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There is a strong and growing trend to do mathematics via diagrammatic algebra, which involves constructing and manipulating equations whose elements are diagrams drawn in the plane. The manipulations are diagram rewrites. The diagrams are considered not as figures illustrating a `string of symbols' algebraic expression; instead, they are considered to be algebraic objects in their own rights. An archetypal diagrammatic algebraic theory is the theory of skein modules.

Notation of "algebra as strings of symbols" was pioneered by Al-Qalasadi in the fifteenth century. Previous to Al-Qalasadi, equations were often written as paragraphs of text, as in Al-Khwarizmi's Compendious Book on Calculation by Completion and Balancing.

Question: Who invented diagrammatic algebra? And when?

The first diagrammatic algebraic text I know is Louis Kauffman's 1987 paper State models and the Jones polynomial. It contains passages such as: Kauffman diagrammatic algebra

A subquestion is: Is this indeed the origin of diagrammatic algebra?

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Conway came close to such manipulations (see p. 338): – Ian Agol Jun 3 '14 at 5:42
I'd say God invented it. – Fernando Muro Jun 3 '14 at 6:38
Penrose, with his spin networks? Or Feynman? Otherwise, have a look at – David Roberts Jun 3 '14 at 7:02
I would not perhaps place Feynman diagrams in this category. They are a diagrammatic representation of physical processes with precise computational rules to obtain numbers out of them, but they are not a diagrammatic algebra. I would however put Roger Penrose's diagrammatic tensor calculus in this class. That must date back to the 60s/70s. – José Figueroa-O'Farrill Jun 3 '14 at 7:07
Ronnie Brown told me around 1987 about the idea related to this but not yet implemented. All alphabets are one-dimensional, i. e. based on concatenation of symbols. It seems that at least in some areas of homotopy theory one may benefit from introducing (at least) two-dimensional alphabets in which symbols can be put together on the plane - not just left-right but also above-below... – მამუკა ჯიბლაძე Jun 3 '14 at 7:15
up vote 14 down vote accepted

I'd say C.S. Peirce, from about 1882 until his death, was a very early practitioner with his Existential Graphs. This is a tripartite diagrammatic algebra or calculus that deals with propositional logic, first-order logic with equality, and modal logic. I (with Gerry Brady, who is an expert on Peirce) have argued here and there their kinship with Feynman diagrams and string diagram calculus.

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In this direction then Venn diagrams are around 1880 – მამუკა ჯიბლაძე Jun 3 '14 at 8:44
Any chance you could add a bit more detail to this? At first glance it's hard for me to see string diagrams here. Certainly's Peirce's work was influential on Kauffman and so there's some relationship, but could you give an example of something in Peirce's work which is clearly diagrammatic algebra? – Noah Snyder Jun 3 '14 at 8:47
@NoahSnyder You might try as a start. The actual "strings" in Peirce's Beta all called "ligatures", and weakening rules for ligatures that are mentioned in Peirce's "permissions" can be interpreted nowadays as units and counits for diagonal and projection maps in cartesian bicategories. Actual calculations in Peirce's work might be more prominently displayed in the case of Alpha, but I don't have actual page numbers to hand (maybe later). – Todd Trimble Jun 3 '14 at 11:47
@მამუკაჯიბლაძე I'm wondering whether Venn diagrams would count. To me they are somewhat static representations, especially useful for reading off disjunctive normals forms; to the best of my knowledge one does not perform substitution or surgery rules on them to perform calculations. – Todd Trimble Jun 3 '14 at 11:52
And Todd and Geraldine's work on Peirce's existential graphs inspired Kate Ponto and Michael Shulman's 'Duality and traces in indexed monoidal categories', as explained here…. – David Corfield Jun 3 '14 at 16:56

Frege's formalisation of first-order logic from the end of the 19th century uses a two-dimensional formalism, see e.g. here.

Konrad Zuse, inventor of the first computer, also developed the first high-level programming language, the Plankalkül which uses a 2-dimensional notation


Programming languages where indentation is syntactically relevant are making a comeback, e.g. Python and Haskell.

Przytycki, in his Classical roots of knot theory traces the history of braid diagrams, including this one of Gauss'.

Gauss braid

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Thanks! Could you elaborate on the sense in which these are diagrammatic algebra? In particular, where are diagram rewrite rules used for computation? – Daniel Moskovich Jun 3 '14 at 9:21
@DanielMoskovich I'm not sure what exactly counts as diagrammatic algebra. The rewrite semantics of programming languages is typically given by compilers that translate programs in the source language to machine language. The latter can easily be given a rewrite semantics. Programming language theory has come up with various graphical calculi for computation for example Girard's geometry of interaction and proof nets (warning: poor Wikipedia articles) and their many descendants ... – Martin Berger Jun 3 '14 at 9:42
... The abstract theory of such rewrite calculi has been studied in e.g. term rewriting, graph rewriting. I don't know if anyone has produced a diagrammatic algebra-oriented analysis of the Plankalkül capturing it's computational dynamics, but it surely could be done. I've been offering a modern analysis of the Plankalkül as a final year project to my students for years, but interest in the history of computation is low. – Martin Berger Jun 3 '14 at 9:42

As I mentioned in a comment, Conway was very close to using diagrammatic algebra in his paper on enumeration of knots and links: he shows skein relations for links differing by tangle replacement defined by symbols, then shows the tangles immediately following in a figure. It's plausible that Conway used diagrammatic algebra notation in his notes, but chose the presentation in the paper for typesetting reasons (it might be worth consulting him).

I found an exposition of Conway's polynomial by Cole Giller from 1982 that actually does use the diagrammatic version of skein relations. It's clear that Kauffman was aware of Giller's exposition (he has alternate expositions from 1979 and 1981 that uses a similar convention to Conway's, but cite Giller's preprint from 1979, even though it didn't appear until 1982), so it's possible that Kauffman subsequently followed Giller's convention. It might be worth looking at the references in Giller's paper to see if there's any earlier papers using diagrammatic algebra.

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Thanks! I hadn't known about Cole Giller's exposition. – Daniel Moskovich Jun 5 '14 at 13:02

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