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The mathematics of Charles Sanders Peirce

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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.
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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. ...
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Chapter
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Chapter
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Chapter
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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.
Presentation
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.]