Thank you Andre for pointing out to my elevator calculus, that I actually invented in 1968 - 1969 while working in my Thesis (published as SLN 145). The same thing happened with elevators that with string diagrams, when I suggested to Mac Lane to write the thesis with the elevators he said to me that elevators were fine for private calculations, but not for publishing. He gave me two reasons: One was that notations were important things, and to introduce a new notation you should be an established mathematician. The second, more important, that he considered unquestionable, was that printers will not accept manuscripts with elevators. I continuously used elevator calculus instead of diagrams in order to find proofs of equations in tensor categories. Or instead of pasting diagrams when calculating in 2-categories (1-arrows in the role of objects, composition in the role of tensor, and 2-cells in the role of arrows). But translate to diagrams or pasting diagrams for publishing. With augmented experience with LateX, I and my students started to publish with elevators. For those that may be curious or interested, here are three links where elevators are explained and used: https://arxiv.org/abs/1406.5762v1 https://arxiv.org/abs/1110.6411v2 https://arxiv.org/abs/1110.5293 Best regards, Eduardo. On 5/6/17 13:45, Joyal, Andr? wrote:
Some more comments.
I always imagined that Penrose was inspired by Feynman's diagrams, but Max's story is casting doubts on this idea; Penrose may have been chiefly concerned with the syntax of tensor calculus.
Let me point out that Eduardo Dubuc invented an "elevator calculus" in the early 70's which is a form of the string diagram notation. It was never published.
Best regards, Andr?
________________________________________ From: Ross Street [ross.street@mq.edu.au] Sent: Friday, May 05, 2017 6:48 PM To: Aleks Kissinger Subject: categories: Re: History of string diagrams
On 4 May 2017, at 1:19 AM, Aleks Kissinger <aleks0@gmail.com<mailto:aleks0@gmail.com>> wrote:
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).
Some random comments:
The person who told me of the Penrose-Rindler reference and the earlier
R. PENROSE, Applications of negative dimensional tensors, in ``Combinatorial Mathematics and its Applications,'' (D.J.A. Welsh, Ed., Academic Press, 1971) 221--244
was Iain Aitchison who found a coloured string-diagram Pascal-triangle-like algorithm for producing the n-cocycle condition arising from the orientals and their cubical analogues. While Iain's more recent
The geometry of oriented cubes, arXiv:1008.1714v1 [math.CT]
has incredible diagrams in comparison with 1984 technology, the string versions are not there.
Speaking of Roger Penrose, Max Kelly used to tell the following story about their time (mid 1950s) in Cambridge. Max thought Roger must be very visually impaired. Two reasons:
1. When Max first met him he was wearing very thick glasses. It turned out Roger was conducting an experiment to test whether one would adapt to wearing lenses that inverted the world. After a few days apparently the brain adjusts and it believes everything is the right way up.
2. Looking over Roger's shoulder on lectures using tensors, Max noticed that Penrose was not using the usual notation at all. He was using the string notation instead. When Max asked why, Roger said that all the i_1, j_2, 1_1, . . . sub- and super-scripts were impossible to read, whereas the connecting strings made it clear.
Who knows what lies in one's subconscience! However, I think the string notation Max used when talking about his work with Eilenberg on extraordinary natural transformations (not the more general Set-based dinatural transformations Dubuc and I wrote about) arose quite independently of Max's Penrose experience. Sometimes when Graeme Segal was in Sydney, I was around while he and Max discussed comparisons of the Eilenberg-Kelly string diagrams
(which do not appear in their paper: A generalization of the functorial calculus, Jour. Algebra 3 (1966) 366--375)
and string diagrams in physics.
Best wishes, Ross
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