Filosofía y Lógica

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the shape and evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. Most interestingly, the context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence. And a careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets (and of the dichotomic conception of set theory) in Hilbert’s early axiomatics, and detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom).

The celebrated “creation” of transfinite set theory by Georg Cantor has been studied in detail by historians of mathematics. However, it has generally been overlooked that his research program cannot be adequately explained as an outgrowth of the mainstream mathematics of his day. We review the main extra-mathematical motivations behind Cantor’s very novel research, giving particular attention to a key contribution, the Grundlagen of 1883, where those motives are articulated in some detail. Evidence from other publications and correspondence is pulled out to provide clarification and a detailed interpretation of those ideas and their impact upon Cantor’s research. Throughout the paper, a special effort is made to place Cantor’s scientific undertakings within the context of developments in German science and philosophy at the time (philosophers such as Trendelenburg and Lotze, scientists like Weber, Riemann, Vogt, Haeckel), and to reflect on the German intellectual atmosphere during the 19th century.