Date of Award
Doctor of Philosophy
Kant distinguished between sensible and intellectual representation. The intellect represents mathematical objects as composed of their parts and so the continuum must be represented intellectually as a collection of punctual parts. However, an influential line of argument, advanced by Aristotle, Kant, and others contends that the continuum cannot be composed of parts, and so not determined by the intellect. Thus an intuition of space and time must be used in addition to intellection to determine mathematical objects.;The semantic tradition, in contrast, holds that intuition is not needed in order to determine objects. The closely related approach of transfinite set theory and the development of measure theory, topology, and mathematical logic has precluded the need for intuition of space and time by constructing continuua out of sets (of well distinguished objects). However this refutation of Kant is not decisive if Leibniz's infinitesimal calculus is taken seriously. For, underlying his calculus is the idea that each curve is locally straight and contains infinitesimal elements indistinguishable from zero. Hence, since there are no such objects in the universe of sets, such a curve cannot be a set. Thus a "Leibnizian puzzle" can be formulated with the consequence that intuition is needed in order to determine such a continuum.;However, this puzzle can be resolved by noting that it is possible, using the concepts of category theory, to widen the notion of set to that of variable set varying smoothly over a space. In such a model each curve is locally straight and the infinitesimal calculus can be developed. Thus the semantic philosophy can be extended to solve Leibniz's puzzle.
Hagen, Gregory Ralph, "Leibniz's Puzzle And The Smooth Continuum: A Study In The Philosophy Of Mathematics" (1996). Digitized Theses. 2593.