NMAK15016U CHANGED: History of Mathematics 2 (Hist2)

History of geometry 1770-1920.

During the indicated period geometry underwent a development that had wide ranging influence on our understanding of

1. what mathematics is and

2. the nature of space.

It is often mentioned as one of the few examples of a revolution in mathematics.' Arround 1830 the discussion of the parallel postulate led to the creation of non-Euclidean geometry. Whether this geometry was consistent remained an open question until Gauss' and Riemann's works on differential geometry made it possible to create a model of the non-Euclidean plane as a surface of constant negative Gauss curvature. This model was also interpreted in projective geometry that was also devellopped in the 19th century. The century ended with different new attempts to give axiomatic descriptions of geometry, among which Hilbert's is the most famous. The considerations concerning non-Euclidean geometry was not only an exercise in axiomatics. For all the actors it was also a question of understanding the nature of (physical) space. The discovery of non-Euclidean geometry led to a rejection of Kant's opinion that geometry (for Kant this meant Euclidean geometry) was an a priori but synthetic intuition. Instead various empirical, conventional or formalistic epistemologies were put forward. The mathematical and philosophical considerations of the 19th century created a background for the revolutionary ideas that Einstein put forward in his special and in particular general theory of relativity. In the course all these interacting subjects will be discussed.
Students are required to take an active part and give seminars.

During the course the student will learn to investigate the history of a piece of mathematics, to analyze a mathematical text from the past, and to use the history of mathematics as a background for reflections on philosophical and sociological questions regarding mathematics. Moreover the course will give the students a more mature view on the mathematical subject in question. The course will be particularly relevant for students who aim for a career in the gymnasium (high school) but all mathematics students can benefit from it.