Who Thinks Abstractly?: From Modern Geometry to Modern Algebra with Emmy Noether

This chapter considers the work of Emmy Noether, a major figure in the history of modern mathematics. In geometry, the primary field of her earlier work, her work followed the tradition, defined by the role of group theory, established by Klein, Lie, and others; in algebra, the primary field in her later work, her work followed the tradition established by Galois and later Dedekind. The chapter does not aim to offer a comprehensive analysis of her work, but only a perspective on it defined by the this study’s Pythagorean view of modern mathematics and its role in physics. While equally Pythagorean, classical physics and relativity are geometrical in spirit, as was Noether’s work on her theorems relating symmetry and conservation laws, and quantum theory was algebraic in spirit, as was Noether’s later work in abstract algebra or algebraic topology, pursued apart from any connections to physics. Section 6.1 is introductory. Section 6.2 discusses her work and the nature of abstract thinking in mathematics, via Hegel’s essay “Who thinks abstractly?” Sect. 6.3. considers Noether’s theorems, both her original, geometrically grounded, theorems, Noether’s Noether’s theorem, and more algebraic versions of these theorems in quantum theory. Sections 6.4 and 6.5 address Noether’s work in abstract algebra and algebraic topology, respectively, which was instrumental in establishing contemporary algebra and topology.