Actual Infinities of Different Kinds in Richard Kilvington’s Philosophical and Theological Texts

Abstract Recent studies on Richard Kilvington’s philosophical and theological texts reveal that he was one of the most ingenious and influential thinkers of his times, even though his followers frequently adopted and adapted his ideas without mentioning his name. Some of his ideas were, however, too much ahead of his times, and thus were either overlooked or harshly criticized by traditionally minded philosophers. This was surely the fate of Kilvington’s concept of infinity. In his question Utrum omnis creatura sit certis limitibus circumscripta (from his commentary on the “Sentences”) Kilvington introduces a distinction between absolute infinity (simpliciter), and relative infinities (secundum quid). According to him there can exist relative infinities regarding quality (qualitative), quantity (quantitative), or number (discretive). A closer look at Kilvington’s analyses reveals that he considers all such infinities as actual. Some of the reasonings Kilvington included in the above-mentioned question he first discussed in the question Utrum continuum sit divisibile in infinitum (from his commentary on De generatione et corruptione). The article presents the sophisticated Kilvington’s accounts on infinities of different kinds, focusing especially on the reasonings related to his concept of infinitum discretive, that are surprisingly similar to the presentations of the properties of infinite sets given much later by David Hilbert and by Georg Cantor.