Comprehending the Connection of Things: Bernhard Riemann and the Architecture of Mathematical Concepts Arkady Plotnitsky Abstract This chapter is an essay on the conceptual nature of Riemann’s thinking and its impact, as conceptual thinking, on mathematics, physics, and philoso- phy. In order to fully appreciate the revolutionary nature of this thinking and of Riemann’s practice of mathematics, one must, this chapter argues, rethink the nature of mathematical or scientific concepts in Riemann and beyond. The chapter will attempt to do so with the help of Deleuze and Guattari’s concept of philosophical concept. The chapter will argue that a fundamentally analogous concept of concept is also applicable in mathematics and science, specifically and most pertinently to Riemann, in physics, and that this concept is exceptionally helpful and even neces- sary for understanding Riemann’s thinking and practice, and creative mathematical and scientific thinking and practice in general. 1 Introduction This chapter is an essay on the conceptual nature of Bernhard Riemann’s thinking and its impact, as conceptual thinking, on mathematics, physics, and philosophy. In order to fully appreciate the revolutionary nature of this thinking and of Riemann’s practice of mathematics, one must, I argue, rethink the nature and structure, architecture, of mathematical or scientific concepts in Riemann and beyond. I shall attempt to do so here with the help of Gilles Deleuze and Félix Guattari’s concept of philosophical concept, as defined in What Is Philosophy?[8], the culminating work of Deleuze’s philosophy, on which I shall comment presently. I argue that a fundamentally analo- gous concept of concept is also applicable in mathematics and science, specifically and most pertinently to Riemann, in physics, and that this concept is exceptionally helpful and even necessary for understanding Riemann’s thinking and practice, and creative mathematical and scientific thinking and practice in general. While I shall address Riemann’s work in physics, I shall, given my scope, be less concerned with A. Plotnitsky (B ) Theory and Cultural Studies Program, Purdue University, W. Lafayette, IN 47907, USA e-mail: plotnits@purdue.edu © Springer International Publishing AG 2017 L. Ji et al. (eds.), From Riemann to Differential Geometry and Relativity, DOI 10.1007/978-3-319-60039-0_11 329