Steven Savitt

John Earman's new book,World Enough and Space-Time, is a brisk account of the controversy between space-time absolutists and relationists. The book is intended, one is told, to be “appropriate for use in an upper-level undergraduate or beginning graduate course in the philosophy of science”, but Earman's no-holds-barred approach to the mathematics of space-time theories will have bludgeoned most philosophical readers, undergraduate or beyond, into submission long before it is revealed that Pirani and Williams “have studied the integrability conditions for Born-rigid motions in curved space-times and have shown that space-times of Petrov types II, III, and N do not admit of nonrotating Born-rigid motions”. I say this sadly, because Earman's book is a discerning review of an important literature, and most of its main arguments can be grasped even if some technical details remain out of reach. The more you reach for those details, the more compelling the book will become.

Hilary Putnam argued that the special theory of relativity shows that there can be no temporal becoming. Howard Stein replied by defining a becoming relation in Minkowski spacetime. Clifton and Hogarth extended and sharpened Stein’s results. Game over? To the contrary, it has been argued that the Stein-Clifton-Hogarth theorems actually support Putnam’s contention, in that if an apparently minimal condition is put on the becoming relation, then these theorems entail that the becoming relation must be the universal relation. I recount this dialectic in some detail and then try to define and defend a becoming relation based on a present that does indeed consist of more than one point or event but still satisfies the sort of objectivity requirements that Stein-Clifton-Hogarth require of a becoming relation. This present is not a global hyperplane or surface, however; it is a local structure. I close with some methodological remarks about the relation between the present and the real and about the importance of the specious or psychological present.

Barry Dainton wrote Time and Space “to provide an introduction to the contemporary philosophical debate that presupposes little or nothing by way of prior exposure to the subject, but that will also take the interested and determined reader quite a long way”. He has achieved much of what he intended in this difficult enterprise. He covers the major arguments in a fair-minded way, writes clearly, and has found a good illustrator to provide a host of diagrams that his student readers in particular will find helpful. The book also has a helpful glossary and list of web resources, as well as a brief bibliography and index.

The distinction between presentism and eternalism is usually sought in some formula like ‘Only presently existing things exist’ or ‘Past, present, and future events are equally real’. I argue that ambiguities in the copula prevent these slogans from distinguishing significant opposed positions. I suggest in addition that one can find a series of significant distinctions if one takes spacetime structure into account. These presentisms and eternalisms are not contradictory. They are complementary elements of a complete naturalistic philosophy of time.

One of the questions that is addressed, from various perspectives, is the origin of time-asymmetry. Given the time-symmetry of the dynamical laws, all inferences about the future that are derivable from a dynamical theory are matched by inferences about the past. For Huw Price, who discusses the origins of cosmological time asymmetry, this is reason to treat all time-asymmetric cosmological theories with caution. He dismisses both the inflationary model and Stephen Hawking’s proposal to account for time-asymmetry with his famous “no boundary condition.” Instead, on the basis of the fact that we have no a priori reasons to distinguish between initial and final conditions, he advocates Gold’s time-symmetric model for the universe, in which the thermodynamical arrow of time is tied to the expansion of the universe, so that in the contracting phase towards the big crunch, entropy decreases.

Wittgenstein's remarks in his Tractatus on mathematics are quite obscure. Benacerraf and Putnam wrote, "In his Tractatus Loqico-Philosophicus, Wittgenstein maintained, following Russell and Frege, that mathematics was reducible to logic." On the other hand, Max Black claims, "Wittgenstein does not regard mathematics as reducible to logic, in the manner of Whitehead and Russell." I offer a detailed commentary upon Wittgenstein's remarks, concluding that his views most likely do not follow those of Frege and Russell. I reject a criticism of Wittgenstein presented by Black but find severe shortcomings in the view I take Wittgenstein to be presenting.