This folder contains links to papers related to Laws of Form and Cybernetics.

See Box Algebra Exercise. This exercise is an introduction to the mathematics of G. Spencer-Brown and Charles Sanders Peirce.

See Abstract Comics1. Here is an artistic approach to box algebra by Andrei Molotiu.

See Abstract Comics2. Here is an another artistic approach to box algebra by Matthieu Baillif.

See Recursive Forms. Hand-drawn notes by LK on forms and fractals, circa 1984.

See Linear Algebra . Notes on Linear Algebra and Iterants by LK, circa 1996.

See Spin Nets . Slide show on q-deformed spin nets and topological quantum computing.

For some notes about knots in the the seven color map on the torus see the pdf file The Knot in the Seven Color Map.

  • LOF Book in progress related to Laws of Form.
  • LOF Notes on Syllogisms.
  • LOF Diagrammatic Wheel of Syllogisms.
  • LOF Discussion Paper on Laws of Form, Topology and Quantified Syllogisms.
  • Extension Unpublished LK paper proving that Extension is a single intial for Laws of Form. See also ROBBINS.
  • Zeroid. Book in progress on infinitesimal calculus.
  • Non-Duality Slide show from Conference on Science and Non-Duality, San Rafael, CA, October 2009.
  • BOOLEAN NOTES Notes by LK on Boolean Algebra and Multiple Valued Logic, circa 1979.
  • Complex Numbers Notes by LK on Complex Numbers, circa 1982. These notes include standard material about complex numbers and quadratic and cubic equations, and also an interpretation of complex numbers as formalizations of oscillations so that the square root of minus one is an oscillation between plus one and minus one. The notes also emphasize that solutions to quadratic equations can be obtained by recursion and via infinite continued fractions.
  • KELLEY Excerpt on basic mathematics from book by John Kelley, part of the telvision course "Continental Classroon" that Kelley gave in 1960.
  • SIGN Sign and Space -- A Mathematical Essay.
  • TimeSpace A sequel to Sign and Space, concise and mathematical.
  • SELFREF Self Reference and Recursive Forms.
  • LUHMANN Talk dedicated to Nicolas Luhmann by Heinz von Foerster, Translated by LHK and DBS.
  • REL Special Relativity and a Calculus of Distinctions.
  • ARITH Arithmetic in the Form.
  • McCulloch Link to pioneering cybernetic paper by Warren McCulloch on the nature of number.
  • Cantor Essay on Cantor's proof that the powerset of a set has higher cardinality than the set itself.
  • Beyond Cantor The immortal lecture by Professor Hilbert Squarepunkt refuting Cantor's diagonal argument.
  • Halt! A concise proof of the undecideability of the halting problem.
  • COMPLEX Complex Numbers and Algebraic Logic, Proceedings of the Tenth International Conference on Multiple Valued Logic(1980), IEEE Computer Society Press, 209-213.
  • Autopoeisis Classical paper on autopoesis by Maturana, Uribe and Varela.(FV).
  • CalculusForSelfReference Paper on Self-Reference by Francisco Varela (FV).
  • NetworkSynthesisLK paper - Using Re-entry Calculus from Laws of Form for circuit design and self-reference.
  • Design Counters Notes on circuit design.
  • Balance GameA supplement to the Network Synthesis paper showing how to structure circuit design as a game of balance and transition on the circuit graphs.
  • JK-FlipFlopA supplement to the Network Synthesis paper showing how a circuit in Chapter 11 of Laws of Form is a variation on the standard JK Flip Flop. This makes a direct connection between the circuit design in Laws of Form and the basic technology for computer circuits.
  • DeMorgan Algebras LK paper on completeness, wave-form interpretation and recursion in DeMorgan algebras (four-valued logic), Proceedings of the eighth international symposium on Multiple-valued logic (1978).
  • Form Dynamics LK and FV paper - Spatial and Temporal Dynamics of Forms.
  • Imaginary Values LK paper - Imaginary Values in Mathematical Logic,Proceedings of the Seventeenth International Conference on Multiple Valued Logic, May 26-28 (1987).
  • ROBBINS LK paper on the Robbins problem from Proceedings of 20th Intenational Symposium on Multiple Valued Logic (1990).
  • Simplified solution to the Robbins Problem using Box Algebra RobbinsSolution
  • Simplified solution to the Robbins Problem using Box Algebra RobbinsSolutionDetails
  • KNOT AUTOMATA LK Paper on knot automata and problems about asychronous circuit design from Proceedings of 24th Intenational Symposium on Multiple Valued Logic (1994)..
  • EigenForm Slides for a talk given in Vienna in November 2006 -- the "Heinz von Foerster Lecture on Cybernetics".
  • EigenForm LK paper on Eigenform.
  • Reflexivity LK paper on Eigenform and Reflexivity.
  • ReflexivityPhysics LK paper on Eigenform, Reflexivity and Physics.
  • Peirce LK paper on the Mathematics of Charles Sanders Peirce.
  • TimeParadox LK paper on time, paradox, multiple-valued logic.
  • Flagg Resolution LK paper on the Flagg Resolution.
  • VirtualLogic LK Kybernetes paper on Virtual Logic.
  • Bios Paper on Bios with Hector Sabelli.
  • MathematicalBios A second paper on Bios with Hector Sabelli.
  • "KnotLogic" Paper by LK on foundations of mathematics and logic in relation to knots. Discusses knot set theory, lambda calculus, magmas, braids, tangles, electricity, pregeometry and more. This paper is published in the book "Knots and Applications" edited by LK and published by World Scientific Press.
  • "Protein Folding Topology" Paper by LK and Yuri Magarshak on topological invariants of protein folding. This paper is published in the book "Knots and Applications" edited by LK and published by World Scientific Press.
  • Iconic Slides for a talk given in Brussels, December 2008 at Trends in Logic VI and for the talk I gave at the Short Course in Quantum Compputing at the AMS Meeting in Washington D.C. on January 4, 2009. Shows how the Fibonacci model, universal for topological quantum computation is constructed from the "logical particle" implicit in C. S. Peirce's Sign of Illation and in the Mark of G. Spencer-Brown, and the relationship of this logical particle to the Jones polynomial and the Temperley-Lieb recoupling theory. This structure leads to quantum algorithms for computing colored Jones polynomials and Witten-Reshetikhin-Turavev invariatns at roots of unity.
  • Fibonacci Quanta ANPA paper on quanta in LOF and the Fibonacci model.
  • BioLogic -- What is the relationship of Logic and Biology? Slides for a talk given at the Wollic Conference in Rio de Janeiro in July 2007.
  • NonCommutative Worlds Slides for a talk given at the Newton Institute in Cambridge, UK in December 2006 -- Conference on NonCommutative Geometry. See Newton Institute.
  • NonCommutative Worlds The LK Paper.
  • Discrete Non-Commutative ElectromagnetismPaper by LK and Pierre Noyes giving a discrete interpretation to the Feynman-Dyson derivation of electromagnetism from a commutator calculus.
  • Discrete DiracPaper by LK and Pierre Noyes giving a discrete interpretation for the Dirac Equation and a solution to the Feynman Checkerboard Problem.
  • Knots, Braids and Elementary Particles Slides for a talk given at the Red Raider Conference in Lubbock, Texas, October 2008.
  • "Map Theorem" Paper by LK on G. Spencer-Brown's approach to the Four Color Theorem.
  • "Map Colorings,Networks,Knots" Paper by LK on map coloring (vector cross product reformulation, Penrose formula), q-deformed spin networks, knot invariants, quantum group and three manifold invariants.
  • "Penrose" Scan of fundamental paper "Applications of Negative Dimensional Tensors" by Roger Penrose.
  • "Impasse Group" Paper by Irving Kittell (BAMS 1935) articulating a group of recoloring operations on a map that is all colored except for a five-sided region.
  • "Tutte Polynomial Signed Graphs" pdf file of paper by LK about a Tutte polynomial for signed graphs that extends the classical Tutte polynomial and includes the bracket polynomial for knots and links as a special case.
  • "Whitney's Theorem on Map Coloing" fundamental paper by Hassler Whitney.
  • "Snarks" a copy of the seminal paper by Rufus Isaacs constructing uncolorable, non-planar cubic graphs.
  • "Petersen"A paper on the Petersen Perfect Matching Theorem.
  • "Tutte"A paper by Tutte on the Four Color Theorem.