Several mathematical theories, as for instance Newton’s theory of fluxions and fluents, Frege’s theory of the foundations of arithmetic, or Peano’s theory of natural numbers were first formulated in a logically inconsistent form. Only after some period of time consistent formulations of these theories were found. The paper analyzes several historical cases of this “initial inconsistency”. It suggests distinguishing three kinds of inconsistency according to the “distance” of the proposed inconsistent theory from its consistent variant. These three kinds correspond to whether re-formulations, relativizations or re-codings are needed for turning the inconsistent theory into a consistent one.
inconsistency, non-standard analysis, theory of fluctions