Taking seminal mathematical concepts and theories as their starting point, the contributors to this volume define myriad forms of interaction between mathematical research and historiographical problems. In the process, they pose such important questions as "Can an independent historian of mathematics make fruitful contributions to the development of mathematics?" and "Where did twentieth-century mathematics go wrong?". This provocative collection of papers evolved from a History of Mathematics Symposium organized in Tokyo in conjunction with the International Congress of Mathematics held in Kyoto, Japan. The venue gave excellent cause to develop yet a further line of study rarely given its due in such anthologies: the mathematical traditions in the East. On this subject, papers discuss how the intermingling of cultures contributed to the introduction of Indian mathematics to the Islamic world, and Western mathematics to Japan.