This book is based on an in-depth filmed conversation between Howard Burton and James Robert Brown, Emeritus Professor of Philosophy at the University of Toronto. This wide-ranging conversation addresses a central theme in current philosophy: Platonism vs. Naturalism and provides accounts of both approaches to mathematics. The Platonist-Naturalist debate over mathematics is explored in a comprehensive fashion and also sheds light on non-mathematical aspects of a dispute that is central to contemporary philosophy. Thought experiments stand as a fascinating challenge to the necessity of data in the empirical sciences. Are these experiments, conducted uniquely in our imagination, simply rhetorical devices or communication tools or are they an essential part of scientific practice? This book also surveys the current state of this debate and explores new avenues of research into the epistemology of thought experiments. This carefully-edited book includes an introduction, Mathematical Metaphysics, and questions for discussion at the end of each chapter: I. Introducing Platonism - An explanation for what mathematics really is II. Attacks and Defenses - Platonism under fire III. Seeing With the Mind’s Eye - Two revealing examples IV. Platonism Bounces Back - From quantum physics to the mind-body problem V. The Philosophical Life - On sociological divides and being a team player About Ideas Roadshow Conversations: This book is part of an expanding series of 100+ Ideas Roadshow conversations, each one presenting a wealth of candid insights from a leading expert through a focused yet informal setting to give non-specialists a uniquely accessible window into frontline research and scholarship that wouldn't otherwise be encountered through standard lectures and textbooks.