How did Frege and Hilbert differ concerning the interpretation of mathematical theories

Frege hoped that the theories of mathematics could be given bases of self-evident axioms. As far as possible, the aim was the reduction of mathematics to logic; the axioms grounding mathematical theories should be logical principles, and the rules of inference used to move from the axioms to the theory should be rules of logic. Frege did not believe, however, that the reduction to logic was achievable for all theory; the basis of axioms for geometry, for example, would be the self-evident truths of geometry, rather than self-evident logical principles. Although Frege allowed that some axioms be principles of logic, and others not, he maintained that all axioms should be axioms in the classical sense: they should have a truth-value, they should be true, and we should be able to take their truth for granted. Further, the terms of the axioms should not, in general, be left undefined. He recognised the necessity of leaving some terms undefined—for it is not possible to create a language with every term defined, without at some stage giving a circular definition,—but he held that the undefined terms, the primitive terms, should in some sense have their meanings fixed. This point will be made clearer when the contrast to Hilbert‟s view is made below.