Abstract
Several mathematical theories, such as Newton's theory of fluxions and fluents or Peano's theory of natural numbers were originally formulated in an 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 distinguishes three kinds of inconsistency by measuring the "distance" of the inconsistent theory from its consistent form.
They correspond to whether re-formulations, relativizations or recodings are needed for turning the inconsistent theory into a consistent one. The paper argues that inconsistencies of different kind have different cognitive backgroud and thus should have different role in education.