Analysing the number concept in Weierstraß’ lecture notes: conceptions disregarded and remained unknown

Abstract

The motivation for undertaking this research came from reading Kossak’s 1872 publication about Weierstraß’ conception of numbers. It became clear that Weierstraß was developing a general notion of complex numbers that aligned closely with the concepts found in the eighteenth-century French arithmetic textbooks, an area already investigated by one of us. This prompted a deeper examination of the arithmetic framework that Weierstraß had introduced as the foundation for his lectures on analytic function theory. Our inquiry led to a comprehensive assessment of the lecture notes produced by students who attended these courses, which constitute the essential sources for understanding the development of Weierstraß’ foundational approach from the early 1860s to 1886. Rigorous notions of irrational numbers were developed in the second half of the nineteenth century by one French mathematician and three German mathematicians: Charles Méray, Georg Cantor, Richard Dedekind, and Carl Weierstraß. Nowadays, Dedekind’s approach to irrational numbers largely dominates the modern understanding of the real number system, while Weierstraß’ conception has been mostly forgotten. Weierstraß did not develop his theory with the aim of establishing a rigorous notion of real numbers, as is commonly reported by those who are aware of his achievements. Instead, his goal was to create a new notion of complex numbers—not merely the “standard” complex numbers introduced by Gauß in 1831, but a more general concept. To achieve this, he introduced his own method for constructing irrational numbers. This broader conception, which is little known today, provided the numerical foundation for Weierstraß’ preferred field of mathematical research: elliptic and Abelian functions. His primary concern, which he reformulated over time, came from Gauß’ 1831 observation about the impracticality of extending the number system to what Weierstraß called general complex numbers, later known as hypercomplex numbers.