 The central difference between Frege and Hilbert over the nature of axioms, i.e., over the question whether axioms are determinately true claims about a fixed subject-matter or reinterpretable sentences expressing multiply-instantiable conditions, lies at the heart of the difference between an older way of thinking of theories, exemplified by Frege, and a new way that gathered strength at the end of the nineteenth century. Perhaps most clearly illustrated in Dedekind 1888, the central idea of the new approach is to understand mathematical theories as characterizing general “structural” conditions that might be had in common by any number of different ordered domains. Just as, in algebra, the axioms for a group give general conditions that can be satisfied by any manner of object whatsoever under appropriate relations, so too on the new view the axioms of geometry specify multiply-instantiable conditions. Viewing theories from this modern perspective, it is entirely appropriate to take axioms as Hilbert does, since reinterpretable sentences are the right vehicles to express the multiply-instantiable conditions in  question.[8]  From the point of view of the earlier fixed-domain conception of theories, on the other hand, such reinterpretable sentences are entirely inappropriate as axioms, since they fail to fix a determinate subject-matter. On this question, i.e., the issue of the fixed-domain (Fregean) vs. multiply-instantiable structure (Hilbertian) conception of mathematical theories, the jury is still out: this debate continues to animate contemporary philosophy of mathematics (see entry on  philosophy of mathematics).  