Patterns in diagrams, patterns in formulas

This talk is not about patterns in nature, but about patterns in the abstract mathematical formulas and diagrams we use to reason. I'll start from the question of diagrammatic reasoning in mathematics: can one reason rigorously using diagrams as one does with sentences or symbolic formulas? I'll describe recent work that aims at showing that one can indeed do so, at least in some cases; I'll then explain why the open-ended possibility of noticing patterns in diagrams raises a challenge for efforts at fully capturing diagrammatic reasoning rigorously. Finally, I'll show that this discussion forces us to reconsider the role of formulas as well: the appearance of unexpected patterns in our symbolic calculations can sometimes prove crucial, too.