de Freitas, E. & Sinclair, N. (2017). Concepts as generative devices. In E. de Freitas, N. Sinclair, & A. Coles (Eds.), What is a mathematical concept? (pp. 76-89) Cambridge, UK: Cambridge University Press. 

This article presents six claims about concepts based on Gilles Chatelet’s writing about the flexibility and generative potential of mathematical concepts. It highlights the important tension between the logical (rule following, axiomatic) and ontological (creative) dimensions of mathematical activity.

1.) Concepts are not merely metaphors or representations. 

2.) Concepts are not mental constructs abstracted from the material world. 

3.) Concepts are vibrant and indeterminate, having one foot in the virtual and one in the actual. 

4.) Concepts operate as both logical and ontological devices. 

5.) There is no a priori logical ordering between mathematical concepts. 

6.) Concepts emerge from aesthetic-political acts.