Mathematical logician Arthur Collings takes George Spencer Brown’s 1969 book ‘Laws of Form’ as a starting point for further mathematical applications and models. This talk was presented at the interdisciplinary 2019 conference ‘Laws of Form: A Fiftieth Anniversary’ at Liverpool University, UK. Speakers from surprisingly different disciplines were invited to talk about this complex book of maths and logic fifty years after its first publication, ranging from theoretical, mathematics to physics, literature, mysticism and business. It could be said the book brought out in a time of counterculture and innovation still holds a cult following, and it receives claims of being incomprehensible and simultaneously holding the key to understanding the universe. In 2019, West organised the exhibition ‘Alphabetum III’ on the enigma of Laws of Form.

In his talk on advanced algebra, Collings introduces his research by pointing out the first page of Laws of Form, which has Chinese characters inserted from the Tao Te Ching, an ancient Chinese text regarded as the foundation of Buddhist Daoism. The characters say ‘The beginning of heaven and earth has no name’ - before there are distinctions there are no names. Can we imagine something before the beginning, before distinction?

Based on Spencer-Brown’s ‘mark of distinction’, a symbol that could reduce all algebra to one sign, Collings proposes the ‘imaginary mark’ as a similar mathematical symbol that may nest itself in an equation but cancels itself out in a cycle of four. The motivation behind his research is to develop a calculus like Laws of Form, but one that emphasises this cyclical pattern of cancellation. The mark can stand for any value of distinction, yet Collings mentions the familiar pattern of the arithmetic of a clock. It can stand for the rotation of one turn, marking quarter hours, where after four orders the mark returns to its beginning. Many people are not aware that they know modular arithmetic of order-four cancellations in everyday life, such as seasons, the calendar or traffic circles. Collings’ BF (Brown 4) Calculus furthermore, is ‘containment-based’, which poses an extension to Laws of Form which was not already present or applied in the initial calculus. Namely, it employs a notation without reference to ordered pairs, but builds a system upon the single symbol of the imaginary mark to make sets of juxtapositions and subsequently generates new consequences within the same equation, none of which exist in the Primary Algebra. From this extension of Laws of Form that provides a new axiom set through the imaginary mark, an entirely new normal form is derived, and BF Calculus is shown to be functionally and axiomatically complete.

Interpreting the BF calculus is challenging, because of many possible interpretations of such abstract variables. Collings illustrates the parable of the elephant and the blind men, where the names of ‘different’ mathematical systems can reference the same underlying form. The man holding the elephant’s trunk thinks he’s holding a hose; the one that touches the skin finds a canvas, etc. Grasping a logical structure so complex, yet at the same time so primary, shows the attraction and ongoing relevance of this book.

Arthur M. Collings is an independent researcher and lives in Red Hook, New York. He is currently Treasurer of the American Society for Cybernetics. He is a cartographer and conservationist by profession and is Vice President of Dutchess Land Conservancy, a conservation land trust in New York's Hudson Valley.

Text: Yael Keijzer