Belief Revision vs. Conceptual Change in Mathematics

In his influential book Conceptual Revolutions (1992), Thagard asked whether the question of conceptual change is identical with the question of belief revision. One might argue that they are identical, because “whenever a concept changes, it does so by virtue of changes in the beliefs that employ that concept”. According to him, however, all those kinds of conceptual change that involve conceptual hierarchies (e.g., branch jumping or tree switching) cannot be interpreted as simple kinds of belief revision. What is curious is that Thagard’s interesting question has failed to attract any serious response from belief revision theorists. The silence of belief revision theorists may be due to both wings of their fundamental principle of informational economy, i.e., the principle of minimal change and the principle of entrenchment. Indeed, Gärdenfors and Rott conceded that their formal theory of belief revision “is concerned solely with small changes like those occurring in normal science” [8]. In this paper, I propose to re-examine Thagard’s question in the context of the problem of conceptual change in mathematics. First, I shall present a strengthened version of the argument for the redundancy of conceptual change by exploiting the notion of implicit definition in mathematics. If the primitive terms of a given mathematical structure are defined implicitly by its axioms, how could there be other conceptual changes than those via changing axioms? Secondly, I shall examine some famous episodes of domain extensions in the history of numbers in terms of belief revision and conceptual change. Finally, I shall show that there are extensive and intricate interaction between conceptual change and belief revision in these cases.