Hilbert, Duality, and the Geometrical Roots of Model Theory

Abstract

The article investigates one of the key contributions to modern structural mathematics, namely Hilbert’sFoundations of Geometry(1899) and its mathematical roots in nineteenth-century projective geometry. A central innovation of Hilbert’s book was to provide semantically minded independence proofs for various fragments of Euclidean geometry, thereby contributing to the development of the model-theoretic point of view in logical theory. Though it is generally acknowledged that the development of model theory is intimately bound up with innovations in 19th century geometry (in particular, the development of non-Euclidean geometries), so far, little has been said about how exactly model-theoretic concepts grew out of methodological investigations within projective geometry. This article is supposed to fill this lacuna and investigates this geometrical prehistory of modern model theory, eventually leading up to Hilbert’sFoundations.

On consistency and existence in mathematics

“…One only needs to apply a reversible one-one transformation and lay it down that the axioms shall be correspondingly the same for the transformed things' (IV/4, p. 42). Here Hilbert does indeed appear to anticipate the fact that isomorphic structures are elementarily equivalent -a fact which is further confirmed by his invocation of the 'principle of duality' from projective geometry as an illustration (on which see Eder and Schiemer 2018). This appears to be compatible with the contemporary understanding of structures as isomorphism types.…”

Frege and the origins of model theory in nineteenth century geometry

“…It should also be mentioned that, in spite of passages like the one quoted earlier, there is still room for disagreement about how Hilbert understood his method of proving independence and consistency in detail at various times. See Eder and Schiemer (2018) for further discussion. 26 Frege held introductory courses in complex analysis and seminars on advanced topics such as elliptic functions and Abelian integrals, and in geometry he lectured on both synthetic and analytic geometry.…”

Frege on intuition and objecthood in projective geometry

“…andLorenat (2015) for a discussion of the duality controversy. Some of the issues that are discussed in this section have been investigated inEder (2019), which in turn is partly based on joint work with Georg Schiemer [seeEder and Schiemer (2018)]. 28 Chasles later clarified that really all that matters is that incidence is preserved.…”