Notation

A repo to track the historical evolution of notational systems in arts and sciences.

Antiquity

Petroglyph

Petroglpyhs by ancient humans are probably the first ways of creating a map of communication with fellow beings.

Hieroglyphics

Levantine

Cretan maze symbol

Arithmetica

Square of Opposition

Mesopotamian

Asian

Chinese

Indian

Mesoamerican

Mayan

Middle Ages

Carolingian Miniscule

8th century script that became the calligraphic standard.

Zā’irjah

Paper

Medieval Music Notation

Neume

Daseian Notation

Oresme

Llull

Pasigraphy

Real Character

John Wilkins

Blissymbols

Memory Palace

Unicorn Tapestry

Abacists

Liber Abacci

Leon Battista Albert

Alberti Cipher Disks

Enlightenment

Napier

Stifel

Circular slide rule based on Bürgi’s logarithm tables

https://klaustruemper.com/2018/02/17/circular-slide-rule-based-on-burgis-logarithm-tables-of-1620/

Bombelli

Bacon Ciphers

First equation: Recorde

Descartes

Vives

Nicolaus Reimers

Bartholomäus Keckermann

Alsted

Weigel

Sturm

Gottfried Leibniz

Ars Combinatoria

Llull’s work would influence a key figure in the history of science: Gottfried Leibniz. In his dissertation on combinatorics, De Arte Combinatoria, influenced by Descartes’ idea and Llull’s rotating wheels, he proposes an alphabet of human thought.

Binary notation

Leibniz did work with binary arithmetic.

He turned to I Ching for his inspiration. He used 0 to denote the broken line representing chaos and 1 to denote the straight line representing order in the ancient text.

Differentiation notation

In print, the notation first appeared before public in Nova methodus pro maximis et minimis, itemque tangentibus, qua nee fractas, nee irrationales quantitates moratur, & singulare pro illis calculi genus in Acta Eruditorum (Pages 467-473) in 1684.

There is also an upside down ± symbol present in the text which is curious.

Integration notation

Leibniz purportedly made use of the integral sign in his private notebooks (LH 35, 8, 8).

This notation first appears in print for public in De Geometria Recondita et analysi indivisibilium atque infinitorium in Acta Eruditorum (Pages 292-300) in 1686.

Instead of the italic long s, the serif version can be found to represent the symbol in print.

Weise

Samuel Grosser

Lange

Lambert

Ulrich

Steinbart

Johann Gebhardt Ehrenreich Maaß

Made triangle diagrams based on Lambert’s line diagrams

Johann Gottfried Kiesewetter

Used circle diagrams to illustrate rules of conversion

Ploucquet Diagrams

Kant

Mellin

Newton

Euler Diagrams

Industrial Age

George Boole

Venn

Jevons

Marquand

Hamiltonian Notation

De Morgan’s Spicular Notation

Modern Age (1800 - 1940)

Cayley

Arthur Cayley was the first person to coin the ideas of finite group and trees. It is also very interesting that he played around with visual notations to convey ideas about these algebraic structures.

Group Multiplication Table

Trees

Cayley Graph

Lewis Carrol Notation

Frege

Begriffsschrift

• Original Paper

Gottlob

A programming language to play around with Begriffsschrift notation:

Charles Pierce

https://arxiv.org/ftp/arxiv/papers/1108/1108.2429.pdf

https://mulpress.mcmaster.ca/russelljournal/article/download/2056/2081/

Jan Łukasiewicz

He seems to have a logical matrix in his book and also, need to create a catalog of the notations he has employed in his other works such as many valued logics.

Stamm

Stam seems to be the first person to publish work on Sheffer Stroke and Pierce Arrow: https://twitter.com/rrrichardzach/status/1251532455829319680

Gentzen

Sequent Calculus

Research who brought in the sequent calculus deduction method to the forefront of computer science deduction methods.

Post

Truth Tables

Russell

Truth Tables

Wittgenstein

Truth Tables

Stanisław Leśniewski

Ideogrammatic notation

This one needs deeper investigation as it is much close to box-X notation of Charles Peirce, XLA notation of Shea Zellweger, and Randolph diagrams. Much interesting about this idea is that he had a certain philosophical grounding and use of brackets to complement this notation for operators.

The Logical Systems of Leśniewski

https://link.springer.com/book/10.1007%2F978-3-319-00482-2

Behmann (1922)

Inverted representation of T for falsehood.

Ramsey (1927)

http://www.columbia.edu/%7Eav72/papers/JANCL_2003.pdf

Space Age (1940 - 1970)

Cybernetics

McCullough Pitts Notation

Randolph Diagrams

Randolph Diagrams were used to notate Boolean operations in a 2 by 2 grid. This can be extended to more than one truth values.

These might have precursors in X-frame notation of Peirce in “A Proposed Logical Notation (1903)”. Detail from this paper

Karnaugh Maps

Karnaugh maps are used to notate Boolean algebra. This is an improvement upon Veitch Chart which is a rediscovery of Marquand Diagrams introduced by Allan Marquand.

Marquand Diagrams

Martin Gardner

Logic Machines and Diagrams

A book surveying logical machines and diagrams

The Propositional Calculus with Directed Graphs with Frank Hararay

APL

Plankalkul

Direct expression via simulation

A quasi arithmetical notation for syntactic description - Yehoshua Bar Hillel (1953)

Information Age (1970 - Now)

Language builders

Feynman Diagrams

Physics and Feynman’s Diagrams

John Barwise

APL - Iverson

John Cage Notations

Esoteric languages

Befunge

Brainfuck

Piet

Billiards Ball Computer

A small exposition here.

Diagrammatic Algebra for Concurrency

Diagrammatic Algebra: From Linear to Concurrent Systems

Picturing Resources in Concurrency

Geometry of Interaction

• Paper

GoI Visualizer

Unker non-linear writing system

2020

Linear Logic in Existential Graph Notation

Adele Lopez (2020)

Konstantin Osmei (2020)

ZX Calculus Animation

Craig Gidney

Quantum Circuit Simulator

Correlation Surface

Adinkras for Supersymmetry

Dominic Hughes

Logic without Syntax (2005)

First-order proofs without syntax (2019)

Intutionistic proofs without syntax (2019)

Hest programming language

Ivan Reese (2019)

Jamie Vicary

Homotopy.io

A web based proof assistant for globular n-categories. Considered to be the successor to Globular

Globular

Opetopic

A visual editor for opetopes.

Discopy

String Diagrams

Joe

Article

Johannes Drever

Form Bakery

Drever’s playground for George Spencer-Brown’s notation from Laws of Form.

Jules Hedges

The Art of String Diagrams

Peter Selinger

Survey of Graphical Languages for Monoidal Categories

Resources

A History of Mathematical Notation - Florian

Art of Memory - Rossi/Yates

The Notation of Medieval Music

Numerical Notation: A Comparative History - Stephen Chrisomalis

Umberto Eco

Enlightening Symbols - Joseph Mazur

The Development of Peirce’s Logic and Semeiotic Theory of Notation

Logic Machines and Diagrams — Martin Gardner

Sign-creation and man-sign engineering

Notation as a Tool for Thought - Iverson

History of Binary and Other Nondecimal Numeration

Heaviside - On Operators in Physical Mathematics

His take in simplifying Maxwell’s equations could also be helpful in understanding the intellectual framework shift that helped in changing the perspective on functions.

Computer Science Metanotation - Guy Steele

A History of Truth-Values - Jean-Yves Béziau

A History of Logic Diagrams (Amirouche Moktefi, Sun-Joo Shin)

Susanne Langer on Sheffer’s Notational Velocity:

Susanne Langer and the Woeful World of Facts - Giulia Felapi (2017)

Facts: The Logical Perspective of the World

Philosophy in a New Key

Feeling and Form

Irving Anellis

The Historical Sources of Tree Graphs and the Tree Method in the Work of Peirce and Gentzen

A good paper tracing the history of trees in Mathematics

Jon Barwise and John Etchemendy

Visual information and valid reasoning

Janice Glasgow, Harinarayanan, Chandrasekharan

Diagrammatic Reasoning: Cognitive and Computational Perspectives

Jens Lemanski

Means or Ends: On the Valuation of Logic Diagrams

Periods in the Use of Euler-Type Diagrams

Amirouche Moktefi, Francesco Bellucci, Ahti-Veikko Pietarinen

Diagrammatic Autarchy: Linear Diagrams in the 17th and 18th Centuries

Amirouche Moktefi, Shin

A History of Logic Diagrams (2012)

What is a logical diagram?

Legg (2013)

An Eleventh-Century Venn Diagram

A.W.F. Edwards (2006)

Euler-Diagramme: Zur Morphologie einer Repräsentationsform in der Logik

Peter Bernhard (2001)

The Remarkable Diagrams of Johann Maass

P. Bernhard (2007)

Math Origins: The Logical Ideas

Earliest uses of symbols of operation

Earliest uses of symbols of operation: Part 2

The History of Mathematical Symbols

Contributors to the Universal Language

The Search for the Perfect Language

Umberto Eco

Tools

Rune Generator

A fun tool to generate a rune like language: https://watabou.itch.io/rune-generator

Visual Lambda Calculus

Books

Books that take a largely diagramattic approach in its pedagogy. For more information check out

Diagramattic Immanence (2015)

Rocco Gangle

Diagrammatology

Frederik Stjernfelt

Plant Form

Dynamics

Mathematical Reasoning with Diagrams (2001)

Mateja Jamnik

The Philosophical Status of Diagrams

Mark Greaves (2002) Has a compilation of Euler style and other logic diagram sunder the heading ‘Early diagrams for Syllogistic Logic’

Talks

Spatial Thinking is the Foundation of Thought

A tour de force of the different kinds of notational devices and spatial thinking tools that humans have employed over the course of history to make sense of the world around them. Note to self: I need to take the spectrum of ideas presented there and incorporate them as categories in this document.

Discussions

Who invented diagrammatic algebra?

Compendiums

Fred Hohman’s curation to improve mathematical notation

List of links of techniques to help enhance mathematical notation

Catalog of notations by Katherine Ye