Silvia de Toffoli (Stanford University) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "Notations and Diagrams in Algebra". Abstract: The aim of this talk is to investigate the roles of Commutative Diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that, differently from other mathematical diagrams, CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation, that goes beyond the traditional bipartition of mathematical representations into graphic and linguistic. It will be argued that one of the reasons why CDs form a good notation is that they are highly ‘mathematically tractable’: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of a ‘diagram chase’. In order to draw inferences, experts move algebraic elements around the diagrams. These diagrams present a dynamic nature. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture.