynamical state of the gravitational field at a given time. Basically, a constraint defines, among the viable trajectories in the configuration space, the ones that correspond to the effective dynamics of the system under scrutiny. They are conditions that viable solutions must satisfy.

82 trajectories correspond to single gauge orbits and thus encode a different temporal parametrization. A correspondence map between two diverse dynamical trajectories can be obtained with a gauge transformation. The trajectories corresponding to the models � and � (the ones which figure in the hole argument) are so related, and “this shows that our approach respects the general covariance of general relativity in the sense that it is indifferent to changes of foliation, time function, and identification map. Changing any of these simply carries us from one dynamical trajectory to a gauge-related one” (Belot and Earman, 2004, pp. 225-226). Again, this implies that, according to a substantivalist interpretation of GR, diverse models correspond to distinct physical states of affairs. In fact, “substantivalists are committed to a literal construal of the gauge-theoretic formulation of general relativity. And, like any literal interpretation of a gauge theory, substantivalism implies that the theory is indeterministic” (Belot and Earman, 2004, p. 228). The only way to avoid the indeterminism of the hole argument resides in accepting that diffeomorphic models represent the same physically possible situation. A relationalist account rejects the idea that two possible physical systems with the same geometric structure (though differently related to existent spacetime points) correspond to distinct physical situations. Thus, if one aims to give a deterministic interpretation of GR, a relational account of dynamics has to be preferred vis-à-vis a substantivalist one. In this case, only gauge invariant quantities are accorded physical meaning. The existence of gauge degrees of freedom shows that the theory contains surplus variables. However, it is still controversial how to obtain the reduced phase space of GR77, so much so that “until some progress is made on these technical questions, a dark cloud hangs over the programme of providing gauge-invariant interpretations of general relativity” (Belot and Earman, 2004, p. 229). In addition, the gauge-invariant interpretation of GR seems to imply that change does not exist, for any two points of the manifold are connected via a gauge transformation and thus correspond to the same physical state of affairs. In the subsequent section I will expand on this latter issue.

77 Remember that, in order to provide a gauge-invariant interpretation of the principle of general covariance, one needs to start with the configuration space and define the so-called reduced phase space, whose points are the gauge orbits of the original manifold. A gauge-invariant approach to general covariance is thus equivalent to a literal interpretation of the reduced phase space.

83 4.4. Gauge invariance and Quantum Gravity

One of the main questions arising from an in-depth analysis of the principle of general covariance is whether or not it allows any tenable notion of change. If general covariance is understood as a principle of gauge invariance, then it looks as if GR is inadequate to accommodate any form of change. The possible lack of change would eventually raise vital concerns about the nature of time. Importantly, the issue under scrutiny in this context is not the (alleged) existence of a moving now, the definition of a temporal direction, or the link between temporal relations and causal relations. Rather, it is the very possibility of the dynamical evolution of variables. If only gauge-invariant variables are conceived as physically real, then for any model there are no evolving physical real quantities. A gauge-invariant interpretation of GR entails that no evolution in time of the relevant physical parameters obtains. And this undermines a genuine notion of change. There are scholars, such as William G. Unruh (1991) and Karel Kuchar (1992), who are dissatisfied with this kind of interpretation of GR. They argue that this theoretical approach is unable to provide a fruitful description of ordinary temporality and thus reject the idea that QG should eventually rest upon such an interpretative foundation78. There are others, such as Barbour and Rovelli (1991, 1991a, 2011), who believe time does not ultimately appertain to the structure of reality and thus implement alternative descriptions of dynamics which do not refer to any temporal parameter. All these scholars agree that understanding the ultimate meaning of general covariance is crucial for a viable formulation of a theory of quantum gravity. What they do not concur on is the role that time should be accorded in this scenario. While in the context of GR the notions of time and change can be banned, it might be the case that in the canonical quantization of GR these notions should (or must) be found or imposed upon the intrinsic structure to the phase space of GR. According to Below and Earman (2004, p. 233) there are thus two alternative options. First, one can embrace the consequences of

78 According to Unruh (1989), there are two roles which time plays in quantum theory which should be represented also in the context of QG. First, time is the ordering parameter of the other variables, in the sense that the latter display diverse properties with respect to the former. Second, it “divides the sets of possible observations into sets which are internally mutually contradictory” (Unruh, 1989, p. 1182). In this respect, time is the parameter which allows the occurrence of contradictory phenomena (Unruh, 1988).

84 a gauge-invariant interpretation of GR and thus recover classical physics starting from an underlying, changeless and atemporal, structure. Second, one can reject this interpretation of GR and look for alternative strategies to implement the principle of general covariance in QG. In the first case, the challenge posed by the principle of general covariance is to provide an account of the theory where neither time nor change play any fundamental role, but which is nevertheless able to recover ordinary experience time and to offer a workable quantization programme for QG. In the second case, one has to include a temporal parameter to the conceptual structure of GR prior to quantization. This second option comes in two diverse forms. On the one side, there are scholars, such as Robert Beig (1994) and Arthur E. Fischer and Vincent Moncrief (1996), who disprove the idea that GR lacks a preferred splitting of spacetime into space and time. According to them, break- ing the general covariance of GR by the introduction of a preferred foliation allows to introduce a temporal variable, both in classical and quantum contexts. On this view, the principle of general covariance of GR is regarded as an “artefact of a particular formulation of the theory” (Belot and Earman, 2004, p. 240). On the other side, there are others, such as Kuchar (1992, 1993) and Ioannis Kouletsis (2008), who put forward a more traditional understanding of the general covariance of GR while rejecting that it is a principle of gauge invariance. Three methods for quantizing GR can thus be defined, each of which corresponds to a different interpretation of the general covariance of GR and a different approach to QG. From a diverse orientation towards the principle of general covariance divergent positions about the existence and nature of change arise. And this interpretative divergence leads to three inequivalent theories of QG, namely the Evolving Constants approach, the Constant Mean Curvature approach and the Internal Time approach. While I cannot expand on these models in this context, let me at least sketch their key aspects, with a specific focus on the status of spacetime that each of them entails. Of course, if one of these proposals will eventually lead to an empirically successful theory of quantum gravity, the subsequent interpretation of GR should be favoured. The Evolving Constants approach, whose most cogent formulation is due to Carlo Rovelli (1991; 1991a; 1991b; 1991c), conceives the general covariance of GR as a principle of gauge invariance. According to this approach, only the variables which commute

85 with the constraints are considered as physically real. This means that only gauge-invariant parameters are accorded physical meaning. Spacetime is typically conceived in a relationalist fashion. As spacetime points fail to commute with the constraints of the theory, they cannot be physically real. The Constant Mean Curvature approach states that any viable model of GR comes equipped with a preferred foliation as well as a preferred temporal parametrization. In this approach, there is a privileged notion of simultaneity and a privileged parametrization of time. However, this parametrization is contingent upon the dynamics of the system rather than externally imposed. This means that the time that figures is not the absolute Newtonian time. Still, using this temporal parametrization, GR can be written as a Ham- iltonian system. In this regard, GR describes the temporal evolution of the geometry of space. As a consequence, space is generally conceived in a relational manner. The Internal Time approach is the one proposed by Kuchar (1988, 1992, 1993). According to him, there is a difference between the vectorial and the scalar constraint of the gauge-theoretic formulation of GR. In particular, he claims that “the observable quantities of general relativity must commute with the vector constraint, but that they need not commute with the scalar constraint” (Belot and Earman, 2004, p. 244). While the vectorial constraint encodes the diffeomorphism invariance of general covariance, the scalar constraint generates the dynamical change of physical systems. In Kuchar’s (1992, p. 293) words “two points on the same orbit of [the scalar constraint] are two events in the dynamical evolution of the system. Such events are physically distinguishable rather than being descriptions of the same physical state”. Hence, there are physically real quantities which do not commute with the scalar constraint of GR79.

4.5. Rovelli’s Evolving Constant approach

In this last section my intention is to expand on the conceptual structure and relevance of Rovelli’s formulation of the Evolving Constant approach. In particular, my objective here

79 While Belot and Earman’s (2004) survey of the problem of time in the context of GR and QG is highly recognized, Dean Rickles (2008) disagrees that a proper interpretation of the principle of general covariance is linked to a subsequent attitude towards the ontological status of spacetime. According to him, the nature of gauge orbits can be conceived both in a relational and substantivalist fashion. In this respect, Rickles (2008, p. 141) avers that “general relativity offers good support to neither relationalism nor substantivalism”.

86 is to explore whether or not, under a certain vantage point, Rovelli’s account of time can be squared with the notion of processual becoming. For while the gauge-invariant interpretation of the principle of general covariance seems to faithfully encode GR’s main lesson, it poses challenging questions around the nature and structure of classical physical time and ordinary experience time. According to Rickles (2008, p. 141), there are two ways of setting the problem of time. The first is in terms of states: distinct spacetime regions are connected via gauge transformations and, within the gauge invariant paradigm, they should be representing one and the same state of affairs. The second is in terms of observables: no gauge invariant quantity is able to distinguish distinct spacetime regions. Taken together, these aspects form the so-called frozen formalism of classical GR. The problem, which is currently addressed in a variety of approaches towards QG80, is thus how to recover the familiar structure of space, time and spacetime at the macro, low-energy, level. Rovelli’s (1991; 1991a; 1991b; 1991c; 2002; 2011) Evolving Constant approach, which moves from a gauge-invariant interpretation of the principle of general covariance, stems from the attempt to address precisely this type of concern. It states that QG describes a fundamentally timeless reality but argues that this theoretical framework can host a viable notion of dynamics and change. Rovelli’s idea is that of implementing a one-parameter family of observables that can constitute the types of changing properties displayed at the classical macro level. The point is thus to substitute intrinsic properties (such as “the mass of this object is x”) with relational properties (such as “the mass of this object is x with respect to y”). These quantities, which are gauge invariant, are thus

80 The recovery of classical space, time and spacetime is particularly crucial in the emergent approaches towards QG. Emergent Gravity (EG) is becoming increasingly popular among scholars. In this view, low- energy macroscopic properties are said to be emergent from underlying interacting constituents. Drawing from the work of such scholars as Jeremy Butterfield & David Isham (1999), Butterfield (2011, 2011a), Karen Crowther (2014), here I take emergence to be a relation between different physical theories, in which the lower-energy domain exhibits, as compared to the higher-energy one, both novel and autonomous behaviour. In this framework, novelty is conceived as robust behaviour displayed by the macro-system but absent in the underlying microscopic one. On the other hand, autonomy is intended as the emergent theory’s imperviousness to changes in the microscopic system. As Crowther (2014, p. 70) clarifies, autonomy is not intended in absolute terms, rather in quasi-autonomous ones, “meaning that the (emerging) level is independent of much or most of the high-energy physics”. This kind of approach requires a few steps to be taken, namely: define the fundamental constituents of spacetime, explore their dynamics and describe how, in some approximation, classical spacetime eventually emerges. According to the physicist B.L. Hu (2009, p. 3) reality is a complex system composed of several levels in which there are “emergent rules or laws governing the organization and dynamics of the basic constituents and new modes of interaction at every level of structure”. As a consequence, the universe displays processes with emerging features, whose links need proper analysis.

87 constants of motion, while at the same time encoding variation: “The change we normally observe taking place is to be described in terms of a one-parameter family of constants of motion, {�(�)}∈ℝ, an evolving constant of motion” (Rickles, 2008, p. 161). This approach is particularly interesting from a philosophical standpoint, as it conceives individuals (objects, aggregates, bodies) in an anti-essentialist fashion. Gauge-invariant quantities encode individuals’ variation, while individuals do not display any stable or permanent property. In this respect, change over time is analogous to variation through space. While, technically, it is hard to select this one-parameter family of constants of motion, Rovelli’s project of conceiving dynamics as a process of interaction between gauge-invariant quantities is still very appealing. In other words, Rovelli maintains that “the observables of general relativity and quantum gravity are relative quantities expressing correlations between dynamical variables” (Rickles, 2008, p. 162). For a better understanding of this approach, it is necessary to expand on Rovelli’s (2002) distinction between partial and complete observables. This distinction is valuable insofar as it relates with several issues concerning observability. First, whether or not time can be conceived as an observable in quantum mechanics. Second, what quantities should be accorded observability in GR. Third, whether or not the notion of observability should be conferred only to those quantities which commute with the scalar constraint (the so-called Wheeler-DeWitt operator) in quantum gravity. The notion of observability is directly connected with that of measurability. A measure gives us information about the state of a physical system. In the gauge-invariant interpretation of the gauge theory one distinguishes between gauge-invariant quantities (observables) and gauge-dependent quantities (non-observable). As Rovelli goes on to explain, there are several issues concerning the notion and status of observable. In particular, it is not clear what relation time holds with observability and measurability. Accord- ing to him, “the observables in quantum gravity are relative quantities expressing correlations between dynamical variables” (Rovelli, 2002, p.1). As a preliminary step, says Rovelli, one needs to characterize two distinct notions of observability. A partial observable is a physical quantity for which a measurement procedure can be established. A complete observable is a physical quantity whose value (or probability distribution in the case of quantum theory) can be predicted by the relevant theory. On this view, partial observables can be measured but not predicted, while

88 complete observables are correlations between partial observables, and they can be both measured and predicted. The main point under scrutiny is how it is possible to derive complete observables (such as the total energy of a system) from partial observables (such as spacetime coordinates in GR). In order to address this question, Rovelli offers a further classification in terms of dependent and independent variables. This distinction can be conceived in the following way. Consider as two partial observables q and t (position and time); then, if it is possible to write q in terms of t, namely as q(t), but this function is not reversal, namely t(q) is not admissible, then q is a dependent partial observable and t is an independent partial observable. While in pre-relativistic theories the distinction between dependent and independent partial observables holds, Rovelli (ibid., pp. 3-4) argues that in the context of GR the situation is completely different81:

The key difference between general relativistic physics and pre-GR physics is the fact in general relativist physics the distinction between dependent and independent partial observables is lost. A pre-GR theory is formulated in terms of variables (such as q) evolving as functions of certain distinguished variables (such as t). General relativistic systems are formulated in terms of variables […] that evolve with respect to each other. General relativity expresses relations between these, but in general we cannot solve for one as a function of the other. Partial observables are genuinely on the same footing.

On this view, the theory describes the evolution of partial (gauge variant) observables with respect to each other82. All these partial observables are equivalent. None of them can be taken as an independent, privileged parameter vis-à-vis others. In the context of

81 On the one hand, in a non-relativistic context, where the spacetime structure is taken as fixed, no distinction between these two definitions of observable can be grasped. On the other hand, in a generally relativistic context, where the spacetime structure is dynamical, the difference between these two notions of observability arises. 82 In the context of GR, Rovelli (2002) offers an instructive example to grasp the idea that no partial observable is independent (and thus, somehow, privileged) with respect to others. Consider a very accurate clock which is mounted on a satellite of the GPS system. This clock transmits its local time, whose signal is collected by the launching base and compared with the time of an equally accurate clock mounted on the base. As a result of the different positions of the clocks, the timing shows a discrepancy due to generally relativistic effects. Rovelli calls � and � respectively the signal received from the satellite and the local clock reading. General relativity can then be employed to predict the relation between these two partial observables �(�, �) = 0. In this case, two partial observables can predict another one, �, which is then a complete observable, according to the classification above mentioned. A question naturally arises: which one of them should be referred as the independent variable? One might say that � should be favoured, as it refers to our commonly reference time. However, � corresponds to an accepted standard of time. Clearly, none of them can be chosen as the independent one.

89 QG, operators corresponding to physical observables must commute with the Wheeler- DeWitt constraint operator. This operator encodes the temporal evolution. This means that physical observables must be time-independent or time-invariant. The point is then how to describe evolution by means of invariant quantities. In particular, I want to scrutinize the role that time is accorded in this theoretical framework. The issue is that the coordinate time and the physical evolution of systems are two independent concepts. In order to get the evolution of systems one needs to start with the extended configuration space, consider a function of the variables under scrutiny and thus determine the dynamics. The dynamics is referred to the mutual variation of these partial observables. While none of them displays a well-defined evolution (qua partial, and thus not predictable observable), the correlation between these partial observables is well defined and thus independent of the temporal parametrization. One of these partial observables is arbitrarily taken as a reference clock. This procedure is of practical interest but indicates nothing about the ontological status of this variable. Spacetime coordinates are indeed partial observables, meaning that they cannot be predicted. They can just be used to localize complete observables. Rovelli (2002, p. 5) states that “quantum theory deals with the relation between partial observables. It can deal with the relation between physical variables and (gauge-fixed) coordinates (�⃗, �). But not with the value of the coordinates alone. Therefore, it is meaningless to search for the quantum theory of the (�⃗, �) alone”. Mechanical systems are represented by the space of partial observables, which describes the kinematics of the theory, and a function that determines the dynamics. Classi- cal dynamics deals with partial observables. These relations are contingent upon a specific set of parameters, which mark the time-independent states of the system. The space of these states is the phase space. By fixing a subset of partial observables, the other partial observables can be written is terms of this subset. And this allows to define the complete observables of the theory, whose value can be uniquely predicted. This formulation of mechanics is valuable insofar as it does not require any notion of external time. The extended configuration space is the space of partial observables. While this space, together with its associated phase space, are generally regarded as lacking a physical significance, Rovelli argues that this is not the case. On the contrary, he contends that the

90 they represent the space of non-predictable observables, thus playing a central role in the general structure of mechanics, both at the classical and quantum level. In the last part of this section I intend to connect the analysis of this chapter with the overall economy of the present project. Remember that my intention was to explore certain physical theoretical frameworks and to analyse which kind of connection these models have to temporal becoming and ordinary experience time. To this purpose, I have defended a position in which both temporal experience and temporal becoming are conceived as processual, i.e. as extended phenomena. In the case of Barbour’s shape dynamics, I have shown that a theoretical model which rebuts processuality severely undermines physical becoming. In this case, I want to expand on the conceptual consequences of Rovelli’s theoretical approach. In particular, I would like to show that, under a certain vantage point, Rovelli’s relational account of dynamics accommodates processual becoming. The temporal structure of reality has been deeply transformed following the elaboration of both STR and GR. This structure reveals that the ordering of events cannot be traced in terms of a unique directed sequence. Rather, the unfolding of events displays much more complex patterns in which the notions of “past”, “present” and “future” can only be defined locally. However, this does not mean that a genuine concept of becoming is lacking. Quite the reverse, reality has a temporal structure which ultimately expresses becoming, while becoming “is different and more complex than a naïve oriented one- dimensional succession of instants” (Rovelli, 2019, p. 1326). As Rovelli goes on to explain, physics is a theory about the happening or coming- into-being of events, not about things. It describes motion, evolution, change. On this regard, “becoming is primary both in the phenomenology of our experience and in our physics” (Rovelli, 2019, p. 1331). This means that reality is characterized by the unfolding of processes, where a process is “what happens to a system S between interactions with other physical systems” (Rovelli and Vidotto, 2014, p. 41). What is particularly relevant here is that both human temporal cognition and physical evolution are mapped in terms of interaction/correlation between physical systems. Both phenomena can be conceived as local processes. On the one hand, ordinary experience time is a process in which information coming from the “external” environment is elaborated, integrated and re-uti- lized for various purposes. On the other hand, physical time describes the local properties

91 of a local event as intertwined to a more complex chain of local events with different local properties. Rovelli advocates a dynamical representation of the world. While the unfolding of events is not organized with respect to a common, privileged, temporal parameter, the Einstein’s field equations are nevertheless evolution equations, thus encoding dynamics and change. Becoming is real and things happen rather than simply exist in their fixed spacetime locations. Interaction generates evolution and becoming, where the latter is always marked locally. These local patterns, which evolve dynamically, are not independent to one another. They are all partially related within the causal structure of the 4-dimensional spacetime geometry of GR, while “the ensemble of all events of the world cannot be objectively arranged into a single simple succession of global instants” (Rov- elli, 2019, p. 1332). Becoming is a complex, multifaceted, multi-layered phenomenon encoding the correlation between events. On this view, relativity does not reject temporality but shows its underlying causal structure. Realizing that temporality displays a much more complex structure does not imply that temporal becoming does not exist. Surely, some aspects of common-sense intuitions about the nature of time do not figure in the relativistic context, namely directionality or the idea of a moving-now. Still, “the fact that so many aspects of experiential time depend on approximations, and on complex structures, does not alter the fact that what elementary physics describes is happening, not entities” (Rovelli, ibidem). In this respect, concludes Rovelli, “the four-dimensional spacetime is only a car- tography of the relations between multiple local becomings” (Rovelli, 2019, p. 1334). All in all, it seems plausible to assert that Rovelli’s relational account of dynamics opens the door to a conceptual validation of processual becoming. The extended configuration space is the context in which “isolable” parts (aggregates, individuals, bodies) unfold as a result of processes of interaction among other “isolable” parts. Processual becoming encodes the dynamical evolution of systems with respect to each other. It describes the correlation among partial observables. Processual becoming is thus the order of partition. In addition, the advantage of a process-based account of reality is that the conceptual gap between human temporal cognition and physical time is sharply reduced. Both ordinary experience time and physical time come out as the result of dynamical processes.

92 4.6. Concluding remarks

In this Chapter I have offered a concise and simplified survey of one of the most challenging projects within current theoretical physics, namely QG. The quest for a micro theory of gravity arises from the inconsistencies of both QFT and GR. Many and at time incompatible approaches have been developed for this purpose. In this Chapter, I have concentrated on canonical QG, a theoretical perspective which starts with GR and operates via quantization procedures à la Dirac. Classical observables are thus replaced by corresponding quantum operators in the Hilbert space. In order for a proper quantization to be implemented, the interpretation of the principle of general covariance is crucial. As I have discussed above, different approaches towards the ultimate significance of general covariance lead to different types of quantization in quantum gravity. In the last section I focused on Rovelli’s relational account of dynamics to demonstrate that his theoretical approach is consistent with processual becoming.

93 Conclusions

I would like to close the analysis that I have carried out so far with a short recap of the major steps that I had to take for the purposes of the present work. The first two Chapters laid out the theoretical backdrop for the investigations of Chapter 3 and 4. Indeed, a few conceptual tools were necessary to undertake a comparison between Barbour’s and Rov- elli’s physical theories. To clear the path to an orderly analysis of the main elements of these influential proposals, I had to analyse what it is that they refute of the most widespread understandings of time within physics and the philosophy of physics. To this end, I made the point that the metaphysical position which befits the ultimate outcomes of relativistic physics, that is, the block universe view, is potentially affected by a few frictions. The main tenet of the block view is that the universe can be conceived as a four-dimensional entity in which no ontological distinction between past, present and future events can be drawn. All events are given once and for all in their respective location within the spacetime manifold. To put it otherwise, events, which are conceived as point-like, form a dimensional block of spacetime. They are ordered to one another in keeping with relations of “earlier than”, “later than” and “simultaneous with”, while these relations are unchanging – for time is one of the dimensions of the block. In the light of this, I concluded that, amongst its various formulations, the most consistent version of the block view is the one that does not allow for dynamism. Based on the work of various critics, along with considerations made by Barbour and Rovelli themselves, I foregrounded two main problems that beset this metaphysical view. I obviously did not mean to claim that these problems are not resolvable in order to strengthen the block universe model – as in fact there are promising attempts to achieve this aim (see e.g. Peterson and Silberstein, 2010; Romero, 2012, 2013). Rather, I wanted to identify the points of departure of theories that, as the block view does, do not take time to be fundamental but are dissatisfied with the two issues that I identified. These are the following: first, the representation of events as point-like – which the critics of the block view at best consider as a useful approximation to modellize the universe for the sake of the theory; second, the recurring reappearance of dynamical features when the level of explanation goes from the total set of events to the single event as well as the connection between events. Whereas the most consistent advocates of the block view try

94 to strip the universe of every dynamical feature, at any level, most promoters of the block universe, whether consciously or not, are torn between the lexicon of being and the lexicon of becoming. To give some flesh to the theoretical relevance of these frictions, Chapter 2 focused on the nature of events and the way they are ordered according to the dynamical block view of the universe. My aim was to show that one of its consequences is that it robs becoming of its physical meaningfulness to confine it to the sphere of the human perspectival experience. In the physical realm, only the language of “occurring” and “being” applies, while “coming into being” and “becoming” only arise within human (tensed) experience. To bring to the surface the impasse that this modellization of events incurs I cast light on the irreducibly processual nature of events. It is here that I teased out the meaning of processuality. For the notion of process entails that no entity (or event) whatsoever is isolable from the nest of relations this entity (or event) is the bearer of; while the set of relations giving life to individual entities (or events) depends on the nature of these entities (or events). This lays stress on the non-separability and mutual dependence between events and relations – a non-separability that excludes the possibility of there being self-identified events, separate from one another, and only connected by external relations of precedence, successiveness and simultaneity. Again, I did not so much intend to deny the plausibility of physical theories that envisage a becomingless world, as to put on the table the conceptual tensions that have led Barbour and Rovelli to draw from different premises for their physical enterprises. Barbour opts for a wholesale rejection of events and the related notion of becoming, whereas Rovelli puts into question the point-like nature of events and ties becoming to the correlation between partial observables. My interest in Barbour’s theoretical enterprise was driven by his seminal reformulation of Newtonian mechanics in a fully background independent fashion. While this is relevant to theoretical physics per se, for the purposes of my work it was his peculiar version of relationalism that counted in the first place. Barbour can be regarded as the heir to the fathers of relationalism, namely, Leibniz and Mach. As some scholars point out (see e.g. Anderson, 2017; Mercati, 2018), the Leibniz-Mach-Barbour relationalism can be considered as the purest form of background independent relationalism. Therefore, Barbour gets out the predicament of the four-dimensional block by seriously questioning

95 the fundamentality of the four-dimensional requirement. What is primary is space, and in particular, configurations. My conclusion was that, while this view successfully dispenses with becoming, it reinstates time as an ordering principle obtained as a distillation of all the changes in the universe. Rovelli’s approach is somewhat specular to Barbour’s. For, while he deeply challenges the nature of time as a global, privileged, superimposed parameter, his theoretical framework nevertheless comes equipped with a genuine notion of becoming as the outcome of the interaction between systems. While there is no way in which physics can do without dynamics, coordinate time is unobservable because of general coordinate invariance. Physics deals with the evolution of variables, out of which one can be arbitrarily chosen as the reference one. Still, all physical variables are on an equal footing and are mutually correlated. Endorsing a dynamic view of the universe means that physics has to implement processualism, whereby no individual entity can be singled out without reference to the chain of relations that actualize it. Relations can thus be conceived as modes, that is, specific ways in which entities are. These entities do not have any existence in distinction to their ways of existence, while their ways of existence cannot be in isolation from these entities. One can only draw a conceptual distinction between entities and their modes, but hardly an ontological one. To sum up, the objective of the present work was to scrutinize physical paradigms that aim to clarify the ultimate role of time and temporal becoming in foundational physics – where time and temporal becoming are to be treated as non-interchangeable, distinct phenomena. The main conclusion is that we are left with two alternatives. First, one can discard dynamism and embrace a fully static account of the universe. In this case, I mounted the argument that the most consistent physical framework is Barbour’s, whose pure type of relationalism disposes of becoming and allows conceiving time as an abstraction from the variation among the configurational space. Second, one can favour dynamism and defend a processual account of the universe. In this latter case, the most appealing theoretical model is Rovelli’s, whose version of relationalism encodes a strong notion of becoming and grounds entities’ individuality on the set of relations these entities display. On this view, physics is the study of the dynamics within which the interaction among systems gives life to individuated entities.

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