Hi Pawel, we definitly have had plenty of experiences at the quantum computing end making it clear that diagrams reduce the respect when preducing actual results, and that translating to the Hilbert space model, which causes loosing generality, actually increases likeliness of acceptance. I also felt that in Theoretical Computer Science symbolic categories also improve the acceptance likeliness as compared to diagrams, although this may have been changing recently in part thanks to your efforts. That said, a lot of people do appreciate the idea of an entirely diagrammatic formalisms, in particular researchers with a multi-disciplinary tendency, and in quantum foundations a diagrammatic default for crafting theories is in the process of becoming the standard, so there they have been very succesful, artly thanks to leading researchers (without a category theory background) such as Lucien Hardy picking them up.
On 2 May 2017, at 15:50, Pawel Sobocinski <sobocinski@gmail.com> wrote:
Dear Categorists,
I would like to ask for comments about the history of string diagrams as graphical notation for the arrows of higher and monoidal categories. For the sake of precision, I mean the (various kinds of) graphical notation where there is a "dimension flip", i.e. given a (weak) n-category, the n-cells are drawn as points (0-dimension), the n-1 cells as lines (1-dimension) etc. This includes, as a special case, string diagrams as notation for the arrows of symmetric monoidal categories (Joyal and Street), which have found a number of applications (quantum mechanics, computer science, engineering, linguistics, ...) in recent years. We now also have impressive online tools, such as Jamie Vicary's Globular, that allow both type-setting and computing with string diagrams.
It seems to me that there aren't very many historical notes available: Peter Selinger's "A survey of graphical languages for monoidal categories" is a nice survey but it's quite terse on the historical aspects. In the historical notes that I've come across, string diagrams are often mentioned in the same breath with Penrose tensor diagrams, Feynman diagrams, and proof nets, but while there are of course similarities, there are also clear differences owing to the categorical nature of string diagrams; for example, string diagrams are usually quite strictly "typed" with domain and codomain determined by dangling wires in the case of monoidal categories (or, in higher dimensions, surfaces).
I'm interested in the history of the use of the notation, as well as the surrounding "sociological" aspects. Through overheard gossip, I believe that the notation was a quasi-secret "house style" in some groups, used for calculations, but carefully exided from formal publications. But maybe this is a bit overblown, and the printing technology simply wasn't there? Or were there particularly conservative editors who were not comfortable with publishing diagrammatic calculations?
In any case, it seems strange that we have had to wait until the 1990s for this notation to actually start making it into papers. Many calculations in earlier works were quite clearly worked out with string diagrams, then painstakingly copied into equations. Sometimes, clearly graphical structures were described in some detail without actually being drawn: e.g. the construction of free compact closed categories in Kelly and Laplazas 1980 "Coherence for compact closed categories". From personal experience, some papers become much more readable after being redrawn into almost comic books: Carboni and Walters' 1987 "Cartesian bicategories I" comes to mind.
I'm reminded of quote by E.J. Aiton from his biography of Leibniz (which I came across in Peter Gabriel's Matrices, géométrie, algèbre linéaire):
"Owing to the reluctance of printers to accept books on mathematics, because of the difficulties of type-setting and the small number of potential readers, the statement of results in letters, especially when these were registered in the Royal Society or the Paris Academy, provided a means of establishing a claim to invention, rending possible publication at a later date. The most precious possessions of a mathematician were, of course, the original methods by which new results could be obtained. While communicating results, in order to establish his possession of a general method, to which he might refer in impenetrably opaque terms, he took pains to eliminate any dues that would enable his correspondent to guess the method..."
I'd appreciate any comments -- both personal and more summative. I'll be happy to compile any information sent to me personally, or to the list, and make it available online. I'm especially interested in:
* Who came up with the notation? When was it first used? Was it rediscovered independently by several groups? * Was there an effort to keep it a "house secret"? * Was there any institutional resistance to the use/publishing of string diagrams?
Finally, I'd like to take the opportunity to advertise the 1st Workshop on String Diagrams in Computation, Logic, and Physics, which I'm organising with Aleks Kissinger, and which will take place at the Jericho Tavern in Oxford, September 8-9, 2017. More information is available at http://string2017.cs.ru.nl, and we will soon send out a formal call for papers.
Best wishes, Pawel.
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