Using the creative fiction of conversations in the afterlife, this book presents a thorough exploration of the views of Georg Cantor, Gottfried Leibniz, and Bertrand Russell on the infinite. An unusual feature is that the vast majority of text is taken verbatim from the writings of these thinkers themselves, adroitly juxtaposed in such a way as to form natural dialogue.

The first dialogue explores Cantor’s interpretation of the actual infinite as transfinite, showing how Leibniz’s syncategorematic interpretation (involving infinite multiplicity without infinite totalities) provides a consistent alternative. The second and third explore Russell’s philosophy of mathematics across two periods of his development, at the turn of the century and some twenty years later. They show Leibniz as being able to offer lively rejoinders to Cantor’s and Russell’s criticisms of his philosophy. In this way the book provides a unique and accessible perspective on the foundations of mathematics, treating issues such as the paradoxes of set theory, intensional versus extensional conceptions of sets, the interpretation of infinite aggregates as distributive wholes, the role of possibility in mathematics, how to understand mathematical existence, the definitions of number and continuity, the treating of classes as fictions, the foundations of the calculus, the status of the axiom of infinity, the philosophy of space and time, the solution of Zeno’s paradoxes of motion, and much more.

This book is a valuable and entertaining read for researchers and advanced students in philosophy of mathematics, mathematical logic, and the history of the foundations of mathematics.

Richard T. W. Arthur is Professor Emeritus at McMaster University, living in Toronto. Specializing in early modern natural philosophy, mathematics, and the foundations of physics, he focuses on time and the infinite. He holds degrees in physics (BA, Oxford, 1972) and philosophy (MA, McGill, 1976; PhD, Western Ontario, 1981). Arthur has taught applied mathematics and philosophy in Canada, Nigeria, the United States, Italy and England. He has authored or co-authored eleven books and over 80 articles and chapters, primarily on seventeenth-century philosophy, including works on Leibniz, Newton, and Descartes. His notable books include "Leibniz" (Polity Press, 2014) and "Leibniz on Time, Space, and Relativity" (Oxford University Press, 2021). In philosophy of physics, he published "The Reality of Time Flow" (Springer, 2019). He recently published an edition of translations of Leibniz’s scientific articles and three co-authored books, Leibniz on the Foundations of the Differential Calculus (with David Rabouin), Writings on the Metaphysics of the Infinite (with Osvaldo Ottaviani),and Russell on Leibniz (with Nicholas Griifin).