See http://mathoverflow.net/questions/168888/who-invented-diagrammatic-algebra

The thesis (in German) can be download here:

]]>Yes, it should be . Fixed.

]]>Zhen, I’m not sure I really want to argue about it here, but it seems to me a major point of string diagrams would be the recognition that the meaning of a string diagram, whether in a general monoidal category or just in vector spaces, is invariant under certain types of deformation of string diagrams (e.g., the yanking moves for adjunctions) — that seems to be the whole point which makes their manipulation so effective. And it seems to me that that point is definitely in need of proof, even when restricted to just the vector space case.

But I’m happy to shut up and let Qiaochu keep talking!

]]>@Todd: If Penrose proved anything, it probably isn’t be in _Road to Reality_. But I’m not sure there’s anything that even needs proving in the context of finite-dimensional vector spaces: all the objects in question can be regarded as arrays of numbers, and the diagram is just a way of keeping track of all the indices and contractions. Of course, maybe there’s some subtlety I’m missing…

]]>@Zhen: (as an aside, I’m sure you know that Geometry of Tensor Calculus I covers a number of doctrines besides braided monoidal categories). I have to admit that I’ve looked only cursorily at Penrose’s work. Does he prove theorems that place his notation on a rigorous footing? It was my impression that this is precisely a value of Joyal and Street’s work. (Of course, Feynman diagrams were another precursor of string diagrams. It could even be said that Peirce’s existential graphs were yet another precursor.)

]]>To the best of my knowledge, the calculus of string diagrams as applied to braided monoidal categories was first formalised by Joyal and Street in their 1991 paper, in which the soundness of the calculus is also proven. But if you are only using string diagrams for, say, the category of finite-dimensional vector spaces, then it is probably not unfair to only mention Penrose.

Penrose’s diagrammatic notation has some other clever features: for example, the metric tensor is written as an upside down U, highlighting its naturality of its use in lowering indices. Even determinants can be expressed reasonably efficiently… that’s all in _Road to Reality_.

]]>I’m not familiar with the history of string diagrams and I didn’t want to say something incorrect about Joyal and Street’s contributions. I also didn’t want to talk too much about category theory until later.

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