Abstract

What is the role of infinity in mathematics? Historians of the calculus and philosophers of mathematics are both concerned with this question, although they are blissfully ignorant of their respective contributions. This dissertation tries to show by example that philosophers and historians of mathematics have something to offer one another. It focuses on the conceptual changes in the development of the calculus. ;Classical analysis resulted from a struggle between the desire to solve problems and the desire to insure consistency. Thus I have considered the works of past philosophers and mathematicians in order to explain why the calculus has its present form. The starting point is Greek geometry and the complementary philosophy of Aristotle which provide a logically satisfactory, if impoverished, approach to continuity. More powerful mathematical theories had to deal with two concepts which the ancients would not accept: the actually infinite and instantaneous velocity. The change began in the high middle ages and it is typified by the work of Oresme. By inventing the calculus, Newton greatly advanced the mathematical theory of continuity without really solving any logical difficulties. Classical analysis of the nineteenth century provided a sort of solution based on infinite aggregates, but not without inviting the paradoxes of set theory. ;The result of this: If we accept classical analysis as a mathematical theory which has solved the logical problems surrounding continuity and infinity, then it follows that the concepts of the calculus must lack the sort of foundation in logic or experience which is ordinarily sought