[[./img/notation.png]]
** Petroglyph
Petroglpyhs by ancient humans are probably the first ways of creating a map of communication with fellow beings.
[[./img/petroglyph.jpg]]
[[./img/petroglyph-2.jpg]]
Hieroglyphics Levantine Cretan maze symbol Arithmetica Square of Opposition Mesopotamian Asian Chinese * Indian * Mesoamerican Mayan
** Carolingian Miniscule 8th century script that became the calligraphic standard.
** Zā’irjah
[[./img/zā’irjah.png]]
*** [[http://www.alpha60.de/research/scrambling_truth/DavidLink_ScramblingTruth2010_100dpi.pdf][Paper]]
** Medieval Music Notation *** [[https://en.wikipedia.org/wiki/Neume][Neume]] [[./img/neume.jpg]]
*** Daseian Notation [[./img/daseian.jpg]]
** Oresme
** Llull [[./img/ars-combinatoria-llull.jpg]] [[./img/llull-tree.jpg]]
** Pasigraphy
** Real Character John Wilkins
** Blissymbols
Memory Palace Unicorn Tapestry Abacists Liber Abacci
** Leon Battista Albert
*** [[https://en.wikipedia.org/wiki/Alberti_cipher][Alberti Cipher Disks]] [[./img/alberti-disks.jpg]]
** 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
[[./img/vives.png]]
** Nicolaus Reimers
[[./img/reimers-1.png]] [[./img/reimers-2.png]]
** Bartholomäus Keckermann [[./img/keckermann.png]]
Alsted Weigel [[./img/weigel.jpg]]
Sturm Gottfried Leibniz
*** Ars Combinatoria
[[./img/ars-combinatoria.png]]
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.
[[./img/leibniz-binary-arithmetic.png]]
He turned to [[http://www.leibniz-translations.com/binary.htm][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.
[[./img/leibniz-binary-notation.png]]
*** Differentiation notation
[[./img/dx-in-print.png]]
In print, the notation first appeared before public in [[https://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-differential-calculus][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 [[https://digitale-sammlungen.gwlb.de/][private notebooks (LH 35, 8, 8)]].
[[./img/first-appearance-of-integral-in-leibniz-manuscript.png]]
[[./img/integral-in-print.png]]
This notation first appears in print for public in [[https://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-integral-calculus][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 [[./img/grosser.jpg]]
** Lange [[./img/lange.jpg]]
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 [[./img/ploucquet.jpg]]
** Kant [[./img/kant.jpg]]
** Mellin [[./img/mellin.jpg]]
** Newton
** Euler Diagrams
George Boole Venn Jevons Marquand Hamiltonian Notation De Morgan’s Spicular Notation
** 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 [[./img/cayley-group-multiplication-table.png]]
*** Trees
[[./img/cayley-tree.png]]
*** Cayley Graph [[./img/cayley-graph.png]]
** Lewis Carrol Notation
** Frege
*** Begriffsschrift
[[./img/begriffsschrift.png]]
**** [[http://www.attoparsec.com/artifacts/gottlob/index.html][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.
[[https://amzn.to/30vGFgD][The Logical Systems of Leśniewski]] [[Leśniewski's Systems of Logic and Foundations of Mathematics][https://link.springer.com/book/10.1007%2F978-3-319-00482-2]]
*** Behmann (1922)
Inverted representation of T for falsehood.
*** Ramsey (1927)
**** [[Facts and Propositions][http://www.columbia.edu/%7Eav72/papers/JANCL_2003.pdf]]
Cybernetics McCullough Pitts Notation
** Randolph Diagrams
[[./img/randolph.png]]
Randolph Diagrams were used to notate Boolean operations in a 2 by 2 grid. This can be extended to more than one truth values.
[[./img/randolph-multiple-truth-values.png]]
These might have precursors in X-frame notation of Peirce in “A Proposed Logical Notation (1903)”. Detail from this [[https://arxiv.org/ftp/arxiv/papers/1108/1108.2429.pdf][paper]]
[[./img/x-frame-notation.png]]
** 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
*** [[https://monoskop.org/images/e/e6/Gardner_Martin_Logic_Machines_and_Diagrams.pdf][Logic Machines and Diagrams]] A book surveying logical machines and diagrams
*** [[https://www.archim.org.uk/eureka/archive/Eureka-48.pdf][The Propositional Calculus with Directed Graphs with Frank Hararay]]
[[./img/gardner-propositional-graphs1.png]] [[./img/gardner-propositional-graphs2.png]]
APL Plankalkul Direct expression via simulation [[https://www.jstor.org/stable/410452][A quasi arithmetical notation for syntactic description - Yehoshua Bar Hillel (1953)]]
** John Barwise
** APL - Iverson
** [[https://monoskop.org/images/9/92/Cage_John_Notations.pdf][John Cage Notations]]
** Esoteric languages
*** Befunge
*** Brainfuck
*** Piet
** Billiards Ball Computer
[[./img/billiards-ball-logic-1.png]]
[[./img/billiards-ball-logic-2.png]]
[[http://archive.is/yWuli][A small exposition here.]]
[[./img/billiards-ball-exposition.gif]]
** Diagrammatic Algebra for Concurrency
[[https://dl.acm.org/doi/pdf/10.1145/3290338?download=true][Diagrammatic Algebra: From Linear to Concurrent Systems]] [[https://ora.ox.ac.uk/objects/uuid:cd021f28-bb43-4aa1-8cde-7d81d4e202e9/download_file?file_format=pdf&safe_filename=corrected-thesis-rp.pdf&type_of_work=Thesis][Picturing Resources in Concurrency]] [[./img/resources-in-concurrency.png]]
** Geometry of Interaction
*** [[https://koko-m.github.io/GoI-Visualiser/][GoI Visualizer]] [[./img/goi-visualizer.png]]
** [[https://s.ai/nlws/][Unker non-linear writing system]]
2020
[[./img/unker.png]]
** [[http://adelelopez.com/visual-linear-logic][Linear Logic in Existential Graph Notation]]
Adele Lopez (2020)
[[./img/visual-linear-logic.png]]
** Konstantin Osmei (2020)
*** [[https://twitter.com/konstantinosmei/status/1265204171856384004][ZX Calculus Animation]]
[[./img/konstantin-zx.gif]]
** Craig Gidney
*** [[https://algassert.com/quirk][Quantum Circuit Simulator]] [[./img/quantum-circuit.png]]
*** [[https://twitter.com/CraigGidney/status/1234414447038386176][Correlation Surface]] [[./img/correlation-surface.jpg]] [[./img/braid-twist-braid-hadamard.png]]
** Adinkras for Supersymmetry
[[./img/adinkras.png]]
** Dominic Hughes
[[http://boole.stanford.edu/~dominic/papers/lws/lws.pdf][Logic without Syntax (2005)]] [[https://arxiv.org/pdf/1906.11236.pdf][First-order proofs without syntax (2019)]] *** [[https://hal.inria.fr/hal-02386878/file/icp.pdf][Intutionistic proofs without syntax (2019)]]
** Hest programming language
Ivan Reese (2019)
[[./img/hest.gif]]
** Jamie Vicary
*** [[https://homotopy.io][Homotopy.io]] [[./img/homotopyio.jpg]]
A web based proof assistant for globular n-categories. Considered to be the successor to Globular
*** [[http://globular.science/][Globular]]
[[./img/globularscience.png]]
** [[http://opetopic.net/][Opetopic]]
A visual editor for opetopes.
[[./img/opetopic1.png]] [[./img/opetopic2.png]] [[./img/opetopic3.png]]
** [[https://github.com/oxford-quantum-group/discopy][Discopy]]
** String Diagrams
*** Joe
[[https://joemathjoe.wordpress.com/2020/06/23/a-different-string-presentation-of-monads/][Article]]
[[./img/monads-as-pipes-1.png]] [[./img/monads-as-pipes-2.png]]
*** Johannes Drever
**** [[https://drever.github.io/form-bakery/][Form Bakery]] [[./img/form-bakery.png]]
Drever’s playground for George Spencer-Brown’s notation from Laws of Form.
*** Jules Hedges
*** [[https://julesh.com/the-art-of-string-diagrams/][The Art of String Diagrams]]
*** Peter Selinger
**** [[https://arxiv.org/pdf/0908.3347.pdf][Survey of Graphical Languages for Monoidal Categories]]
[[./img/monoidal-categories-atlas.png]] [[./img/survey-of-monoidal-categories-1.png]] [[./img/survey-of-monoidal-categories-2.png]]
** [[https://www.sciencedirect.com/science/article/pii/B9780444529374500113][A History of Logic Diagrams (Amirouche Moktefi, Sun-Joo Shin)]]
** Susanne Langer on Sheffer’s Notational Velocity:
*** [[https://eprints.soton.ac.uk/402453/1/__filestore.soton.ac.uk_users_lb8_mydesktop_ePrints_Susan%2520Langer%2520and%2520the%2520Woeful%2520World%2520of%2520Facts%2520.pdf][Susanne Langer and the Woeful World of Facts - Giulia Felapi (2017)]]
*** [[http://krmcdani.mysite.syr.edu/langer.pdf][Facts: The Logical Perspective of the World]]
*** Philosophy in a New Key
*** Feeling and Form
** Irving Anellis
*** [[https://link.springer.com/chapter/10.1007/978-3-319-24756-4_3][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 [[https://dl.acm.org/doi/10.5555/115665.115667][Visual information and valid reasoning]]
** Janice Glasgow, Harinarayanan, Chandrasekharan [[https://mitpress.mit.edu/books/diagrammatic-reasoning][Diagrammatic Reasoning: Cognitive and Computational Perspectives]]
** Jens Lemanski *** [[https://www.researchgate.net/profile/Jens_Lemanski/publication/318743956_Means_or_End_On_the_Valuation_of_Logic_Diagrams/links/597ae44ca6fdcc61bb3bd68a/Means-or-End-On-the-Valuation-of-Logic-Diagrams.pdf][Means or Ends: On the Valuation of Logic Diagrams]]
*** [[https://www.researchgate.net/publication/318614644_Periods_in_the_Use_of_Euler-Type_Diagrams][Periods in the Use of Euler-Type Diagrams]]
* Amirouche Moktefi, Francesco Bellucci, Ahti-Veikko Pietarinen ** [[http://ceur-ws.org/Vol-1132/paper4.pdf][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)
*** [[https://www.maa.org/press/periodicals/convergence/math-origins-the-logical-ideas][Math Origins: The Logical Ideas]]
*** [[http://www.luigigobbi.com/EarliestUsesOfSymbolsOfOperation/][Earliest uses of symbols of operation]]
*** [[http://www.luigigobbi.com/EarliestUsesOfSymbolsOfOperation/Part2of2.htm][Earliest uses of symbols of operation: Part 2]]
*** [[https://web.archive.org/web/20060821061824/http://www.roma.unisa.edu.au/07305/symbols.htm][The History of Mathematical Symbols]]
*** [[http://wp1.fuchu.jp/~d-logic/en/uni.html][Contributors to the Universal Language]]
*** [[https://amzn.to/2CNZlPj][The Search for the Perfect Language]] Umberto Eco
** Rune Generator A fun tool to generate a rune like language: https://watabou.itch.io/rune-generator
** Visual Lambda Calculus
Books that take a largely diagramattic approach in its pedagogy. For more information check out
** [[https://amzn.to/2BfQNAr][Diagramattic Immanence (2015)]] Rocco Gangle
** [[https://link.springer.com/book/10.1007/978-1-4020-5652-9][Diagrammatology]] Frederik Stjernfelt
[[./diagrammatology.jpg]]
[[https://amzn.to/32zMNpr][Plant Form]] [[https://amzn.to/3hfzGOs][Dynamics]] ** [[https://www.cl.cam.ac.uk/~mj201/book.html][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’
** [[https://www.youtube.com/watch?v=RhFRzgQNg0k][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.
** [[https://mathoverflow.net/questions/168888/who-invented-diagrammatic-algebra][Who invented diagrammatic algebra?]]
Compendiums
[[https://github.com/fredhohman/awesome-mathematical-notation-design][Fred Hohman’s curation to improve mathematical notation]] List of links of techniques to help enhance mathematical notation
[[https://github.com/hypotext/notation][Catalog of notations by Katherine Ye]]