Poincaré and All

The protagonist of the chapter is Henri Poincaré, an outstanding French mathematician whose contribution to the development of mathematics is extremely rich. For the argument presented in the book, however, his understanding of the idea of space, presented by him at the beginning of the twentieth century, is of the most important significance. The chapter discusses the development of understanding the phenomenon of space that takes place not only within geometry but also in art. The history of geometry is also the story of the development of the idea of space since Euclid. This idea turned out to be necessary for the creation of the image of the world, and this belief appeared about a whole range of philosophers such as Giordano Bruno, Isaac Newton, Gottfried Leibniz, and Immanuel Kant. The breakthrough took place in the nineteenth century, when geometry, and with it space, were completely deprived of the obligation to represent the world and become completely autonomous. The chapter goes on to describe Poincaré’s idea of space as part of his overall approach to geometry. He constructed an image of a completely abstract and speculative space. At the same time, Poincaré introduces the concept of a qualitative approach, i.e. without the need for metric coordinate systems, which inaugurates a new field of geometry: topology. Poincaré also interpreted space as a dynamic structure, which introduced one of the most important constructions of modern physics, also used in other sciences, known by names such as configuration space, state space, phase space or space-time. In this way, a cognitive category was created, which enabled a possibility of the universal and abstract description for a selected phenomenon, in our case, knowledge.