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This essay explores the Mathematics of Charles Sanders Peirce. We concentrate on his notational approaches to basic logic and his general ideas about Sign, Symbol and diagrammatic thought. In the course of this paper we discuss two notations of Peirce, one of Nicod and one of Spencer-Brown. Needless to say, a notation connotes an entire language and these contexts are elaborated herein. The first Peirce notation is the portmanteau (see below) Sign of illation. The second Peirce notation is the form of implication in the existential graphs (see below). The Nicod notation is a portmanteau of the Sheffer stroke and an (overbar) negation sign. The Spencer-Brown notation is in line with the Peirce Sign of illation. It remained for Spencer-Brown (some fifty years after Peirce and Nicod) to see the relevance of an arithmetic of forms underlying his notation and thus putting the final touch on a development that, from a broad perspective, looks like the world mind doing its best to remember the significant patterns that join logic, speech and mathematics. The movement downward to the Form ('we take the form of distinction for the form.'[9, Chapter 1, page 1]) through the joining together of words into archetypal portmanteau Signs can be no accident in this process of return to the beginning.

Content uploaded by Louis Kauffman

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... The goal would not be to break new ground in mathematics-let us leave this to the mathematicians-but rather to apply to math what we know about the human mind and make math easier to learn and use, more ergonomic. The point of departure for this project is Spencer-Brown's (1969) seminal adaptation of Charles Peirce's iconic logic (Roberts, 1973;Kauffman, 2001;Shin, 2002) that became the cornerstone of iconic mathematics (Kauffman and Varela, 1980;James, 1993;Kauffman, 1995;Bricken, 2019aBricken, ,b, 2021. ...

Mathematics is a struggle for many. To make it more accessible, behavioral and educational scientists are redesigning how it is taught. To a similar end, a few rogue mathematicians and computer scientists are doing something more radical: they are redesigning mathematics itself, improving its ergonomic features. Charles Peirce, an important contributor to ordinary symbolic logic, also introduced a rigorous but non-symbolic, graphical alternative to it that is easier to picture. In the spirit of this iconic logic, George Spencer-Brown founded iconic mathematics. Performing iconic arithmetic, algebra, and even trigonometry, resembles doing calculations on an abacus, which is still popular in education today, has aided humanity for millennia, helps even when it is merely imagined, and ameliorates severe disability in basic computation. Interestingly, whereas some intellectually disabled individuals excel in very complex numerical tasks, others of normal intelligence fail even in very simple ones. A comparison of their wider psychological profiles suggests that iconic mathematics ought to suit the very people traditional mathematics leaves behind.

... Although Kauffman (2001) makes no reference to the NOR units or gates, Fig. 1 is instrumental in showing that and how the cross refers to both the logical NOR in general (here represented by the outermost cross on the right-hand side of the figure) and to specific, the most familiar of which are neither a nor b (bottom-left corner of the outer square), b not a (top-left corner), a and b (top-right corner), and a not b (bottom-right corner). Thus, we find that the NOR is its default operation, neither … nor …, and contains what it is not. ...

The field of paradox studies keeps struggling to put the notion of paradox into the very centre of organizational life and managerial decision-making, with mixed success. We argue that this research ambition can be realized much more effectively by anchoring the field in three interrelated conceptual approaches which build on paradox as the paradigmatic point of departure. These approaches include Spencer Brown’s form calculus, Niklas Luhmann’s systems and organization theory, and the traditional Indian logical construct of tetralemma. In the proposed argument, paradox constitutes the very identity of organizations as (re-entries of) distinctions drawn in the environment; it is actualized in every act of organizational decision communication, as well as in the process of the continual vanishing and renewal of such acts. In this conception of organizational life, the key challenge is to debunk false distinctions by using tetralemmatization strategies that entail a radical questioning of the problematic observational perspectives.

... This universality and the negation included in the operation have particularly interested mathematicians and philosophers. In Kauffman (2001), this history of the reception is traced from the Peirce arrow of Charles Sanders Peirce, the Sheffer stroke of Henry Maurice Sheffer, the Nicod arrow of Jean George Pierre Nicod to the cross of George Spencer-Brown. This series could be extended to include Edward Stamm, as can be seen from a quotation from George Spencer-Brown published in Roth (2017) from the typewriter manuscript "Design with the NOR" , which was written in 1961. ...

This paper aims to show how a sociological description – a swarm analysis of the Nazi dictatorship – initially made with the means borrowed from George Spencer-Brown’s Calculus of Indications, can be transformed into a digital circuit and with which methods and tools of digital mathematics this digital circuit can be analyzed and described in its behavior. Thus, the paper also aims to contribute to a better understanding of Chapter 11 of “Laws of Form.” The analysis uses methods of automata theory for finite, deterministic automata. Basic set operations of digital mathematics and special set operations of the Boolean Differential Calculus are used to calculate digital circuits. The software used is based on ternary logic, in which the binary Boolean logic of the elements {0, 1} is extended by the third element “Don’t care” to {0, 1, −}. The paper confirms the method of transforming a form into a digital circuit derived from the comparative functional and structural analysis of the Modulator from Chapter 11 of “Laws of Form” and defines general rules for this transformation. It is shown how the indeterminacy of re-entrant forms can be resolved in the medium of time using the methods of automata theory. On this basis, a refined definition of the degree of a form is presented. The paper shows the potential of interdisciplinary approaches between sociology and information technology and provides methods and tools of digital mathematics such as ternary logic, Boolean Differential Calculus and automata theory for application in sociology.

Advances in digitalization have changed our apprehension of technology from discrete devices and application software as bounded artifacts, to dynamically evolving social-material entanglements in Digitalized Work Arrangements (DWA). This development makes studying DWAs increasingly difficult and challenges us to advance our methods that define how we can study, observe, and conceptualize DWAs. In this essay, we draw on the mathematical-logical formalism of the Laws of Form (LoF) (Spencer-Brown, 1969) to analyze how six illustrative IS studies conceptualize the social and the material to arrive at distinct perspectives on DWAs. Our analysis reveals three archetypes that capture these studies' conceptualizations and that inform a discussion of how IS research can extend qualitative methods beyond those six works. We offer two contributions. First, we provide novel insights and explanations to key conceptualizations in the study of DWAs. Second, we present the LoF as a pre-ontological and pre-theoretical formalism that enables commensurability of methods and development of novel qualitative empirical methods. Specifically, we demonstrate how the formalism helps articulating the distinctions we draw to refine our object of study and to critically examine and reconstruct other researchers' reasoning.

Thomas A. Sebeok has left semiotics a comprehensive theoretical apparatus for studying semiosis across species and across systems (biological and artificial). Uniting the notions of form , sign , and model into an integrative purview of meaning-making, known as modeling systems theory, Sebeok has provided a conceptual and terminological apparatus for studying all forms of meaning in terms of the fundamental “standing-for principle” that undergirds all semiotic theories. This essay revisits the Sebeokian perspective, delineating its main components in a retrospective way, highlighting its value not only to semiotics but to computer science as well. Above all else, Sebeok has made it possible concretely to establish specific interconnections between semiotics and cognate disciplines in ways that are relatively free of terminological complexities and ambiguities which have beset semiotics in the past.

This workbook of "études" offers a collection of experimental texts for communal dialogue and discovery that crosses multiple academic disciplines, including: foundations of physics, metaphysics, theoretical biology, semiotics, cognitive science, linguistics, phenomenology, logic & mathematics, poetry and theology. Each étude probes limits, horizons and boundaries by implicitly bring into relation foundational issues that characterize different academic disciplines or systems of meaning formation. Some formal techniques are deployed the études. Most notable is the use of the “logic of three” to overcome falsely totalizing images and inexorable dualities. This technique involves a particular kind of attunement to the “betweenness” of mediation that is not common in modern science. The attunement draws on the formal and precise movements of analysis in concert with the metaphoric and singular movements of synthesis.

After Thomas Sebeok’s proposal of global semiotics in the 70s, an attempt to move beyond anthroposemiotics to the realm of zoosemiotics, phytosemiotics, endosemiotics, and, ultimately, to the all-encompassing realm of biosemiotics was made. Semiotics was then established as a serious candidate as the transdisciplinary base of science and humanities –particularly from the triadic and pragmaticist semiotic proposal of C. S. Peirce. However, the semiotic attempt to explain the fundamental aspects of living systems from the standpoint of meaning production and reproduction demonstrate that in order to explain the meaning-making process in living organisms a systemic, biological, cybernetic and informational approach was also needed. The integrative visions have discovered some basic similarities among these theoretical perspectives from which it has been possible to recognize complementarities among them. At the same time, it also made possible to identify variations at the very bottom of each approach, which resulted in a complex task of theoretical integration. Thus, in order to uncover these tensions and complementarities, I will focus my attention in the process of communication in an attempt to move from cybernetics to semiotics and further on to cybersemiotics considering some aspects of biosemiotics, first and second-order cybernetics, Peircean semiotics, and information theory. The goal of this chapter is to overcome the problem of defining the limits and boundaries of communication as a physical, biological, and social phenomena and its nature as an academic field by proposing communication as a transdisciplinary concept from the point of view of cybersemiotics (Vidales, Commun Soc 30:45–67, 2017b), from which it is also possible to address the process of communication, explained in what Brier (Cogn Semiotics 4:28–63, 2009) considers to be the levels of cybersemiotics, and the consequences it may have for the explanation of meaning-making processes in living systems.

The analysis takes up the conjunction of semiotics and cybernetics as a problem in theory construction in the human sciences. From a philosophical perspective, this is also the ontological problem of communicology: the disciplinary study of human communication. My analysis suggests current conceptions of “semiotics” and “cybernetics” are misunderstood because “information” is assumed as synonymous with “communication” and that the axioms of “mathematics” are identical to those of “logics”. The evidence contained in the misunderstandings is a conflation of reductionist ecology ideas about the “environment” differentiation of (1) human beings [apperceptive organic life], (2) animals [perceptive organic life], and machines [inorganic and constructed mechanisms]. The communicological view argues that a correct understanding of these issues requires a competence in logics and linguistics to determine the metatheory criteria for choosing evidence among humans, animals, and machines. The domain thematic is the phenomenological synergism of human embodiment as expression and perception. In this context, my criterion for evidence is the structure or form of a pure concept of reason (choice making judgment) that is given a priori in consciousness, the notion demonstrated by Immanuel Kant: A notion is a rule that you know before you experience it as a result.

In his 1926 life-size poster “Der Mensch als Industriepalast” (Man as Industrial Palace), physician and infographics pioneer Fritz Kahn draws on the state of the art, science, and technology of the second industrial revolution to depict core processes of the human organism as if it were a factory: tubes filled with chemicals, a metabolic division of labor in assembly lines, a mechanical pump for the heart, a camera for the eye, and a telephone switchboard for the central nervous system. The higher brain compartments, however, are still Gutenberg Galaxies of books and humans, devoid of any industrial technology. At the time, computers were clearly wetware. At the turn of the next industrial revolution, George Spencer Brown was one of these human computers in a brain compartment of a British radio transmission equipment manufacturer, Mullard Equipment Limited, whose Technical Publications Department features as sponsor and potential publisher of his typescript Design with the NOR. This 1961 typescript shows that Spencer Brown’s 1969 book Laws of Form adds a veritable piece of hardware to the libraries of industrial man, whose updated wetware can now think like a transistor.

The relationship of G. Spencer-Brown's Laws of Form (1972) with multiple-valued logic is discussed. The calculus of indications is presented as a diagrammatic formal system. This leads to domains and values by allowing infinite and self-referential expressions that extend the system. The author reformulates the Varela/Kauffman calculi for self-reference, and gives a completeness proof for the corresponding three-valued algebra.

Harvard University, September 5-10, 1989. Handout.
[See Publication: Shea Zellweger, 1997, Untapped potential in Peirce's iconic notation for the sixteen binary connectives. In Nathan Houser, Don D. Roberts, and James Van Evra (editors), Studies in the Logic of Charles Sanders Peirce, 334-386. Bloomington: Indiana University Press.]