Re-Formulations as a Third Pattern of Change in Mathematics

Abstract

The changes that were analyzed in the previous two chapters were of a global nature. Each of them consisted in the rebuilding of the syntactic or semantic structure of wide are as of mathematics. Therefore they happened rarely. They were not the fruit of the work of one mathematician but rather the result of the work of a whole series of them. The resulting language that emerges from their collective effort is usually a compromise that unites the innovations originating from many different authors. In contrast to this, the changes that will be analyzed in the present chapter are of a local nature. Usually they are related to a single definition, theorem, proof, or axiom. They happen often and form the content of the everyday work of mathematicians. These changes consist in the reformulation of a problem, a definition, a proposition, or an axiom and therefore I suggest calling them re-formulations. The hyphen in the name means that it is not an arbitrary reformulation but one which introduced a change important from the epistemological point of view. We can formulate a problem, a definition, a proposition, or an axiom in a particular language in many different ways. In some contexts these formulations may be equivalent and we can use them as synonyms.