Abstract

Infinitesimals have been the objects of considerable controversy. They were influential components of important mathematical advances such as the development of the calculus, only to fall into disfavor because the concept of the infinitesimal was thought to be vague, to lack rigor, or perhaps even to be self-contradictory. It is the purpose of this essay to subject the concept of the infinitesimal to close scrutiny and attempt to determine whether this past assessment has been fair, or whether, on the other hand, infinitesimals can be made respectable and can play a proper and useful role in the ongoing process of mathematical discovery. ;A variety of examples which argue for the plausibility of infinitesimals are examined, and the standard objections to infinitesimals are considered and rejected as unconvincing. The method of limits in the calculus is examined to see if it provides an adequate, and more rigorous, substitute for infinitesimals, only to discover that it is in many respects less intuitively satisfying. The logicist construction of numbers is analyzed to see why it does not permit the introduction of infinitesimals, only to find that it is lacking as a general development of the concept of number. Cantor's realm of the transfinite is discussed, in the hope that an adequate account of the large infinite will lead to the infinitesimal as well. We find, however, that Cantor denounced infinitesimals; when we examine the basis for this rejection, we find it, too, unconvincing. ;A modern debate on continuity and infinitesimals, carried on in the Journal of Philosophy by Jose Benardete and Henry Kyburg, Jr., is discussed in detail because of its relevance to the issue; we find there an intriguing argument developed by Benardete for the reality of infinitesimals and rejected by Kyburg. This leads us to a consideration of two recent formal mathematical systems, the non-standard analysis of Abraham Robinson and the construction of numbers by John Conway. The existence of these systems settles the question of the logical consistency and possibility of infinitesimals in the affirmative. ;We conclude that not only are infinitesimals logically possible, but that there are a number of good reasons to believe in their actual existence. Our pursuit of infinitesimals leads us into the nature of true continuity, perhaps settling some issues there but raising the question of the nature of continuity ever more strongly