The concept of axiom in Hilbert’s thought

David Hilbert is considered the champion of formalism and the mathematician who turned the axiomatic method into what we know nowadays. Hilbert compares in 19001 the axiomatic method with the genetic one, and he argues that only the former is capable of giving a true foundation to mathematics. In this paper I will analyze the key concept of the axiomatic method, namely that of axiom. To understand the concept of axiom in Hilbert’s work means to delve into the modern axiomatic method. We will also explore the goals of Hilbert’s use of the axiomatic method and its differences from previous practice. An important difference between the axiomatic method at the beginning of the 20th century and the ancient one is the role of intuition. 20th century mathematics needed to account for the abstract features that were introduced during the 19th century. On the other hand, previous mathematics was rather based on intuitive skills that were not considered sufficient anymore. In Hilbert’s foundational writings we can find a notion of formal system that is already similar to the one we use today, and in the Grundlagen der Geometrie2 (1899) the new features are implicit yet. Later on, formal systems gain some structural characters that depend on the use of a new formal logic. Hilbert spells out the tools of logic inferences, proofs are regarded as finite objects, theorems are now considered valid only if they are deducible from the axioms with a finite number of admitted inferences, etc. As a consequence we can find a deep change in the concept of axiom.