During the past few months there have appeared many books on time's "arrow". Huw Price's book is a significant contribution, remarkable for its scope. There are chapters on the second law of thermodynamics, on the arrow of radiation, on cosmology and on quantum mechanics. At all these levels, the arrow of time plays an important role.

How is this possible? Both classical and quantum dynamics are time reversible; the future and the past play the same role in them. This fact may appropriately be called the time paradox. Price suggests a radical solution to this paradox. According to him, we "need to think about time's puzzles from a new viewpoint, a viewpoint outside time". We have to move to an "atemporal block universe" view of time. According to him, we have to change our habit of explaining the present state of the universe in terms of its earlier state - thus introducing a temporal bias. Price writes: "We are creatures in time, and this has a very great effect on how we think about time and the temporal aspects of reality." We should attain an "untainted perspective, the Archimedean view of reality - 'the view from nowhere'" (to quote Thomas Nagel). And again: "We have to distinguish how the world actually is from how it seems to be from our particular standpoint." To achieve this, says Price, we have to get rid of anthropocentrism "which infects science". We need a new Copernican revolution to solve the time paradox.

But Price's proposal seems to this reviewer a self-defeating position. It would mean going back to Parmenides, the ancient Greek philosopher: "Nor was it ever, nor will it be; since now it is, all together one, continuous." Or as Einstein said, "Time is an illusion." But if there is no flow of time, then what about life and human existence? After all, even for Parmenides there was a well-defined moment when he began to write his famous poem. There can be no creative acts without the flow of time, something Price seems to recognise since he writes that we are "creatures in time". Yet if there is a flow of time, it would isolate us -not only us but all forms of life -from an atemporal nature. Which would then lead to a form of Cartesian dualism, and the distinguishing of res extensa from res cogitans. Such a conclusion is hardly satisfactory for the modern mind.

Price's book can, however, be read in a different way. The same physicists who accept that the basic laws of physics are time symmetric try to explain temporal asymmetry by special considerations, often based on statistical arguments. Price points out that this explanation involves a "double standard". He comments: "It takes an argument which could be used equally well in either temporal direction and applies it selectively, in one direction but not the other. The result is that the asymmetry they get out is just the asymmetry they put in."

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I agree with this. "Either there is no temporal asymmetry at any stage, or it is there from the beginning." But then Price goes on to consider only the first of these two possibilities, which leads him to his radical conclusion that the flow of time is associated with an anthropomorphic view of the universe.

If instead we consider the second possibility, the time-reversible laws of classical and quantum mechanics must be extended when we deal with situations in which we expect to find time asymmetry. While the standard formulation of classical or quantum mechanics consists of dynamical groups involving time symmetry, in important situations associated with chaos or in large systems formed by many interacting units, we find that we obtain semigroups that include the distinction between past and future. Knowing the state of the system at time t, we may ask what is its state at t+1 or t-1 (as Price does). In terms of groups there is no difference, but in terms of semigroups we find dissipative processes (such as diffusion, chemical reactions), and the state at t+1 differs from the state at t-1.

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This is not the place to go into a description of dynamical semigroups. What I want to emphasise here is that Price's attempt to eradicate the flow of time does not seem to lead to any new insight into the difficulties of modern physics. We can, of course, consider only a few examples of his approach.

In chapter two, Price considers "The lessons of the second law". He describes the numerous attempts to derive the increase of entropy despite the time reversibility of the laws of dynamics. As already stated, his critique is right. But is this the whole story? Transport equations involving diffusion or heat conduction are irreversible processes in which past and future play different roles. They are as "real" as, say, planetary motion; for example, the diffusion coefficient can be measured to as many places of accuracy as experimental devices permit. Moreover, we now know that irreversible processes play an essential role in life. Is life then the result of an anthropomorphic illusion? The difficulty is to relate these irreversible processes to the time-reversible laws of dynamics. To dismiss them seems a strange attitude. Physics as a whole just is not time symmetric as assumed by Price.

Chapter four concerns "Arrows and errors in contemporary cosmology". Again there are many interesting remarks, such as that "cosmology is farther away from an adequate explanation of temporal asymmetry than many cosmologists believe". But I again disagree with Price's conclusions. Time symmetry implies the existence of inverse time transformations. More precisely, if there exists a transformation which moves a system from t to t*, there must also exist an inverse transformation which brings the system back from t* to t. But we know there exists a class of systems (called curiously "exact systems") that do not admit inverse transformations. Singularities are an example. In classical physics (the situation is different in quantum theory), the big bang is associated with a singularity. According to the initial condition (the "critical mass"), there may or may not exist inverse transformations. If the universe is expanding forever, there is only one time direction. What, then, is the reason to favor a priori time-symmetric universes that admit both a "big bang" and a "big crunch"?

Price even goes so far to favour the cosmological view of T. Gold. According to Gold, time's flow will be inverted if the universe contracts. But to my knowledge, there is no indication that irreversible processes such those involved in hydrodynamics, chemistry or biology depend directly on the cosmological state of the universe. Shall we see dinosaurs appear again on earth for reasons of time symmetry? Price's appeal to cosmology corresponds also to the weakness of the well-known attempt by Roger Penrose to associate time asymmetry with the cosmological Weyl tensor. Whatever the role of cosmology in irreversibility, the fact is that on earth we observe both time-reversible and time-irreversible processes. Therefore it is essential to find the distinction between the two types of dynamics that lead to such different behaviour. This goal is being pursued by various groups including ours. Price seems to ignore the problem.

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In his "directions for further work", Price exhorts us to the "exploration of models incorporating advanced action especially in quantum mechanics". What is new here? Advanced action has been used repeatedly in the past; it played an important role in Dirac's so-called lambda process to renormalise the mass of the classical electron, published in 1938. But how advanced action might help to solve the main puzzle of quantum theory associated with the so-called measurement problem is not clear to me. The fact is that in some situations we have to use the quantum wall-function description, while in others (such as measurement) we have to include a transition from wave functions to statistical ensembles. We have then a "collapse" of the wave function. But I believe Price is correct when he criticises the current interpretations of the collapse as the effect of the "environment" or of the time ordering introduced by the observer (first the preparation of the wave packet, then its measurement).

In summary, this is an interesting book, written with great clarity and conviction. There is indeed a "time paradox". And there are indeed many "explanations" of it in the literature that invoke a double standard. But the radical solution proposed by Price seems to be a return to Parmenides, and the negation of the "narrative element" in nature that appears so clearly at all levels of observation. What seems more reasonable is to look instead for the dynamical "roots" of irreversible processes and to investigate the limitations of classical and quantum dynamics when applied to "unstable" dynamical systems. Anyway, the merit of Price's book is that it formulates the problem clearly, even if one disagrees with the solution proposed.

Ilya Prigogine, a Nobel laureate in chemistry, is director, Solvay Institutes for Physics and Chemistry, Free University, Brussels, and director, Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems, University of Texas, Austin.