Leibniz’s Philosophy of Infinity: Comparisons within and across Taxonomies

In this dissertation, I analyze the distinction between different types of infinity that Leibniz identifies throughout his philosophical and mathematical works. By attending to these differences, I show how Leibniz’s rejection of infinite number as a contradictory notion does not entail the impossibility of infinitely small lines. Chapter 1 explains the differences between three grades of infinity that Leibniz identifies in a 1676 taxonomy on the infinite and how this taxonomy was an attempt to avoid paradoxes of the infinite. It also contains a description of a separate taxonomy of the infinite that Leibniz gave in 1706, in which Leibniz explicitly bans any type of infinity that is a whole composed of infinity many distinct parts. Chapter 2 treats five places where infinity lines arise in Leibniz’s mathematics: infinite number, the composition of the continuum, infinite series, infinitesimals and their bounded infinite counterparts, and unbounded infinite lines. Looking at the different ways Leibniz evaluates each of these concepts, we see that infinitesimals and bounded infinite lines stand on firmer conceptual footing then the others from Leibniz’s point of view. Chapter 3 argues that Leibniz’s claims that infinitesimals are “fictions” or “impossible,” this is in reference to a specific type of impossibility that he calls the impossible per accidens. Unlike the absolutely impossible, this type of impossibility is not rooted in contradiction, but conflict with metaphysical principles that bar their existence in the order of created things. Hence, Leibniz’s stance towards infinitesimals allows them to be perfectly coherent entities for the purposes of geometric reasoning, despite his ban on their existence within the physical world