Two linked lectures examine how mathematics and philosophy have mutually influenced the notion of the finite and the infinite, surveying ancient positions that treat the infinite either as an aggregate of finite things or as a distinct metaphysical reality and noting later Neo‑Platonic and Alexandrian departures. The author argues that modern symbolic logic and the reconception of number and space detach mathematics from pure intuition, prompting a reshaping of metaphysical inquiry. The second study considers pragmatist theories of knowledge and urges that scientific advances require continual restatement of metaphysical problems and attention to their ethical and epistemological consequences.