Imaginary Values

摘要

George Spencer-Brown's work Laws of Form makes a new beginning in mathematics, logic and epistemology by starting with the idea of distinction and the idea of indication. A calculus with two values ensues and from this beginning a new kind of value beyond the True and the Falsecomes forth that can be called imaginary. This paper is an explication of specific aspects of imaginary values in relation to the original calculus of indications of Spencer-Brown. We consider two new arithmetics that extend this calculus by using concepts of almost unmarked and almost markedin an extended hierarchy of unmarked< almost unmarked< almost marked< marked. This aspect of extra values leads to the notions of Heyting and Co-Heyting algebras. We explain these relationships via a topological calculus of boundaries. We later review calculi involving the re- enteringmark and discuss a new way to relate the square root of negation to laws of form via a four-fold operator and its relationship with the work of Art Collings. The paper ends with a short exposition of fractal structures in relation to the re-entering mark. All these extensions of laws of formare seen as examples of how mathematics becomes a new vehicle for reason and understanding.