Mathematics

Mathematics is the “language of nature,” a privileged mode of expression in science. We think it latches onto something essential about the physical universe, and we seek theories that reduce phenomena to mathematical laws. Yet, this attitude could not arise from the philosophies dominant before the early modern period. In orthodox Aristotelianism, mathematical categories are too impoverished to capture the causal structure of the world. In the revived Platonism of its opponents, the natural world is too corrupt to exemplify mathematical perfection. Modern mathematical science required a novel tertium quid, due to Pietro Catena.

In Defending the Axioms: On the Philosophical Foundations of Set Theory, Penelope Maddy introduces two methodologically equivalent but philosophically distinct positions, termed Thin Realism and Arealism, which presumably respect set-theoretic practice. Further, Maddy concludes that for her idealized naturalistic inquirer, there is no substantive difference between the two positions. However, I argue that Thin Realism loses its tenability due to the presence of foundational pluralism in broader mathematical practice. In turn, this presents a naturalistic way to undermine Maddy’s conclusion that these are two equally admissible ways of describing the underlying constraints of mathematical practice for the philosophical naturalist.

This note scouts a broad class of explanations of central importance to contemporary computer science. These explanations, which I call 'limitative' explanations, explain why certain problems cannot be solved computationally. Limitative explanations are philosophically rich, but have not received the attention they deserve. The primary goals of this note are to isolate limitative explanations and provide a preliminary account of what makes them explanatory. On the account I favour, limitative explanations are best understood as non-causal mathematical explanations which depend on highly idealized models of computation.

Wilhelm (2021) has recently defended a criterion for comparing struc- ture of mathematical objects, which he calls Subgroup. He argues that Subgroup is better than SYM∗, another widely adopted criterion. We argue that this is mistaken; Subgroup is strictly worse than SYM∗. We then formulate a new criterion that improves on both SYM∗ and Sub- group, answering Wilhelm’s criticisms of SYM∗ along the way. We con- clude by arguing that no criterion that looks only to the automorphisms of mathematical objects to compare their structure can be fully satisfactory.