On the role of functions in the internal organization of mathematics. The case of Klein's Erlangen Program

Enetz Ezenarro
ILCLI & Teachers University School at Vitoria-Gasteiz
Seminar on Logic and Foundations of Symbolic Systems


After the failure of Bourbaki's structuralism to explain the unity and internal coherence of contemporary mathematics, new proposals are needed to achieve, at last in part, their aim of finding the essential organising principle of mathematics. My purpose in this talk is to briefly analyse Klein's Erlangen Program [Yaglom (1988): Felix Klein and Sophus Lie. Evolution of the idea of symmetry in the nineteenth century, ch. 7], a case in which a profound reorganisation, systematisation and simplification took place in geometry, focusing on some concrete functions and their algebraic structures. This shift of attention in geometry opened the door to a more fundamental conception of the field and inspired the creation of new research areas. Furthermore, we will try to show that successive generalisations that have been made since then have contributed to solidify that viewpoint [Sharpe (1997), Differential Geometry. Cartan's Generalization of Klein's Erlangen Program]. It can't be considered, as we are going to see, that it has been an isolated case. I believe that these cases show the central role that particular functions, understood in their broadest sense, can play, whenever an effective systematisation is needed in any mathematical area. My point is that the ubiquity of functions as main organisational tools may be a major characteristic of contemporary mathematics. This is an important lesson we can learn from Klein's Erlangen Program.