As a mathematician and topologist, Louis Kaufmann examines how’s ‘Laws of Form’ from 1969, a theoretical book on math and logics, can be explained without having to really use mathematics. To do this, he dedicates his talk on the concept of ‘making the distinction’, seeing as George Spencer-Brown (photo) developed the mark of distinction as crux for introducing any field, such as mathematics, logics, science but also sociology or even mysticism, at all. What is the nature of a distinction and when does it occur? The premise is that in the construction-phase of an observation the observer and his construct, the object, are one and can’t be distinguished. Here the observer and whatever is outside of the observer to look at are not only interchangeable, but in their form identical. This is almost a mystical statement, but also the most concrete thing one could say about language and the observer, and those modalities all interact as Laws of Form is read again and again.

From the one form consequently a ‘first distinction’ is made. Even from describing it as I have now I have outlined a form from my second-order observation, imagining the observer and their object as one possible entity at which I am looking. As a rule, this exercise can be called ‘autopoiesis’, a Greek term that refers to a system capable of reproducing and maintaining itself by creating its own parts and eventually further components. In a mathematical sense there is no definition to distinction, because a definition in mathematics is already a certain form of distinction. Rather, Spencer-Brown’s theory begins self-referentially: the idea of distinction is taken as given, and that one can’t make an individuation without drawing a distinction in the first place. It is a circular statement where the form of the distinction is taken for the form, at will. Any interaction with something in the world (or in thought) makes many distinctions, moving from state to state, of non-existence to existence to non-existence again. It relies on an entirely virtual capacity in our brain or perception, that continually allows us to zoom out and conceptualise, outline a new inner and outer form, like a Mandelbrot fractal. Every element of a domain can interact with another and create a new element, sitting inside itself. So in a sense, every element of the domain is a transformation of the domain and we have no stops on what constitutes an entity. To demonstrate: a circle makes a distinction in the plane between its inside and its outside, by finding its mark (such as the drawn line) and agreeing upon its difference. However, it could also be a symbol standing for a distinction. Or, it could be written and become its name. Another thought experiment: if I’m an entity, and you are another entity, and we have a conversation, this interaction becomes a new entity and its own ‘form’, a sign for and in itself.

Just thinking of the properties of signs and indications encompasses thinking about Laws of Form semiologically, without making any calculi – we don’t have to be mathematicians to try it out.

Louis Hirsch Kauffman is an American mathematician, topologist, and professor in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago. He is known for the introduction and development of the bracket polynomial and the Kauffman polynomial.

Text: Yael Keijzer