Philosophy of Mathematics seminar

Silvia De Toffoli (Pavia)

How to prove things with diagrams  

In my talk, I will focus on diagrams in contemporary mathematics. I will advance two main theses, which I will support with the aid of case studies. (1) Some types of diagrams form mathematical notational systems and play non-redundant roles in proofs. They not only serve as illustrations but are themselves genuine reasoning tools. That is to say, we can think with diagrams. (2) Some diagrams are essential to the proofs in which they figure. Although we can find diagram-free counterparts of diagrammatic proofs, given plausible criteria of identity for proof, any such counterpart would be a different proof from the original one. In sum, inter-transformability does not imply inter-translatability. Donald Davidson once said that a picture is “not worth a thousand words, or any other number. Words are the wrong currency to exchange for a picture.” I will show that words are also the wrong currency to exchange for a mathematical diagram. I will argue that in order to appreciate the effectiveness of diagrams, it is not sufficient to consider only their informational content and how such content can be put into words. We must also consider the articulation of such content and why it matters in practice. This can be done, for instance, by evaluating how such articulation facilitates extracting information, carrying out specific inferences, and performing calculations.