Modern approaches to mathematical notation

Question

I'm interested if there is literature on projects which try to improve formal notation, especially for doing mathematics on an advanced level. For example, I'm thinking along the lines of diagrammatic ideas like Penrose notation, but maybe even for more basic and common topics.

edit: I initially posted this on the linguistics board, and there I conjectured it might be rather suited for the cognitive science board too. I have to point out that just moving it here over night was a very bad move - Now that it's here I can't essentially ask an analog question again. Of course, if I wanted to post it on the math board, the question would have been formulated very differently. The question is not about mathematicans introducing handy notation on the fly while they work on their topic. One has a rough feel for which answers might be delivered within the math community itself. The reason is that it's pretty clear how things go down: On the math board the question can't compete with stuff like "I'm a high schooler and I discovered this fact about complex numbers" (upvote, upvote, upvote), which lead to this thread being sucked into the abyss after 10 hours, where it will not receive any more answers.

Answer 1

You might find the articles

1. (R. BROWN and T. PORTER), The intuitions of higher dimensional algebra for the study of structured space, Revue de Synthèse, 124 (2003) 174-203. here

2. (R. BROWN and T. Porter), Category theory and higher dimensional algebra: potential descriptive tools in neuroscience, Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1 (2003) 80-92. arXiv:math/0306223

relevant to the notation question.

Also I suggest a web search on string diagrams in category theory.

Answer 2

Concerning diagrammatic notation like Penrose's, it can be helpful once you get used to it, but is somewhat bulky. The tendency in math is usually away from diagrammatic notation and toward more compact symbolic notation. A good example of this is Frege's two-dimensional notation for logical formulas as developed in his work Begriffsschrift. His notation hasn't caught on, and was replaced by notation developed by his successors like Peano and others.

I don't know about literature on improving diagrammatic notation, so the comment that follows does not directly answer your question, but there are some papers that analyze the use of diagrams in math; see for example Sherry's article the role of diagrams in mathematical arguments