Alex,
Your observations about existential graphs are a good starting point for several topics.
Re Jon Awbrey: I've known him for many years. He's developing a system that begins with EGs and connects with many mathematical issues. But I've been relating a much broader range of Peirce's theories to the full range of issues in the latest developments of AI and cognitive science.
Re Boutbaki: They started from a totally different direction, and they discovered a version of "squashed" existential graphs. They define variables by starting with a linear formula with existential quantifiers. Then they draw arcs above the line to connect each quantifier with the place in each function or relation where a variable would appear. Finally, they choose a letter as the name of each arc. Then they insert the name of the arc at each end point of each arc. Finally, they erase the arcs to get a more familiar formula.
To map their squashed EGs to Peirce's notation, (1) convert each formula to a version with just the operators for AND, NOT, and EXISTS; (2) Erase all the AND operators and assume that the blank regions represent AND. (3) Replace each NOT operator with a shaded region. (4) pull the squashed EGs apart to full two dimensional graphs with shaded ovals for negation. (5) If some of the arc lines cross, move to 3D to avoid any crossing.
And voila: You now have an existential graph. The Bourbaki demonstrated that all of mathematics can be specified by EGs.
But please read the following article: "The ignorance of the Bourbaki" by Adrian Mathias, ttps://www.dpmms.cam.ac.uk/~ardm/bourbaki.pdf
Individually, the members of the Bourbaki were brilliant mathematicians, the books they produced contain a great deal of important insights and mathematical results. But their goal was mistaken, and their method had some serious flaws. The article is only 12 pages long, and it is well worth reading.
And by the way, note the huge number of mathematical theories they related. Tha's only a finite number, but there is no limit to the number that could be developed -- that implies infinity.
Just look at Wolfram's Mathematica for the huge number of theories that have been implemented in computable forms that can be used for practical applications. Unlike LLMs, those theories are very precise, and they don't make stupid mistakes. Nobody calls them AI. They call them mathematics.
John
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From: "Alex Shkotin" <alex.shkotin(a)gmail.com>
John,
A few more thoughts.
It is very interesting to compare your approach with this [1] project of Jon Awbrey as you have the same root: Pierce's EG.
By the way, even such a formalist as N. Bourbaki, in order to avoid variables bound by a quantifiers, turned a formative construction into a graph. In this graph, occurrences of quantifier variables are replaced by the sign □, and are directly connected to their quantifier by an edge. This saved N. Bourbaki from writing an algorithm for binding a quantifier variable to its own quantifier.
Alex
[1] https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/
