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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://www.academia.edu/community/5Nxkbx

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

• 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://www.academia.edu/community/5Nxkbx

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

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

the road from here.

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://www.academia.edu/community/VoWPd2

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

• 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

<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

the road from here.

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

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