How to Prove Things with Diagrams

For over a century, philosophers and mathematicians have held deep suspicions about diagrams in proofs, regarding them as at best heuristic aids, and at worst as sources of error. Silvia De Toffoli overturns these beliefs, showing that diagrams can be not only legitimate but also essential components of rigorous mathematical proofs.

The use of diagrams and other visual representations is widespread in mathematics as well as in many other domains of human discourse. There is no reasonable doubt that they can be of enormous heuristic value; they improve understanding by making certain properties salient, which would otherwise remain unnoticed. Some diagrams, however, do more. Beyond mere static illustrative devices, they are used dynamically for thinking. Diagrams, in this sense, are tools of reasoning. This is particularly evident in the case of mathematical diagrams, which are best conceived as elements of genuine notational systems.Even in mathematics, the most abstract domain of theoretical knowledge, thinking is mediated by heterogeneous representational formats. How to Prove Things with Diagrams offers a systematic epistemological analysis of mathematical diagrams focusing on their role in proofs and grounded in actual mathematical practice. It thus contributes to the philosophy of mathematical practice, offering fresh insights into how we prove things in mathematics. How to Prove Things with Diagrams engages with case studies from contemporary mathematics: knot theory, homological algebra, and category theory to challenge long-standing prejudices against diagrams, revealing them as powerful, reliable, and often irreplaceable instruments of mathematical reasoning.