Introduction and welcome

In 1977, having previously studied the mathematical foundational work of Boole, Frege, Russell, Whitehead, Wittgenstein and Gödel, Graham was given a copy of George Spencer-Brown's Laws of Form and quickly recognized it as a work of the highest order.

Over the following years, Graham thoroughly analysed the calculus presented in Laws, and in 1983 sent his demonstration of the primary algebra from a single initial-equation to Spencer-Brown who invited him to his home in Cambridge, their discussions initially focusing on formal algebras and Spencer-Brown's work on the four-colour conjecture. A friendship quickly developed with Graham becoming Spencer-Brown's principal friend and confidant for more than three decades, frequently visiting him at his successive homes in Cambridge, London and Horningsham.

Graham is the leading authority on the primary algebra, with Spencer-Brown once remarking, "You have made the field your own", and is the custodian of Spencer-Brown's personal effects, notebooks, unpublished papers, poetry, miscellaneous writings, and photographs. These documents and several hundred hours of their discussions and other audio recordings are being digitised with the intention that they will form an archive for possible future publication online. Graham initiated the Laws of Form conferences now being generously hosted by the University of Liverpool.

In addition to formal algebras, Graham's interests include cryptography, having written a detailed description of, and a computer program which emulates, the logical operations of the bombe, the machine devised by Alan Turing and Gordon Welchman, which broke the German Enigma machine on an industrial scale; the philosophy of physics; binocular stargazing; and chess. Graham has an MSc in Computing Science from Birkbeck, University of London.

Introduction to Laws of Form

Video

Laws of Form by G. Spencer-Brown is a remarkable book that asks the reader to start as near the beginning as he or she can. "We take as given the idea of distinction and the idea of indication and that we cannot make an indication without drawing a distinction. We take therefore, the form of distinction for the form." This is how the book begins.
"We see now that the first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical." That is how the book ends (with an injunction to return to the beginning of the book and reenter it once again).
In between, there develops a remarkable mathematics that is based on a single sign of distinction, the mark < >. The mark is seen to make a distinction in its own indicational space and the mathematics of the book evolves naturally from the idea of distinction. Throughout, there is an insight that the idea of distinction can be taken as the moving foundation for evolving knowledge both in the direction of complexity and in the direction of simplicity. This talk will be an introduction to Spencer-Brown's thought and to his injunctions for understanding in terms of the imagination of such a possibility as a distinction. As the end quote above indicates, this journey necessarily involves the one who takes it.

Louis H Kauffman is Professor of Mathematics Emeritus at the University of Illinois at Chicago. He is a graduate of MIT (undergraduate) and Princeton University (PhD in Mathematics). Kauffman works in knot theory and its relationships to other fields, including combinatorics, algebra, low dimensional manifolds, physics and natural science. He has long been interested and working with extensions and variations of Laws of Form and working with the principle that mathematics is about what a distinction could be if there would be a distinction. Kauffman is a fellow of the American Mathematical Society and a recipient of the Warren McCulloch and Norbert Wiener awards of the American Society for Cybernetics.

More info: homepages.math.uic.edu/~kauffman

The Primary Algebra: Operations and Structures

Graham's presentation on Laws of Form will include some of the results of his research into the primary algebra. It will be shown how to factorize the operations of the algebra and will define which operations are required to form a complete initial-set, and in the case of incomplete initial-sets, which operations are necessary to complete them. It will be shown how the factorized operations can be recombined to yield single-initial equations. Graham is compiling his work in formal algebras into book form.

The Pen Factory