A Geometric Model of Reduction and Emergence

My special purpose in this contribution is to use the hierarchy of geometries as Felix Klein conceived it to illustrate the relationship between the various tiers of the hierarchy of natural sciences. I want to try to clarify the concept of ‘emergence’, by which I mean the emergence at each tier of the hierarchy of concepts peculiar to and distinctive of that tier, and not obviously reducible to the notions of the level immediately above or higher still. One such notion, as we shall see, is that of a ‘circle’ in Euclidean geometry. The contextually distinctive notions of biology at the organism level include, for example, ‘heredity’, ‘infection’, ‘immunity’, ‘sexuality’ and ‘fear’.