Paradoxes of the Infinite: Classic Themes and Recent Results

In this talk, I will give a survey of recent work I have done—some published, some unpublished—on the historical, mathematical, and philosophical problems related to the assignment of "sizes" to infinite sets. I will focus in particular on infinite sets of natural numbers. The historical part of the presentation will take its start from Greek and Arabic contributions to the possibility of measuring infinite sets according to size and sketch some developments spanning the period between Galileo and Cantor. In the systematic part of the talk, I will discuss recent theories of numerosities that preserve the part-whole principle in the assignment of sizes to infinite sets of natural numbers and show how the historical and mathematical considerations yield benefits in the philosophy of mathematics. In particular, I will discuss (1) an argument by Gödel claiming that in extending counting from the finite to the infinite, the Cantorian solution is inevitable; and (2) consequences for neo-logicism.

Paolo Mancosu is the Willis S. and Marion Slusser Professor of Philosophy at the University of California, Berkeley. He has made significant contributions to the history and philosophy of mathematics and logic, especially the philosophy of mathematical practice, mathematical explanation, the history of 20th century logic, and neo-logicism. His most recent book, Abstraction and Infinity (Oxford Unversity Press, 2017), concerns the use of abstraction principles in the philosophy of mathematics. He previous books include Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford University Press, 1996), From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s (Oxford University Press, 1998), The Philosophy of Mathematical Practice (Oxford University Press, 2008), The Adventure of Reason. Interplay between Philosophy of Mathematics and Mathematical Logic: 1900–1940 (Oxford University Press, 2010), and Inside the Zhivago Storm. The Editorial Adventures of Pasternak’s Masterpiece (Feltrinelli, 2013).