Re: History of string diagrams

[Aleks Kissinger <aleks0@gmail.com>]

A short note: This idea that string diagrams are, due to technical
issues, only useful for private calculation, is said explicitly by
Penrose. Penrose and Rindler's book "Spinsors and Spacetime" (CUP
1984) has an 11-page appendix full of all sorts of beautiful,
carefully hand-drawn graphical notation for tensors and various
operations on them (e.g. anti-symmetrization and covariant
derivative). On the second page, he says the following:

"The notation has been found very useful in practice as it grealy
simplifies the appearance of complicated tensor or spinor equations,
the various interrelations expressed being discernable at a glance.
Unfortunately the notation seems to be of value mainly for private
calculations because it cannot be printed in the normal way."


Best,

Aleks


On 2 May 2017 at 16:50, 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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