A debate about the axiomatization of arithmetic: Otto Ho¨lder against Robert Graßmann

This article provides a detailed discussion of Otto Ho¨lder's ideas on the foundations of mathematics. It focuses on a paper he published in 1892 written in reaction to a book published in 1891 by Robert Graßmann, which Ho¨lder saw as an attempt to axiomatize arithmetic. Ho¨lder's paper is important for at least three reasons: First, it represents what might be called Ho¨lder's research manifesto on the foundations of mathematics, containing a wealth of ideas which Ho¨lder gradually developed in a variety of publications until the end of his life. Second, Ho¨lder's analysis of R. Graßmann's foundational ideas provides an important assessment of the contribution of Hermann and Robert Graßmann to the axiomatization of arithmetic, a contribution which, though often mentioned, is itself still not widely acknowledged and not fully understood. Third, the effort of exposing the weak spots in R. Graßmann's ideas led Ho¨lder to formulate the main problems confronting formal axiomatics: independence of the axioms, consistency, completeness, and the issue of the relationship between pure mathematics and its applications. The first part of this paper presents R. Graßmann's ideas on the foundations of mathematics as outlined in two closely related works published in 1872 and 1891. The second part focuses on Ho¨lder's analysis of these ideas. [Copyright &y& Elsevier]