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

Synthese 196 (8):3389-3407 (2019)
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Abstract

Kim :1099–1112, 2013) defends a logicist theory of numbers. According to him, numbers are adverbial entities, similar to those denoted by “frequently” and “at 100 mph”. He even introduces new adverbs for numbers: “1-wise”, “2-wise”, and so on. For example, “Fs exist 2-wise” means that there are 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.

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Namjoong Kim
Gyeongsang National University

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