Supervenience and Necessity: A Response to Balaguer

In Platonism and Anti-Platonism in Mathematics, Mark Balaguer attempts to show that there is (1) one and only one defensible version of platonism, (2) one and only one defensible version of anti-platonism, and (3) no fact of the matter as to which is true. His arguments depend essentially on the notion of supervenience, yet he rejects metaphysical necessity. I argue that he cannot use logical, conceptual, or nomological necessity to explicate supervenience. Balaguer must either give up the arguments that make use of supervenience or accept metaphysical necessity. I also consider and reject a possible response to my arguments.