Of Numbers and Electrons

The sciences are full of theories which, in the course of making detailed claims about the physical world, say things which entail that there are mathematical entities like numbers and sets. According to an influential tradition stemming from Quine (1948) and Putnam (1972), good scientific reasoning—induction, broadly construed—requires us to believe some such theory, or some disjunction of such theories. And it is because of this that we ought to believe that there are mathematical entities. The belief that there are numbers is, according to this tradition, on a similar epistemological footing to the belief that there are electrons, viruses, quasars, etc.1 Some will regard this analogy as unhelpful because they think that we can know that there are mathematical entities in the same way—whatever it is—that we know that all dogs are dogs, or that all bachelors are unmarried.2 Others may regard this analogy as unhelpful because they think that we can directly perceive that there are mathematical entities—such as sets of dogs standing in the street in front of us—just as we can directly perceive that there are red things; no inductive inference of any sort is required. Still others will regard the analogy as unhelpful because they think the epis-