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Ryan Schwiebert (Apr 30 2024 at 14:22):As I learn about string diagrams and their related monoidal categories, I can't help be reminded of the "pictures" almost all of us started out with while doing Euclidean geometry.
I think historically many people are dismissive of proofs relying only on pictures since "they can mislead us." And yet I see in the recent material, suitably defined diagrammatic languages are robust enough to constitute proof.
So, I think it is a very natural question to ask "can Euclidean constructions be rephrased in string diagrams? Is there a version of string diagrams that represents straightedge-compass constructions? Does it perhaps have deficiencies or complexities that lead to the "mistakes" people make with geometric proofs by pictures?"
I would not be surprised at all if Euclidean style constructions were not suitable for string diagrams.
Ryan Schwiebert (Apr 30 2024 at 14:25):IIRC, if one operates in analytic-mindset, constructions could be viewed as solution sets to polynomial equations, and the use of equations and constraints sounds like we would be drifting toward categories. But when I started this question, I had mainly the classical synthetic-mindset in mind.