Back to the Philosophy of Arithmetic
摘要
This chapter advances a nominalistic and empirically grounded account of the semantics and epistemology of arithmetic, challenging the traditional view that the truth and objectivity of arithmetical statements require the existence of numbers as abstract objects. Building on the discussion from previous chapters, it argues that numerical reference, truth, and epistemic features such as objectivity, necessity, and apriority can be explained by the cognitive and social practices of counting and calculation. Drawing on a Peircean framework of reference, the chapter proposes that numerals can function referentially even in the absence of ontologically robust numerical entities. The core thesis is that the epistemic roles ascribed to numbers are instead fulfilled by procedural structures inherent in arithmetical practices. This approach preserves the standard interpretation of arithmetic while avoiding ontological commitment to abstract mathematical objects.