Bad company objection to Joongol Kim's adverbial theory of numbers

Abstract

Kim (Synthese 190(6):1099-1112, 2013) defends a logicist theory of numbers. According to him, numbers are adverbial entities, similar to those denoted by "frequently" and "at 100mph". He even introduces new adverbs for numbers: "1-wise", "2-wise", and so on. For example, "Fs exist 2-wise" means that there are (at least) two Fs. Kim claims that, because we can derive Dedekind-Peano axioms from his definition of numbers as adverbial entities, it is a new form of logicism. In this paper, I will, however, argue that his theory is vulnerable to an analogue of the so-called Bad Company objection to neo-Fregeanism. This means that we cannot be sure that numbers are actually given to us by Kim's definition; for, we don't know whether it is indeed a good definition. So, unless Kim, or somebody else, provides a demarcation criterion between good and bad adverbial definitions, Kim's theory will remain incomplete.