Branch points of algebraic functions and the beginnings of modern knot theory
type: Item
instance of: scholarly article
title: Branch points of algebraic functions and the beginnings of modern knot theory
zbMATH Open document ID: 0855.01014
author: Moritz Epple
published in: Historia Mathematica
publication date: 1997-01-26T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 01A55
Mathematics Subject Classification ID: 01A60
zbMATH DE Number: 851479
MaRDI profile type: Publication
full work available at URL: hmat.1995.1031
OpenAlex ID: W1982930875
cites work: Foundations of geometry (Festschrift 1899). Edited and commented by Klaus Volkert
cites work: A polynomial invariant for knots via von Neumann algebras
cites work: Q3243297
cites work: Q4182695
cites work: Q4756475
cites work: Q4760119
cites work: Q3998730
cites work: Geschichtliche Entwicklung des Homologiebegriffs
cites work: Q3715247
cites work: Q3963208
cites work: A history of algebraic and differential topology 1900--1960
cites work: Q4883908
cites work: Historical evolution of the concept of homotopic paths
cites work: Q3788741
cites work: Q4774196
cites work: Max Dehn
cites work: Singular Points of Complex Hypersurfaces. (AM-61)
cites work: Q3228041
cites work: Klein, Hilbert, and the Gottingen Mathematical Tradition
cites work: Q3687352
cites work: Q4883879
cites work: Riemann, Betti and the birth of topology
cites work: Q5597484
DOI: 10.1006/HMAT.1995.1031
description: scientific article; zbMATH DE number 851479