This book is based on an in-depth 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 isII. Attacks and Defenses - Platonism under fireIII. Seeing With the Mind’s Eye - Two revealing examplesIV. Platonism Bounces Back -From quantum physics to the mind-body problemV. The Philosophical Life - On sociological divides and being a team playerAbout Ideas Roadshow Conversations:This book is part of a series of 100 Ideas Roadshow Conversations. Presented in an accessible, conversational format, Ideas Roadshow books not only explore frontline academic research featuring world-leading researchers but also reveal the inspirations and personal journeys behind the research.