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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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All content in this area was uploaded by Louis Kauffman on Sep 11, 2015

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... Remarkably, although formal logic and mathematics are often deemed quintessentially abstract, versions of them already exist that are to a large extent iconic rather than symbolic (for iconic logic, see Roberts, 1973;Kauffman, 2001;Shin, 2002;Dau, 2011;Peirce, 2020;Peirce, 2021a,b;Bricken, 2023;Kramer, 2023;for iconic mathematics, see Spencer Brown, 1969;Kauffman, 1995;Bricken, 2019a,b;Bricken, 2021;Kramer, 2022). ...

This article lays out the foundation of a new language for easier written communication that is inherently reader-friendly and inherently international. Words usually consist of strings of sounds or squiggles whose meanings are merely a convention. In Icono , instead, they typically are strings of icons that illustrate what they stand for. “Train,” for example, is expressed with the icon of a train, “future” with the icon of a clock surrounded by a clockwise arrow, and “mammal” with the icons of a cow and a mouse—their combination’s meaning given by what they have in common. Moreover, Icono reveals sentence structure graphically before, rather than linguistically after, one begins reading. On smartphones and computers, writing icons can now be faster than writing alphabetic words. And using simple pictures as words helps those who struggle with conditions like dyslexia, aphasia, cerebral palsy, and autism with speech impairment. Because learning its pronunciation or phonetic spelling is optional rather than a prerequisite, and because it shows what it says, Icono is bound to be easier to learn to read—and then easier to read—than any other language, including our own.

... This final section illustrates one path along which such a playing out might proceed, using Peirce's logic graphs (Kauffman, 2001) and George Spencer Brown's (1972) ...

This thesis is comprised of three papers that together with substantial supporting material outline the semiotic basis, the musical intuition, and the practical implementation of a device
that can be used to study music as a theory of cognition. Existing approaches of music cognition typically apply extra-musical cognitive approaches to validate and explore music theory and practice. These approaches, however, together with the generally descriptive nature of music analysis, leave little room for the conversation to flow in the opposite direction, where music theory and practice directly inform underlying theories and philosophies of cognition. The semiotic requirement for effecting such an inversion is a common metric that can act as a bridge between the domains, the material for which is provided here by Zipf’s law, a statistical distribution frequently observed in a broad range of
contexts from social science to neurobiology. By way of illustration, I apply a semiotic concept called the “semiotic eigencycle” (where the prefix eigen- refers to a stably self-
referential system) to show how lower-order pitch class information may be transformed into higher order structural information using a psychologically and music-theoretically
tractable mechanism. The significance of such a tool is that it allows acts of music making to serve as constructive models of cognition in the form of second-order cybernetic processes
(which is to say processes that combine the model and observer in a single system). This affords the analyst a means to study the relationship between structures inherent in musical
surfaces and their effect on structures produced by behaviour and perception with a tool that represents a precisely defined and falsifiable cognitive-theoretical account of the mind.

Most people, evidence suggests, have a hard time thinking straight. Symbolic logic is a tool that can help remedy this problem. Unfortunately, it is highly abstract and uses symbols whose meanings depend on unintuitive arbitrary conventions. Without sacrificing rigor, iconic logic is more concrete and uses icons that resemble what they stand for and whose meanings are thus easier to picture, process, and remember. Here I review and critique iconic existential graphs and concept diagrams—the former link iconic logic to iconic mathematics; the latter expand popular Euler or Venn diagrams, and to some degree, their user-friendliness has been empirically investigated. I lay out how expertise in perception, cognition, and genetics can inform and improve this empirical research to help render iconic logic more ergonomic. After all, logic is a tool, and tools should not only suit their use but also their user.

Euler’s demonstration of the impossibility of traversing the Könisgberg network without having to double back on one of its paths (previous chapter) made it possible to flesh out a hidden mathematical principle of connected networks, which laid the foundation for graph theory and topology. Reconstructing his proof allowed us to see how Euler was guided by poetic logic—whereby he initially used his fantasia to envision the geographical map in an image schematic (outline) way, converting his inner vision into a diagrammatic model, via his ingegno, from which he could then use logical reasoning to establish why the network was impossible to traverse as such and, as a result, what this implied more generally. This episode in mathematical history (among many others) brings out how the fantasia is much more than the brain’s ability to generate spontaneous mental imagery from perceptual input—it is a form of insight thinking that interprets the input and then sparks an abduction (a flash of insight), which led Euler to convert the insight into a graph (a model), which highlighted the structural features of the original map, removing extraneous information from it. In other words, the graph is the end product of poetic logic, with Euler’s ingegno leading him to devise something new that enfolded something significant. This chapter deals with this faculty of poetic logic, as it manifests itself in problem-solving, discoveries, inventions, conjectures, and proofs.Keywords
ingegno
conjecture discovery invention Four-Color Theorem Fermat’s Last Theorem Collatz conjecture

Today’s information technology is becoming ever-more complex, distributed and pervasive. Therefore, problematizing what we observe as Information Systems (IS) researchers is becoming ever-more difficult. This chapter offers a new perspective for qualitative empirical research in the IS field. It looks at how we can possibly study dynamically changing, evolving, spatially and temporally distributed phenomena that evade our accustomed concepts and assumptions about the locus of agency. Or asked differently: How can we formally approach phenomena evading our concept of ‘identity’?
Using the mathematical-logical framework of the Laws-of-Form, formulated in 1969 by George Spencer-Brown, the chapter introduces the notion of distinction to capture the manifestation of concepts. It provides a short overview and illustrates how it can be used on sample concepts drawn from IS sociomateriality research.
The chapter advances qualitative methodology by suggesting a formal notation to communication analysis that is reflective of technologies’ complex nature. Applying the framework not only alters the epistemological boundaries for how to experience and study the ‘digital’, but also helps to build a bridge between deep technological insights, our immediate, unbiased and mundane experience of technologies, and how we speak about them.

The calculus of design is a diagrammatic approach towards the relationship between design and insight. The thesis I am evolving is that insights are not discovered, gained, explored, revealed, or mined, but are operatively de—signed. The de in design neglects the contingency of the space towards the sign. The — is the drawing of a distinction within the operation. Space collapses through the negativity of the sign; the command draws a distinction that neglects the space for the form's sake. The operation to de—sign is counterintuitively not the creation of signs, but their removal, the exclusion of possible sign propositions of space. De—sign is thus an act of exclusion; the possibilities of space are crossed into form.

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.]