Paradoxes of the Infinite

Abstract

Abstract In the introduction to Paradoxien des Unendlichen Bernard Bolzano remarked that most paradoxical results found in mathematics rest on the concept of the infinite. The seventeenth century provided many of the paradoxes of the infinite that constitute the topic of Bolzano’s treatise. If one restricts attention only to those paradoxes that generated foundational discussions, two classes emerge from the plethora of surprising results provided by seventeenth-century mathematicians and philosophers. First are the paradoxes having to do with what Bolzano would have called the general theory of magnitudes, especially the composition of continuous quantities. These will be analyzed with reference to the theory of indivisibles (Cavalieri, Galileo, Torricelli, Tacquet, and Leibniz). Second are the paradoxes relating to the theory of space. These will be investigated with reference to Torricelli’s cubature of an infinitely long solid and the varied philosophical reactions generated by this result, or plane versions of it. The use of infinity in the Leibnizian calculus is the topic of chapter 6.