What Are Mathematical Practices? The Web-of-Practices Approach

This chapter can be considered as made up of two parts, a general discussion of the notion of mathematical practice and the limits of its use, comprised by the first three sections, and a particular case study that is presented in order to exemplify the idea of the web of practices, which occupies the remaining three. The presentation of my approach to the notion of mathematical practice is brief and synthetic but more articulated theoretically than in a previous book (Ferreirós 2016). Considerations in Sect. 3 about mathematical cognition and ethnomathematics have to do with the “limits” of mathematical practice, i.e., with practices that do not fall under that rubric. The case study concerns the evolution of the function concept, which offers a complex example of the web-of-practices approach. We start with early Modern developments that led to the eighteenth-century idea of function (as “analytical expression”) and the nineteenth-century “logical” idea of function, showing how complex is the spiderweb of interrelated mathematical and scientific practices knotted around the function concept. In the last section, I consider the difficult question of the most basic practices and cognitive experiences that form the basis of the idea of function. I propose that two conceptions – a dynamic and an algebraic or operational one – coalesced to give rise to the early Modern idea of function.