Hayden Kajercline

According to structuralism in the philosophy of mathematics, the natural numbers are individuated purely by their structural interrelations. A related metasemantic view, which I call axiomism, holds that the meanings of our arithmetical terms are determined just by our acceptance of categorical axioms for arithmetic. Against both structuralism and axiomism, I present the case of the Dyadians. These speakers accept principles identical to our Peano axioms. Nevertheless, it seems clear that they use terms like “13” and “natural number” with different meanings from us. English speakers and Dyadians refer to numerical structures that are isomorphic but distinct, a possibility that axiomists and structuralists cannot adequately account for. This basic problem has been noted before, going back all the way to Russell’s Principles of Mathematics (1903), but it has often been brushed off. My primary aim in this paper is to clarify the costs of doing so. In the final section of the paper, I suggest a remedy. Contra structuralism and axiomism, we should acknowledge that the natural numbers, and the meanings of our basic arithmetical terms, are essentially tied to counting. This is also reminiscent of proposals from Russell, Frege, and the neo-Fregeans. However, I identify two senses of the transitive verb “to count” that are often conflated in this context. In only one of the two senses is counting essential to the natural numbers or our concepts of them.

What is assertion? In this paper, leading theories are surveyed and found to be too restrictive. For every effect, intention, norm, or commitment that has been proposed as essential to assertion, we can find cases where assertions apparently lack this feature. The underlying problem is that proponents of these “strong” theories have focused only on the role of assertion in specific kinds of linguistic practices, regarded as typical or paradigmatic. However, the versatility of assertoric speech reveals this to be a dubious strategy for identifying the essential nature of the speech act. In the second part of the paper, I try out an alternative approach. The idea is to first identify only the most basic thing a speaker does when they make an assertion. The “Minimal Theory” of assertion then claims that this is all there is to asserting. I consider various ways we might be compelled to go beyond the Minimal Theory, but argue that, once appropriately refined, it stands as an adequate account of the speech act.