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. With almost all of the text taken verbatim from the writings of these thinkers themselves, it consists in three dialogues. The first is between Cantor and Leibniz on the infinite, exploring the philosophical underpinnings of Cantor’s transfinite set theory, and showing how Leibniz’s syncategorematic interpretation of the actual infinite provides a consistent alternative. Dialogues 2 and 3 are imaginary conversations that Russell has with Leibniz at two distinct junctures in his philosophical development, around 1900 and 1914. These dialogues provide a unique 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. The result is a highly original perspective on the issues involved.
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It is under consideration with Birkhäuser, but not yet published.