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.

Try math.SE, or stackoverflow may have people who work on math notation software. — prash, Jun 25 '13 at 23:24
Category theory's commuting diagrams are somewhat like this. — Eric Stucky, Jun 26 '13 at 01:00
I would have said commuting diagrams too. Also, exact sequences. (Penrose notation is unreadable to me.) — Billy, Jun 26 '13 at 01:31
The [tag:reference-request] tag should not be used as the only tag on a question; see tag-wiki and meta. Please include tags pertaining to mathematical content in future questions. — Lord_Farin, Jun 26 '13 at 06:54
@Lord_Farin: the question had some other tags, which were dropped when it was moves here. — Nikolaj-K, Jun 26 '13 at 07:47
Nick: I hope you will elaborate somewhat on just what it is you are hoping to find out. — MJD, Jul 03 '13 at 15:26
@MJD: It's pretty involved, but maybe I manage to formulate it one day. — Nikolaj-K, Jul 03 '13 at 15:36
Maybe today would be a good day to try, since you already posted the question. — MJD, Jul 03 '13 at 15:46
I am not qualified to answer this question for real, but when I looked up Penrose notation I found this question with an answer pointing to A survey of graphical languages for monoidal categories, which you might find of interest. — dfan, Jul 03 '13 at 16:51
@MJD: Thanks for the bounty, but I don't have everything together yet. I have a private project going on, which is about fitting math representation to the human cognition. How can our inner workings be relieved? It's a motivation behind "What is an efficient nesting of mathematical theorems?". In turn, related to this are my notes I started two months ago in form of a wiki http://axiomsofchoice.org/. Also see the graph. — Nikolaj-K, Jul 03 '13 at 21:27

score 3 · Answer 1 · edited Jun 26 '13 at 06:56

You might find the articles

(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
(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.

Thanks for the answer! This is a link to the second paper mentioned. — Nikolaj-K, Jun 26 '13 at 08:02

Mikhail Katz · Answer 2 · 2013-07-03T15:46:03.587

1

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

edited Jul 03 '13 at 15:46

answered Jun 26 '13 at 08:26

Mikhail Katz

47,573

Do you think that the decision for using strings instead of diagrams has other reasons than practical ones. Like narrow space. The Begriffsschrift is over 100 years old, but category theories like to use diagrams. I wonder about which notation let one understand the concepts better. – Nikolaj-K Jun 26 '13 at 08:47
I personally like diagrams. For example, a commutative diagram illustrates the idea better than its linear formulation in terms of traditional composition notation. On the other hand, I am a differential geometer by training, while the standard (linear) logical notation was definitely not developed by geometers :-) In category theory, a certain number of leading people are algebraic geometers, which might explain the preference for the diagrammatic approach. – Mikhail Katz Jun 26 '13 at 08:51
I assume many logic people think geometrically too today. "Sheaves and logic", right? – Nikolaj-K Jun 26 '13 at 09:00
On the other hand the successor to the begriffsschrift is the proof trees that one often sees: http://i.imgur.com/zn6e4lS.png for example. I think the trend away from diagrammatic notation toward linear symbolic notation was partly driven by the technology used to do mathematics. Diagrams are harder to typeset and print than lines of symbols, and the resistance to diagrams would have varied as printing technologies evolved and changed, maybe reaching a local maximum around 1980, just before the introduction of desktop publishing. – MJD Jul 03 '13 at 15:18

Modern approaches to mathematical notation

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