Branch points of algebraic functions and the beginnings of modern knot theory
The main point of this article has nothing to do with the title. It asserts that new mathematics is invented by a process of differentiation together with an elimination of the context. In less bald terms, mathematics changes by someone seeing the essence of a problem in an existing mathematical field has nothing to do with that particular field, that the problem can be formulated in terms which have nothing in common with where it first occurred. The problem is then removed from the existing field where it occurs and becomes a topic for study in itself abstracted or extracted from all the previous context, physical or mathematical or whatever. This is exemplified by Wirtinger's formulation, influenced by Felix Klein's views of the integrity of mathematics and its contexts, of a problem in complex algebraic function theory as a particular problem in group theory but expressed in function theoretic terms in 1895. This is then taken up by Tietze in 1908, and abstracted by Dehn into combinatorial topology, and in particular into (combinatorial) knot theory in 1910. Article details the relations between all the people involved -- those mentioned above and Hilbert, among many others -- and their mathematics, and their views of mathematics, including Klein's critique of abstraction: here in function theory, but also regarding the Hamilton-Jacobi differential equations. It is not only the doing of mathematics which is discussed, but also how mathematics is done. In passing, it reconstructs Wirtinger's contribution to these theories as the prime mover in this part of knot theory.
scientific article; zbMATH DE number 851479
scientific article