Infinity, Causation, and Paradox by Alexander R. Pruss (review)

Pruss defends the thesis of causal finitism, the view that "nothing can be affected by infinitely many causes." Pruss's strategy is to examine in detail various paradoxes that attend causal infinitism and to argue that the best way of dealing with such paradoxes is not to offer individual resolutions of them but to "kill" them uniformly through the adoption of causal finitism.

In chapter 1 Pruss treats briefly an important competitor to causal finitism, namely, "full finitism," the view that an actually infinite number of things, whether causally related or not, cannot exist. Full finitism admittedly kills all the paradoxes, too; but Pruss deems the cost too high: he argues that full finitism is incompatible with classical mathematics' embrace of actual infinites and implies that the future course of events is finite, unless one adopts the implausible "growing block" theory of time. If one rejects full finitism, that naturally raises the question why specifically causal finitism is true, a question Pruss addresses but finally leaves open in chapter 7.

Causal finitism is the conjunction of the theses that infinite causal cooperation is impossible and that infinite causal regresses are impossible. Before discussing the paradoxes of causal infinitism, Pruss in chapter 2 focuses on nonparadoxical arguments against infinite regresses. He argues that such regresses are inevitably vicious, which renders them implausible. This sets the stage for the discussion of causal paradoxes in chapters 3 through 6.

In chapter 3 Pruss discusses nonprobabilistic paradoxes such as the famous Thomson's Lamp and his own Grim Reapers paradox (which he unfortunately emasculates by offering a "non-violent version" in which lamps replace the deadly Grim Reapers).

The lengthy chapters 4 and 5 treat paradoxical lotteries and decision theoretical paradoxes respectively. Pruss argues that if causal infinitism is true, then one can construct infinite fair lotteries in which there are denumerably infinitely many tickets sold, each one of which has a zero or infinitesimal probability of winning. Such lotteries lead to unacceptable consequences, such as a participant's expecting to be surprised, puzzles concerning symmetry between two participants, Bayesian manipulation of rational agents, and the improvement of every participant's chances of winning. In chapter 5 Pruss considers paradoxes in probability and decision theory that are independent of infinite fair lotteries. Interestingly, he admits that some of the paradoxes discussed can be killed by full finitism but not by causal finitism, so that they require on his view individual resolutions.

In chapter 6 Pruss discusses paradoxes that can be proved using the Axiom of Choice, such as the Banach-Tarski paradox. Using the resources of an infinite multiverse (which is consistent with causal finitism), Pruss argues that one can causally compute the values of a choice function by means of an Axiom of Choice machine, resulting in compelling rationality paradoxes akin to Banach-Tarski. Since the Axiom of Choice is in Pruss's view true, causal infinitism would imply that the construction of such a paradoxical machine is possible. Hence, we have reason to reject causal infinitism.

In chapter 7 Pruss refines causal finitism and also discusses its competitors, including, once again, full finitism. Causal finitism turns out to be a family of theories, one's preferred theory to be determined by how one answers a number of questions left open in the book. Chapter 8 provides an interesting discussion of whether causal finitism entails the discreteness of time and space (the answer given is that it does not). Finally, in chapter 9 we have an exploration of the theological consequences of causal finitism. Pruss argues that given that every contingent thing has a cause, there must exist at least one necessarily existent, uncaused cause, a conclusion congenial to theists.

Pruss's book is well organized, densely packed, and meticulously argued. The most technical sections are marked with an asterisk to alert the nonspecialist reader. Much of the book can be understood and evaluated only by a professionally trained mathematician...