Conceptual Metaphor and the Cognitive Foundations of Mathematics: Actual Infinity and Human Imagination*

The Infinite is one of the most intriguing ideas in which the human mind has ever engaged. Full of paradoxes and controversies, it has raised fundamental issues in domains as diverse and profound as theology, physics, and philosophy. The infinite, an elusive and counterintuitive idea, has even played a central role in defining mathematics, a fundamental field of human intellectual inquiry characterized by precision, certainty, objectivity, and effectiveness in modeling our real finite world. Particularly rich is the notion of actual infinity, that is, infinity seen as a “completed,” “realized” entity. This powerful notion has become so pervasive and fruitful in mathematics that if we decide to abolish it, most of mathematics as we know it would simply disappear, from infinitesimal calculus, to projective geometry, to set theory, to mention only a few.