John Cusbert

A proof of a mathematical theorem tells us that the theorem is true (or should be accepted), but some proofs go further and tell us why the theorem is true (or should be accepted). That is, some, but not all, proofs are explanatory. Call this intra-mathematical explanation and it is to be contrasted with extra-mathematical explanation, where mathematics explains things external to mathematics. This chapter focuses on the intra-mathematical case. The authors consider a couple of examples of explanatory proofs from contemporary mathematics. They determine whether these proofs share some common feature that may account for their explanatoriness. The authors conclude with two plausible, but competing, accounts of mathematical explanation and suggest that there might be more than one kind of explanation at work in mathematics.

Existence comparativism holds that it can sometimes be better or worse, for a given person, that that person exist rather than not. The dominant argument in the literature on this issue is the Metaphysical Argument, which purports to show that existence comparativism is metaphysically incoherent. The argument of this chapter is that the Metaphysical Argument fails. Even if existence cannot be personally better than non-existence, the Metaphysical Argument cannot be the reason for this, since the argument proves too much. Denying its first premise means holding that personal betterness comparisons between a fixed pair of possible worlds are contingent; some recent work has taken this course. Denying the second premise means holding that A can be better for S than B even when S does not exist. This chapter presents the case for denying the second premise. Contrary to something of a consensus in the literature, this is not absurd in general metaphysical terms. And there is a particular analysis of personal betterness comparisons that explains in more detail how one outcome can be better than another for an individual who does not exist. The chapter therefore concludes that existence comparativism, whether true or false, is metaphysically coherent.

This paper examines the stability of objective chance. I defend the stable chance thesis : that in any given possible world, any pair of intrinsic duplicate physical setups with the same chances of being subject to the same external influences must yield the same chances. I argue that SCT compares favourably to rivals in the literature. I then consider a challenge to SCT involving time travel and causal loops. I argue that SCT survives this challenge, but that such cases expose chance as less stable than we might unreflectively have thought. In particular, the chances associated with a physical setup are sensitive to the way that this setup is embedded in a wider causal structure.

The A-theory and the B-theory advance competing claims about how time is grounded. The A-theory says that A-facts are more fundamental in grounding time than are B-facts, and the B-theory says the reverse. We argue that whichever theory is true of the actual world is also true of all possible worlds containing time. We do this by arguing that time is uniquely groundable: however time is actually grounded, it is necessarily grounded in that way. It follows that if either the A-theory or the B-theory is actually false, then it is necessarily false.

In population ethics, Narveson’s dictum states: morality favours making people happy, but is neutral about making happy people. The thought is intuitively appealing; for example, it prohibits creating new people at the expense of those who already exist. However, there are well-known obstacles to accommodating Narveson’s dictum within a standard framework of overall betterness: any attempt to do so violates very plausible formal features of betterness. Therefore, the prevailing view is that the dictum is off-limits to consequentialists, who are thereby committed to the unsavoury normative consequences of denying it. We argue against the prevailing view, by showing that Narveson’s dictum can be accommodated within “multidimensional” consequentialism. The key move is to deny the normative preeminence of overall betterness, instead taking moral decision-making to rest directly on “respects” of betterness. The multidimensional approach permits a consequentialist account of Narveson’s dictum in which betterness is well-behaved. It also yields a new way to think of the connection between goodness and rightness, thus revealing new terrain in the space of possible moral theories.

Temkin's Essentially Comparative View of moral ideals says that goodness is comparison set dependent: the goodness of an outcome is relativized to a set of outcomes. This view does not entail that betterness is intransitive; indeed, it provides the resources for maintaining transitivity. However, it does entail that the structure of goodness is more complex than is standardly supposed. It thereby demands a modification of the standard connection between goodness and decision. I set out this challenge, canvas some options, and show that Temkin vacillates between two of them.

The A-theory and the B-theory advance competing claims about how time is grounded. The A-theory says that A-facts are more fundamental in grounding time than are B-facts, and the B-theory says the reverse. We argue that whichever theory is true of the actual world is also true of all possible worlds containing time. We do this by arguing that time is uniquely groundable: however time is actually grounded, it is necessarily grounded in that way. It follows that if either the A-theory or the B-theory is actually false, then it is necessarily false.

I argue that the past can be objectively chancy in cases of backwards causation, and defend a view of chance that allows for this. Using a case, I argue against the popular temporal view of chance, according to which chances are defined relative to times, and all chancy events must lie in the future. I then state and defend the causal view of chance, according to which chances are defined relative to causal histories, and all chancy events must lie causally downstream. The causal view replicates the intuitively correct results of the temporal view in cases of ordinary forwards causation, while correctly handling cases of backwards causation. I conclude that objective chance is more closely related to the direction of causation than it is to the direction of time.