Abstract

Contemporary philosophers of mathematics are deadlocked between two alternative ontologies for numbers: Platonism and nominalism. According to contemporary mathematical Platonism, numbers are real abstract objects, i.e. particulars which are nonetheless “wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal.” While this view does justice to intuitions about numbers and mathematical semantics, it leaves unclear how we could ever learn anything by mathematical inquiry. Mathematical nominalism, by contrast, holds that numbers do not exist extra-mentally, which raises difficulties about how mathematical statements could be true or false. Both theories, moreover, leave inexplicable how mathematics could have such a close relationship with natural science, since neither abstract nor mental objects can influence concrete physical objects. The Thomist understanding of quantity as an accident inhering in concrete substances breaks this deadlock by granting numbers a foundation in extra-mental reality which explains why numeric expressions are relevant in natural science and how we come to know the truth or falsity of mathematical statements. Further, this kind of moderate realism captures the semantic advantages of the Platonists and the ontological parsimony of the nominalists (since for Thomists quantity is not a separate, free-standing ontological addition). Despite these advantages, the Thomistic understanding of number has been out of favor since the work of Cantor, who argued that actual infinities (regarded by Aristotelians as impossible) are required for modern developments in mathematics. This paper frees philosophers of mathematics to embrace the advantages of Thomistic realism by showing that Cantor’s understanding of actual and potential infinity is not the same as Aquinas’s, and that potential infinities (in Aquinas and Aristotle’s sense) can do all the work Cantor claims for actual infinities in modern mathematics.