Bridging Theories with Axioms: Boole, Stone, and Tarski

In discussions of mathematical practice the role axiomatics has often been confined to providing the starting points for formal proofs, with little or no effect on the discovery or creation of new mathematics. For example, quite recently Patras wrote that the axiomatic method “never allows for authentic creation” (Patras 2001, 159), and similar views have been popular with philosophers of science and mathematics throughout the 20th century. Nevertheless, it is undeniable that axiomatic systems have played an essential role in a number of mathematical innovations, most famously in the discovery of non-Euclidean geometries. It was Euclid’s axiomatization of geometry that motivated the investigations of Bolyai and Lobachevsky, and the later construction of models by Beltrami and Klein. Moreover, it was not only through the investigation and modification of given systems of axioms that new mathematical notions were introduced, but also by using axiomatic characterizations to express analogies and to discover new