Abstracta in Time: A Metaontological Reappraisal of Mathematical ‘Existence’

This paper investigates how abstracta, such as numbers or functions, can be said to emerge at particular times, as in the claim that “complex numbers did not exist before the sixteenth century.” Standard accounts encounter well-known difficulties. Platonism posits timeless entities and struggles to explain historical emergence, while nominalism denies mathematical existence and fails to explain applicability. The paper develops a neutralist metaontology that interprets temporally indexed existence claims as statements about when a concept is available within a community’s inferential and representational practices. Unlike deflationary or thin-object views, neutralism distinctively formalizes these claims through the operator EXt(x), which captures the historical emergence of abstracta without positing timeless entities. Formally, ‘x exists at t’ is true iff EXt(x), where EXt(x) specifies that x is licensed by the rules, resources, and acceptance conditions operative at t. This framework provides a systematic account of temporally indexed existence claims in mathematics, illustrated by historical cases such as zero and complex numbers, and improves on both Platonist and nominalist approaches.