Jeremy Heis

The consensus for the last century or so has been that diagrammatic proofs are not genuine proofs. Recent philosophical work, however, has shown that (at least in some circumstances) diagrams can be perfectly rigorous. The implication of this work is that, if diagrammatic reasoning in a particular field is illegitimate, it must be so for local reasons, not because of some in-principle illegitimacy of diagrammatic reasoning in general. In this talk, I try to identify some of the reasons why geometers in particular began to reject diagrammatic proofs. I argue that the reasons often cited nowadays — that diagrams illicitly infer from a particular to all cases, or can’t handle analytic notions like continuity — played little role in this development. I highlight one very significant (but rarely discussed) flaw in diagrammatic reasoning: diagrammatic methods don’t allow for fully general proofs of theorems. I explain this objection (which goes back to Descartes), and how Poncelet and his school developed around 1820 new diagrammatic methods to meet this objection. As I explain, these new methods required a kind of diagrammatic reasoning that is fundamentally different from the now well-known diagrammatic method from Euclid’s Elements. And, as I show (using the case of synthetic treatments of the duals of curves of degrees higher than 2), it eventually became clear that this method does not work. Truly general results in “modern” geometry could not be proven diagrammatically.

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