Who Needs Mathematical ontology?

Mr Cristian Soto (University of Melbourne)

Various contenders in the philosophy of mathematics debate claim that we need a mathematical ontology. Among others, the first type of contender, for sure, is the full-blown mathematical platonist, who maintains that there is an objective, mind-independent mathematical realm; the second is the Quine-inspired mathematical realist, who argues that we need to postulate the reality of at least those mathematical entities which are successfully applied to science; the third candidate, in turn, is the advocate of the indispensability argument, who counsels us to endorse the reality of those mathematical bits which are indispensable to the pursuit of specific scientific tasks; yet, a fourth candidate is the aristotelian mathematical realist, who attempts to persuade us that mathematical ontology comprises all those patterned and structural aspects of physical systems; and lastly the fifth group are those well-known mathematicians, like Gödel, who have actually believed and claimed that there exist a mathematical realm, whose properties and relational features are discovered and described by mathematics.

Opposing these views, in this paper I argue that neither scientific nor mathematical practice give us reason for believing in the reality of abstract mathematical entities. My argument addresses these two dimensions: as to scientific practice, we need not add a mathematical ontology to our scientific ontology, whereas, as to mathematical practice, I propose that mathematical methodology does not involve claims about mathematical ontology. (This latter point will be the weakest part of my argument given my lack of serious professional training in mathematics). Hence, as far as scientific and mathematical practices are concerned, I shall put forward the view that we need not a mathematical ontology after all – and that that is a good thing not to have.