The past decades, there has been an increased philosophical interest in mathematical practice. These philosophers intend to answer questions related to the activities of working mathematicians. One of these questions is what aims direct mathematical research. After all, mathematicians often express the desire for good or beautiful mathematics. These types of reflections indicate the need for an understanding of aims that go beyond justification. In this paper, we explore potential ways in which philosophy can clarify what these aims are. Furthermore, we stress the importance for a multidisciplinary approach to this problem.

Explanations in mathematics are not yet well understood. I discuss Steiner’s theory of mathematical explanation, then attempt to improve it by assimilating its core intuitions to Woodward’s counterfactual theory of explanation. The theory that results deals successfully with many cases, but it fails to handle certain types of explanatory asymmetry. To fix this, I draw on Woodward’s interventionist theory of causation and develop a quasi-interventionist theory of mathematical explanation. According to this theory, the asymmetry of mathematical explanations is subjective in the sense that it does not depend on the objective structure of mathematics itself; but I argue that this is not a problem, since the same subjectivity can be found in causal explanation.

In Gracely's decision theoretic puzzle, the Devil's Offer, a subject is offered to take part in a lottery in which she stands to win eternal bliss or eternal torment. Her chances of winning the lottery increase as time elapses. Expected utility maximization seems to lead to the paradoxical situation in which the subject indefinitely postpones the lottery, resulting in the least desirable outcome. However, as various authors have pointed out with respect to decision problems similar to the Devil's Offer, the reasoning that leads to the paradoxical conclusion is not valid. In particular, distinguishing between a synchronic and diachronic version of decision problems in which the number of choice options is infinite, allows for a fine-grained analysis of the requirements of rational decision making. The aim of this paper is twofold. I will show that Arntzenius et al.'s (2004) analysis of problems that are structurally similar to Gracely's decision problem can be adapted to show that the latter can be handled within Bayesian decision theory. Furthermore, I will discuss a variation on the diachronic version of the Devil's Offer, in which the subject is unaware of the fact that she is confronted with infinitely many choices. This modified version seems to be a genuine case in which the principle of utility maximization leads to a paradoxical result. However, I will show that this paradox is only apparent and that the problem can also be resolved within a broadly Bayesian conception of rationality.

Recently, the Elementary Process Theory (EPT) has been developed as a set of fundamental principles that might underlie a gravitational repulsion of matter and antimatter. This paper presents set matrix theory (SMT) as the foundation of the mathematicallogical framework in which the EPT has been formalized: it is, namely, objectionable to use Zermelo-Fraenkel set theory (ZF) as such. SMT is a generalization of ZF: whereas ZF uses only sets as primitive objects, in the framework of SMT nite matrices with set-valued entries are objects sui generis, with a 1 1 set matrix [x] being identical to the set x. It is proved that every set that can be constructed in ZF can also be constructed in SMT: as a mathematical foundation, SMT is thus not weaker than ZF. In addition, it is shown that SMT is more suitable than ZF for the intended application to physics. The conclusion is that SMT, contrary to ZF, is acceptable as the mathematical-logical foundation of the framework for physics that is determined by the EPT.