Axiomatic systems and the Hilbert-Frege controversy

Jean-Louis Hudry, University of Edinburgh (10 Mar 2004)

First SPPA Seminar Day, University of Edinburgh

In his 'Grundlagen der Geometrie' (1899), Hilbert provides us with an account of Euclid's geometry, defined as a system of axioms and no longer as a collection of definitions, postulates and common notions. Thus, Euclid's geometry is transformed into an Euclidean model divided into five groups of axioms. Each group pertains to a particular relation, such as a relation of betweenness for the Axioms of Order or a relation of completeness for the Axioms of Continuity.

This paper aims to assess Hilbert's justifications of axiomatic systems and Frege's objections to them. The concepts of an Euclidean model are only defined by the relation/axiom to which they belong, and do not have proper (intuitive or whatever) meanings since they do not refer to any definition or object outside the axiom system. Such concepts have only implicit definitions derived from the axioms alone. Consequently, any system of things is acceptable, and a system 'love, law, chimney sweep' or 'table, chair, mug' (Hilbert's instances) is as implicitly meaningful as a system 'point, line, plane'. Systems of things are interchangeable only if they belong to consistent and complete models. Consistency is defined as the absence of contradiction, such that an axiom is true if not contradicting the rest of the axioms. If there exists a model in which an axiom contradicts other axioms, then the axiom is said to be independent. For instance, the model of an axiom system P & A is consistent if and only if A is an independent axiom. Furthermore, completeness implies that two models are isomorphic (identical in structure), meaning that there is a one-one correspondence between the elements that preserves all relations in either model.

Frege rejects the Euclidean model and claims that it is composed of pseudo-axioms and pseudo definitions that do not refer to proper geometric objects. He appeals to the notion of thought or sense, to which a truth condition is applied, and an axiom is true only if it asserts something that is already laid down by a true thought. In other words, an axiom cannot be a definition, since a merely assertive proposition does not define a concept; only a thought does. As well, Frege accuses Hilbert of confusing first-level concepts with second level ones, and of neglecting the fact that only first-level concepts, under which objects fall, can be laid down by real definitions and asserted by real axioms. If points, lines and planes are first-level concepts about geometric objects, then they do not belong to a second-level system of things, meaning that they are not interchangeable with 'love, law, chimney sweep' or 'table, chair, mug'.

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