Is It the Same Twelfth? Questioning an Unquestioned Principle

This paper explores new methodological avenues. It investigates the theory component of praxeologies, a component that has often been overlooked. It goes back to a series of research, both in the French and the Anglo-Saxon contexts, in the domain of school arithmetic, including Douady and Perrin-Glorian pioneer didactic engineering and works related to multiplicative reasoning in cognitive psychology. These works were not developed within the ATD framework. The paper interprets them in terms of praxeologies and investigates their theory component. That is, it attempts at considering alternative foundations of numbers, namely quantities instead of sets. This enables to shed a new light on issues like multiplicative reasoning, and actual foundations of numbers in successful pioneer didactic engineering works and in East Asia.