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- HS:
- Evan Patterson’s “Knowledge Representation in Bicategories of Relations” is also drawn up in terms of string diagrams, as a way of explaining the W3C RDF and OWL standards. So it looks like we have a nice route from Peirce to RDF via string diagrams. Or the other way around: whichever route one prefers to travel.

Dear Henry,

I opened a topic on Relation Theory in the Logic stream of Category Theory Zulipchat to discuss the logic of relative terms and the mathematics of relations as they develop from Peirce’s first breakthroughs (1865–1870). As I have mentioned on a number of occasions, there are radical innovations in this work, probing deeper strata of logic and mathematics than ever before mined and thus undermining the fundamental nominalism of First Order Logic as we know it.

Regards,

Jon

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- JM:
- In category theory, we have this perspective that we should focus attention on maps, on the relationships between objects, rather than on the objects themselves. What’s your favorite examples of people giving a schpiel about this? Blog posts, snippets from books or papers, or even just giving your own take right now, are all welcome.

My first “abstract algebra” course in college (U Mich, 1970), the last project our instructor assigned us was to “do something creative”, a piece of creative writing, painting, sculpture, or other objet d’art, reflecting on one of the topics covered in the course.

I wrote a science fiction story about two species of creatures, the *Sets* and the *Mappings*. No way I can remember all the details but I recall it explored a theme of duality between the two forms of life and the way ideas about “things in themselves” evolved over time into ideas about “that which changes into itself”.

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Dear John, Robert, Edwina,

This discussion reminds me of the time I spent the big bucks buying a copy of Stjernfelt’s *Diagrammatology* which ran to over 500 pages with many sections in very small print and had just over 50 diagrams in the whole book.

So I think the real “versus” being dealt with here is not so much the difference between “thinking in diagrams” and “thinking in words” as the difference between “thinking in words about thinking in diagrams” and “thinking in words about thinking in words”.

Those of us, the very few, who have actually been working on “moving pictures” from the very get-go, have learned to see things somewhat differently.

*Normative science rests largely on phenomenology and on mathematics; metaphysics on phenomenology and on normative science.*

❧ Charles Sanders Peirce • *Collected Papers*, CP 1.186 (1903)

Syllabus • Classification of Sciences (CP 1.180–202, G-1903-2b)

Regardez,

Jon

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- CN:
- Recently a few of us have been using the “cartesian bicategories of relations” of Carboni and Walters, in particular their string diagrams, as syntax for relations. The string diagrams in question are more or less a directed version of Peirce’s lines of identity. They’re usually described in terms of commutative special frobenius algebras. I suspect the reason we keep finding commutative special frobenius algebras is that they support lines of identity in this way.

Dear Chad, Henry, …

Chaos rules my niche of the world right now so I’ll just break a bit of the ice by sharing the following links to my ongoing study of Peirce’s 1870 Logic Of Relatives.

- Peirce’s 1870 LOR • Overview • Part 1 • Part 2 • Part 3 • References

See especially the following paragraph.

To my way of thinking the above paragraph is one of the most radical passages in the history of logic, relativizing traditional assumptions of an absolute distinction between generals (universals) and individuals. Among other things, it pulls the rug out from under any standing for nominalism as opposed to realism about universals.

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- JM:
- I find it very frustrating not to be able to draw crosses and expressions within emails or Word documents. Does anyone know of any software or apps that can do this? If not, with so many computer scientists on this group, could someone produce something?

Dear John, All …

People with backgrounds in computing, combinatorics, or graph theory would immediately recognize Spencer Brown’s expressions are isomorphic to what graph theorists know and love as “trees”, more specifically “rooted trees”, with a particular manner of attaching letters to the nodes to be described later. In those fields there’s a standard way of mapping trees to strings of parentheses and letters. That operation is called “traversing the tree” when one passes from trees to strings and the reverse operation is called “parsing the string” when one passes from strings to trees.

The transformation of Spencer Brown’s simple closed figures in the plane or his formal expressions of “crosses” into rooted trees, together with the further transformation of those two forms to “pointer data structures” in computer memory, is discussed in the following post on my blog.

There’s a more formal presentation of logical graphs, working from the axioms or “initials” I borrowed with modifications from Peirce and Spencer Brown, in the following blog post.

Those two pieces are combined and extended in the following article.

The program I developed all through the 80s using those data structures in its logic module is documented so far as I’ve done to date on the following page.

Regards,

Jon

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- LC:
- As someone who has worked on, teaches, and uses the CoI [Calculus of Indications] to make classical syllogistic logic much easier to practice and more visually intuitive than any of the visualisations we have to date, I would be very interested in finding out more about your work in applying GSB’s work to logical tables, particularly if it does a similar thing.

Dear Leon,

Gauging the gap between entry-level formal systems like propositional calculi and calculi qualified to handle quantified predicates, functions, combinators, etc. is one of my oldest research pursuits and still very much a work in progress. When I point people to the live edges of my understanding, the places where I break off in my searches, I usually end up numbering those episodes of risk-taking under the heading of “Failures to Communicate” — but it doesn’t stop me from trying. So I’ll take a chance and post a few links along those lines in a little while but it may avert a measure of misunderstanding if I mention the main forces setting me on my present path.

I had already been studying Peirce’s *Collected Papers* from my first couple of years in college, especially fascinated by his approach to logic, his amphecks, his logical graphs, both entitative and existential, his overall visual and visionary way of doing mathematics. And then a friend pointed me to the entry for Spencer Brown’s *Laws of Form* in the first *Whole Earth Catalog* and I sent off for a copy right away. My computer courses and self-directed programming play rounded out the triple of primary impacts on the way I would understand and develop logical graphs from that point on.

*To be continued …*

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- SB:
- From what I’ve noticed there are two kinds of mathematical thinking: manipulating abstract syntax, versus direct experience/perception of concrete mathematics. These two are intertwined in various ways, but in my experience people generally excel in one of these two styles of thinking and not the other. I think that many famous collaborations between two mathematicians are divided along these lines.

Dear Simon.

Susan Awbrey and I have worked a lot and written a little on a variety of “two-culture” and “cognitive style” questions from a broadly pragmatic perspective informed by the work of C.S. Peirce, John Dewey, and like-minded thinkers. The three dimensional spaces of Peirce’s triadic sign relations afford a perspective on the ways diverse thinkers can specialize their thought to different planes or facets of a sign relation’s full volume. Various issues along these lines are discussed in the following paper.

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- HS:
- I place Logic within Mathematics and modal logic is a field of Logic,

and so of mathematics. You will find that modal logic comes up a lot

working with machines, programs, and all state based systems.

Dear Henry,

Just by way of personal orientation, I tend to follow Peirce and assorted classical sources in viewing logic as a normative science whereas mathematics is a hypothetical descriptive science. That gives a picture of their relationship like the one I drew in the following post.

*Normative science rests largely on phenomenology and on mathematics; metaphysics on phenomenology and on normative science.*

❧ Charles Sanders Peirce • *Collected Papers*, CP 1.186 (1903)

Syllabus • Classification of Sciences (CP 1.180–202, G-1903-2b)

The way I see it, then, logic is more an application of mathematics than a subfield of it.

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- HS:
- If one were to think about maths and children’s education one would need to look at the needs of other subjects too. It should be easy for people here to work out how cats ties in with physics and biology — having a maths of open systems could help a lot there. But one would also want to help maths tie in with the humanities. In France children sometime after 13 or so read Voltaire’s
*Candide*published 1759, where Voltaire makes fun of Leibniz’ idea that we live in the best possible world, by having Candide go around the world and witness all the suffering known at the time. It would be good if the maths department then also gave some introduction to fragments of contemporary modal logic, so that the children could see that the “best possible world” idea is abandoned by contemporary modal logics.

Dear Henry,

I’ve never found much use for modal logic in mathematics proper since mathematics is all about possible existence, in the sense of what is not inconsistent with a given set of premisses. Of course, one can entertain modal logic as an endeavor to construct mathematical models of natural language intuitions about possibility, contingency, necessity, etc. but that is an application of mathematics to an empirical domain.

As far as best possibilities go we certainly do a lot of work on optimization in math and its applications to the special sciences and engineering. For instance, a lot of physics begins with skiers on snowy slopes and their contemplation of gradients. That very sort of thinking by Leibniz led to his personal discovery of differential calculus.

Regards,

Jon

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