A Century of Time

I

Time has always been difficult to understand, but in the twentieth century our un-understanding has become clearer, partly as a result of the failure of sustained attempts to give non-temporal accounts of time, partly under the stimulus of questions posed by the physical sciences.

We find it difficult to talk coherently about time. We experience time as passing, but need to nail it down, if we are to talk about it to different people on different occasions. Various metaphors suggest themselves---time as an ever-rolling stream, time as the passage from potentiality through actuality to unalterable necessity, time as a fleeting present reality balanced precariously between a future that is not yet real and a past that no longer is. But each metaphor has its own puzzles and paradoxes: how fast does time flow? what sort of modalities are involved, and how can they change into one another? is reality a property the present can acquire and then lose? We want to talk accurately and scientifically, and to eschew metaphors, concentrating on features that stay reliably the same from one person to another and from one occasion to another. But if we confine our discourse to non- metaphorical accounts of what stays reliably the same from one person to another and from one occasion to another, we leave out the temporality that is peculiar to time.

II

In the first decade of the twentieth century McTaggart argued that time was unreal on philosophic grounds. 1 He claimed---correctly---that time essentially involved tense, and---incorrectly---that tenses were inherently self-contradictory. Every event, he said, had to be both future, present and past, but these were incompatible. 2 He concluded that only tenseless discourse was free from contradiction, but since this left out the specifically temporal features of time, no consistent account of time could be given. In discussing this contention, we need to make three distinctions. We need, first, to distinguish two claims: a strong one that tensed terms---those belonging to the A series in McTaggart's terminology---are sufficient for all temporal discourse; and a weaker one that the A series is a perfectly consistent part of temporal discourse which also includes tenseless discourse---the B series in McTaggart's terminology. We need, secondly, to distinguish the claim that the A series is not self-contradictory from the exercise of giving an adequate philosophical exegesis of it. And, thirdly, we need to distinguish the tensed uses of verbs from a tenseless one, first noticed by Plato, 3 and now widely used in mathematics and the natural sciences. I adapt a suggestion of Professor Smart, who italicised verbs used in a tenseless way; 4 I not only italicise them but put them into the infinitive, saying that two and two be four, in contrast to the locutions that it was cold last night, is getting dark now, and will be warm next summer. Once we make this distinction, we no longer have even the appearance of inconsistency in our philosophical exegesis of tensed language. It is indeed true, as McTaggart claims, that every event at some time be future, at some time be present and at some time be past (there is no objection to using untensed language to talk about time), and this may well be a key part of our account of temporality. But it does not follow from this that at some time it be both future, present and past. The A series is a perfectly consistent part of temporal discourse. It may still be that if the B series is forbidden us, then we may be unable to talk philosophically about time; and indeed, life would be awkward if we had no dates, no calendars and no diaries, but discourse exclusively in terms of the A series would not be self-contradictory. It would still be true to say that for any event it was once future, either was or is or will be present, and will be past, which is no more self-contradictory than to say that I was once a boy, either was or is or shall be in employment, and shall be an old-age pensioner. McTaggart seeks to generate an inconsistency by following down an infinite regress of ever more complex tenses, but if we stick to tensed language there is not even an appearance of inconsistency. McTaggart tries to generate one by construing these tensed locutions tenselessly, and each time the ensuing inconsistency is resolved by distinguishing the tenses, construing the resulting, more complexly tensed, locution tenselessly again. That does, indeed, produce an infinite regress. But it is one generated by McTaggart's seeking an all-time philosophical exegesis, not by the incoherence of tensed discourse. At most it could show that we cannot give a tenseless account of tensed discourse. Maybe not. But that is a far cry from showing that tensed discourse is itself incoherent, or that the combination of tensed and tenseless discourse we ordinarily use is in any way defective.

Although McTaggart's argument is fallacious, it has been defended by a number of distinguished thinkers. 5 An underlying unease about tenses and temporal indicators such as ‘now’, ‘yesterday’ and ‘in three weeks time’, has been widespread. 6 They are token-reflexive, or indexical, terms, like ‘I’, ‘this’ and ‘here’; they depend on the context of utterance, and so are seemingly subjective. Russell called them ‘egocentric particulars’. If we would be completely objective, we should purge our discourse of all such terms, using only dates to secure specific temporal reference, embedded in a language that was itself either timeless or omnitemporal, so as to disengage ourselves from the transitory flux of the here and now, and be spectator of all time.7 The view from “Nowhen”, is the view that scientists do, and philosophers should, adopt.8

The urge is powerful but misguided. In order not to be bogged down in subjectivity, we do not need to go to a Platonist extreme of timeless, impersonal and unlocated truths; and if we do confine ourselves to them alone, we cannot give a complete account. Rather than ignore every feature that is not the same from every point of view, we need to register the importance of points of view, and in considering each, consider also how its perspective would alter if any other point of view were adopted. We need to conjugate. Verbs conjugate in tenses much as they do in persons. If I now say “Yesterday I was in Grantchester at ten to three”, then yesterday at ten to three, in order to say the same thing, I should have needed to say “I am in Grantchester now”, much as you would have to say “Yesterday you were in Grantchester at ten to three”, or, speaking then, “You are in Grantchester now”. Provided we conjugate correctly, we compensate for the different temporal or personal standpoints being adopted. In the Special Theory we similarly compensate for the different frames of reference being adopted. We express the underlying sameness of structure manifested in different frames of reference by covariance. Although I in my frame of reference assign to a particle a different energy from the one you do in yours, our assignments vary systematically with the difference between our frames of reference. We do not have to confine ourselves to invariant magnitudes, such as spacetime separation, though they naturally are of great importance. Similarly in ordinary discourse, we do not need to confine ourselves to timeless, impersonal and unlocated discourse, in order to transcend our individual temporal and personal standpoints, although dates and proper names, often being more precise and usually more widely understood than token-reflexives, are important. But not all-important: if we confine ourselves to timeless, impersonal and unlocated discourse, we leave out the links with experience that make our discourse relevant. If I am in a strange town and see a map posted, I need the arrow with the caption “You are here”. It is not enough to have a complete guest list---we sometimes have to ask “who are you?”, and need an answer in the first person. So, too, with time. A dating system is no good, unless we can know what the date is now; a story that begins ‘Once upon a time’ is as clearly fictional as one beginning ‘A certain man went down from Jerusalem to Jericho’. It matters very much whether an unpleasant experience is going to happen to me, or is already over,9 just as I am much more alarmed if it is going to happen to me rather than to someone else who has been brain-washed to have all my memories and to think he is J.R. Lucas.

McTaggart argued that the A series, though inherently self-contradictory, was essential to time. Most of those who would purge our discourse of tenses, however, have held that we can achieve an adequate account of time without recourse to tenses. The claim is suspect. If we confine ourselves to timeless, impersonal and unlocated discourse, we may leave out not only the links with experience that make our discourse relevant, but also those that make it meaningful: a tenseless account of time may be no more temporal than an impersonal account of personality is personal. But time is connected with many other fundamental concepts, and it may be that we can adequately characterize time in terms of change, cause, or experience.

Aristotle famously defined time as the number (arithmos---he should have said metron, measure, for he was normally sensitive to the distinction) of change with respect to before and after.10 ‘With respect to before and after’ is significant. He could not define time simply in terms of change, for then we should not be able to distinguish one and the same poker's being red at one spatial extremity and black at another from its being red at one time and black at another time. In our normal way of thinking time is connected with the possibility of change, against a background of continuing identity over time: we think of things that may in some respects be different at different times, while in other respects remaining the same. It is natural to go further, and claim that experience would not be possible unless it could be experience of things that could be recognised, in spite of change, as being the same things as herebefore. But Walker has constructed a model of a non-temporal, purely spatial experience in which a pattern could be recognised, in spite of interruption, as being the same pattern; essentially, a spatial parameter could differentiate one instance of an underlying entity from another, while a putative spatial continuity could secure identity.11

Time is intimately connected with cause. Granted an asymmetric transitive relation of causal influenceability, Robb developed a theory of “conical order”, giving rise to the light-cone structure of Minkowski spacetime. He needed to assume that space has more than one dimension and that there is a finite maximum speed at which causal influence can be propagated---the speed of light---and then was able to define orthogonality, and even the entire metric, and show that spacetime had a Lorentz signature.12 His work shows how far we can go in developing a causal theory of time: we can derive the general structure of spacetime and even its metric, in a way that makes it perspicuous why timelikeness is different from spacelikeness, and can be characteristic of only one dimension.13 But the asymmetry of the relation of causal influenceability can be questioned: why should teleological explanations---explanations that look towards the future---be ruled out?14

We can accommodate teleological explanations, and distinguish them from ordinary causal ones, if we ground our concept of causality in that of agency---my bringing it about that a desired effect should occur. Our ordinary experience is not just a passive solipsistic experience but of ourselves being agents deliberating about what to do in the future and owning up to what we have done in the past. To be a person is to be an agent---ego ergo ago---and to use the first person singular is to allow the use of the second person by others, with the possibility of communication, in which we ask questions in order to elicit answers. Equally the use of the first person plural presupposes communication between us whereby we reach agreement about the reasons we shall act on. Communication presupposes time.15 And though Walker's purely passive experience is describable, the description is one of successive acts of attention as the subject identifies first one and then another instance of pattern or colour supposedly continuing behind an intervening item of the visual field. Moreover no auditory experience could be non-temporal, nor tactile experiences of throbbing or vibration. It seems to be a concomitant of consciousness to be aware of time passing, and we talk of time standing still at times of great exaltation or extreme terror. The meagreness of the non-temporal experience described reveals the importance of temporality in experience as we ordinarily experience it. We may well be able to characterize time in terms of experience, provided it is reasonably rich, or in terms of causality, provided that it is in turn explained in terms of agency, but these are inherently personal concepts, not free from the “taint” of token-reflexivity; and a full account of what it is to be a person is ineliminably temporal, with present, past and future as necessary parts of our conceptual structure.

III

During the twentieth century physics often seemed to require a tenseless account of time. Minkowski's spacetime lent strong support to a “block universe”, in which time was a fourth dimension, analogous to the three dimensions of space, with the future already in existence, just waiting to occur. We picture world-lines in Minkowski spacetime as already timelessly existing and ourselves as going along a world-line encountering our pre-determined future as it takes place---in Weyl's words “The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the lifeline of my body, does a section of this world come to life as a fleeting image in space which continually changes in time.”16 Again, the integration of time with space into spacetime suggests that time is not an independent dimension of special importance, but only on a par with the dimensions of space. And, finally, Einstein's Principle of Relativity seems to make the distinction between past, present and future entirely relative to our arbitrary choice of a frame of reference.

These arguments have been accepted by many, but can in each case be countered. The first is not so much an argument as the illusion of a picture. Although Minkowski spacetime, a four-dimensional manifold with Lorentz signature, seems more integrated and solid than a Cartesian product of a three- dimensional space with a one-dimensional time, exactly the same picture can be imagined if we think of a world-line in Newtonian space-and-time---the Galilean transformation can be seen as just a special case of the Lorentz transformation in which the speed of light is infinite. Both are available, both misleading, both non-obligatory.

Much more important than the picture is the integration itself, and the fact that whereas temporal duration was an invariant quantity under the Galilean transformations, it is not under the Lorentz transformation, which preserves only spacetime separations, but not either spatial ones or temporal ones separately. But even though Minkowski spacetime can be described as a four-dimensional manifold, its metric (and hence also the topology based on it) is radically unlike that of a Euclidean four-space. The Lorentz signature (+++—) marks a fundamental difference between timelike directions and timelike separations on the one hand and spacelike directions and spacelike separations on the other. Mathematicians are tempted to discount this difference between timelike and spacelike separations---they can transform Minkowski spacetime into a simple four-space by setting t=ix4. True, but that is not a transformation of no consequence; a transformation that converts simple exponential functions into periodic ones is clearly a highly significant one. If there were some fundamental periodicity which underlay the fundamental rhythm of the universe, some ubiquitous chronon say, then we might look to some cis(x4) [short for cos(x4) + isin(x4)] to explain it in terms of a manifold more fundamental than time. But so far as we know, there is no preferred period, even though there are many rhythms all in harmony with one another; and the details of any such explanation are altogether obscure. However much time is treated as a fourth dimension, it remains in all applications a significantly un-spacelike one.

Although time is not spacelike, the integration of time and space has an important bearing on the direction of time. The directedness (or anisotropy) of time has long been a problem for physicists. Clearly, tenses confer a direction on time, but it is possible to have time directed without distinguishing past, present and future, and most physicists, though unwilling to countenance there being any fundamental distinction between temporal instants or intervals, have acknowledged that there is a fundamental anisotropy in time.17 But the anisotropy of time is difficult to reconcile with the fact that Newtonian mechanics, and likewise electromagnetism, are time-reversible: if we substitute —t for t in the equations, the result is the same. Yet that is not how things happen---trees grow, plates break, fires radiate energy, milk mixes with tea, but not the reverse: we can tell at once if a film is being run backwards. It was a great problem for classical physics how to re-insert “the arrow of time” into a physics that was apparently isotropic with respect to time. The Second Law of Thermodynamics---that entropy increases---could be explained as a feature of boundary conditions. Causally antecedent conditions were more precisely specifiable than their effects, and so had higher negative entropy. If events were generally explicable, and if the sort of explanation that every event ought to have was a causal explanation, then just as the direction of explanation led to conditions of higher and higher negative entropy, so the passage of time led to conditions of higher and higher positive entropy. But discomfort remains: the concept of explanation seems dangerously anthropocentric, and in the long run entropy should decrease just as often as it increases. Twentieth-century physics, however, embeds the anisotropy of time more deeply in the structure of the universe. For a reversal of the direction of time would involve a change of parity, since the time-like dimension is integrated with a number of space-like dimensions that are all on a par. Professor Dummett could make warriors to have been brave and rain dances to have been performed, provided that their hearts were on the right and they danced their reels counter-clockwise.18 Indeed, there could be an instantiation of the laws of physics in which time was reversed, but it would be a looking-glass world that was, granted orientability, inaccessible from ours.19 Indeed, the invariance goes deeper, involving charge as well as time and parity: but the implication is the same; although it may be convenient on occasion to think of an electron moving backwards in time rather than a positron moving forwards, the difference between the two directions of time is as great as that between positive and negative charge.20

The anisotropy of time argues against other speculative possibilities that can be envisaged, such as (++——) or (+++——), which involve a two-dimensional time. But then we should lose the directedness of time; on a two-dimensional surface I can rotate an object so as to have any particular pointer pointing in the opposite direction. Still, it might be objected, the orientation would be preserved, provided the two-dimensional space was orientable; so, once again, it seems, Professor Dummett could make warriors to have been brave and rain dances to have been performed, provided that some super time also went backwards. But, quite apart from there being no warrant for more than one time-like dimension, it is not some two- dimensional orientation but the direction of our time that is in issue. And what has really been shown is that if time is anisotropic, then it must also have only one dimension.

The distinction between past, present and future seemed to be undercut by the Special Theory, which made them relative to the frame of reference. According to one frame of reference a distant event, say on Betelgeuse, would lie in the future, but according to another in the past; whether it was future or past would not be something intrinsic to the event, but would depend on how the frame of reference adopted was moving. So there could not be any modal or ontological difference between future and past---we could not say that the future was open and the past fixed, or that the future was unreal and the present real.21 But that was to extrapolate the Principle of Relativity beyond its proper sphere. The Special Theory ascribes dates to distant events in such a way as to harmonize the symmetries of the Maxwell equations. So far as the Special Theory is concerned, distant events are distant. I cannot be there to know what happens when; I can only ascribe a position and a date on the strength of causal influences transmitted with a velocity no greater than the speed of light. If we ascribe them in accordance with the Lorentz transformation, then the laws of electromagnetism will come out the same whatever inertial frame of reference we adopt. That is a good reason for adopting the Lorentz transformation as our way of correlating positions and dates in different inertial frames of reference, but it does not follow that that is the way we must ascribe positions and dates to distant events, and that no other ascription could be correct.22 The point can be illustrated by a backward glance at Newton's belief in Absolute Space. Newtonian mechanics is relativistic. Galileo's transformations come out the same in any uniformly moving frame, so that it is impossible to tell, within Newtonian mechanics, whether we are moving with respect to Absolute Space or not. Newton recognised this, but maintained, correctly, that Absolute Space was nevertheless a coherent notion, and opined that the centre of the solar system might be at rest in it. Later, when electromagnetic theory was being developed, Michelson and Morley sought to determine the rest frame of the ether, and if they had succeeded, we should have undoubtedly taken that as being at rest in Absolute Space. Similarly now, if some superluminal velocity of transmission of causal influence were discovered, we should be able to distinguish frames of reference, and say which were at rest absolutely and which were moving. If, for instance, we were able to communicate telepathically with extra-terrestial beings in some distant galaxy, or if God were to tell us what was going on in Betelgeuse now, then we should have no hesitation in restricting the Principle of Relativity to the phenomena of electromagnetism only. Hence it cannot be an absolute principle foreclosing absolutely any possibility of absolute time.

These are not mere possibilities. In many of the models that cosmologists use---solutions of the field equations of the General Theory---there is a world-wide cosmic time that flows, if not evenly and uniformly, at least generally and universally. There are thus also preferred hyperplanes (not necessarily flat) of simultaneity constituting a world-wide present and separating a real unalterable past from a possible future not yet actualised. The Special Theory is countered not just by the General Theory, but more fundamentally by quantum mechanics. It is difficult to make sense of quantum mechanics, and the interpretation I adopt is rejected by many. But it is natural to try and interpret it realistically, and to construe the collapse of the wave packet as a real event in which the many possible eigen-vectors, with their associated eigen-values, give way to a single eigen-vector with one definite eigen-value. If this is so, there is a definite moment of truth when possibilities become definitely true or definitely false. There is a fact of the matter, quite independent of whether we know it or not, and of how and when we know it. Knowledge may be unable to travel faster than the speed of light, but reality does not have to travel at all. Galaxies may be thousands of light-years away, and we shall be able to assign a date to a particular collapse only thousands of years after the event, and our assignment may well depend on an idiosyncratic choice of frame of reference, but nonetheless there will have been a definite moment at which the event occurred quite independent of any frame of reference. There is a world-wide tide of actualisation---collapse into eigen-ness---constituting a preferred foliation by hyperplanes (not necessarily flat) of co-presentness sweeping through the universe: a tide which determines an absolute present. In this, quantum mechanics goes beyond anything that thermodynamics and cosmology suggest, which, although witnessing to there being a definite directedness in time, do not pick out any particular time as pre-eminently real---the moment of truth when possibilities become actual or elsefade away. Quantum mechanics, however, does. Although we still do not understand it, and its interpretation is still much disputed, it now appears to be irremediably probabilistic, and not only insists on the arrow being kept in time, but distinguishes a present as the boundary between an alterable future and an unalterable past.

Quantum mechanics tells against projectivism. Projectivists acknowledge the temporal character of experience and the ineliminably tensed aspect of discourse, but attribute it to an anthropocentric viewpoint we cannot help taking up. Hume made out that the necessity of causal connexion was not something we discovered in the world, but something we imposed on our experience of the world, and Mellor holds that tense is likewise projected by us onto an untensed system of dates so that we can act in a timely fashion when the situation calls for it.23 Tenses are like secondary qualities. We paint them on the world, but are led to do so by reason of our being entropy-increasing organisms and agents that need to know when the time is ripe for action. Such explanations cannot be dismissed out of hand, but as Mackie admits, when explaining our ascriptions of moral terms as being merely the projection of our attitudes, such theories are “error” theories: they claim that our untutored accounts of our experience and of what we are doing when we make temporal or moral judgements, are deeply and systematically wrong. We could all be mistaken, but the burden of proof lies heavily on those who would convict us all of error. If we resist Hume's making out that cause is merely a necessity in our mental processes brought about by long conditioning, we should equally resist the contention that the temporality of time resides solely in the mind's eye as the agent wonders where next to increase disorder. Although explanations that explain experience away can be cogent, a recalcitrant empiricism demands that our theories of the world should explain how things actually appear to us. Experience seems to be temporal. And the most natural explanation of experience being temporal is that reality is, and our experience simply mirrors the temporality of the world, a temporality witnessed, on a realist construal, by our most fundamental physical theory, quantum mechanics.

Although the Special Theory made time seem spacelike, and the General Theory made it rather functional and material, quantum mechanics is kind to time. In quantum mechanics time is mostly an independent variable, not under pressure to be anything else. Much more important, however, is the way quantum mechanics supports a tensed understanding of time, with a distinguished present and inherent directedness. Directedness rules out two-dimensional time. Directedness alone does not rule out the possibility of cyclic time---one can go round and round the equator, either from East to West or from West to East---but if there is some real difference between past and future, then the Gödel solutions,24 though mathematically possible, are not really so. Thus quantum mechanics redresses the balance. We no longer need feel obliged to construe time in a non-temporal way in order to be truly scientific and philosophically respectable.

IV

Philosophers' suspicion of tense has been increased by the difficulty of giving an account of its logic. Tenses modify verbs, and we ought to be able to give an account of tensed discourse in terms of modal logic, making use of the operators [_] and <> (representing the square and diamond). If past and future are not to be the same, the Brouwerian thesis
B |- [_]<>p ---> p
must not hold: and then we can illuminatingly introduce [_]-1 to represent the other tense operator, somewhat akin to Prior's H, with G being [_], F being <>, and P being <>-1. If the present is to be distinguished from the past and the future, the axioms traditionally called T and what we might call T-1 cannot be axioms:
[_]p ---> p
[_]-1p ---> p
should hold. [Or, more familiarly:
Gp ---> p
Hp ---> p
should hold.] If past pasts are past and future futures are future, 4 and 4-1 must be theses.
<><>p ---> <>p [or |- FFp ---> Fp]
<>- 1<>-1p ---> <>-1p [or |- PPp ---> Pp] Tense logic on these lines (though not in exactly this form) was developed by Prior.25 His untimely death prevented him from developing it further, and in particular developing a mixed logic accommodating both tenses and dates. Prior was reluctant to distinguish time from the course of events, and so speaks of branching time, where we should more naturally speak of branching courses of events.26 We can distinguish there being only one past course of events but many possible courses of events in the future by stipulating that [_]-1 (since [_] represents the strong future operator, and [_]-1 the strong past), satisfies besides the axiom 4-1 a further axiom 4.3-1:
[_]-1([_]-1p ---> [_]-1q) V [_]1([_]-1q ---> [_]-1p)
[orH(Hp ---> Hq) V H(Hq ---> Hp) ]
or some principle of trichotomy, such as Prior's
Pp & Pq ---> (P(p&q) V P(p&Pq) V P(Pp&q))
or, more intuitively,
FPp ---> (p V Pp V Fp)
in order to secure linearity.

It is difficult to give a satisfactory semantics of tense logic. In spite of locutions such as ‘coming true’ and ‘moment of truth’, we feel that ascriptions of truth should be omnitemporal. But ascriptions of truth to future contingents seem to lead to fatalism. Aristotle was worried by the argument that if it is true now that there is going to be a sea battle tomorrow, then when tomorrow comes there must be a sea battle, because otherwise the truth-value of what was by then a past statement would be altered.27 The problem was much discussed in the Middle Ages without any simple solution commending itself. Three-valued logics in the twentieth century have proved equally unappealing. Up to a point we can treat future tensed statements modally, distinguishing firm predictions of what must happen from mere assertions of future possibility, but many future-tensed statements fall in between. Ryle, following Ockham, argued that the only truth that could be ascribed to them was “valedictory truth”, and that, contrary to appearances, past ascriptions of truth to future-tensed statements were not really past and unalterable, but dependent on what actually happens. 28 But that solution, though eminently suitable for conjectures and bets, is too peremptory for many serious predictions. If we are to take account of the openness of the future and the unalterability of the past, we are led to a “tree semantics”, developed during the last two decades of the twentieth century, 29 in which possible states of the universe are portrayed as linked together in a tree-like structure, the trunk representing the unique unalterable past, and the branches the possible future courses of events. As time passes, branches, representing possibilities not actualised, drop off, and the trunk grows longer. A prediction is true at the time it was made if the situation predicted obtains in all the branches then available. We can distinguish between the truth conditions for predictions, difficult to satisfy at the time when the prediction is made, from the truth conditions for conjectures, which are easier to satisfy since they hold only after the event.

The lengthening of the trunk can be seen also in terms of a different account, which holds that the future is unreal, and only the present and past are real. The passage of time is then the accretion of fresh slices of reality. 30 The ontological account is slightly less good than the modal account, because it does not distinguish between unreal futures that are quite impossible and unreal futures that are quite possible but not yet realised in actuality. Nevertheless, the ontological account captures important insights about individuals: future individuals do not exist; there are no moral problems over abstinence as there are with abortion---if I fail to sow my wild oats, there are no possible people to complain that they have been deprived of existence, whereas an embryo arguably has a right not to be terminated. Prior explores the logic of the modal and ontological accounts in chapters VII and VIII of his Past, Present and Future, and the two have been combined into a single treatment by Wegener. 31

Quantum mechanics seems to support a modal understanding of time, with time as the passage from the open future through the actuality of the present to the unalterable fixity of the past; and this in turn supports an ontological picture, with the passage of time being marked by the accretion of unalterable truth. These characterizations of time seem profound. They enable us to give a non-circular definition of terms like ‘before’ and ‘after’, or ‘earlier’ and ‘later’, terms which advocates of tenseless accounts need, but cannot give content to in a purely tenseless way.

V

Cosmologists speculating about the origin and destiny of the universe have raised difficult questions about time. In the middle of the twentieth century continuous creation was in favour, drawing support from the homogeneity of time which holds that no date should be special. But the empirical evidence of the background radiation told against it. We are still picking up the echoes of the Big Bang, when our universe exploded into existence. It was natural for thinkers to wonder whether the beginning of the universe was also the beginning of time, or whether there had been time before the Big Bang, and if so, mindless of Augustinian dangers, to ask what had been happening then.

Positivists deny that there could be time without change, and conclude that before there was a universe in which change could happen, there could not have been any time. But although we may need change to measure time, change is not a necessary concomitant of time. We can have time passing without there being any change. 32 The mere fact that the universe had a beginning does not of itself constitute a beginning of time too. Time may have begun then, or may have begun earlier, or may have been beginning-less.

Beginning-less time can be understood in two ways: metrically and topologically. If time can be measured, then either there is some finite number of years (or some other suitable isochronous intervals) that have elapsed since the beginning of time, or there is not: if the former, time began that number of years ago; if the latter, time had no beginning but has been flowing from everlasting. It is important to specify that the intervals be isochronous; otherwise the condition becomes vacuous---there are infinitely many finite intervals between 1AD and 2AD of the form
[1+1/2(n+1), 1+1/2n) with n running from 0 to 00 (infinity) and more generally, any finite magnitude can be transformed into an infinite one by a suitable strictly monotonic function. Isochronous intervals are determined by periodic processes, the rotation of the earth, its orbiting round the sun, the vibration of a caesium atom. It is a contingent fact that these processes all keep in step; it could be that there were two families of processes, each in internal harmony, but out of step with each other, 33 the infinite past of the one constituting only a finite past as measured by the other, with accompanying difficulties in deciding which, if either, was more fundamental. But, so far as we know, all natural processes all keep in step: there is “a natural rhythm to the universe”, which would support an unambiguous extrapolation backwards beyond the Big Bang, provided we can attach meaning to natural processes or a natural rhythm then. Many cosmologists deny it, holding that the Big Bang was a spacetime singularity at which all the laws of nature broke down. But we might view it along the lines of Plato's Timaeus, in which the Demiurge creates in accordance with pre-existent patterns: we might hold that quantum mechanics did not just happen to be true in our universe, but be in some relevant modality necessarily true, so that even if caesium atoms did not exist, we could still counterfactually count the number of vibrations that would have occurred. In that case it would be a perfectly definite question whether there had thus far been only a finite number or an infinite number of possible caesium vibrations, and hence whether time had a beginning or was beginning-less.

If there is no natural rhythm to time apart from natural processes, and the Big Bang was a singularity when there were no regularities of natural process, then we cannot extrapolate metrically beyond the Big Bang. But it does not follow that we cannot extrapolate at all. Shoemaker's argument for a period of time in which nothing happened would no longer be available, but we can distinguish in Prior's tense logic the thesis that time had a beginning:
Pp V P–Pp
from the thesis that change had a beginning:
Pp ---> P(Hp V H–p).
The former cannot follow from the latter, since if we interpret all Prior's tense operators as null operators, the former is not a tautology whereas the latter is. Hence we can assert that change had a beginning without being committed thereby to holding that time had a beginning.

Changeless time is not a contradiction in terms, but is a difficult concept none the less. Many thinkers have refused to attach any real sense to it:

<T>ime ceases to be well-defined in the very early universe just as the direction ‘north’ ceases to be well-defined at the North Pole of the Earth. Asking what happens before the Big Bang is like asking for a point one mile north of the North Pole. 34
But if we are seriously concerned with that question, we shall distinguish different strands of meaning in our concept of northness. In one cartographic sense, ‘north’ does just mean ‘in the direction of the North Pole’, and has no application at the North Pole. But long before we knew the earth was round and had a North Pole, we could identify north by means of the Sun and the Pole Star. It was an underlying and deeper sense in which north is determined by the axis of the Earth's diurnal rotation. In the superficial sense, one cannot get more towards the North Pole than the North Pole itself, and the question is a nonsense: but the persistent questioner, who is trying to articulate a deeper sense of direction, can be answered in terms of the astronomical north. A point one mile north of the North Pole is a point at latitude 90ø N, and altitude 5,280ft. Even though some of the physical parameters we correlate with time during most of the universe's history cannot be extrapolated back before the Big Bang, so that we cannot ask quasi-temporal questions of them, it does not follow that we cannot ask temporal questions at all. For time is not connected only to physical parameters, but is linked with other non-physical concepts too, notably mental ones. I can hypothetically imagine a disembodied intelligence at the time of the Big Bang and wonder what his experience would have been like.

That it is conceptually possible for there to have been time before the Big Bang does not establish that there actually was. It could be that time did begin with the Big Bang. Or it could be that had I, safely disembodied, been around, I should have witnessed an earlier universe ending with a big crunch. It could be that, as Barrow mused, the stars were all stationary in a tranquil state, 35 and had been so since the beginning of time, a beginning defined topologically, an instant preceding all others, and so preceded by none. Or it could be that the set of antecedent instants was a topologically open interval with no first instant, no beginning. In each case there are difficulties. Leibniz posed the question, if time existed before the Creation, “Why did not God create everything a year sooner?”, and, reckoning that it was unanswerable, concluded that time could not have had any antecedent existence. 36 But Leibniz was appealing to a strong principle of symmetry, presupposed in determining the isochronous intervals of metric time, but not a necessary concomitant of a time in which there were no processes. Kant objected to beginning-less time, but his objection, if valid, would equally rule out 2AD following 1AD, for every interval of time comprises an infinite number of smaller intervals, so that its conclusion is preceded by a completed infinity of previous items. 37 He also objected to time's having a beginning on the ground that the first instant could have no past, and so could not have been the boundary between past and future. More modernly, we might complain that no explanation of how things were then could be framed in terms of how they had been at some earlier date. But these are not the only explanations, nor the best. Leibniz 38 would have approved the speculations of modern cosmologists who explain why the universe came into existence not by efficient causes, but as being the best, in a somewhat recherché sense of ‘best’, 39 instantiation of a number of rational desiderata.

We can, at present, only speculate. We may speculate that the universe just wobbled into existence as the result of a quantum fluctuation, or we may attribute it to the rational decision of a personal creator. We are equally uncertain, as the second millenium ends, about the future, whether the universe will expand, albeit at a diminishing rate, or whether after some finite number of millenia we shall all be engulfed in a Big Crunch. We do not know the answers, but the questions have become clearer, and the unsuccessful efforts of those who would account for time entirely tenselessly have helped show the way to a properly temporal understanding of time.

To return from footnote to text, click on footnote number

1. McTaggart, (1908).
2. McTaggart (1908, pp.457-474). Reprinted in McTaggart (1934), and in Le Poidevin and MacBeath, (1993, pp.23-34). See also McTaggart (1927, ch.xxxiii).
3. Timaeus, 37e.
4. Smart (1963, p.133).
5. Dummett (1960). Mellor (1981, pp.98-102). Reprinted with corrections in Le Poidevin and MacBeath, (1993, pp.47-59). Le Poidevin (1991, ch.6).
6. See for example Quine (1960, pp.170f., 191-194, 208, 226-227), Quine (1965, p.6) and Smart (1963).
7. Republic VI, 486a8.
8. Price (1996, p.4).
9. Prior (1959, pp.12-17). Reprinted in Prior (1976, pp.78-84).
10. Physics IV, 11, 220a25.
11. Walker (1978, ch.3, part 3, pp.34-41).
12. Robb (1914), (1936) and (1921).
13. For the application of Robb's approach to the General Theory, see further Malament (1977) and Sklar (1977).
14. Price (1996) argues that such explanations may well be allowable.
15. This is contested by MacBeath (1983).
16. Weyl, (1949, p.116)
17. Denbigh (1981).
18. Dummett (1964).
19. Nerlich (1976 and 1994, esp.ch.2).
20. The possibility of a mirror-image world continues to fascinate us. See Gardner (1964 and 1982).
21. Putnam (1967). Rietdijk (1966) and (1976). Lango (1969).
22. For further discussion of these arguments, see Stein (1968) and (1970). See also Capek (1961) and (1968), Costa de Beauregard (1968), Sorabji (1980, pp.114-119), and Torretti (1983, pp.249-251).
23. Mellor (1981). Mellor (1998).
24. Gödel (1949), (1951)
25. Prior (1957). Prior (1967). Prior (1968). Prior (1976). See also Butterfield (1984).
26. There are problems in assigning the same dates to different branches, but these can be overcome. See Lockwood (1997, pp.40-45).
27. De Interpretatione, ch.9, 18a28-19b4.
28. Ryle (1954). This solution was earlier put forward by William of Ockham and Ferdinand of Cordova, and endorsed by the Pope. See Prior (1967, p.121), and Williams (1991, p.124). The alternative, given below, is credited to Epicurus, Aquinas, Peter de Rivo and C.S.Peirce.
29. The best account is given by McCall (1994). See also Stone (1997).
30. Broad (1923, ch.2, pp.66- 67, 73). See also Godfrey Smith (1978) and Lloyd (1978).
31. Wegener and &O/slashhrstrø (1997). The formalism has been improved and simplified in a paper by Wegener (with Øhrstrøm), entitled ‘A New Tense Logic for Created Truth’, forthcoming.
32. This has been shown most convincingly by Shoemaker (1969). See also Newton-Smith (1980, ch.2, sections 5&6. pp.19-28), Williams (1986) and Le Poidevin (1991, ch.6).
33. As suggested by (Milne 1948, pp.224-225); see also (Swinburne 1996).
34. Hawking, (1988, $4.5, 68).
35. Barrow (1970, p.3).
36. Leibniz (1716, Third Letter to Clarke, $6/1956, pp.26-27).
37. Kant (1781/1929, A426-429, B454-457/pp.396-397).
38. Leibniz (1686) $$XIX,XXII; see also Margenau (1950, $19.11, 422-5).
39. ‘Best’ is a state of minimum excitation for a spatially homogeneous and isotropic universe closed in all spacelike dimensions. The Euclidean- functional integral prescribes, analogously to the principle of Least Action, one reasonable candidate for the ground- state wave function in a “minisuperspace”. See Hartle and Hawking (1983), Vilenkin (1988).

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My reply to Professor Mellor's Transcendental Tense I at the joint meeting of the Aristotelian Society at Watford in July 1998.

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