Infinitesimals

J. L. Bell

pp. 285-315

in: Synthese 75 (3), 1988.

Abstract

The infinitesimal methods commonly used in the 17th and 18th centuries to solve analytical problems had a great deal of elegance and intuitive appeal. But the notion of infinitesimal itself was flawed by contradictions. These arose as a result of attempting to representchange in terms ofstatic conceptions. Now, one may regard infinitesimals as the residual traces of change after the process of change has been terminated. The difficulty was that these residual traces could not logically coexist with the static quantities traditionally employed by mathematics. The solution to this difficulty, as it turns out, is to regard these quantities asalso being subject to (a form of) change, for then they will have the same nature as the infinitesimals representing the residual traces of change, and will become,ipso facto, compatible with these latter.