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By Saul A. Basri

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In his persistent attempt to solve the consistency problem, Hilbert held prima facie dramatically different positions. First, around 1900, he pursued a kind of Dedekindian logicism and, almost 20 years later, he took quite seriously Russell’s attempt of founding mathematics in logic. Second, in 1920 and 1921, he adopted Kroneckerian constructivism for a restricted mathematical base theory that was then to be expanded stepwise to fully cover mathematical practice; each step was to be secured through a consistency argument.

Contrasting this special form of contentual axiomatics with its general existential form, Bernays calls it sharpened axiomatics (verschärfte Axiomatik). The philosophical significance of consistency proofs is thus to be seen and assessed in terms of the objective underpinnings of the frames within which reductions are achieved. A crucial question is, consequently, what kind of procedures can be viewed as generative arithmetic ones. The (clauses of ) elementary inductive definitions of syntactic notions, like formula or proof, were initially viewed in that light.

Bernays, already in the 1930s, stated this view clearly, when considering mathematics as “the science of idealized structures”. My considerations try to bridge the gap between the structuralism of mathematical practice and the structuralism of philosophical analysis. The latter is almost exclusively concerned with structures that are categorically fixed; indeed, it mostly deals with epistemological and metaphysical issues for natural numbers. Let me note that models of complete ordered fields (“real numbers”) are also unique up to isomorphism; but their domains are not accessible.

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