Translation equation and Sincov’s equation – A historical remark

Abstract

Gottlob Frege (1848 – 1925), the world famous logician was also a pioneer in iteration theory. His habilitation thesis “Rechnungsmethoden, die sich auf eine Erweiterung des Grössenbegriffes gründen” (“Methods of Calculation based on an Extension of the Concept of Quantity”) published 1874 yields a foundation of iteration theory and dynamical systems in one and also in several variables. He considers there the translation equation $$f(s,f(t,x))=f(s+t,x)$$ f ( s,f ( t,x ) ) = f ( s + t,x ) and all the three so-called Aczél-Jabotinsky equations connected with the differentiable solutions of it. By this way Frege e.g. recognized also the importance of the infinitesimal generator of a dynamical system. A comprehensive presentation of this matter may be found in Gronau [4]. Frege treated in this connection also Sincov’s equation $$ \Psi(z,x)=\Psi(z,y)+\Psi(y,x) $$ Ψ ( z,x ) = Ψ ( z,y ) + Ψ ( y,x ) and gave its general solution almost 30 years before Sincov. The history and background of Sincov’s equation is described in Gronau [5]. Here we give a detailed description of the connection between the translation equation and the Sincov equation.

Keywords

• iteration theory
• gottlob frege
• history of mathematics