Helmut and Jon, HR
I think, the problem with bringing together Peirce and conventional mathematics is, that Peirces monism is one of time / change, and the conventional mathematical monism is one of space / permanence.
Peirce would not say that. Charles learned mathematics from his father Benjamin, he edited his father's book on linear algebra, and he added some important theorems to it. They both made a clear distinction between pure mathematics and applied mathematics. Peirce's use of mathematical relations in semiotic is not a theory of mathematics. It's an application of mathematics to analyze and represent signs and patterns of signs. JA
My guess is that Peirce's category theory, when taken at its full promise and broadest historical perspective, will find its place in a line of inquiry extending from Aristotle's Categories up through category theory in its present-day mathematical sense
No. They are different in kind. Aristotle's categories were inspired by the kinds of words in Greek. When Theophrastus was asked whether A's categories classified what exist or the ways of talking about what exists, he replied "both". He claimed that Aristotle believed that the things that exist fall into the same hierarchical patterns as the words that describe the things that exist. Kant developed his 12 categories as a replacement for Aristotle's. And Peirce discovered his triads by analyzing systematic patterns in Kant's table of 4 x 3. In short, Peirce's categories are patterns of patterns -- metalevel patterns. For examples, see the slides on "Patterns of logic and ontology": http://www.jfsowa.com/talks/patolog1.pdf That was for a 5-day short course I taught in 2013. Slides 13 to 22 summarize Aristotle's ontology. Note slide 17, which shows how each category has a characteristic question. When applied to a particular individual, such as George Washington, each question leads to an answer that describes one aspect of GW. The slides in patolog4.pdf go into more detail about a wider range of ontologies. Slides 21 to 27 survey the developments from the Scholastics to John Wilkins (1668). Slide 28 shows Kant's table, and slide 29 discusses Peirce's observations about Kant's table. Then slide 30 is a revised version of Wilkins' categories (slide 26). The top node is Being, the left branch is labeled Signs, and the right branch is labeled Physics. The text below the diagram explains how the signs on the left are related to the things on the right. Summary: Peirce's categories classify signs, Aristotle's categories classify aspects of the world, and Slide 30 shows how the two classifications are related. Re mathematical category theory: Many mathematicians believe that the term 'category theory' was a poor choice. The focus of category theory is on the mappings or morphisms. The things that are mapped could be mathematical structures of any kind. Some mathematicians call it a "theory of arrows" -- the symbols that represent the maps. John
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