Peircers, Discussion arose in the Laws Of Form Group about computational explorations of George Spencer Brown's calculus of indications.
Readers of Peirce are generally aware that Spencer Brown revived certain aspects of Peirce's logical graphs, focusing on the Alpha level interpretable for Boolean Algebra and Propositional Calculus but adding some hints of potential extension and generalization. He used what amounts to Peirce's “entitative” interpretation of the graphical forms in his exposition but was clear about the abstract character of the forms themselves, as evidenced by their admitting dual interpretations, dubbed “entitative” and “existential” by Peirce. In computational context the question naturally arises how to code the abstract formal structures used by the calculi of CSP and GSB into the relatively concrete forms that a computer can process. I began my response to that question as follows ... Theme One • A Program Of Inquiry : 6 https://inquiryintoinquiry.com/2018/02/28/theme-one-%e2%80%a2-a-program-of-inquiry-6/ {A plaintext transcript follows, but see my blog copy if the ASCII figure below gets mushed in this email.) Programs are algorithms that operate on data structures (Wirth). How do we turn abstract graphs like those used by C.S. Peirce and Spencer Brown into concrete data structures that algorithms can manipulate? There are many ways to do this, but one very efficient way is through the use of “pointer data structures”. The full documentation of my Theme One Program is still in progress, but here's a link to a page of exposition, describing the family of graphs used in the program, how to code the graphs as strings of parentheses, commas, and letters, and how the program parses the strings into pointer structures that live in computer memory. Theme One Program • Exposition http://intersci.ss.uci.edu/wiki/index.php/Theme_One_Program_%E2%80%A2_Exposition Here's a link to a suitable point of entry for our present purpose: Theme One Program • Expository Note 2 http://intersci.ss.uci.edu/wiki/index.php/Theme_One_Program_%E2%80%A2_Exposition#Expository_Note_2 2. Painted And Rooted Cacti And Conifers ========================================= Figure 1 depicts a typical example of a painted and rooted cactus (PARCA). o a | d o---o o \ / b c | o----o----o b e \ / \ / \ / \ / @ a c e Figure 1. Painted And Rooted Cactus The graph itself is a mathematical object and does not inhabit the page or other medium before our eyes, and it must not be confused with any picture or other representation of it, anymore than we'd want someone to confuse us with a picture of ourselves, but it's a fair enough picture, once we understand the conventions of representation involved. Let V(G) be the set of nodes in a graph G and let L be a set of identifiers. We very often find ourselves in situations where we have to consider many different ways of associating the nodes of G with the identifiers in L. Various manners of associating nodes with identifiers have been given conventional names by different schools of graph theorists. I will give one way of describing a few of the most common patterns of association. • A graph is “painted” if there is a relation between its node set and a set of identifiers, in which case the relation is called a painting and the identifiers are called paints. • A graph is “colored” if there is a function from its node set to a set of identifiers, in which case the function is called a coloring and the identifiers are called colors. • A graph is “labeled” if there is a one-to-one mapping between its node set and a set of identifiers, in which case the mapping is called a labeling and the identifiers are called labels. • A graph is said to be “rooted” if it has a unique distinguished node, in which case the distinguished node is called the root of the graph. The graph in Figure 1 has a root node marked by the “at” sign or amphora symbol “@”. The graph in Figure 1 has eight nodes plus the five paints in the set { a, b, c, d, e }. The painting of nodes is depicted by drawing the paints of each node next to the node they paint. Observe that some nodes may be painted with an empty set of paints. The structure of a painted and rooted cactus (PARC) can be encoded in the form of character string called a painted and rooted cactus expression (PARCE). For the remainder of this discussion the terms cactus and cactus expression will be used to mean the painted and rooted varieties. A cactus expression is formed on an alphabet consisting of the relevant set of identifiers, the paints, together with three punctuation marks: the left parenthesis, the comma, and the right parenthesis. To be continued ... Jon On 2/26/2018 7:18 AM, Jon Awbrey wrote:
Peircers, I started learning programming about the same time I first ran across C.S. Peirce's Logical Graphs and Spencer Brown's Laws of Form in the late 60s and naturally tried each new language and each new set of skills I learned on writing processors and theorem provers for the propositional calculus levels of their graph-theoretic formalisms. Using previous work I had done in Lisp, I spent the 80s developing a series of Pascal programs that integrated aspects of sequential learning with aspects of propositional reasoning over an extension of the CSP-GSB systems. I applied the program to some observational data sets from one of my advisor's research projects and got an M.A. in Quantitative Psych out of it. People looking for contemporary applications stemming from the general Peircean paradigm may find some of the directions in this work of interest. The following blog post updates a list of links to what documentation and exposition I've put on the web so far: Survey of Theme One Program • 2 http://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-%e2%80%a2-2/ This is a Survey ( http://inquiryintoinquiry.com/surveys/ ) of previous blog and wiki posts on the Theme One Program that I developed all through the 1980s. The aim of the project was to develop fundamental algorithms and data structures to support an integrated learning and reasoning interface, looking toward the design of an Automated Research Tool (ART) that could do double duty as a medium for Inquiry Driven Education (IDE). I wrote up a running pilot version of the program well enough to get a Master's degree out of it, but I'm still getting around to writing up the complete documentation. Wiki Hub ======== * Theme One Program ( http://intersci.ss.uci.edu/wiki/index.php/Theme_One_Program ) Documentation in Progress ========================= * Theme One Program • Pascal Source Code ( https://www.academia.edu/5210987/Theme_One_Program_Pascal_Source_Code ) * Theme One Program • User Guide ( https://www.academia.edu/5211369/Theme_One_Program_User_Guide ) * Theme One Program • Exposition ( http://intersci.ss.uci.edu/wiki/index.php/Theme_One_Program_%E2%80%A2_Exposition ) Applications ============ * Applications of a Propositional Calculator : Constraint Satisfaction Problems ( https://www.academia.edu/4727842/Applications_of_a_Propositional_Calculator_Constraint_Satisfaction_Problems ) * Exploratory Qualitative Analysis of Sequential Observation Data ( http://intersci.ss.uci.edu/wiki/index.php/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data ) References ========== * Awbrey, S.M., and Awbrey, J.L. (May 1991), “An Architecture for Inquiry : Building Computer Platforms for Discovery”, Proceedings of the Eighth International Conference on Technology and Education, Toronto, Canada, pp. 874–875. ( https://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery ) . * Awbrey, J.L., and Awbrey, S.M. (January 1991), “Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry”, Poster presented at the Annual Sigma Xi Research Forum, University of Texas Medical Branch, Galveston, TX. * Awbrey, J.L., and Awbrey, S.M. (August 1990), “Exploring Research Data Interactively • Theme One : A Program of Inquiry”, Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training, Society for Applied Learning Technology, Washington, DC, pp. 9–15. ( https://www.academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry ) .
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