One of my interests is in the intersection between images and
prose. The note by Peirce "Proto-Graphical Syntax (Comment 7.1)
is prescient. It echo's a smasiograhic image that has been
discussed recently on the web. The image below shows two houses,
and the tree shapes represent people.
In the image a girl complains to her ex-lover that his new relationship is not useful, and he is missing an opportunity to have children with her. This image was created by a Yukaghir girl for a local game similar to Pictionary.
> whether I might be comparing apples and oranges in lumping philosophical and mathematical categories under the same head... there are many differences in the categorical paradigms different observers developed over the centuries.
There are two kinds of people: lumpers and splitters. On this issue, one could argue for a single lump, as you do. The mathematical category theory is so general that it could be applied to the philosophical theories.
But the kinds of problems that the mathematical theory was designed to solve are very different from the philosophical problems that Aristotle, Porphyry, Kant, and Peirce were trying to solve.
I believe that the pedagogical issues tip the balance in favor of the splitters. It's easier to explain the philosophical issues without bringing in the mathematical theory, and it's easier to explain the mathematical theory without bringing in the philosophical issues. The only people who could understand the application of the mathematical theory to the philosophical issues are those who already have a deep understanding of both. That group is very small, very sophisticated, and they don't need an elementary tutorial.
Jon A, List
I strongly agree with your emphasis on Peirce's mathematics and logic, which were the foundation for his way of thinking, writing, and research from early childhood to the end. But we should also emphasize that all exact thinking in every field is diagrammatic and mathematical. That includes phenomenology and the derivation of the semeiotic categories by the application of mathematics to phenomenology.
To emphasize that point, I'll cite Leonardo da Vinci, who is often called a renaissance genius. He was certainly a genius who lived in the Renaissance, but the essence of his genius was his exact diagrammatical -- hence mathematical -- reasoning.
Although Leonardo had never studied mathematics, he was one of the most creative *applied* mathematicians of all time. Just today, I came across a computerized analysis of a bridge design that had not been built. The analysis showed that his design was structurally sound and that it included safety features that were not re-invented for another 300 years. Unlike the infamous Tacoma Narrows bridge, Leonardo's design could withstand high winds. See https://techxplore.com/news/2019-10-leonardo-da-vinci-bridge.html
That article shows the complexity of the design and the computerized construction (with a 3-D printer) that was required to build and test a model.. This is one of many examples of Leonardo's ability to visualize complex 3-D structures that are (a) mathematically correct, (b) physically sound, and (c) fit for purpose as engineering designs.
Leonardo's diagrammatic reasoning was also the foundation for his studies of anatomy and their application in his art. He dissected many animals and human cadavers in order to study the structure of the bones, muscles, and flesh that determined the shapes and motions of their bodies. The precision of his diagrammatic analysis enabled him to produce the most lifelike art of his day (or even of all the days since then).
In his discussions of phenomenology, Peirce had a high regard for artists who are able to analyze their perceptions with mathematical precision. That is the essence of phenomenology. Peirce also had a low opinion of "metaphysicians" who did not have the ability to do diagrammatic reasoning.
Another non-mathematician who was a genius at precise diagrammatic reasoning was Michael Faraday. He could visualize electrical and magnetic fields and their dynamic interactions. To develop his famous equations, Maxwell spent many hours working with Faraday in order to translate Faraday's phenomenological insights to partial differential equations.
Moral of the story: Phenomenology requires the utmost precision in diagrammatic reasoning and in the translation of that reasoning to engineering designs, anatomical drawings, artistic creations, and formal notations in science, mathematics, music, logic, and computer science.
For more examples of the importance of visualization in mathematics itself, see http://jfsowa.com/talks/ppe.pdf .