# Mathematical Duality in Logical Graphs • Discussion

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### Jon Awbrey

May 4, 2024, 11:32:31 AMMay 4
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Mathematical Duality in Logical Graphs • Discussion 1
https://inquiryintoinquiry.com/2024/05/04/mathematical-duality-in-logical-graphs-discussion-1/

Re: Mathematical Duality in Logical Graphs • 1
https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

Re: Laws of Form • Lyle Anderson
https://groups.io/g/lawsofform/message/109

Re: Brading, K., Castellani, E., and Teh, N., (2017),
“Symmetry and Symmetry Breaking”, The Stanford Encyclopedia of
Philosophy (Winter 2017), Edward N. Zalta (ed.).
https://plato.stanford.edu/archives/win2017/entries/symmetry-breaking/

Dear Lyle,

Thanks for the link to the article on symmetry and symmetry breaking.
I did once take a Master's in Mathematics, specializing in combinatorics,
graph theory, and group theory. When it comes to the bearing of symmetry
groups on logical graphs and the calculus of indications, it will take
careful attention to the details of the relationship between the two
interpretations singled out by Peirce and Spencer Brown.

Both Peirce and Spencer Brown recognized the relevant duality,
if they differed in what they found most convenient to use in
their development and exposition, and most of us will emphasize
one interpretation or the other as a matter of facility or taste
in a chosen application, so it requires a bit of effort to keep
the underlying unity in focus. I recently made another try at
taking a more balanced view, drawing up a series of tables in
parallel columns the way one commonly does with dual theorems
in projective geometry, so I'll shortly share more of that work.

Resources —

Logic Syllabus
https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions
https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Regards,

Jon

cc: https://mathstodon.xyz/@Inquiry/112383472875561906

### Jon Awbrey

May 5, 2024, 12:00:42 PMMay 5
to Cybernetic Communications, Laws of Form, Structural Modeling, SysSciWG
Mathematical Duality in Logical Graphs • Discussion 2
https://inquiryintoinquiry.com/2024/05/04/mathematical-duality-in-logical-graphs-discussion-2/

Re: Interpretive Duality in Logical Graphs • 1
https://inquiryintoinquiry.com/2024/04/22/interpretive-duality-in-logical-graphs-1/

Re: Mathematical Duality in Logical Graphs • 1
https://inquiryintoinquiry.com/2024/05/03/mathematical-duality-in-logical-graphs-1/

Re: Laws of Form • Lyle Anderson
https://groups.io/g/lawsofform/message/139

<QUOTE LA:>

Definition 1. A group (G, ∗) is a set G together
with a binary operation ∗ : G × G → G satisfying
the following three conditions.

1. Associativity. For any x, y, z ∈ G,
we have (x ∗ y) ∗ z = x ∗ (y ∗ z).

2. Identity. There is an identity element e ∈ G
such that ∀ g ∈ G, we have e ∗ g = g ∗ e = g.

3. Inverses. Each element has an inverse, that is,
for each g ∈ G, there is some h ∈ G such that
g ∗ h = h ∗ g = e.

</QUOTE>

Dear Lyle,

Thanks for supplying that definition of a mathematical group.
It will afford us a wealth of useful concepts and notations
as we proceed. As you know, the above three axioms define
what is properly called an “abstract group”. Over the
course of group theory's history that definition was
gradually abstracted from the more concrete examples
of permutation groups and transformation groups initially
arising in the theory of equations and their solvability.

As it happens, the application of group theory I'll be developing
over the next several posts will be using the more concrete type
of structure, where a “transformation group” G is said to “act on”
a set X by permuting its elements among themselves. In the work
we do here, each group G we contemplate will be acting on a set X
which may be taken as either one of two things, either a canonical
set of expressions in a formal language or the mathematical objects
denoted by those expressions.

What you say about deriving arithmetic, algebra, group theory,
and all the rest from the calculus of indications may well be
true, but it remains to be shown if so, and that's aways down

Resources —

Logic Syllabus
https://inquiryintoinquiry.com/logic-syllabus/

Logical Graphs • First Impressions
https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/

Logical Graphs • Formal Development
https://inquiryintoinquiry.com/2023/09/01/logical-graphs-formal-development-a/

Regards,

Jon