Steven Savitt

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.

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.

J. M. E. McTaggart’s anti-passage argument (the argument that time is “unreal) has misled philosophers of time for almost a century. The present paper shows that the clearest formulation of this argument, that of D. H. Mellor in Real Time II, is unsound when its premises are interpreted so that it is valid and invalid when it so interpreted that it is sound). This argument need mislead us no longer. The crucial item in the interpretation of the premises is the copula ‘is’, as in ‘E is past’. The copula may be either tensed or tenseless. While this ambiguity of the copula has been noted before, its implications for McTaggart’s argument had not been adequately exploited. This paper does that job.