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Hilbert’s Geometry and Mathematical Truth

David Hilbert’s account of geometrical truth is partially motivated by his 1899consistency 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.

Author Information:

Eileen Nutting

Philosophy

University of Kansas