I have read a mathematical-philosophical book by Prof. Fumiharu Kato in Tokyo Institute of Technology（東京工業大学の加藤文元教授）, called “Bernhard Riemann’s Mathematics and Thoughts（リーマンの数学と思想）” (Kyoritsu Shuppan/共立出版). It supposes to be the 4th of a series of books describing mathematics by Riemann in the publisher. The main theme is the rise of metaphorical structure in nature proposed by Riemann himself. For mathematical detail please refer this site (only in Japanese). Before Riemann, there were two ideas of dualism: intrinsic provision of positions and extrinsic provision of metrics. Riemann, however, sublimated these ideas to introduce intrinsic provision of existence. This proposition cruised humanity from an idea of geometry a priori (due to the famous philosopher, Immanuel Kant) to an idea of manifolds a posteriori. More in detail, before Riemann the mathematics was dominated by formula deformations, and this methodology more or less bogged down by that age. Riemann (and some others) treated ideas of sets (in its primitive form), topologies and invariants to get over the obstacle, and this worked very fine, according to the development of mathematics aftermath. This transition is remarked as “from science of quantity to science of idea”, requiring construction of the ideas for carving out the logics. The idea had already appeared in intrinsic Gauss curvature independent of coordinates with an idea of absoluteness, but Riemann went far beyond. He came up with ideas similar to modern topology and differentiable manifolds. This ensures superiority of relationships of elements over their entity, probably leading to structuralism in later ages, which was obviously inspired by mathematics. It is also notable that Riemann was also careful about application of observable scales, which is very important for interpreting actual data set for natural science. Overall, this book is a nice introduction to a mathematical world related to the rise of modern.
