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Time's arrows today: Recent physical and philosophical work on the direction of time
Pergamon
Stud. Hist. Phil. Mod. Phys. Vol. 21, No. 2, pp. 221-221,
1996
Published by Elsevier Science Ltd. Printed in Great Britain 1355-2198/96 $15.00+0.00
ESSAYREVIEW Time’s Arrows Today: Recent Physical and Philosophical Work on the Direction of Time Kenneth G. Denbigh” S. F. Savitt (ed.), Time’s Arrows Toduy (Cambridge University Press, 1995) ISBN O-521-461 1l-l (hardback) zZ37.50,U.S.$49.95. The book’s title tempts me to ask: Have today’s arrows changed since yesterday? So soon? But joking apart I think it is worth my asking: Just what is meant by ‘Time’s Arrow?‘. Soup cools, radium atoms decay, and so on and so forth. Yet the cooling of the soup etc. is only meaningful if it is judged relative to some other change. Wittgenstein in his Tractatus put the matter very clearly: ‘The description of the temporal sequence of events is only possible if we support ourselves on another process’ (Wittgenstein, 1922, 6.3611). Which reference process shall it be? The rising and setting of the Sun might be thought of as a possibility. Yet the very words ‘rising’ and ‘setting’ are only meaningful relative to some other process. Does the ultimate reference process have to be my own consciousness of ‘earlier than’ and ‘later than’? Fortunately this seems always to be in agreement with other People’s. Even so this is a minefield of subjectivity! Leaving all that, I want first to point out that the Second Law can be formulated in a way which makes no separate reference to ‘earlier than’ and ‘later than’. This was done by Schriidinger when he worked in Dublin. Not many thermodynamicists seem to have noticed his results, perhaps because he published his paper on the subject in the Proceedings of the Royal Irish Academy (Schriidinger, 1950). We consider a film strip showing photographs of two adiabatically isolated systems, A and B, undergoing internal changes. We do not know which end of *19 Sheridan Road, London SW19 3HW, U.K. 13552198(96)00006-8 221
222
Studies in History and Philosophy of Modern Physics
the strip was the earlier in the taking, but the photographs display readings of temperature, etc., from which we can calculate the entropies of A and B. Let i and j refer to the numbering of any two frames, the numbering being consecutive from one end of the strip to the other. Let S denote the entropy. Barring the occurrence of exceedingly improbable fluctuation phenomena the result of our study of the strip shows that the systems A and B invariably change their entropies in parallel with each other. It follows that:
This is true because the two brackets have the same sign, positive or negative, due to the parallelism. The foregoing expression contains the essence of the Second Law even though it says nothing about Time’s Arrow. But the temporal awareness of the observer can easily be brought in. If that observer is me, let t,, and tM, refer to the clock times at which I see the frames numbered i and j. (And of course by convention we create the clocks so that greater clock times correspond to ‘later than’, as this is to the human observers.) Considering system A, what I observe is that if tMi>tMj then S,,>S,,, and if t,,
CL, - qNM, - hq,)2 0. since here again the two brackets have the same sign. If the left-hand side of this inequality is divided by the positive quantity (tM, - tMJ2 we obtain
(x4,- &/MM, - tM,>3 03 or in the limit dS/dt 2 0, which is usually understood as the statement of the Second Law. And it also expresses Time’s Arrow in terms of entropy increase. In summary, it is the parallelism of the entropy changes that provides an objective statement of the Second Law, just as much as that higher entropy states occur later in consciousness. Notice however that the concept of entropy is less significant than that of irreversibility, for there are several processes that are irreversible without being entropic with any certainty (Denbigh, 1989). Also it is the notion of irreversibility that directs our attention to where the ‘arrow’ of time is really to be seen. It lies in the fact that no processes exist that can be completely reversed when all effects on the environment, however small, are allowed for, and when exceedingly improbable fluctuations are disregarded. In short, there is a prevailing irreversibility in the world. So much for my introductory remarks, no doubt familiar to many of this book’s readers. Let me now turn to the contents of the book itself.
Time’s Arrow
223
The preliminary 19 pages give a comprehensive survey of the whole book by its editor, Steven Savitt. One of the survey’s most interesting features is his identification of Seven different Arrows of Time. Perhaps the most secure of them, in my view, is the spreading outwards into infinite space of gas molecules, released by the breaking of a phial. This instance of irreversibility was first pointed out by E. A. Milne, and its theory was developed by Hill and Grtinbaum, and by Oliver Penrose and Percival. A somewhat similar irreversible ‘outwards spreading’ process was put forward by Popper; it is the expansion of circular waves on a pond. Another helpful feature of Savitt’s survey is his summary of competing views about whether or not quantum mechanics (QM) is time-reversal-invariant. William Unruh begins his chapter by remarking on how radically different the notion of time is in general relativity as compared to Newtonian theory. Time is now linked to gravity, but he wishes to dispense with the idea of a gravitational force. ‘I want’, he says, ‘to describe gravity not as a force but as the unequable flow of time from place to place.’ He goes on to discuss QM in relation to time. The ‘so-called’ collapse of the wave packet is not dynamics; it is the way in which new information ‘is incorporated into a theory of insufficient cause’ (p. 44). But this need not imply that QM is inherently time-asymmetric. Unruh supports Aharonov et al. (1964) in their view that QM does not pick out a direction in time. Huw Price’s intention in the following chapter is to extract as much as is possible concerning ‘Time’s Arrow’ from what, at present, is believed to be true in cosmology. ‘Nothing in physics tells us that one end of the universe is objectively the start and the other end objectively the finish’ (p. 70). Indeed there is little scope for a low entropy Big Bang that does not commit us to a low entropy Big Crunch (p. 67). This was pointed out originally by Thomas Gold, and is discussed by Price in relation to views held by Roger Penrose and by Hawking. In a short paper, Anthony Leggett remarks that the problem of Time’s Arrow is characterised by a unique ‘slipperiness’, since it is difficult to find questions to ask that are really meaninigful (p. 97). Nevertheless, he explores the possibility that there is a deep connection between the problem of time’s arrow and the problem of the quantum measurement paradox. Philip Stamp’s article of 48 pages covers much the same ground as Leggett’s, but in greater detail; and a large part of it concerns Leggett’s published researches. He takes his point of departure from the fact that important advances have recently been made in the understanding of QM at the macroscopic level. At very low temperatures certain quantum systems undergo a pronounced ‘discretization’ of their energy levels, together with the formation of energy gaps. This permits a huge reduction in the rates of processes, and allows a corresponding greater degree of control. As a consequence, ‘with each
224
Studies in History and Philosophy of Modern Physics
new advance, more and more irreversible processes become reversible in the laboratory, underlining the practical nature of our distinction between reversible and irreversible physics’. Stamp proceeds to review the work of the last 10 yr on decoherence and dissipation. Under certain circumstances decoherence can be dramatically reduced, as in superconductors and magnets. He goes on to say that the ‘quantum arrow’ is no different from the thermodynamic arrow, although the details are different. Stamp argues that Schrbdinger’s wave-function formulation of QM displays the connection with the Second Law better than does the formulation due to Heisenberg. Storrs McCall, in the next chapter, investigates the possibility that spacetime has a branched structure, like that of a tree, To be sure this idea has been mooted previously in the philosophical literature about time, and I think had not been widely accepted. Even so, there are some interesting and original features in McCall’s proposal. He refers to what he calls ‘the flow of time’. This follows from the idea that ‘at the first branch point, one and only one is selected to become part of the past. The unselected branches vanish, so that the first branch point moves up the tree in a stochastic manner and the tree “grows” by losing branches. This progressive branch attrition is what in the model constitutes the flow of time’. This is all very well, but in my view the indiscriminate use of verbs can give rise to important errors. For instance, in the passage just quoted the supposed ‘flow of time’ depends on the verbs ‘to select’, ‘to become’, ‘to vanish’, ‘to move’ and ‘to grow’. All of these already make important assumptions about the meaning of time. A little later in the chapter (p. 157) the author says ‘although the branched model changes in the sense that it suffers progressive branch loss, [. . .] it does not change in time. Branch loss is instead what constitutes the flow of time’. Thus in McCall’s usage the word ‘flow’ already implies time. It is not something separate; there is not ‘time’ together with its ‘flow’, as could be said if we were talking about the flow of water, or whatever. An alternative view, I think, is that the moment at which a branch is lost corresponds to the subjective ‘now’. This indeed is how ‘now’ has been understood by some of the philosophers who have already adopted the branching model. If we are prepared in this way to use the branching theory for dealing with subjective time, and for subjective time only, McCall shows very ingeniously how quantum probabilities can be given precise values and how they can be used to explain the EPR paradox. Even so, is it right to bring subjective time into physics? I should have liked to have had ‘time enough’ for studying this paper more carefully. (And indeed ‘time enough’ is just what one does need when thinking about time!)
Time’s Arrow
225
The next chapter by Roy Douglas is severely mathematical, but has an important feature in common with McCall’s_this is the view that spacetime has a stochastic branching structure. He derives it from ‘the stochastic branching implicit in quantum mechanics’ (p. 175). Where the present author differs from McCall is in regard to determinism, which Roy Douglas defends. The mathematical basis for Douglas is topology, and it results in a many-worlds interpretation of QM. Regretfully I am not at all qualified to make any comments. But let me mention a couple of things said by the author. One of them is a severe criticism of McCall’s chapter on the same lines as my own; he says that the ‘losing of branches’ in McCall requires the latter to postulate a second ‘time parameter’. The other is Douglas’s rejection of the indeterminism involved in the ‘losing of branches’ as proposed by McCall-a losing which the latter treats as a random process. I come now to two fine chapters by Lawrence Sklar. The first poses the question (p. 192) why it is that the world shows ‘irreversible behaviour when the underlying dynamical laws seem [. _ .] to be completely symmetrical in time [. . .I’. The author outlines attempts by Boltzmann and the Ehrenfests to resolve this problem. More recently, further attempts have been made by Krylov, Jaynes, Oliver Penrose and Prigogine, but Sklar thinks that none of these will answer the problem. I will not go into detail since this chapter is a reprint of a paper published by Sklar in 1986. In the second of these two chapters he returns to the same problem, but looks at it from a different standpoint. This is concerned with the manifest differences between time and space, even though they have been so closely bracketed as ‘spacetime’. The asymmetry of time is much more than something based on subjective experience; it is also objectively based on entropy increase. Does space have any similar asymmetry? There certainly exists the distinction between ‘up’ and ‘down’ but this can be clearly accounted for by the local direction of the gravitational field (p. 219). Indeed, ‘the direction of the gravitational force serves as a reduction basis for the updown distinction’ (p. 220). Accordingly, Sklar goes on to ask whether or not a similar reduction basis is available for the distinction between ‘before’ and ‘after’. There are instances that seem to negate this possibility. For instance, ‘the direction of causal determination is always from past to future’, but this seems to have nothing to do with entropy! Eddington is quoted as having held similar views. And Sklar puts his own position very clearly: ‘[. . .] whatever the relation of temporal “afterness” is, it is not an entropically defined relation [_. .], whereas “down-ness” is indeed defined solely in terms of the local gravitational field’ (p. 225). For this reason Sklar is tempted to suggest that the time of perception may have to be distinguished from the time of the physical world. But a page later
226
Studies in History and Philosophy of Modern Physics
he gives reasons for withdrawing from that suggestion: ‘If what we mean by “time”, when we talk of the time order of events of the physical world, has nothing to do with the meaning of “time”, as meant when we talk about the order of time in our experiences, then why call it time at all?’ (p. 226). This is an excellent chapter, and Sklar has been singularly honest in ending it inconclusively. In his final paragraph he writes: ‘We cannot just identifv time order with entropic order, at least if we mean by time order perceived time order. The perceived “afterness” of events and the entropic relations of events are, as Eddington claimed, too unalike to be identified’. Furthermore, it does not help to make a distinction between humanly perceived time order and time order as it appears in physical theory. The chapter that follows Sklar’s is by Martin Barrett and Elliott Sober. It is not about Time’s Arrow but instead develops a biological analogue of the thermodynamic entropy. It is based on Khinchin’s treatment of statistical mechanics. In the field of population genetics use is made of Wahlund’s Principle, which concerns the coupling of populations that were previously isolated. The number of alleles of each type is conserved, and this constancy is the analogue of energy in statistical thermodynamics. Two final chapters are by Paul Horwich and John Earman, and are about closed causal chains and time travel respectively. As is well known, Gijdel pioneered the discovery that quite a number of solutions to Einstein’s field equations display closed timelike curves. There are other instances of what appear to be closed causal chains. For example there is Feynman’s proposal that positrons are simply electrons going backwards in time, and then going forwards again. Eat-man in his chapter gives much attention to the grandfather paradox. ‘Kurt travels into the past and shoots his grandfather at a time before grandpa became a father, thus preventing Kurt from being born, with the upshot that there is no Kurt to travel into the past [. . .I’. Earman discusses the paradox first from the standpoint of determinism and free will, and later from the standpoint of closed timelike curves and time travel per se. The paradox is not resolved, but neither does Earman conclude that there is any prospect of proving that time travel is impossible. By way of a comment on the foregoing, it is surely the case that ‘time’ is a package of at least three quite distinct notions: there is the moment we call ‘now’ or ‘the present’, and this is an aspect of consciousness; there is Time’s Arrow, related to irreversibility; and thirdly there is ‘spacetime’, which belongs to general relativity. How, one wonders, can such very different concepts be welded together? The getting of a better understanding of this peculiar complexity of time requires a continued input of new ideas. This is exactly what is of value in books such as the present one, for here the reader is stimulated by the sharp contrasts
227
Time’s Arrow
existing between the various chapters. May Time’s Arrows Today achieve the large readership it deserves! References Aharonov, Y., Bergman, P. G. and Lebowitz, J. L. (1964) Physicul Review B134, 1410. Denbigh, K. G. (1989) British Journal for the Philosophy of Science 40, 501. Schriidinger, E. (1950) Proceedings of the Royal Irish Acudemy A53, 189. Wittgenstein, L. (1922) Tractatus Logico-Philosophicus (London: Kegan Paul).
Stud. Hist. Phil. Mod. Phys. Vol. 21, No. 2, pp. 221-221,
1996
Published by Elsevier Science Ltd. Printed in Great Britain 1355-2198/96 $15.00+0.00
ESSAYREVIEW Time’s Arrows Today: Recent Physical and Philosophical Work on the Direction of Time Kenneth G. Denbigh” S. F. Savitt (ed.), Time’s Arrows Toduy (Cambridge University Press, 1995) ISBN O-521-461 1l-l (hardback) zZ37.50,U.S.$49.95. The book’s title tempts me to ask: Have today’s arrows changed since yesterday? So soon? But joking apart I think it is worth my asking: Just what is meant by ‘Time’s Arrow?‘. Soup cools, radium atoms decay, and so on and so forth. Yet the cooling of the soup etc. is only meaningful if it is judged relative to some other change. Wittgenstein in his Tractatus put the matter very clearly: ‘The description of the temporal sequence of events is only possible if we support ourselves on another process’ (Wittgenstein, 1922, 6.3611). Which reference process shall it be? The rising and setting of the Sun might be thought of as a possibility. Yet the very words ‘rising’ and ‘setting’ are only meaningful relative to some other process. Does the ultimate reference process have to be my own consciousness of ‘earlier than’ and ‘later than’? Fortunately this seems always to be in agreement with other People’s. Even so this is a minefield of subjectivity! Leaving all that, I want first to point out that the Second Law can be formulated in a way which makes no separate reference to ‘earlier than’ and ‘later than’. This was done by Schriidinger when he worked in Dublin. Not many thermodynamicists seem to have noticed his results, perhaps because he published his paper on the subject in the Proceedings of the Royal Irish Academy (Schriidinger, 1950). We consider a film strip showing photographs of two adiabatically isolated systems, A and B, undergoing internal changes. We do not know which end of *19 Sheridan Road, London SW19 3HW, U.K. 13552198(96)00006-8 221
222
Studies in History and Philosophy of Modern Physics
the strip was the earlier in the taking, but the photographs display readings of temperature, etc., from which we can calculate the entropies of A and B. Let i and j refer to the numbering of any two frames, the numbering being consecutive from one end of the strip to the other. Let S denote the entropy. Barring the occurrence of exceedingly improbable fluctuation phenomena the result of our study of the strip shows that the systems A and B invariably change their entropies in parallel with each other. It follows that:
This is true because the two brackets have the same sign, positive or negative, due to the parallelism. The foregoing expression contains the essence of the Second Law even though it says nothing about Time’s Arrow. But the temporal awareness of the observer can easily be brought in. If that observer is me, let t,, and tM, refer to the clock times at which I see the frames numbered i and j. (And of course by convention we create the clocks so that greater clock times correspond to ‘later than’, as this is to the human observers.) Considering system A, what I observe is that if tMi>tMj then S,,>S,,, and if t,,
CL, - qNM, - hq,)2 0. since here again the two brackets have the same sign. If the left-hand side of this inequality is divided by the positive quantity (tM, - tMJ2 we obtain
(x4,- &/MM, - tM,>3 03 or in the limit dS/dt 2 0, which is usually understood as the statement of the Second Law. And it also expresses Time’s Arrow in terms of entropy increase. In summary, it is the parallelism of the entropy changes that provides an objective statement of the Second Law, just as much as that higher entropy states occur later in consciousness. Notice however that the concept of entropy is less significant than that of irreversibility, for there are several processes that are irreversible without being entropic with any certainty (Denbigh, 1989). Also it is the notion of irreversibility that directs our attention to where the ‘arrow’ of time is really to be seen. It lies in the fact that no processes exist that can be completely reversed when all effects on the environment, however small, are allowed for, and when exceedingly improbable fluctuations are disregarded. In short, there is a prevailing irreversibility in the world. So much for my introductory remarks, no doubt familiar to many of this book’s readers. Let me now turn to the contents of the book itself.
Time’s Arrow
223
The preliminary 19 pages give a comprehensive survey of the whole book by its editor, Steven Savitt. One of the survey’s most interesting features is his identification of Seven different Arrows of Time. Perhaps the most secure of them, in my view, is the spreading outwards into infinite space of gas molecules, released by the breaking of a phial. This instance of irreversibility was first pointed out by E. A. Milne, and its theory was developed by Hill and Grtinbaum, and by Oliver Penrose and Percival. A somewhat similar irreversible ‘outwards spreading’ process was put forward by Popper; it is the expansion of circular waves on a pond. Another helpful feature of Savitt’s survey is his summary of competing views about whether or not quantum mechanics (QM) is time-reversal-invariant. William Unruh begins his chapter by remarking on how radically different the notion of time is in general relativity as compared to Newtonian theory. Time is now linked to gravity, but he wishes to dispense with the idea of a gravitational force. ‘I want’, he says, ‘to describe gravity not as a force but as the unequable flow of time from place to place.’ He goes on to discuss QM in relation to time. The ‘so-called’ collapse of the wave packet is not dynamics; it is the way in which new information ‘is incorporated into a theory of insufficient cause’ (p. 44). But this need not imply that QM is inherently time-asymmetric. Unruh supports Aharonov et al. (1964) in their view that QM does not pick out a direction in time. Huw Price’s intention in the following chapter is to extract as much as is possible concerning ‘Time’s Arrow’ from what, at present, is believed to be true in cosmology. ‘Nothing in physics tells us that one end of the universe is objectively the start and the other end objectively the finish’ (p. 70). Indeed there is little scope for a low entropy Big Bang that does not commit us to a low entropy Big Crunch (p. 67). This was pointed out originally by Thomas Gold, and is discussed by Price in relation to views held by Roger Penrose and by Hawking. In a short paper, Anthony Leggett remarks that the problem of Time’s Arrow is characterised by a unique ‘slipperiness’, since it is difficult to find questions to ask that are really meaninigful (p. 97). Nevertheless, he explores the possibility that there is a deep connection between the problem of time’s arrow and the problem of the quantum measurement paradox. Philip Stamp’s article of 48 pages covers much the same ground as Leggett’s, but in greater detail; and a large part of it concerns Leggett’s published researches. He takes his point of departure from the fact that important advances have recently been made in the understanding of QM at the macroscopic level. At very low temperatures certain quantum systems undergo a pronounced ‘discretization’ of their energy levels, together with the formation of energy gaps. This permits a huge reduction in the rates of processes, and allows a corresponding greater degree of control. As a consequence, ‘with each
224
Studies in History and Philosophy of Modern Physics
new advance, more and more irreversible processes become reversible in the laboratory, underlining the practical nature of our distinction between reversible and irreversible physics’. Stamp proceeds to review the work of the last 10 yr on decoherence and dissipation. Under certain circumstances decoherence can be dramatically reduced, as in superconductors and magnets. He goes on to say that the ‘quantum arrow’ is no different from the thermodynamic arrow, although the details are different. Stamp argues that Schrbdinger’s wave-function formulation of QM displays the connection with the Second Law better than does the formulation due to Heisenberg. Storrs McCall, in the next chapter, investigates the possibility that spacetime has a branched structure, like that of a tree, To be sure this idea has been mooted previously in the philosophical literature about time, and I think had not been widely accepted. Even so, there are some interesting and original features in McCall’s proposal. He refers to what he calls ‘the flow of time’. This follows from the idea that ‘at the first branch point, one and only one is selected to become part of the past. The unselected branches vanish, so that the first branch point moves up the tree in a stochastic manner and the tree “grows” by losing branches. This progressive branch attrition is what in the model constitutes the flow of time’. This is all very well, but in my view the indiscriminate use of verbs can give rise to important errors. For instance, in the passage just quoted the supposed ‘flow of time’ depends on the verbs ‘to select’, ‘to become’, ‘to vanish’, ‘to move’ and ‘to grow’. All of these already make important assumptions about the meaning of time. A little later in the chapter (p. 157) the author says ‘although the branched model changes in the sense that it suffers progressive branch loss, [. . .] it does not change in time. Branch loss is instead what constitutes the flow of time’. Thus in McCall’s usage the word ‘flow’ already implies time. It is not something separate; there is not ‘time’ together with its ‘flow’, as could be said if we were talking about the flow of water, or whatever. An alternative view, I think, is that the moment at which a branch is lost corresponds to the subjective ‘now’. This indeed is how ‘now’ has been understood by some of the philosophers who have already adopted the branching model. If we are prepared in this way to use the branching theory for dealing with subjective time, and for subjective time only, McCall shows very ingeniously how quantum probabilities can be given precise values and how they can be used to explain the EPR paradox. Even so, is it right to bring subjective time into physics? I should have liked to have had ‘time enough’ for studying this paper more carefully. (And indeed ‘time enough’ is just what one does need when thinking about time!)
Time’s Arrow
225
The next chapter by Roy Douglas is severely mathematical, but has an important feature in common with McCall’s_this is the view that spacetime has a stochastic branching structure. He derives it from ‘the stochastic branching implicit in quantum mechanics’ (p. 175). Where the present author differs from McCall is in regard to determinism, which Roy Douglas defends. The mathematical basis for Douglas is topology, and it results in a many-worlds interpretation of QM. Regretfully I am not at all qualified to make any comments. But let me mention a couple of things said by the author. One of them is a severe criticism of McCall’s chapter on the same lines as my own; he says that the ‘losing of branches’ in McCall requires the latter to postulate a second ‘time parameter’. The other is Douglas’s rejection of the indeterminism involved in the ‘losing of branches’ as proposed by McCall-a losing which the latter treats as a random process. I come now to two fine chapters by Lawrence Sklar. The first poses the question (p. 192) why it is that the world shows ‘irreversible behaviour when the underlying dynamical laws seem [. _ .] to be completely symmetrical in time [. . .I’. The author outlines attempts by Boltzmann and the Ehrenfests to resolve this problem. More recently, further attempts have been made by Krylov, Jaynes, Oliver Penrose and Prigogine, but Sklar thinks that none of these will answer the problem. I will not go into detail since this chapter is a reprint of a paper published by Sklar in 1986. In the second of these two chapters he returns to the same problem, but looks at it from a different standpoint. This is concerned with the manifest differences between time and space, even though they have been so closely bracketed as ‘spacetime’. The asymmetry of time is much more than something based on subjective experience; it is also objectively based on entropy increase. Does space have any similar asymmetry? There certainly exists the distinction between ‘up’ and ‘down’ but this can be clearly accounted for by the local direction of the gravitational field (p. 219). Indeed, ‘the direction of the gravitational force serves as a reduction basis for the updown distinction’ (p. 220). Accordingly, Sklar goes on to ask whether or not a similar reduction basis is available for the distinction between ‘before’ and ‘after’. There are instances that seem to negate this possibility. For instance, ‘the direction of causal determination is always from past to future’, but this seems to have nothing to do with entropy! Eddington is quoted as having held similar views. And Sklar puts his own position very clearly: ‘[. . .] whatever the relation of temporal “afterness” is, it is not an entropically defined relation [_. .], whereas “down-ness” is indeed defined solely in terms of the local gravitational field’ (p. 225). For this reason Sklar is tempted to suggest that the time of perception may have to be distinguished from the time of the physical world. But a page later
226
Studies in History and Philosophy of Modern Physics
he gives reasons for withdrawing from that suggestion: ‘If what we mean by “time”, when we talk of the time order of events of the physical world, has nothing to do with the meaning of “time”, as meant when we talk about the order of time in our experiences, then why call it time at all?’ (p. 226). This is an excellent chapter, and Sklar has been singularly honest in ending it inconclusively. In his final paragraph he writes: ‘We cannot just identifv time order with entropic order, at least if we mean by time order perceived time order. The perceived “afterness” of events and the entropic relations of events are, as Eddington claimed, too unalike to be identified’. Furthermore, it does not help to make a distinction between humanly perceived time order and time order as it appears in physical theory. The chapter that follows Sklar’s is by Martin Barrett and Elliott Sober. It is not about Time’s Arrow but instead develops a biological analogue of the thermodynamic entropy. It is based on Khinchin’s treatment of statistical mechanics. In the field of population genetics use is made of Wahlund’s Principle, which concerns the coupling of populations that were previously isolated. The number of alleles of each type is conserved, and this constancy is the analogue of energy in statistical thermodynamics. Two final chapters are by Paul Horwich and John Earman, and are about closed causal chains and time travel respectively. As is well known, Gijdel pioneered the discovery that quite a number of solutions to Einstein’s field equations display closed timelike curves. There are other instances of what appear to be closed causal chains. For example there is Feynman’s proposal that positrons are simply electrons going backwards in time, and then going forwards again. Eat-man in his chapter gives much attention to the grandfather paradox. ‘Kurt travels into the past and shoots his grandfather at a time before grandpa became a father, thus preventing Kurt from being born, with the upshot that there is no Kurt to travel into the past [. . .I’. Earman discusses the paradox first from the standpoint of determinism and free will, and later from the standpoint of closed timelike curves and time travel per se. The paradox is not resolved, but neither does Earman conclude that there is any prospect of proving that time travel is impossible. By way of a comment on the foregoing, it is surely the case that ‘time’ is a package of at least three quite distinct notions: there is the moment we call ‘now’ or ‘the present’, and this is an aspect of consciousness; there is Time’s Arrow, related to irreversibility; and thirdly there is ‘spacetime’, which belongs to general relativity. How, one wonders, can such very different concepts be welded together? The getting of a better understanding of this peculiar complexity of time requires a continued input of new ideas. This is exactly what is of value in books such as the present one, for here the reader is stimulated by the sharp contrasts
227
Time’s Arrow
existing between the various chapters. May Time’s Arrows Today achieve the large readership it deserves! References Aharonov, Y., Bergman, P. G. and Lebowitz, J. L. (1964) Physicul Review B134, 1410. Denbigh, K. G. (1989) British Journal for the Philosophy of Science 40, 501. Schriidinger, E. (1950) Proceedings of the Royal Irish Acudemy A53, 189. Wittgenstein, L. (1922) Tractatus Logico-Philosophicus (London: Kegan Paul).