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Hilbert’s Geometry and Mathematical TruthDavid Hilbert’s account of geometrical truth is partially motivated by his 1899
consistency proof for Euclidean geometry (relative to arithmetic). His proof contains two
main parts: (1) a full axiomatization of the geometry, and (2) a construction of an
arithmetical model that satisfies the relevant axioms. These two steps raise seemingly
competing desiderata for accounts of mathematical truth. I examine Hilbert’s attempt to
reconcile them in his account of geometrical—and more generally, mathematical—truth.
Hilbert’s task in step (1) was to characterize the space of Euclid’s geometry. This fact
introduces the first desideratum for an account of geometrical truth: it must fit the
intended subject matter. Had Hilbert failed to fully and accurately characterize his
intended subject matter, he would have failed to demonstrate the consistency of
Euclidean geometry. And because the aim was to characterize Euclid’s geometry in a
fully axiomatic way, he had to provide axioms to capture the features of the geometrical
space that, in Euclid’s presentation, emerged from diagrammatically constructive
postulates (think: compass and straightedge).
This full axiomatization of the geometry is required to make step (2) feasible. By
replacing diagrammatically constructive postulates with axioms, Hilbert distanced his
version of Euclidean geometry from the space of diagrammatic constructions. He could
sever the essential tie between geometrical space and physical space. This made it
possible for an arithmetical model (the segment calculus) to exhibit all of the structural
features of Euclidean space. The arithmetical model could satisfy all of Hilbert’s axioms
under an appropriate interpretation, but it would not have been a suitable interpretation of
Euclid’s diagrammatical constructions.
Step (2) introduces a second desideratum for accounts of mathematical truth: axioms
must admit of multiple interpretations. In order to prove consistency, Hilbert needs the
segment calculus to satisfy the axioms of Euclidean geometry that he set out in step (1).
So, an arithmetical interpretation of those axioms must be admissible; the possible
referents of terms like ‘point’ must not be limited to things like points in physical space.
The semantics of geometric terms must not be wholly determined by the intended subject
matter that geometric axioms were set out to characterize.
There is some tension between both wanting axioms to have a privileged interpretation—
in this case, the one that matches Euclid’s geometry—and wanting axioms to admit of
multiple interpretations. I will set out Hilbert’s account of geometrical truth, and
mathematical truth more generally, as designed to navigate between these two desiderata.
University of Kansas