Recently, I have been reading Charles Parsons’s Mathematical Thought and its Objects for a seminar¹. As a result, I have been brought back to seeing how silly caring about mathematical ontology can be. Presumably we can all agree that mathematicians talk about sets as “objects” standing in the relation “membership” to other sets. But Parsons feels the need to question whether there is something more to that to the ontology of set theory. Granted, he is attempting to defend a sort of structuralism about mathematics, but even that seems strange.

Regardless of what mathematical entities “actually are”, it seems that the more important questions are methodological. Mathematicians are not going to stop using certain ways of speaking because some philosophers finally agreed, for instance, that mathematical objects really don’t exist. Rather, more interesting questions involve how transformation occurs within mathematical practice and the standards for the mathematical community. This can come in many forms, including the naturalism/Second Philosophy of Penelope Maddy², or more controversially in the form of conventionalism in the spirit of Carnap or Poincare.

Regardless of what I tend to consider important, however, it is probably a good idea to get used to thinking about mathematical ontology and epistemology, since presumably there is significant potential for me to get ambushed with questions about it later. So back to reading and toting around a red “M“.

¹ Parsons, Charles (2008). Mathematical Thought and its Objects. Cambridge University Press.

² Maddy, Penelope (2007). Second Philosophy: A Naturalistic Method. Oxford University Press.