Time's error: Is time's asymmetry extrinsic?

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  • FERREL Ct tR ISTENSEN

    T IME 'S ERROR: IS T IME 'S ASYMMETRY EXTRINS IC?

    1. T I lE THEORY

    The subject of this paper is the asymmetry of time, also spoken of as its anisotropy or directionality. As it is commonsensically conceived, time's two directions are not only opposite to one another but struc- turally different: the earlier-to-later direction is different from the later-to-earlier one. By contrast, space is ordinarily regarded as being isotropic, the same in all directions. Or, to state this idea in terms that don't sound committed to a "substantivalist" view of temporality, the temporal relation of precedence (or subsequence) is asymmetrical: if e is earlier than jr, then f is not earlier than e. But the nearest spatial analog of that relation is betweenness along a line, which is sym- metrical (in the sense that if x is between y and z then x is between z and y). The only asymmetry had by a one-dimensional region of space is such as might be imposed from the outside - as with "r ight" and "left," which involve relations to an asymmetrical object; or via a sheer convention - say, to arbitrarily call one direction "plus" and the other "minus."

    Now, though space itself is isotropic, the arrangement and move- ment of objects can be such as to display an asymmetry with respect to space. In other words there are asymmetries in space, ones involving the non-spatial aspects of its contents, even though there is evidently no asymmetry of space. This somewhat abstract idea is best under- stood in terms of examples like the familiar north-south asymmetry, which is tied to such features as the direction of the earth's rotation relative to the sun and stars. Another instance is the up-down asym- metry, which has to do with earth's gravity. In everyday language, things are pulled in one of two opposite directions but not the other; in the terminology of field theories, there is a gravitational gradient in the neighborhood of the earth. The interesting thing about the latter asymmetry in space is that untutored common sense takes it to be an asymmetry of space itself. It was an important scientific breakthrough when thinkers first decided that matter tends to move, not simply

    Erkenntnis 26 (1987) 231-248. 9 1987 by D. Reidel Publishing Company.

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    "down," but toward other matter. It is intriguing that we should so naturally take something which is extrinsic to space to be a part of its own character.

    The next question is almost automatic: might it not be the same for time? Might it be that the seeming directionality of time also represents an illusion or inferential error of some sort, resulting from a mere asymmetry in time'? That would mean time in itself is sym- metrical: there is the temporal betweenness relation (the possibility of topologically closed time will be ignored here, for simplicity) but no such thing as temporal subsequence. And as it turns out, many modern thinkers have claimed that this is the case (see, e.g., Davies, 1974).

    Indeed, a second view of this same general sort has also been held: that temporal relations are after all asymmetrical, but dependently so. In other words, temporality is in this respect not absolute; temporal relations are dependent for their nature on other sorts of properties and relations borne by things in time. To explain by way of analogy - indeed, the motivation seems to be the same - consider the widely- held philosophical view that temporal separation is dependent on the existence of change: two states of the universe can't be temporally distant from one another (hence numerically distinct) unless they, or other states temporally between them, are qualitatively different from each other, in some non-temporal respect. Similarly, the Leibnizian "identity of indiscernibles" doctrine makes the topological character of time, as open or closed, dependent on the non-temporal features of things in time. Likewise, then, the claim is sometimes made that time is anisotropic only because of the asymmetrical arrangement of its contents (e.g., van Fraassen, 1967). In what follows, both the no- asymmetry claim and the dependent-asymmetry claim will be covered under the assertion that time's asymmetry is extrinsic.

    It may help make the basic idea more clear to mention one possible consequence of it at this point: the extrinsic asymmetry might be either universal or localized. To explain, note that the up-down anisotropy is merely local: pervasive as it seems in daily life, it is limited to the region of the earth. Our "up" direction, followed into outer space, may become "down" in the vicinity of another massive body, and in between (where gravitational fields are balanced or negligible), neither direction is up or down. Conceivably, then, the earlier-later asymmetry of common experience is limited to our region of time or of space. Indeed, suppose it were so highly spatially

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    localized that different persons could have opposite time-senses: then one would remember events which for another are still in the future. (Given the difficulties that this situation could pose for com- munication, however, it might not be possible for either to tell the other what to expect.) Such conjectures reveal graphically just how revolutionary is the hypothesis we are about to investigate.

    The first question to be answered is what sort of phenomenon might supply the extra ingredient to symmetrical time. One possibility that comes to mind is causation, which is intimately linked to temporality in ordinary thought. Prima facie, it is the right sort of thing to provide an asymmetry, since it is conceived as being not only asymmetrical in itself (if e caused f then f did not cause e) but also asymmetric in time: we do not commonsensically accept the possibility of causation back- ward in time, with the cause later than the effect. True, retrocausation is often considered by creative thinkers - e.g., in putative cases of precognition - but generally as the rare exception rather than the rule. Reflecting on the ways causation may be involved in our experience of time, then, might make clearer how the latter could come out seeming different than it really is in regard to symmetry. Not many scientists and philosophers seem to support this particular view, however. Among those who propose an extrinsic asymmetry for time. in fact, many hold a Humean view of causation; that means any temporal asymmetry it may have must come from thai c~f time. And certain others believe in "necessary connections," but take them to be sym- metrical: neither of two causally joined events is the cause lot the effect) of the other (Griinbaum, 1{~73). Hence wc will turn elsewhere in attempting to answer the question before us, to the sort of thing that is usually invoked to provide time's asymmetry.

    In a word, the standard answer is irreversible processes. Roughly, an irreversible or t ime-asymmetrical type of process is one such that instances of its temporal inverse either (i) never do occur (de facto if not of necessity), or, for the statistical version of the definition, l ii~ occur appreciably less frequently than do its own instances. Let me explain this in terms of the simplest case, a process-type consisting of just two different states. Because of the difference between them, the temporal direction from the one (say, state O) to the other (say H) is distinguishable from that from the H to the C). A time-asymmetrical process-type, then, is one such that the H always or most always occurs on the same temporal side of the O - or in other words, the O

  • 234 FERREL (? | tR ISTENSEN

    (almost) always falls on the same side of the H. A time-symmetrical process-type, by contrast, is one which either has no temporal inverse (e.g., it is of the form HQH) or else has one which occurs with comparable frequency: there are about as many ABCDE's as there are EDCBA's. On reflection, it should be obvious that irreversible processes surround us constantly. One has only to see just about any movie run backward to appreciate this fact. Seedlings turn into sunflowers, but the reverse does not occur.

    There is one highly general sort of irreversible process, however, that is most often proposed as supplying time's asymmetry, namely the kind that involves increase of entropy. It is beyond my purposes to discuss this highly complex subject, beyond reminding the reader of the following: the statistical version of the Second Law of Ther- modynamics states that the entropy (in very crude terms, the degree of disorder) of an isolated system of particles virtually never decreases with increasing time. And when two such systems interact, the com- posite system which has the two of them as its parts will also almost certainly not decrease in entropy; hence if the interaction should result in one of the subsystems decreasing its disorder, it will almost inevit- ably be at the expense of increased entropy in the other. Extrapolating this idea, if such is legitimate, the universe as a whole must be tending towards ever greater disorder.

    So pervasive is this phenomenon in nature - the chemical processes that go on around us and inside our own bodies depend essentially on it - that as long as there has been a statistical theory of entropy, certain scientists and philosophers have wanted to attribute the ear- lier-later asymmetry of time to it. These include Ludwig Boltzmann, the father of statistical thermodynamics, Sir Arthur Eddington, who dubbed entropy "t ime's arrow," and many others. If they are right, of course, we'll have to state the theory a little differently than has been done so far. We should then not say things like "entropy always tends to increase" or "disorder increases with increasing time," for these would normally imply that there is both the entropic asymmetry in time and the earlier-later asymmetry of time as well. Instead we will have to say that entropy tends to increase always in the same temporal direction - or equivalently, that it tends always to decrease in only one direction of time. In other words, in different isolated systems the entropic changes "run parallel," with their higher- (and lower-) entropy ends virtually all on the same temporal side.

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    Though entropic asymmetry is the one most commonly associated with the view that time's directionality is extrinsic, there are evidently other types of irreversible process as well. One that is sometimes mentioned by proponents of intrinsically symmetrical time is the class of "centro-symmetr ic" processes, which may be illustrated by such common occurrences as the spreading of a series of concentric waves on the surface of a pool of water when an object is dropped into it. This is a significant class of physical phenomena, for all sorts of waves - mechanical waves in solid matter, electromagnetic radiation, etc. - are propagated in centro-symmetric fashion. The temporal asymmetry of processes of this sort is revealed in the fact that we often encounter waves spreading out from a common center, but rarely see waves collapsing precisely to a single spot. For example, a piece of wood dropped in a quiet pond will send out a series of circular disturbances, but we would be surprised to see such waves rush in coherently upon a piece of wood and hurl it into the air - unless, that is, the whole thing had been carefully orchestrated in advance. To make this statement in terms that don't presuppose an earlier-later asymmetry for time itself, once again, we may say that the great majority of such processes are oriented in time with their larger concentric rings or spheres in one temporal direction, their smaller ones in the other.

    There may be yet other sorts of irreversible processes which we could discover on reflection. One possibility that has occurred to the author will now be pointed out: we might call it "similarity-after- contact" asymmetry, or more abstractly, "information copying" asymmetry. As an illustration consider the case of a barefoot person stepping on loose damp sand, leaving a footprint (cf. Griinbaum, 1973). The patch of sand and the sole of the person's foot display a very similar (though reversed) shape after the interaction, whereas before only the foot had had that configuration. In most other cases of temporary contact between two bodies, of course, the resulting similarity between them is not nearly so great, but it is very often present. The colliding of two automobiles usually results in one or both of them leaving traces of paint on the other, so that the regions of collsion share color-properties afterward that they didn't before. Even a stick dragged across the ground may impart one of its dimensions (its width) to the resulting furrow. And so on. Eschewing mention of earlier-later asymmetry and causal asymmetry, the general pattern here may be described thus: on one temporal side of a case of

  • 236 I~, ERR E l . ( 'HR ISTENSEN

    contact between two things, they often share a similarity they don't display on the other temporal side, and instances of such processes are oriented in time in such a way that the great majority have their greater-similarity ends on the same side. This notion of a similarity- after-contact occurrence obviously needs further development, but it would seem to represent yet another general kind of irreversible process.

    Now, the idea with which this paper began isn't just about temporal asymmetry in the abstract, but specifically concerns the earlier-later asymmetry. A question that naturally presents itself, then, is what there is to tie that particular sort of anisotropy to irreversible proces- ses. This matter seems especially acute now we're aware that there is a plurality of distinct asymmetries in time. To return to the comparison with the relation of being higher in altitude than, notice that that spatial asymmetry is accompanied by all sorts of others: air pressure, average temperature, cosmic ray density, pollen count and perhaps hundreds more. Yet surely these latter are quite incidental. Gravita- tional asymmetry is the only reasonable candidate for being identified with the up-down anisotropy - indeed, it is responsible for most of the others. (Though cosmic ray density is a function of the amount of atmosphere they have to pass through in order to reach the earth's surface.) Even if there were only one known type of irreversible process, however, one would still want to know exactly what its connection is with the already-familiar concept of subsequence: what is there to identify what is prima facie a different asymmetry with that asymmetry? Unfortunately, few of those who have defended Boltz- mann's thesis have attempted explicitly to answer this question. The standard pattern in articles on the subject begins by wondering where the earlier-later asymmetry comes from, casually identifies it with the time-asymmetry of one or more irreversible processes, and then pro- ceeds to discuss details of the latter. Perhaps the underlying attitude is just a hopeful "This must somehow or other be it." But surely more has to be said. (Of course, a given term from ordinary speech can always be assigned a new use arbitrarily. But such a move in the case of 'earlier than' would only generate confusion.)

    Let us consider various possible claims, then, concerning the nature

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    of the connection between irreversible processes and the antecedently familiar concept of subsequence. To begin with, it could be suggested that the existence of the former is "at bottom" what we mean, in ordinary language, when we talk of the latter. But this would imply that it is self-contradictory to speak of - and hence impossible to imagine - those processes as happening backwards in time, just as it is to speak of or imagine married spinsters. And that is manifestly not the case. The backward-run movie illustrates nicely what can be imagined and coherently described in this regard. Then perhaps it isn't a matter of verbal definition but of (unconscious) inference. Maybe, that is, the human race has come to believe in the directionality of time as the result of observing all the irreversible processes that take place in time, much as it has in the case of the up-down directionalit\' of space. But this suggestion is also clearly wrong, as a matter of psychological history. (Whether such an inference might correctly be drawn will be discussed later.i We don't in any ordinary sense of the word observe that O's always fall on the same temporal side of H's in otherwise symmetrical-seeming time; if anything, we observe that O's precede or follow H's. To be more precise, we see an H while remembering a Q, or have a fresh memory of a O while having an even fresher one of an H, or something of that nature.

    This leads us to the place where the answer evidently must be found: if there is a connection between time-asymmetrical processes and the earlier-later concept at all, it will have to be because the former somehow give rise to such experiences as memory and anti- cipation. This in turn evidently means there must be some irreversible process(es) in our brains that underlie those experiences. So even though we're not aware of those processes (and hence draw no inferences from them, nor employ them in definitions) they are still the ultimate source of our concept of asymmetrical temporality.

    As to just which processes might be inw)lved, of course, at this stage of scientific enquiry there is still too little known about the workings of the brain to do more than speculate. But it seems to me that there are at least two asymmetries in experience for which such a source would have to be sought: that involved in the fact that some memories seem older than others (because "'dimmer," or whatever), and the qualitative difference between memory- images and sensations. One possibility suggests itself immediately in regard to the former: that increasing entropic disorder of "brain traces" (including retrieval

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    mechanisms) with time is the reason why memories fade (cf. Smart, 1968). It seems a lot to hope that the answer should turn out to be so simple and straightforward, given the general complexity of the human organism. But this at least gives us an idea of the general direction research must take, to discover the source of our concept of earlier-later asymmetry.

    Whatever science may ultimately discover in this regard, it must be emphasized that finding such a source for our anisotropic experience of time does not logically require that time itself is symmetrical. The bare fact that our experience, of time is mediated by processes that take place in time doesn't argue that any or all of the structural features of the latter aren't also possessed by temporality in its own right. Indeed, it remains an open question whether the processes involved are asymmetrical just (in part) because time itself is so. This brings us to the question of whether or not the thesis I've been describing is true. It is widely held, in certain circles, but what is its justification? Let us now turn to look at that question.

    2. THE ARGUMENTS

    A number of different motives for holding Boltzmann's theory are to be found - none too explicitly, as a rule - in the extant literature. Some of them are fuzzy intuitions whose evidential status is highly suspect: a general craving for symmetry in nature, for example, or the feeling, inspired by Relativity, that time ought to be as much like space as possible. But a priori preferences cannot substitute for empirical grounds, and nothing in the theories of Relativity actually requires that time be like space in this particular respect, it seems to me, whatever may be so for other respects. Viewing time as a subspace of spacetime might make one wonder how it could be appreciably different from space; but it is a unique and everywhere- distinct subspace, so that there is no logical objection to its having features which the rest of the continuum does not possess. (Weird spacetimes with non-orientable timelike curves might challenge this, but they are controversial in their own right.) It is also true that General Relativity makes the metric and topology of spacetime dependent upon (interdependent with, actually) the mass-energy of its contents, and this is suggestive of yet other types of dependency. But

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    at least as the theory stands, there is nothing in it to affect the directionality of time.

    Another sort of motive for belief in extrinsic asymmetry has already been mentioned, exemplified by the traditional claim that there can be no time without change. It is the wider belief that the nature of time (and space as well) is in general dependent on its contents. That is, any features which temporal relations may have, hence in particular their asymmetry or symmetry, must stem from the non-temporal properties and relations borne by the same relata. So one motive for embracing the thesis under discussion is whatever it is that inspires this more general belief. Now, the latter view has been much discussed by philosophers down through the years. I personally do not share the intuition at all; and when arguments are given for it, their premises (e.g., verificationism) always seem to me at least as debatable as the conclusion. In any case, this particular rationale for extrinsic aniso- tropy is too large an issue to be fruitfully explored here.

    Fortunately, such a philosophical view is by no means held by all who reject intrinsically anisotropic time, so we may turn to arguments that have a more universal appeal. Yet another apparent reason for such rejection is a simple argument from theoretical parsimony. One asymmetry is more economical than two, the reasoning seems to run, and since asymmetry in time is unavoidable, that of time by itself should be eliminated. It seems to me, however, that the gain in economy in this particular case is too small for the argument to have much force. At the very least, it is too weak to outweigh the economy argument on the other side which we will consider shortly. So I will now turn to one last argument, the only other one I know of - and the one that appears to have been most influential in inspiring the view before us.

    The crux of the matter is the following arresting fact: the fun- damental laws of physics, at least as far as they are presently known, are all or virtually all symmetrical in time. Whatever those laws allow to happen in one temporal direction can also happen in the other. To put it another way, if we replace all occurrences of the variable 't ' (including differential ones) with ' - t ' in the mathematical formulation of a given law, the resulting formula retains the same form; it still expresses a law of nature. The sole apparent exception so far dis- covered is an arcane one involving the decay of a certain sub-atomic particle, the neutral K-meson. All the important ones - Maxwell's

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    laws, Newton's or Einstein's equations of motion and gravitation, etc. - are time-symmetrical. It is worth mentioning that even the Second Law of Thermodynamics falls into this category, though that fact will require explanation beyond what was said earlier. It is true that the physical systems we observe around us tend to increase their entropy in only one of the two temporal directions. Hence a system with low entropy will increase it in that direction, and one in a state of maximum disorder will simply remain so, in all probability, in the same direction of time. But since the law is statistical, there is always a non-zero chance that an isolated system will spontaneously return to a lower entropy state. What this means is that over a very long time - longer than the known history of the universe, in fact, for systems of everyday size - one that is isolated from outside influences would periodically both decrease and increase its entropy. Consequently its entropy curve will be symmetrical in time. Hence in this sense, at least, even the Second Law of Thermodynamics is time-symmetrical.

    All this has raised a puzzling question in the minds of many. If time itself is really asymmetrical, how is it that the laws involving time are not'? Wouldn't we expect such a significant structural fact about temporality to be reflected in what can and cannot happen in time? And they conclude that time must likewise be symmetrical after all. Now, the force of this reasoning is difficult to assess. Unless one is already committed to the view that time is in some manner dependent upon its contents, it is hard to know what to make of the idea that the structure of temporality ought to be duplicated in the nature of temporal laws. It is not self-contradictory to hold that time is asym- metrical though the laws of nature are not; to put it schematically, it may just be a brute fact that events of type O are followed by events of type H and that H's are followed by O's. (Note the asymmetry assumed here in the term 'followed'.)

    Something more can be said, however, concerning the premise on which the argument rests. The matter is highly complex, and I feel more than a little uncertainty regarding it, but the reader may consider the following line of thought for whatever it is worth. The question I raise is whether it is really true that the most fundamental laws now known are symmetrical in time. Such is evidently the case for the abstract mathematical formulas in which those laws are standardly expressed, but perhaps that does not settle the matter. To mention one alternative view - though this admittedly raises further controversial

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    issues - perhaps those formulas can in each case be thought of as welding two separate time-asymmetrical laws into a single symmetrical form. Such would be the case, for example, if they are causal laws (involving time-asymmetrical causation, that is); underlying the time- symmetrical formula linking Q's and H's would be two more-fun- damental laws saying that O's cause H's and that H's cause O's, To take a concrete case, the idea is that accelerating charged particles send out electromagnetic radiation, and incoming radiation makes charged particles accelerate. (Note the asymmetry implicit in 'send out,' 'incoming,' and the causal verb 'makes.')

    This suggestion that asymmetrical laws underlie the symmetrical ones will be more clear, and perhaps more plausible, if we look at a couple of special instances. First recall the case of centro-symmetric processes. The laws tell us that concentric waves which expand in either temporal direction are possible. But the conditions under which the two sorts of wave are produced (note the asymmetrical causal verb) seem to be importantly different from one another. For whereas a single object falling into the water will send out ripples in the usual temporal direction, it requires a ring-shaped object falling on the water, or a group of objects hitting at appropriate distances and time intervals (e.g., falling simultaneously in a circle), to produce a con- centric series of ripples oriented the other way in time (see Figure 1). The point here is even more clear when we realize that the latter

    We get . ,t l but not li- t

    and "L t bun t1! 9 IO

    Fig. 1. Waves from impact in water. The circular ripples are indicated by lines; the dots represent the impacts of the objects.

    l

  • 242 FERREk CHIR ISTENSEN

    situation, that of a collapsing circular wave (i.e., it collapses in the standard earlier-later direction and expands in the other), is the aggregate result of a group of waves that expand (in the earlier-to- later direction) but otherwise cancel each other out.

    If this common-sense description is literally correct, now, there is a temporal asymmetry in the manner in which the two sorts of wave pattern lawfully arise, even though they may also both be covered by a single time-symmetrical law. For a second possible illustration of this suggestion, consider the long-term symmetry of the entropy law. Its existence is quite compatible with there being a more fundamental law saying that entropy tends strongly to increase in only one of the two temporal directions. The reason has to do with the fact that entropy is defined only for finite systems, which possess a maximum possible entropy. For once the maximum has been reached, the disorder can't increase any more unless it first decreases again. And even though decreases are highly unlikely, given enough time they will occur in most systems. Hence it follows that in the very long run, for every period of increasing entropy there will be one of decreasing entropy. The entropy curve will be symmetrical even though there is indeed a strong tendency for entropy to increase in only one direction of time.

    It may well be, of course - as my wording in these two examples strongly hints - that such an interpretation of the laws requires a non-Humean view of causation, and it may involve other philoso- phically controversial attitudes as well. But it is not essential to my purposes to hold that this proposition, that asymmetrical laws underlie the symmetrical ones, is actually true; it is sufficient that it might be, for all we know for sure. For that reveals that the main argument for the intrinsic symmetry of time rests on a premise that is by no means certain. And given that its force is unclear even if the premise be accepted, it looks as if the current case for symmetrical time is rather weak. One can go further yet, however; there are also some com- pelling arguments against that claim about time. We will now turn to examine the other side of the issue.

    First off, let us recall that there is evidently a plurality of kinds of irreversible process in nature: those involving entropy change, centro- symmetric processes, "information duplication", K~ decay, and

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    possibly certain others as well. Is it not plausible to suggest that a single asymmetry is responsible for them all, namely that of time itself? For reasons having to do with economy, the ability of a single feature to theoretically explain a diversity of phenomena is in general regar- ded in science as good evidence for the reality of that feature; it has the effect of unifying and organizing our picture of the world. Surely there must be some common reason, one is tempted to argue, for the existence of the various asymmetries in time - what else might it be if not the asymmetry of time itself? Though I haven't seen explicit responses to this argument in the literature, certain theorists seem to have felt its force. For one thing, there have been attempts to eliminate the diversity of irreversible processes by identifying one sort with another - e.g., to show that at a deeper level centro-symmetry is the same thing as entropy increase. These attempts have not generally been endorsed by proponents of extrinsic anisotropy, however. And it seems to me that a complete reduction of this sort, to a single irreversible process, is highly implausible. To consider a mundane but clear example: the similarity of the imprint to the foot, which involves sameness in regard to spatial order, is an entirely different matter lhan difference in the degree of general spatial orderliness. Like any other occurrence, cases of similarity-after-contact are accompanied by in- creases in disorder; the transfer of information from one place to two generally results in some loss of information as well. But to say this is not to reduce either of them to the other.

    Whatever may be the case regarding their seeming multiplicity of kinds, there is a more fundamental argument to be found in the fact that t ime-asymmetrical processes exist at all. Even if there were only one general type of such (say, entropic increase), the point would remain that its instances occur again and again, all around us. They appear to be going on in all parts of the observable universe, and to have done so for billions of years. The question is, where do these ubiquitous asymmetries in time come from, if time itself and the basic laws governing what happens in time are both fully symmetrical? Are the former not instead powerful evidence that the latter are asym- metrical after all? This argument is similar to the main one on the other side; in fact, it turns it upside down. Instead of "How can we reconcile asymmetrical time with time-symmetrical laws'?," it is "How can we reconcile the idea of symmetrical time and t ime-symmetrical laws with the multitude of time-asymmetrical occurrences that

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    experience reveals?" Actually, however, this new argument is a very different sort of challenge. Rather than being based on a fuzzy intuition that there is incongruity between symmetrical laws and asymmetrical time, it has a clearly identifiable source. Let me explain.

    Everywhere in science, for a common pattern to be exhibited by large numbers of independent events is regarded as firm evidence of governance by a common law; that such concurrence should be a sheer accident is taken (for better or worse) to be highly implausible. In particular, then, it can hardly be a coincidence that the countless entropic changes that surround us moment by moment virtually all have the same orientation in time - rather than, say, being arranged randomly in both directions. Moreover, it seems as if whatever laws are responsible must involve the asymmetry of temporality itself. For a law has to determine (at least with probability) what happens in each of its instances; yet if time is symmetrical, there is no aspect of reality to which the law can appeal to decide which way in time a given occurrence should proceed - there is no "difference to make the difference" in respect of temporal direction. To put it schematically, consider a law that says events of type C) are accompanied by events of type H in time: unless time is asymmetrical, there is nothing to determine which side of a given O its H will or probably will fall on. (This same point applies, be it noted, to the earlier-raised suggestion that the asymmetry of causation is responsible for that of time: it provides no explanation for the fact that instances of causation are always oriented the same way in time, rather than sometimes one way and sometimes the other.) I suppose one might simply reject the standard sort of scientific reasoning on which the foregoing argument rests, and accept as a "brute fact" a most colossal series of coin- cidences. Otherwise, it seems to me, it is a powerful argument indeed.

    In fact, most defenders of Boltzmann's thesis have spoken as if they felt the force of the problem. For they have attempted, in disparate ways, to provide an explanation for the irreversible processes within a framework that employs symmetrical time and laws. For example, several have endorsed the following proposed derivation of the exis- tence of t ime-asymmetrical centro-symmetric processes. Though the laws governing wave radiation are indeed time-symmetrical, the ac- count runs, the laws alone won't tell us what happens at any given place and time. For laws are conditional in character, saying only that if a O occurs at time t~ it will be accompanied by an H. To know

  • T IME 'S t-~ R R( ) R 245

    whether a O does in fact happen at t~ we also need ~boundary conditions," which supply the requisite particular facts. But certain sorts of boundary condition, the story continues, are much more common in our world than others. The existence of asymmetries in time is due to there being - as a matter of contingent fact, not a matter of law - a preponderance of one sort of boundary condition rather than another. To be concrete, situations in which a single charged particle is vibrating (or a single object strikes a pond, etc.) are very much more common than cases of a ring of charged particles vibrating simultaneously. For the latter sort of circumstance is very unlikely to happen by chance; its occurrence would virtually require there to be coordination from a single point - which is basically just the other case back again. From all this it is concluded that the statistical fact about boundary conditions is the reason why, in spite of time and its laws being symmetrical, there is asymmetry of centro-symmetric processes ill time. On a second version of the reasoning, the two sorts of boundary condition that are compared are a "'source" and "'sink" at infinity; in the one case tile wave would collapse from infinity to reach the center, while in the other it would expand from a point outward to infinity. This time it is judged that there are no instances of the required boundary condition in the former case, but only in the latter. Hence it is concluded once again that the process will be asymmetrical in time in spite of the symmetry of the governing laws and time itself (see Davies 1974, pp. 119-120; Gri inbaum 1973, p. 264fl.).

    At least, such is their reasoning if I have understood it properly; to me it seems patently fallacious. Though the differing frequencies (including the case of zero frequency) of the two types of boundary condition are indeed part of the reason for the process-asymmetry, they cannot be the entire explanation. For they themselves represent only a numerical asymmetry, not a temporal one; hence they, together with symmetrical time and time-symmetrical laws, can hardly give rise to any asymmetries in time. (Symmetry in, symmetry out.) Figure 2 reveals graphically that this is the case, for when we consider each sort of boundary condition by itself the original objection remains: why is the universe like (a) rather than like (b), or like (c) rather than like (d)? Regardless of how common each sort of boundary condition may be, the asymmetry in time is left unexplained - unless the intrinsic asymmetry of time is surreptitiously assumed, that is.

    There have been numerous other attempts to explain t ime-asym-

  • 246 FERREL CHR|STENSEN

    (a)

    _ l l - I [ _,,I _,,I -,I i -,I l

    -'lL - L t

    (b)

    ,Jl I,, I ~,l t l,,, o ~ I

    (~) "!1 -'I -'1 "1- 9 , I , l , i , ; t l , - i l , . . ! I ' -1~,

    (d)

    "1 ll" "1 !" 9 " I , I " 9 [ I I i l l- i 1-

    Fig 2

    metrical processes without intrinsically asymmetrical time 1 (e.g., Gal- Or, 1972; Gold, 1967; Landsberg, 1982). Unfortunately, most of them are highly technical and complex. To survey and assess them at all adequately would require (at least) another entire article. Here I can only record my conviction that certain of these other accounts also smuggle the asymmetry in, surreptitiously assuming the very thing they are supposed to derive. 2 But beyond this, most of these claims are highly speculative. They involve questions at the frontiers of physics and cosmology, so it is difficult for anyone to know if they have succeeded in their purpose. Indeed, there is a singular lack of consen- sus, among believers in intrinsically symmetrical time, concerning the reason for the multitudinous asymmetries in time. The different puta- tive explanations are mutually incompatible. So for my purposes in this paper, the conclusion must be this: there may well be an adequate

  • T IME 'S ERROR 247

    answer to the argument from irreversible processes, but we simply don't know that yet. The challenge has yet to be clearly answered.

    In the last analysis, it may be possible to solve the problem of time's arrow only through exploring the more general topic of the entire nature of time. Though that can hardly be done here, I must mention one final argument, involving a perennial issue which is crucially relevant to this one. In a word, it is over what is sometimes called "becoming". (Not to be confused with the ubiquitous but nonsensical notion that time "passes".) In clearer terminology, this is the idea that time involves, in addition to (or instead of) the temporal relations assumed in this essay, the temporal "modes of being": having been the case, being (now) the case, and being going to be the case. This concept may well constitute the primary motive of most of those who reject Boltzmann's thesis. For it seems to me that it is simply in- compatible with the idea that time's asymmetry is extrinsic. To focus on what should be the clearest illustration of this fact: the asym- metrical difference between having existed and existing, or that be- tween existing and being going to exist, can in no way be reduced to a mere qualitative asymmetry such as that involving two different degrees of disorder. Once again, however, arguments for this claim will have to wait for another time.

    Let me summarize what has been concluded in the second half of this paper. It seems to me that there are no very strong arguments in favor of the view that time is only extrinsically anisotropic. Moreover, there are some serious arguments in opposition to the claim. The latter are also not conclusive, to be sure, but evidently they have not as yet been adequately answered. All this being so, I find it puzzling that Boltz- mann's thesis of extrinsic temporal asymmetry is accepted so widely without question - it is virtually an article of faith among philosophers and physicists in certain quarters. I suggest that such an attitude is unjustified. At least for now, the assumption that time needs an extrinsic arrow is in error.

    NOTES

    Most of them involve more than the claim that the irreversible processes are governed by the same laws and same types of boundary condition. In addition they posit causal

  • 248 FERREI_ CHRISTENSEN

    links - t ime-symmetrical ones of some sort, necessarily - connecting the individual instances directly or indirectly to a single particular source of asymmetry. (I earlier described the multitude of instances as being causally independent of one another; that appears to be so in daily life, but of course may not be.) Usually this is either the primordial cosmic fireball or the present expansion of the universe. 2 For example, I maintain this is the case with Grf inbaum's claimed derivation of a cosmically pervasive asymmetry for entropic branch systems (Griinbaum 1973, pp. 254ff.; Gold 1967~ pp. 160-166 and 169-171)). For it employs a prior t ime-asymmetrical distinction, between creation and destruction of the branch systems, in its application of the randomness condition. This becomes especially evident when we attempt to derive an asymmetry first, before labelling either temporal side the one of branching off or branching in.

    R F, FE R EN(" [-( S

    Davies, P. C. W.: 1974, The Physics of Time Asymmetry, University of California Press, Berkeley and Los Angeles.

    GaI-Or, B.: 1972, "The Crisis about the Origin of lrreversibility and Time Anisotropy', Science 176, 11-17.

    Gold, T., ed.: 1967, The Nature of Time, Cornell University Press, Ithaca, New York. Gri inbaum, Adolf: 1973, Philosophical Problems of Space and Time, second enlarged

    edition, D. Reidel, Dordrecht. Landsberg, P. T., ed.: 1982, The Enigma of Time, Adam Hilger Ltd., Bristol. Sklar, Lawrence: 1981, 'Up and Down, Left and Right, Past and Future', Nous 15,

    111-129. Smart, J. J. C.: 1968, Between Science and Philosophy, Random House, New York. van Fraassen, Bas: 1967, An Introduction to the Philosophy of Time and Space, Random

    House, New York.

    Manuscript submitted 7 October 1985 Final version received 14 February 1986

    Department of Philosophy University of Alberta Edmonton, Alberta Canada T6G 2E5