Introduction

Abstract

The quoted passages of Vivekananda, Klein, and St. Paul each point to a certain facet of Ramanujan’s work. First, on June 1–5, 1987, the centenary of Ramanujan’s birth was celebrated at the University of Illinois with a series of 28 expository lectures and several contributed papers that traced Ramanujan’s influence to many areas of current research; see the conference Proceedings edited by Andrews et al. [1]. Thus, Ramanujan’s mathematics continues to generate a vast amount of research in a variety of areas. Second, in the sequel, we shall see many instances where Ramanujan made profound contributions but for which he probably did not have rigorous proofs; for example, see Entry 10 of Chapter 13. Third, although St. Paul’s passage is eschatological in nature, it points to the great need to learn how Ramanujan reasoned and made his discoveries. Perhaps we can prove Ramanujan’s claims, but we may not know the well from which they sprung. These three aspects of Ramanujan’s work will frequently be made manifest in the pages that follow.

We take up something—we know it is finite; but as soon as we begin to analyze it, it leads us beyond our reason, and we never find an end to all its qualities, its possibilities, its powers, its relations. It has become infinite. Vivekananda

In a certain sense, mathematics has been advanced most by those who are distinguished more for intuition than for rigorous methods of proof. Felix Klein

For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known. First Corinthians 13: 12