A mathematical apparatus is developed that deals with networks of elements which are connected to each other by well defined connection rules and which perform well defined operations on their inputs. The output of these elements either is transmitted to other elements in the network or – should they be terminal elements – represents the outcome of the computation of the network. The discussion is confined to such rules of connection between elements and their operational modalities as they appear to have anatomical and physiological counter parts in neural tissue. The great latitude given today in the interpretation of nervous activity with regard to what constitutes the “signal” is accounted for by giving the mathematical apparatus the necessary and sufficient latitude to cope with various interpretations. Special attention is given to a mathematical formulation of structural and functional properties of networks that compute invariants in the distribution of their stimuli.

This paper examines the grounding of George Spencer Brown’s notion of a distinction, particularly the ultimate distinctions in intension (the elementary) and extension (the universal), It discusses the consequent notions of inside and outside, and discovers that they are apparent, the consequence of the difference between the self and the external observer. The necessity for the constant redrawing of the distinction is shown to create “things”. The form of all things is identical and continuous. This is reflected in the distinction’s similarity to the Möbius strip rather than the circle. There is no inside, no outside except through the notion of the external observer. At the extremes, the edges dissolve. The elementary and.the universal thus re-enter each other. “Your inside is out and your outside is in.”

Excerpt: If the goal of the teacher’s guidance is to generate understanding, rather than train specific performance, his task will clearly be greatly facilitated if that goal can be represented by an explicit model of the concepts and operations that we assume to be the operative source of mathematical competence. More important still, if students are to taste something of the mathematician’s satisfaction in doing mathematics, they cannot be expected to find it in whatever rewards they might be given for their performance but only through becoming aware of the neatness of fit they have achieved in their own conceptual construction.

Key words: cognition, education

Within the limits of one chapter, an unconventional way of thinking can certainly not be thoroughly justified, but it can, perhaps, be presented in its most characteristic features anchored here and there in single points. There is, of course, the danger of being misunderstood. In the case of constructivism, there is the additional risk that it will be discarded at first sight because, like skepticism – with which it has a certain amount in common – it might seem too cool and critical, or simply incompatible with ordinary common sense. The proponents of an idea, as a rule, explain its nonacceptance differently than do the critics and opponents. Being myself much involved, it seems to me that the resistance met in the 18th century by Giambattista Vico, the first true constructivist, and by Silvio Ceccato and Jean Piaget in the more recent past, is not so much due to inconsistencies or gaps in their argumentation, as to the justifiable suspicion that constructivism intends to undermine too large a part of the traditional view of the world. Indeed, one need not enter very far into constructivist thought to realize that it inevitably leads to the contention that man – and man alone – is responsible for his thinking, his knowledge and, therefore, also for what he does. Today, when behaviorists are still intent on pushing all responsibility into the environment, and sociobiologists are trying to place much of it into genes, a doctrine may well seem uncomfortable if it suggests that we have no one but ourselves to thank for the world in which we appear to be living. That is precisely what constructivism intends to say – but it says a good deal more. We build that world for the most part unawares, simply because we do not know how we do it. That ignorance is quite unnecessary. Radical constructivism maintains – not unlike Kant in his Critique – that the operations by means of which we assemble our experiential world can be explored, and that an awareness of this operating (which Ceccato in Italian so nicely called consapevolezza operativa) can help us do it differently and, perhaps, better.

We discuss the relationship of G. Spencer-Brown’s Laws of Form with multiple-valued logic. The calculus of indications is presented as a diagrammatic formal system. This leads to new domains and values by allowing infinite and self-referential expressions that extend the system. We reformulate the Varela/Kauffman calculi for self-reference, and give a new completeness proof for the corresponding three-valued algebra (CSR).

Key words: George Spencer Brown, multiple-valued logic, self-reference, three-valued algebra

The purpose of this essay is to sketch a picture of the connections between the concept self-reference and important aspects of mathematical and cybernetic thinking. In order to accomplish this task, we begin with a very simple discussion of the meaning of self-reference and then let this unfold into many ideas. Not surprisingly, we encounter wholes and parts, distinctions, pointers and indications. local-global, circulation, feedback. recursion, invariance. self-similarity, re-entry of forms, paradox, and strange loops. But we also find topology, knots and weaves. fractal and recursive forms, infinity, curvature and imaginary numbers! A panoply of fundamental mathematical and physical ideas relating directly to the central turn of self-reference.

This paper is an exposition and extension of ideas begun in the work of G. SpencerBrown (Laws of Form). We discuss the relations between form and process, distinction and indication by the use of simple mathematical models. These models distill the essence of the ideas. They embody and articulate many concepts that could not otherwise be brought into view. The key to the approach is the use of imaginary Boolean values. These are the formal analogs of complex numbers – processes seen as timeless forms, then indicated (self-referentially) and re-entered into the discourse that engendered them. While the discussion in this paper is quite abstract, the ideas and models apply to a wide range of phenomena in mathematics, physics, linguistics, perception and thought.

Key words: self-reference, George Spencer Brown