ARTICLE IN PRESS

Studies in History and Philosophy of

Modern Physics 39 (2008) 82–101

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Special relativity and the future:A defense of the point present

James Harrington

Philosophy Department, Loyola University Chicago, Chicago, USA

Received 12 September 2006; received in revised form 9 April 2007; accepted 7 June 2007

Abstract

In this paper, I defend a theory of local temporality, sometimes referred to as a point-present

theory. This theory has the great advantage that it allows for the possibility of an open future without

requiring any alterations to our standard understanding of special relativity. Such theories, however,

have regularly been rejected out of hand as metaphysically incoherent. After surveying the debate,

I argue that such a transformation of temporal concepts (i) is suggested by the indexical semantics of

tense in a relativistic universe, (ii) when properly understood easily withstands the usual accusations

of metaphysical incoherence and (iii) leads naturally to a metaphilosophical position from which we

can understand and escape the increasing sterility of debates between radical Parmenideans and

radical Heracliteans in the philosophy of time.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Relativity; Space–time; Time; Indeterminism; Becoming; The present

When citing this paper, please use the full journal title Studies in History and Philosophy of

Modern Physics

1. The problem of time in special relativity

Among the many features that both classical physics and ‘‘common-sense’’ attribute totime are the openness of the future and the role of time as a ‘‘global metric.’’ More

see front matter r 2007 Elsevier Ltd. All rights reserved.

.shpsb.2007.06.003

dress: jharr4@luc.edu

ARTICLE IN PRESSJ. Harrington / Studies in History and Philosophy of Modern Physics 39 (2008) 82–101 83

precisely, those of us not in the grip of a particular theory tend to believe both of thefollowing. First, in some sense, the question, ‘‘What is happening right now on the moon?’’must have some determinate answer. Second, that the question ‘‘What will happen to me10 years from today?’’ does not. In this paper, I argue that, although the first of thesebeliefs is false, the second is true. Thus, I will defend a version of what has been called a‘‘point-present’’ theory of time.

More precisely, I will argue that given special relativity, not both of these intuitions canbe correct. Given special relativity then if time is global, the future is closed or,equivalently, determinate. This is not new; it dates back at least to the Putnam–Rietdijk–Stein debate of the late 1960s. What is basically new is that I believe that we should retainonly the second and deny the first assumption. I claim that special relativity provides uswith good reasons for rejecting global time and points us towards an independentlyplausible theory of the open future. However, the resultant theory also requires asubstantial re-evaluation of our temporal concepts in general and of those metaphysicalconcepts connected to them. After recapping some reasonably well-known history in thenext section, I present the argument in three main stages. First, I introduce an indexicalsemantics for tense and argue that the most plausible transition from classical to relativisticconcepts takes local or proper time along worldlines rather than global coordinate time asfundamental.

Second, in the next two sections, I develop my positive account of the openness of thefuture. The principal aim of Section 4 is to develop an account of relative indeterminacy,which will provide us with a plausible account of the openness of the future. One region ofspace–time P is indeterminate relative to another region Q when the causal past of Q failsto determine the state of P. This account of indeterminacy, which is distinct from bothindeterminism and from failure of predictability, provides a plausible account of theopenness of the future in general. Most significantly, if we make certain plausible and quiteweak assumptions, namely that the state of a region of space–time depends only on thecausal past of that region, then, in Einstein–Minkowski space–time, for any point, p, in thespace–time only the causal past of that point is determinate relative to that point. Theremainder of the space–time is indeterminate relative to p. Moreover, in Einstein–Min-kowski space–time, the indeterminacy of the future does not depend on the deterministicor indeterministic structure of the particular causal relations.

From the perspective of contemporary debates in the philosophy of time, e.g. presentismvs. eternalism or three-dimensional (3D) vs. four-dimensional (4D), this position mostresembles an eternalist, 4D perspective. I believe that the 4D space–time manifold is thebasic spatiotemporal entity and that the entire space–time and its contents exist, inwhatever sense space–times exist.1 However, perhaps the most fundamental philosophicalconsequence of this position is that it illustrates just how inappropriate the Platonicmetaphysical structure of those debates is to questions about time; only if we break thehold of the Platonic conviction that only ontology matters will we ever come to areasonable understanding of the metaphysical status of time and temporality. In the lastsection of the paper, I begin to develop this anti-Platonist perspective, which I call neo-Aristotelianism from its affinities to Aristotle’s account of time in Physics IV.

1On presentism see especially Smith (1993). On eternalism see Sider (2001). The first part of Dainton (2001)

provides a useful introduction to these debates.

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2. A new look at some old history

Until the late 1960s, even the advent of special relativity seems to have left these two coreintuitions largely untouched. While all of those paying attention would certainly haveadmitted that global times, i.e. ‘‘planes of simultaneity,’’ would have to be ‘‘relativized,’’they seem to have been confident that any fundamental temporal features of the worldwould be referred to these new, relativized times. The exception here is Godel (1949b) whorecognized fairly early that global time was incompatible with relativity theory. However,although Godel recognized that relativity destroys the possibility of objective global time,he continued to demand that objective temporality must be global. Thus, Godel arguedthat the absence of global time in relativistic space–time is sufficient to disprove theexistence of time itself. Godel thus became the first, as far as I am aware, but not the lastthinker to claim that relativity theory solves all philosophical problems about time bydissolving them.The next fundamental move was made, apparently independently, by Putnam (1967)

and Rietdijk (1966). Both argue that special relativity implies that there can be no objectivedistinction between past, present and future. Maxwell (1985) resurrected this basicargument in the eighties. In response, Stein (1968, 1991) twice claimed that the argumentrested on a fundamental assumption that was inimical to, at least the spirit of, specialrelativity—that assumption being that the boundary between the ‘‘real’’ (Putnam) or‘‘determined’’ (Rietdijk, Maxwell) past and present, and ‘‘unreal’’ or ‘‘undetermined’’future must be a plane of simultaneity, a global time. More precisely, Stein proved thefollowing theorem in his 1991 paper.

Stein’s Theorem. The only reflexive, transitive and Lorenz invariant relations between world

points in Einstein– Minkowski space– time are the trivial relation, the universal relation, and

the relations of past time-like and past causal connection.

The fundamental consequence of this theorem is that the openness of the future relativeto the past for any region of space–time is, given special relativity, logically incompatiblewith the objective global temporal structure.Stein’s theorem leaves us with three possible positions on time and temporality in

Einstein–Minkowski space–time. First, a very strong block universe or Parmenideanposition in which one insists on global times as a necessary condition for objective time andsimply accepts that there is no such thing. This position tends to be popular withphilosophers of physics and physics-influenced metaphysicians who are often drawn toblock universe positions for other reasons as well. While, if the alternative defended belowfails, such positions may be the only remaining possibility, they continue to have theproblems of plausibility and of relation to empirical reality that have plagued Parmenideanpositions going back to, well, Parmenides. Second, one can insist on a strongly Heracliteanposition with a commitment both to the openness of the future and to global times whiledenying standard special relativity and embracing, either some form of neo-Lorentzianism,as with Tooley (1997), or branching space–times, as with McCall (1994, 1976), as areplacement for special relativity. Such positions suffer from at least two severemethodological problems; they seem to get the relationship between physics andphilosophy backwards and to make it impossible to see how to get to general relativity

without the space–time formulation of special relativity.

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These objections are not definitive, but they should be at least sufficient to lead us totake the third alternative seriously, no matter how implausible it seems at first glance. Thatthird alternative is ‘‘point-presentism’’ or, as I prefer, local temporality. As a firstapproximation, local time is the claim that proper time along my worldline is the only andactual time that passes for me and that only my past null-cone and its interior isdeterminate for me. Moreover, I claim that these are actual and objective features of ourexistence as a spatiotemporal entity and not subjective features of our awareness. Thisposition has received different treatments in the philosophy of time literature and in thephilosophy of physics literation. In the philosophy of time literature, where this position ismentioned at all, it is rarely considered in any detail or argued against. Instead, it isgenerally dismissed out of hand. The introduction to Oaklander and Smith (1994) is typicalhere. It has, however, received some attention in the philosophy of physics literature. Inrecent years, several authors, including Shimony (1993), Clifton and Hogarth (1995), Dieks(2006) and Myrvold (2003), have considered versions of the local present and localbecoming. However, all of those treatments take for granted that we have a coherentunderstanding of what it is for an event to ‘‘occur,’’ or some synonym, relative to a regionin space–time or relative to the worldline of an entity in the space–time. However, considerhow problematic this notion of occurrence is in terms of our usual, classical,understanding. The relativistic conception of the occurrence of an event must allow,among other oddities, that relative to any two distinct objects whose worldlines intersect atleast twice, there will be no agreement about when events occur, that is how far in the pastthey occurred. Beyond that such a conception will require that each event occurs one timefor each of the entities whose past it comes to occupy. How is it possible for an event tooccur more than once and yet for each occurrence to be objective?

Thus, my defense of local time involves two distinct projects. First, I must respond tospecific objections to this theory, which have been left un-addressed by previoustreatments, by showing that the relativized conception of the occurrence of an event iscoherent. Those objections fall into two broad classes—semantic objections andmetaphysical objections. The semantic objections are generally versions of Godel’sintuition that time just is global; the existence of a single time for the entire universe just is

a logical or semantic feature of what we mean by time. In Section 3, I introduce anindexical semantics for tense and show that there is no logical requirement that we retainglobal time when we replace the classical temporal order relations with the relativisticorder relations and that there are good methodological and physical reasons to avoiddoing so. The metaphysical objections focus on the status of regions space-like separatedfrom the present. Since any theory of local time must treat the space-like regions asindeterminate in much the same way as the future time-like separated region, objectionsgenerally focus on the claim that treating ‘‘merely spatially separated entities’’ asindeterminate would commit one either to profoundly bizarre solipsism, in which only mypresent self exists, or to a bizarre verificationism, in which the absence of an immediatecausal connection to an entity provides grounds to deny that it exists. In Sections 4 and 5, Iconsider the status of the space-like separated regions in some detail and conclude that ona reasonable statement of the distinction no such pernicious consequences hold.

However, I believe that these explicit objections are not the truly fundamental barriers tothe fair consideration of a local theory of time and temporality. What actually drives theresistance is an implicit understanding that such a radical re-evaluation of the nature oftime—as contingent rather than necessary, relational rather than absolute and local rather

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than universal—requires a nearly equally radical re-evaluation of cherished metaphysicalassumptions about what it is to be real, and thus what it is to do metaphysics. The finalsection of the paper attempts to lay out the terms of such a re-evaluation and renewal ofmetaphysics.2

3. Tenseless time and local time

In recent years, several slightly different versions of token reflexive or indexicalsemantics have appeared in the literature.3 Thus, my goal here is not a general defense ofthe new tenseless theory of time. Rather, I will introduce a particular schema for such asemantics and indicate briefly some of the reasons why I prefer it. I will then deploy thatschema to the particular purposes of this paper: the nature of the transition from classicalto relativistic temporality.The fundamental semantical puzzle about tensed sentences is the difficulty of finding a

schema that makes use only of the temporal order relations (earlier, later, at the same timeas) to account for the dependency of tenses. The schema that I prefer solves this problemby taking the truth of sentences in a context as the fundamental semantic feature ofsentences, roughly following the treatment of demonstratives by Kaplan (1989). Thus, thebasic meta-language for giving the tenseless truth conditions for tensed English sentencesconsists of a tenseless fragment of English with the ability to refer to and quantify overarbitrary regions of space–time, dates, and to discuss the relevant geometrical relationshipsbetween dates, plus the following technical machinery.4 Capital Greek letters will be usedas names for tensed sentences of English.5 For example the meta-language sentence,‘‘S ¼‘The first moon landing is today.’ ’’ says that ‘S’ is a name of ‘The first moon landingis today.’ In addition, the language contains both the function date from occupants ofspace–time to the region of space–time that they occupy and the function ext, from termsin the object language to their extension. Thus, consider the truth conditions for

S ¼ ‘The first moon landing is today.’

Intuitively, it is clear that S is true if and only if someone used it on July 20, 1969. Moreformally,

TC(S).

8cf½S is true in c�2½dateðextð‘the first moon landing’ÞÞ � dayðcÞ�g,

where c ranges over contexts. Let me give one more example. Consider

G ¼ ‘President Bush went to Iraq three weeks ago.’

2Since I first formulated the basic ideas of this section, Steve Savitt has begun to pursue similar ideas from a

different direction. Savitt argues in his 2006 (Savitt, 2006) that the presentism, eternalism debate has become

empty. However, his association of non-ontological temporality with philosophical naturalism takes him in a very

different direction from that pursued here.3See especially Mellor (1981, 1986), Le Poidevin (1991) and Tooley (1997). See also the relevant essays in Le

Poidevin and MacBeath (1993) and Oaklander and Smith (1994).4On one level it might seem simpler here to speak simply of points and regions of space–time. However, part of

the point of the section is that this space–time perspective slots quite nicely into a schema, which allows for other

types of contexts and dates. The terminology of dates comes from Mellor’s work although I use it here differently

than he does.5In the remainder of this section, quotation marks are used according to the usual use mention distinction.

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TC(G).

8cf½G is true in c�2½dateðextð‘President Bush’s trip to Iraq’ÞÞ

is three weeks earlier than c�g. ð1Þ

While a full defense of this schema is beyond the immediate scope of this paper, it hasthree advantages worth pointing out.6 First, even a quick glance at TC(S) or TC(G) shoulddisabuse the reader of any sense that there is some illicit appeal to tensed notions at work.Both of them contain names for tensed sentences of English, but they certainly do notcontain any uses of such a sentence.

Second, the examples make particularly clear the way in which the characteristicbehavior of tenses results from changing relationships between the context of utterance orevaluation and the dates of the subjects of the sentences. This is particularly useful inclarifying the problems with a certain class of objections to tenseless accounts of time, aclass best exemplified by Prior’s ‘‘Thank goodness that’s over’’ (Prior, 1959). In this well-studied argument, Prior suggests that defenders of tenseless theories of time, de-tensers,should be puzzled by certain apparently reasonable human attitudes. In Prior’s example,given that a de-tenser believes that there is no intrinsic difference between a headache atthe time when we suffer from it and the same headache after it ends, he suggests that areasonable de-tenser should not feel relief only after the end of the headache.

Apparently, Prior is deploying a principle that, roughly, it is reasonable to change ourattitudes towards some entity only after detecting some change in the entity.Unfortunately, for Prior, that principle is clearly false. There is at least one situation,other than an actual change in an entity, in which it is rational to change our attitudestowards that entity—when there is a change in our knowledge of the entity. Consider ourattitudes towards a political regime that we had supported because we believed that it hadexemplified certain dearly held political beliefs. We later come to believe that it does not,and in fact never had, exemplified those beliefs, and we cease to support it. There is nochange in the object under consideration, only in us, but it is certainly rational to changefrom support to non-support towards it.

The case is similar with respect to Prior’s headache. For those of us for whom headachesare merely temporary facts of life, we know that our headache will end and so have ageneral feeling of relief for this fact, but we do not know when any particular headachewill conclude in advance of its conclusion, and thus, it is not rational to feel relief for itsending until it has in fact ended. The case of the headache is actually stronger than thatabove because in the case of the headache, there is a change in the relations that we bearto the headache, rather than a merely epistemic change in us. Like all of our attitudesand feelings about something, the feeling of relief depends primarily not on the propertiesthat the thing possesses but on our relations with the thing, and the change from therelation ‘‘is earlier than’’ to the relation ‘‘is later than’’ is a real change in our relation withthe headache.

Third, and most importantly for my purposes here, this schema makes it particularlyeasy to see the advantages in restricting ourselves to local relativistic concepts rather thantheir global coordinate counterparts, for a relativistic theory of tense. To make the issues

6For a more complete discussion of the issues here, see Chapter 3 my dissertation Local time: Foundational

studies for a theory of temporality in a relativistic universe. Unpublished Ph.D Dissertation, The University of

Illinois at Chicago, Chicago.

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as clear as possible, let us consider a simple present tense version of S:

O ¼ ‘The first moon landing is happening now.’

In a classical context, the truth conditions for O are fairly straightforward. O is true in allcontexts, c, such that (some temporal part of) c is simultaneous with (some temporal partof) the first moon landing. When we move to special relativity, things are, of course, nolonger so simple. The transition confronts us with two possible choices. We can parse‘‘now’’ or ‘‘at the same time as’’ either according to the relativistic definition ofsimultaneity and relativize the truth conditions of O to frames of reference or we can parseit as ‘‘here–now’’ and deny that stricto sensu O is true in any context that fails to intersectthe spatiotemporal region occupied by the moon landing.It is obvious that nothing in the logic of the context theory of tense requires the global

reading. In a sense, this is sufficient to establish my point. If a theory has no logical orconceptual impediment and has the advantage of salvaging our intuitions about theopenness of the future, it certainly seems to deserve a full and fair hearing, which it seemsnot to have received to this point. However, the situation is better than that. In addition toits advantage in allowing us to retain an indeterminate future in Einstein–Minkowskispace–time, this choice has two principal advantages and no substantial disadvantages.The first principal advantage is methodological. The local time reading of O transfersnaturally to general relativity in a way that the global reading does not. In particular, thisframework allows us to make good sense of debates about the nature of time and timetravel in universes with closed time-like curves and without any global hyperplanes ofsimultaneity (cf. Earman, 1995; Godel, 1949a).The second advantage is that once we embrace local time, most of the interpretive

problems of special relativity drop away. In special relativity, the only physicallysignificant time is proper time. In a relativistic universe, clocks measure only proper time,and only proper time provides any natural measurement of the rates of physical processes.These phenomena are as empirically well confirmed as anything in physics.7

None of this is new. What appears to be new is an insistence that we ‘‘bite the bullet’’ soto speak. If proper time is the only kind of time that matters, then proper time just is time.Moreover, if that is correct, then we must truly and consistently embrace the consequencesof that truth. Let me just briefly consider two of those consequences. First, the coordinatesystem and simultaneity relation associated with an inertial reference system is privilegedonly with respect to other coordinate systems. It may be geometrically handy, but usingsuch a frame does not give us access to anything about the universe that we cannot equallywell describe in any other coordinate system.8 Second, the temporal distance betweenevents depends on the path one takes between those events. In the classic twin-paradoxcase, neither twin’s path through space–time is privileged. If it took Twin A 5 years andtook Twin B 10 years, then that is what it took them. However, and here is thefundamental point, this does not make the passage of time for either or both of the twins asubjective rather than an objective matter. The fact of the time elapsed along the two pathshas nothing at all to do with the existence of the twins as conscious beings; relativity theory

7For clocks see the classic studies published as Hafele and Keating (1972a, 1972b) and Hafele (1972). On

particle decay see Rossi and Hall (1941).8David Malament’s classic (Malament, 1977) is still one of the best discussions of this feature of inertial

coordinate systems.

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says that we get the same result whether we send twin human beings, twin dogs, twin sets ofatomic clocks or twin samples of radioactive material; and every experiment we can comeup with confirms that, in this at least, relativity theory is correct and common sense iswrong.

4. Relational indeterminacy and the open future

Given the conclusion of the previous section, there is no logical or conceptual bar tolocal temporality. However, the argument so far provides no direct grounds for acceptingthat there is a substantive distinction between the past and the future relative to eachtemporal element, whether global instants or local events. In this section and the next, Iargue that, given certain common and plausible assumptions and one plausible way ofcharacterizing the distinction between a closed past and an open future, we have goodgrounds for believing in an open future in Einstein–Minkowski space–time. The argumentof this section and of the next proceeds in three stages. First, I introduce and explicate adefinition of the open-future vs. closed-past distinction. Second, I demonstrate that, givencertain standard assumptions, the past null-cone and its interior of each point ofEinstein–Minkowski space–time is closed and the rest of the space–time is open relative tothat point. Finally, I argue that this particular way of understanding the distinction allowsus to avoid the standard objections to point-present theories.

What would it be for some region of space–time to be open as opposed to closed?Intuitively, it seems reasonably straightforward that such an open region must be, in somesense, a locus of multiple possibilities. It must be the case that the region could be occupiedby any one of a number of different things, be the site of any one of a number of differentevents or, more generally, be in any one of a number of distinct states. Unfortunately, evengiven a theory that specifies what the possible states of space–time are, this does not get usvery far. In particular, without more specifications of the relationships between regions ofspace–time, all of space–time is either trivially open or trivially closed. Given no additionalinformation, the state of every region of space–time could be anything compatible with aconsistent assignment to the remainder of the space–time; everything is open.Alternatively, a complete specification of the state of space–time clearly fixes the state ofevery region; everything is closed.

Therefore, any non-trivial conception of openness, or indeterminacy, must be relational;a region must be indeterminate relative to the state of some other region of space–time.Can we make this concept any more precise? I believe we can. Start with space–timetheories. Let us say that, abstractly, such a theory consists of a space–time and theassignment of possible values to every point of space–time—in the standard cases,assignments of scalar, vector or tensor fields to the space–time; a specification of all of theavailable types of values to each point of the space–time, I will call the state of thespace–time. Now, let us consider topologically open regions of space–time, down to points.Obviously, given the state of space–time, the relevant values are also determined for eachsuch region. In the absence of a dynamics, there is a meaningful sense in which all possibleassignments of such values are equally allowable. But, of course, this is not what we wantto know. We do not deal in God’s eye views of space–time, except at the most abstractlevels. We deal with regions of space–time.

What we really want to know is given the (full or partial) specification of the state of aregion of space–time, how does that, given a relevant theory, constrain the states of other

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regions of space–time? Obviously, to do this, we need a dynamics which connects the statesof different regions of space–time and connects the various ‘‘elements’’ of the state. Assignequal probability to all dynamically consistent states of the entire space–time. Nowconsider an arbitrary region, an open set in the usual topology, R, of the space–time. Someof the dynamically possible states of the space–time assign the same state to that region;some of them assign different states. Thus, the states of regions inherit the probability thatthey will be in a given state from the initial probability assignment to states of thespace–time. That is, for all possible states r of R, we can derive the probability that R is instate r, probðr � RÞ. But, now consider another region, S. Via the usual definition ofconditional probability, we can define for each state s of S, the probability that S is in sgiven that R is in r, probðs � Sjr � RÞ for each of possible state r of R.Finally, given that our goal is to investigate what regions of space–time are

indeterminate relative to certain other regions in various space–times, the minimalconstraints on the dynamics are the relations of causal connection appropriate to thespace–time under consideration.

Definition 4.1. caus(S,R) if and only if the state of S could causally influence the state of R.

First, note that in these definitions, regions can be replaced with points as the degeneratecase of regions in the usual topology. Second, Definition 4.1 allows us to define a partitionof the space–time, relative to any given region R, into the causal past, C(R), the causalpresent, P(R), and the causal future, F(R), as follows:

Definition 4.2. 8q; R 9pfðq 2 CðRÞÞ 3 ðp 2 R &causqp &�causpqÞg.

Definition 4.3. 8q; R 9pfðq 2 PðRÞÞ 3 ðp 2 R &causqp & causpqÞg.

Definition 4.4. 8q; R 9pfðq 2 FðRÞÞ 3 ðp 2 R &�causqp & causpqÞg.

It will also be useful to have a name for the entire region from which R is causallyaccessible, i.e. the union of C(R) and P(R): call it A(R). We are now ready to say what it isfor the state of one region of space–time to determine the state of another region.

Definition 4.5. S determines R [det(S,R)] if and only if for each possible state s of S, thereexists a state r of R, such that for some Q � S, Q � A(R) and for the state q of Q inducedby s, probðr � Rjq � QÞ ¼ 1.

Finally, we can return to the question that opened this section, what would it be for thefuture to be open? Given the above, it must be that the future is open if it is notdetermined, in the sense of Definition 4.5, by the past. Which past? Given that, accordingto the previous section, the future is defined relative to our changing space–time location, itmust be relative to our past. Thus, define the following two concepts:

Definition 4.6. A point q0 is relationally indeterminate (RI) relative to a point q1 if and onlyif there is no R � Aðq1Þ such that detðR;q0Þ.

Definition 4.7. A point q0 is determinate relative to a point q1 if and only if there is anR � Aðq1Þ such that detðR; q0Þ.

Therefore, we can finally state that the future is open, in the only sense that seems tomatter, if and only if it is RI relative to our changing location. Or, alternatively, thatan event happens at q0 relative to q1 only when the state of q0 is determinate relative to q1.

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In the next section, I will argue that given a standard and natural reading of specialrelativity, future regions of space–time are, in fact, RI.

However, first let us examine these concepts a bit more closely. In the first instance, wemust distinguish relational indeterminacy from two other concepts with which it is oftenconfused—ontological indeterminacy or indeterminism, and predictive indeterminacy.

Definition 4.8. A point q0 is ontologically indeterminate (OI) if and only if there is noR � ½Aðq0Þ � q0� such that det(R,q0).

This, or a closely related, notion of indeterminacy seems to be what Reichenbach reliedon for his conception of the openness of the future.9 However, critics, includingGrunbaum, quickly pointed out that this notion could not serve Reichenbach’s purpose indistinguishing the past from the future. A point that is OI, alternatively which is notdetermined by its own past, is always so. Thus, consider an indeterministic quantummeasurement. Even if the measurement’s outcome is now in our past, it remains true thatthe measurement’s own causal past does not determine its outcome, and thus this relationcannot ground the difference between an open future and a closed past (Grunbaum, 1963,Chapter 10 for an extended discussion). However, we will discover that the logicalconnections between relational and ontological indeterminacy show that Reichenbach waswrong only to the extent that he took indeterminism to be a logically necessary conditionfor becoming. The second conception with which it is possible to confuse relational

indeterminacy is an essentially epistemic notion. This variety of indeterminacy results fromfailure of information gathering or information utilization by particular observers:

Definition 4.9. A point q0 is predictively indeterminate (PI) with respect to an observer Olocated at q1 and having information I, if and only if I is not sufficient for O at q1 to inferthe state of q0 [PI(q0,O,I,q1)].

The reader should note that in this definition, the indeterminacy may arise from twodistinct kinds of failures on the part of the observer. First, predictive indeterminacy couldarise from failures, either accidental or intrinsic to the nature of the observer, to gathersufficient information about a particular event. Second, predictive indeterminacy couldresult from failures, again either intrinsic or accidental, to carry out necessary inferencesfrom information actually possessed.

The confusion of predictive indeterminacy with relational indeterminacy generates muchof the misunderstanding that leads to various charges that, attempting to connect thecausal structure of space–time with becoming, confuse metaphysical with epistemologicalissues or invokes some form of verificationism.10 This arises particularly because we arethe kind of beings who do depend on causal processes for the information that we possess,and thus relational indeterminacy does lead to predictive indeterminacy. Predictiveindeterminacy almost certainly serves in large part to explain our sense that the future isopen. Thus, as with ontological indeterminacy, predictive indeterminacy has a role to play in

9The precise formulation varied somewhat over the course of his career. However, the basic contention that

only the existence of causal indeterminism could ground becoming seems to have been a constant. See, e.g.

Reichenbach (1925, 1953, 1991).10Perhaps the clearest example of the failure to distinguish predictive determinacy from other forms is Popper’s

The Open Universe (Popper, 1991).

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the explanation of becoming but a less central one than has sometimes been attributedto it.In order to examine some of the consequences of these definitions, consider Fig. 1. The

curves intersecting q0 and q1 mark off their respective (and arbitrary) causal pasts. Thearbitrary shape of those regions indicates the independence of the definitions from anysubstantive assumptions about causal connection. Note that C(q1) contains C(q0). Thisallows us to make the following inferences from the definitions. First, we can see that thestate of q0 must be determinate with respect to q1, since a complete specification of thecausal past of q1 includes a specification of the state of q0. Intuitively, even if q0 is notdetermined by the state of its own causal past, q1 has (in principle) direct causal access tothe state of q0, given that by definition causal processes can propagate from q0 to q1. Next,consider the observer O whose world line is represented by the straight line in Fig. 1. Eventhough q0 is determinate with respect to q1, q0 can be PI with respect to O on theassumption that all of O’s information results from the detection of causal processes atq1 if, for example, no actual causal process propagates from q0 to q1; or O is unable todetect those processes; or O is unable to make use of knowledge about the state ofq0 obtained from detection of those processes. Finally, if we assume for the moment thatonly the complete causal past of a point could suffice to determine the state of that point,independently of the direct specification of the state of that point, then q1 is RI withrespect to q0, since C(q0) is only a proper subset of C(q1).These conclusions are instances of more general logical connections between the

definition. First, we should note that any point, q0, RI to another point, q1, is also PI forcausally obtained information with respect to all observers at that point. Given that acomplete specification of the state of the causal past of a point exhausts the possibleknowledge of, or possible information possessed by, an observer at that point, it is clearthat no observer could possess sufficient information to infer the state of a point RI with

Fig. 1. Applications of definitions to arbitrary space–time.

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respect to that observer’s here and now. Similarly, an OI point, q0, is RI for all points, q1,such that q1eC(q0). Clearly, if no specification C(q0) is sufficient to determine the state ofq0, then neither will the specification of any subset of C(q0) that does not contain q0.

5. Application of the definitions to neo-Newtonian space–time

By themselves, the above definitions merely give us the resources to clarify the issues atstake in these debates. Almost all parties to the debate on the force of Stein’s Theoremseem to agree that different relations of causal connectibility are appropriate for differingspace–times.11 The disagreements seem to be about the metaphysical cost and implica-tions of varying the causal connectibility relations in various ways. However, I claim thatthese definitions allow me to show that we can have univocal concepts of determinacyand indeterminacy whose extensions vary when we vary the relation of causal connection;that these concepts have the expected and desired consequences given natural neo-Newtonian and Minkowskian relations of causal connection, respectively; and finally,that they can ground a significant distinction between past and future in at least somespace–times.

The first step in this process is to show how to apply these definitions in classicalspace–time. To that end, we need a definition of the causal past in neo-Newtonianspace–time. Two aspects of neo-Newtonian space–time have natural and immediateinterpretations in terms of classical physics: the simultaneity structure and the affinestructure. The presence of instantaneous action at a distance determines the simultaneitystructure; all and only those time slices of objects (or objects at a time)12 that can exertgravitational force on a given object at a time are simultaneous with it. Newton’s first lawof motion, that force-free objects follow straight lines or affine geodesics in the space–time,determines the affine structure. These two principles exhaust the physical interpretation ofthe space–time since they allow us to define in principle methods for the measurement ofthe intrinsic metric features of the space–time, absolute temporal intervals and absoluteinstantaneous spatial separation.13

Next given the Definitions 4.2–4.4, the simultaneity slices constitute the causal presentfor each point on them. Given this, we can determine which points are causally earlier orlater than each simultaneity slice yielding a definition of the causal past of a point, q0, asthe shaded region in Fig. 2. Given these assumptions, it follows, first that causal effects canreach q0 from all points below the plane of simultaneity, and no causal effects reachq0 from points above the plane of simultaneity. Therefore, the results of the previoussection tell us that all points earlier than, or beneath the current plane of simultaneity inFig. 2, are determinate relative to q0.

Given this, we can consider the various ways that point q1 could be RI to q0 (see Fig. 2).There are essentially three ways in which q1 could fail to be determinate relative to q0.First, there could be a causal influence on q1 that fails to intersect C(q0). Such a causalprocess would have to originate at infinity at a time later than that of q0 and reach q1 in a

11But, see Tooley (1997).12For this particular purpose, these two formulations are equivalent. The point here does not depend, so far as I

can tell, on one’s metaphysical commitments on the nature of temporally extended things.13The fact that classical gravitation is not only instantaneous, but also universal has serious consequences for

the actual application of these methods since this implies that all actual affine geodesics are empty since there is no

actual inertial motion. See Torretti (1996), especially Chapter 1, for a careful discussion of this subtle problem.

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Fig. 2. Relational indeterminacy in classical space–time.

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finite time. This is essentially the so-called ‘‘space invaders’’ problem that bedevils allattempts to formulate determinism in the context of classical space–time. Here I point outonly that the problem disappears in special relativistic context where we tend to postulateexplicit limits on the rate of propagation of causal influences.The second way in which q1 could fail to be determinate is if there is some point

(e.g., q2 in Fig. 2) to the future of q0 which is OI and that has a causal influence onthe state q1. Thus, even if q1 is itself ontologically determinate, there is no subset ofC(q0) sufficient to specify the state of q1. Third, q1 could itself be OI. What thisdemonstrates is that, barring ‘‘space-invaders,’’ indeterminacy can enter a classicalspace–time only through the presence of ontological indeterminacy and only to the futureof some point.This fact about classical space–time combined with the supposed determinism of

classical physics has extremely important consequences in the philosophy of time. Here, Iwill focus on two of them. First, at least as far back as Laplace and Kant, this fact hasdriven philosophers seeking to ground the apparent openness of the future away from thephysics of space and time and towards more radically metaphysical ways of approachingthe problem. Thus, since classical physics seems to bar any access to an account ofindeterminacy in terms of ‘‘failures’’ of causation, this has pushed philosophers to attemptto account for the openness of the future by appeals, for example, to existence as therelevant criterion. This also shows what was right about Reichenbach’s conception ofbecoming. If we are committed to a classical picture of causal structure, then ontologicalindeterminacy seems to be the only way to generate relational indeterminacy in theuniverse and open up the future. However, since Augustine apparently defended such aview 1300 years before the advent of classical physics, the apparent lack of alternatives thatseems to result from classical physics is certainly not the only reason philosophers have hadfor defending an ontological conception of becoming. The implicit reasoning seems to runas follows. If the future is both necessary and determined, then the only way in which it canbe distinct from the past is by virtue either of existence, the ontological view, or someuniquely temporal features of the world, the pure tense view. The problem is, as weshall continue to see, that neither of these conceptions extends naturally to Minkowskispace–time.

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6. Application of the definitions to Einstein–Minkowski space–time

We can now consider how the definitions of Section 4 apply to Einstein–Minkowskispace–time and provide a ground for real becoming. What we will discover is that with adefinition of the causal past appropriate for special relativity, time-oriented Einstein–Minkowski space–time comes equipped with a natural distinction between the past andfuture specified in terms of relational indeterminacy. Next, we will re-visit the connectionbetween relational and ontological indeterminacy. I will argue that, given another sourceof relational indeterminacy, becoming is independent of the presence of ontologicalindeterminacy in the space–time. I then consider two objections to connecting this kind ofa theory to temporal becoming. The first objection directed by Grunbaum againstReichenbach is that failure of causal determination cannot ground becoming because theydo not allow for the necessary ‘‘flow of time’’(cf. Grunbaum, 1963, Chapter 10). Thesecond objection, which will be one of the topics of the next section, is the solipsismobjection mentioned in Section 2.

All standard physical interpretations of Einstein–Minkowski space–time take the causalpast of each point as the past null-cone of that point and its interior (Fig. 3). In addition,P(q) consists only of q itself. Given this definition of AðqÞ, we can determine by inspectionthat any point not itself contained in AðqÞ has points in its causal past not in AðqÞ. Theproblem of the relational indeterminacy of such points therefore depends on how much oftheir causal past must be contained in AðqÞ to determine them. Thus, consider a pointpeAðqÞ. If p is OI, then p is RI to q. If even the entire AðqÞ is not sufficient to determine p,then clearly no subset of it, and thus no subset of AðqÞ, is. Next, consider the status ofpoints that are not OI. If AðqÞ does not itself contain any OI points, then any cross-sectionof AðqÞ will serve to determine p, as long as all of the causal processes leading to p areMarkov processes. In probability theory, a discretely ordered process is a Markov process

Fig. 3. Relational indeterminacy in Einstein–Minkowski space–time.

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if, and only if, the probability of (n)th element of the process conditioned on the ðn�1Þthelement is the same as the probability of nth element conditioned on all previous process.Equivalently, we can say that the ðn�1Þth element screens off (n) from all other elements ofthe process. For continuous space–time processes, this translates in the dependence of ponly on points infinitesimally close to p. However, given that AðqÞ does not contain even acomplete cross-section ofAðpÞ, there are points inAðqÞ not determined byAðqÞ. From thiswe can see that all such p are RI to q.This conception incorporates the space-like regions relative to each point into the

indeterminate future. However, this assimilation is not complete: only the causal future(the future light-cone and its interior) can be affected. What happens in special relativity isthat two different ways of thinking about the present that tend to run together in classicalspace–times become forced apart. One way of thinking about the present is as the regionjust now becoming determinate; in this way of thinking, we treat it as the boundary of thefuture and like the future, still changing and becoming. This aspect of the present has beenemphasized in the previous paragraph. However, the present has another aspect that tendsto assimilate it to the past. From this perspective, because it is becoming right now, it is toolate to do anything about it, just as it is with the past. The space-like regions retain thisaspect of the present, and it distinguishes them from the true future. As the boundarybetween past and future, this expanded present shares some features with both regions.Perhaps the most important aspect of this situation to note is that once we treat space–time

as Minkowskian, the existence of real becoming no longer depends on the existence ofontological indeterminacy. Once we add a causal arrow, the necessary indeterminacy fallsnaturally out of the causal structure of the space–time. This is important because it allows usto respond to one strand of Grunbaum’s objections to this kind of a picture. Once again, inChapter 10 of Philosophical Problems of Space and Time, Grunbaum presents an argumentintended to show that the failure of determinism cannot ground becoming. I take theargument to be as follows: for any given event, either that event is or is not determined by itsown causal past. Whether the event is or is not so determined does not depend on what thehere and now happen to be. That is, what I have called ontological indeterminacy is aparadigmatically tenseless fact about events and thus cannot ground becoming.As we saw above (cf. the discussion of ontological indeterminacy below Definition 4.9)

Grunbaum is correct as far as he goes. Ontological indeterminacy, or indeterminism,cannot fill the role into which philosophers such as Reichenbach have tried to force it.However, relational indeterminacy can. Relational indeterminacy allows a uniquespecification of the past and future relative to each point. Again, Grunbaum complainsthat even so, such a distinction does not serve to pick out a unique now. Again, I agree.However, I do not think that such a unique now is necessary for objective becoming. Thenow is defined only relative to particular occupant of the space–time, including humanbeings. It is exactly the change from RI to determinate that is the objective counterpart tothe subjective sense of temporal passage noted by, e.g. Bergson (1949), Taylor (1992) andso many others.14 Nevertheless is this really a theory of objective becoming? This is thetopic of our final section.

14The only philosopher I am aware of who has come close to making this point is Sir Karl Popper in The Open

Universe (Popper, 1991). However, Popper formulates his conception of indeterminism in terms of predictability,

which camouflages the objective character of the distinction. It also does not allow him to make the crucial

distinction between ontological and relational indeterminacy.

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7. Conclusion: towards a neo-Aristotelian metaphysics of time

Let me recap the conclusions so far. I have argued (Section 3) that the semantics of tensecombined with the physics of relativity provide us with good reasons for adopting localrather than global temporal concepts. Then, in Sections 4 and 5 I have argued that once weadopt this local conception, we have a perfectly natural sense in which Einstein–Minkowski space–time supports a distinction between a closed or determinate past and anopen or indeterminate future. However, this claim has been subject to two principalobjections; objections which on the surface pull in different directions, but which I believeactually flow from a common, but deep, philosophical mistake. Pulling in one direction is aclaim that I have not proved enough. On this objection, I have not provided an account ofan objective past–future distinction; I have merely accounted for the mistaken belief insuch a distinction. From the other direction comes the claim that I have proved too much.In searching for the indeterminate future, so it is claimed, I have made everythingindeterminate and committed myself to some form of radical skepticism or solipsism.

While it is relatively easy to see what is wrong with these particular objections, it is moredifficult to see what the problem is with the motivations behind them. This is because theseobjections really flow from a meta-philosophical position so common as to be largelyinvisible; this is the commitment to a conception of reality, call it the Platonic conception,such that anything real must be real all the way up (or down, pick your metaphor). Untilwe have a clear picture of how this conviction infects the philosophy of time and of itspernicious consequences, we will never have a clear understanding of the nature ormetaphysical role of time and temporality.

Let us first consider the two specific objections. Have I not simply proved that becomingdepends on the existence of human beings and human language and thus is not a realfeature of the universe? However, nothing in my account of the past–future distinctiondepends on human subjectivity. Certainly, if there are no human language users, then thereare no tensed statements in human languages. Nevertheless, the truth makers for thosestatements might very well still exist. Tenses depend not on anything about humansubjectivity but merely on our existence as a certain kind of spatiotemporal entity. On thisaccount it is just as true that my chair has a determinate past and an indeterminate futureas it is that I do. The only difference is that I can state that I do and know that I do, butthis is simply the tautology that only language users and knowers can state or knowanything. It may well be true that from some eternal non-spatiotemporal perspective, aperspective that I can only just glimpse through mathematical physics, the past–futuredistinction dissolves. But, until I can occupy that perspective, not merely glimpse itsexistence, I am still here and now a being with a past and a future.

However, the fact that I can glimpse that perspective and that this plays a role in myaccount of temporality is important for understanding what my precise claims are andavoiding the charge that I fall into radical solipsism or skepticism. The problem of eventsolipsism is the most easily addressed. There are two aspects to the response. First, once weare committed to the space–time perspective, we are committed to the existence of theentire space–time, including the space-like separated regions. In whatever way the pastlight-cone exists, so do the space-like wings at each point. However, this does nothing toaddress the problem of the reality of the supposed occupants of the space-like wings. First,our own past commits us to there being something going on in the regions space-likeseparated from our here and now. Causal processes exist that intersect our own causal

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process and pass out of our region of causal accessibility. We reasonably believe that suchprocesses continue to have effects in those regions just as we would expect. The onlydifference is that the exact nature of those effects is not yet determined for us.Here, I will emphasize once again. This is not an ontological theory of temporal becoming.

I do not believe that there is an ontological difference between the past and the future; theonly possible difference between the past and the future is a difference in their relationship

to us. An ontological difference between the past and the future is neither a necessary nor asufficient condition for temporal becoming. Not necessary, because I have claimed that allwe have a reasonable claim to expect from a theory of becoming is met by the theoryproposed here. Not sufficient, because the mere non-existence of future entities still doesnot explain what it means to claim that the future is a locus of multiple possibilities.15

As far as I can see, there is nothing strange about this. We should probably expect fromthe discussion at the end of the previous section that statements about the space-likeregions relative to any particular here and now are assimilated to the kinds of claims thatwe can make about the future, that is, to predictions. Moreover, we should be used to thefailure of our predictions at this point, to fallibilism. What special relativity does is to takea fallibility usually attributed merely to our own subjective epistemic failures—to predictive

indeterminacy—and shows us that sometimes that failure is not subjective but objective,built into the world and not into us, and arises from true relational indeterminacy.This leads into the charge of skepticism. First, if philosophers have not yet learned to

distinguish fallibilism from skepticism, it will take a better epistemologist than I am toteach them. Second, if this is skepticism, it is of a rather atypical variety. The classicalskeptical arguments begin with our failures; we cannot distinguish dreaming from waking;our perceptions can be in error and thus should never be trusted; and so on. This makesthis position a strange variety of skepticism because it begins with a claim about the

external world. It is based on a theory that starts with all of the things that a classicalskeptic mistrusts and invokes a sketch of a theory about causation that a Humean skepticwould find entirely unjustifiable. That is, it is a positive claim about the nature of theexternal world, not a negative claim about our capacities to know things about that world.This indeterminacy is a fact about the world that tends to manifest itself to us in termsof our epistemic failure, but the fact about the world and not the epistemic failure isprimary.Alternatively, we can think about the situation as follows. We believe that skepticism

is problematic because it seems to have certain pernicious consequences. Philosopherssearch for ways around skeptical arguments because they prohibit us from havingsomething that we, in fact, seem to have, i.e. objective knowledge of the external world.However, while I do not want to legislate language, calling the position developed here avariety of skepticism seems misleading at best. This is because my position does not havethe kind of pernicious consequences usually attributed to skepticism. It cannot preventus from having objective knowledge of the external world because it begins with suchknowledge.Note that nothing here contradicts my response to the previous objection. I can glimpse

the existence of this eternal 4D perspective, and the power of the formal geometricrepresentation of this perspective provides powerful reasons for accepting it as the best

15For a more complete discussion of the meta-philosophical issues raised here and below see my ‘‘What kind of

problem is the problem of time?’’ (Unpublished).

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available account of the fundamental nature and structure of space and time. However,I am still a being within space–time, and, as such, I have a determinate past and anindeterminate future, given that I exist within an appropriate space–time. To really seewhat is going on here we need to move up at least one level of abstraction.

I suggested above (Section 2) that the alternative to the theory of temporality developedhere is a radical Parmenidean or neo-Eleatic conception. Moreover, in the secondobjection above, we can almost hear the shade of Parmenides or Zeno pointing out that ifwe try to fudge, we only end up with non-sense. However, the first objection has almost thesame structure; here, the defenders of radical becoming, the neo-Heracliteans, are pushingan equally strong objection to compromise from the other direction. If the neo-Eleaticsrefuse to admit anything temporal to the universe, the neo-Heracliteans are equallyadamant in refusing admittance to anything atemporal.

However, it is perfectly clear, going back to the responses to the original Eleatics, thatcompromise is possible. Formally, the move advocated here is precisely the response of theIonian physicists, the ancient Atomists and, most importantly, Aristotle in response to theoriginal Eleatic arguments of Parmenides and Melissus. Loosely, an Ionian four elementphysicist, such as Anaximander, accepts as equally real both a chair’s coming intoexistence through changes in the relationship between the essentially changeless andatemporal elements and the changeless and atemporal nature of those elements.16

For our purposes, though, Aristotle made the crucial move when he engaged directlywith the problem of temporality, distinguishing it from the related problem of changethat had preoccupied earlier Greek philosopher–scientists. As is well known, Aristotledefines time as ‘‘the quantity associated with change.’’ This definition hardly bearsrepeating except for an important aspect of Aristotle’s theory of change regularlyoverlooked in this context. Change, for Aristotle, is fundamentally the combinationof Forms, themselves essentially timeless and changeless, with Matter, itself incapableof change, to generate substances. Thus, Aristotle is a realist about time although hedoes not believe that the universe is temporal all the way down. More picturesquely butnot too inaccurately, given Aristotle’s natural teleology, we can envision an Aristotelianuniverse as the essentially temporal expression of the underlying atemporal reality inmatter.

Of course, no contemporary theory of temporality that claims grounding in modernphysics, as this one does, can simply take over Aristotle’s account of change andtemporality: our physics is not Aristotle’s. However, the theory presented above deservesto be called neo-Aristotelian in the following sense. The modern description of space–time,and thus the universe, as fields distributed on a 4D differential manifold, provides aso-called ‘‘God’s Eye view,’’ a glimpse of being qua being sub specie aeternitas. In this, itprovides a perspective not dissimilar to that of theoria in Aristotle’s philosophy. But it isequally true, whether on Aristotle’s account or mine, that we and the things around us arenot eternal. To particular beings within space–time, including but not limited to humanbeings, certain other beings are causally accessible and others are not; are determinate orindeterminate; are past, future or neither.

16For the Eleatics and the Ionian and Atomist response see especially Barnes (1982). This interpretation of this

important moment in the history of philosophy is certainly not without problems. It is, however, clearly plausible,

and more importantly serves to illustrate a theoretical move that I am making, whether or not, the relevant pre-

Socratics actually made it.

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Acknowledgements

Much of the original research for this article was performed while the author was Dean’sScholar at The University of Illinois at Chicago and first appeared in my dissertation,‘‘Local Time: Foundational Studies for a Theory of Relativistic Temporality’’. Jon Jarrett,Nick Huggett and Bill Hart were particularly instrumental in helping me to formulate thekey ideas. Pieces of the argument were presented to various groups over the years and aremuch better because of their comments and questions. Finally, two anonymous referees forStudies in History and Philosophy of Modern Physics made many useful comments. Anyremaining mistakes and infelicities of expression are, of course, solely the fault of theauthor.

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