I regard the whole of arithmetic as a necessary, or at least a natural, consequence of the simplest arithmetical act, that of counting…
Richard Dedekind
Albert Casullo has argued that the propositions of arithmetic could be experientially disconfirmed, with the help of an invented scenario wherein experiences involving our standard counting procedures, as applied to collections of objects, seem to indicate that 2+2≠4. Our best response to this scenario would be, Casullo suggests, to accept the results of our standard counting procedures as correct, and give up our standard arithmetical theory. This suggestion, interestingly enough, is not as bizarre as it initially appears. But indeed a problem lies in the assumption, common to Casullo’s scenario and to his suggested resolution, that our arithmetical theory might possibly be independent of our standard counting procedures. Here I show that this assumption is incoherent, whether the independence at issue is supposed to make room for the genuine possibility that 2+2≠4, or the merely epistemic possibility that we could rationally believe that 2+2≠4: given our standard counting procedures, then (on pain of irrationality) our arithmetical theory follows.