Topics

Animated Logical Graphs


 

Cf: Survey of Animated Logical Graphs • 3
http://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/

Dear Laws of Form Group,

A lot of the work I've done on the C.S. Peirce/Spencer Brown approach
to the “mathematical hypostases underlying logic” currently goes under
the heading of “Animated Logical Graphs”. There's a Survey Page on my
blog which I update from time to time. See the link above and I'll put
a partial transcript below.



I just updated my Survey of blog and wiki posts relating to
Animated Logical Graphs. A great many links went missing
when my old worksite, the InterSciWiki, went offline so
I've been repairing those as I run across them.

Beginnings
==========

Logical Graphs : Introduction
https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/

Logical Graphs : Formal Development
https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/

Elements
========

Logic Syllabus
https://oeis.org/wiki/Logic_Syllabus

Logical Graphs
https://oeis.org/wiki/Logical_Graphs

Minimal Negation Operators
https://oeis.org/wiki/Minimal_negation_operator

Propositional Equation Reasoning Systems
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Examples
========

Peirce's Law
• This Blog
https://inquiryintoinquiry.com/2008/10/06/peirce-s-law/
• OEIS Wiki
https://oeis.org/wiki/Peirce%27s_law

Praeclarum Theorema
• This Blog
https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/
• OEIS Wiki
https://oeis.org/wiki/Logical_Graphs#Praeclarum_theorema

Proof Animations
https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations

Excursions
==========

Cactus Language
https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Futures Of Logical Graphs
https://oeis.org/wiki/Futures_Of_Logical_Graphs

Applications
============

Applications of a Propositional Calculator :
Constraint Satisfaction Problems
https://www.academia.edu/4727842/Applications_of_a_Propositional_Calculator_Constraint_Satisfaction_Problems

Exploratory Qualitative Analysis of Sequential Observation Data
http://web.archive.org/web/20180828161616/http://intersci.ss.uci.edu/wiki/index.php/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data

Differential Analytic Turing Automata
https://oeis.org/wiki/Differential_Analytic_Turing_Automata

Survey of Theme One Program
https://inquiryintoinquiry.com/2020/08/28/survey-of-theme-one-program-3/

Blog Dialogs
============

Animated Logical Graphs
https://inquiryintoinquiry.com/2015/01/08/animated-logical-graphs-1/
https://inquiryintoinquiry.com/2015/01/14/animated-logical-graphs-2/
https://inquiryintoinquiry.com/2015/01/26/animated-logical-graphs-3/
https://inquiryintoinquiry.com/2015/01/27/animated-logical-graphs-4/
https://inquiryintoinquiry.com/2015/01/28/animated-logical-graphs-5/
∙∙∙
https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-30/
∙∙∙
https://inquiryintoinquiry.com/2020/11/12/animated-logical-graphs-45/
https://inquiryintoinquiry.com/2020/11/15/animated-logical-graphs-46/
https://inquiryintoinquiry.com/2020/11/27/animated-logical-graphs-47/
https://inquiryintoinquiry.com/2020/11/30/animated-logical-graphs-48/
https://inquiryintoinquiry.com/2020/12/03/animated-logical-graphs-49/
https://inquiryintoinquiry.com/2020/12/05/animated-logical-graphs-50/
https://inquiryintoinquiry.com/2020/12/20/animated-logical-graphs-51/
https://inquiryintoinquiry.com/2020/12/21/animated-logical-graphs-52/

Regards,

Jon


Dirk Baecker <baecker@...>
 

Dear Jon,

You don't have, by any means, one or two papers which put your ideas together?

best

Dirk

Am 19.01.2021 um 15:24 schrieb Jon Awbrey <jawbrey@att.net>:

Cf: Survey of Animated Logical Graphs • 3
http://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/

Dear Laws of Form Group,

A lot of the work I've done on the C.S. Peirce/Spencer Brown approach
to the “mathematical hypostases underlying logic” currently goes under
the heading of “Animated Logical Graphs”. There's a Survey Page on my
blog which I update from time to time. See the link above and I'll put
a partial transcript below.



I just updated my Survey of blog and wiki posts relating to
Animated Logical Graphs. A great many links went missing
when my old worksite, the InterSciWiki, went offline so
I've been repairing those as I run across them.

Beginnings
==========

Logical Graphs : Introduction
https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/

Logical Graphs : Formal Development
https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/

Elements
========

Logic Syllabus
https://oeis.org/wiki/Logic_Syllabus

Logical Graphs
https://oeis.org/wiki/Logical_Graphs

Minimal Negation Operators
https://oeis.org/wiki/Minimal_negation_operator

Propositional Equation Reasoning Systems
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Examples
========

Peirce's Law
• This Blog
https://inquiryintoinquiry.com/2008/10/06/peirce-s-law/
• OEIS Wiki
https://oeis.org/wiki/Peirce%27s_law

Praeclarum Theorema
• This Blog
https://inquiryintoinquiry.com/2008/10/05/praeclarum-theorema/
• OEIS Wiki
https://oeis.org/wiki/Logical_Graphs#Praeclarum_theorema

Proof Animations
https://oeis.org/wiki/User:Jon_Awbrey/ANIMATION#Proof_Animations

Excursions
==========

Cactus Language
https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Futures Of Logical Graphs
https://oeis.org/wiki/Futures_Of_Logical_Graphs

Applications
============

Applications of a Propositional Calculator :
Constraint Satisfaction Problems
https://www.academia.edu/4727842/Applications_of_a_Propositional_Calculator_Constraint_Satisfaction_Problems

Exploratory Qualitative Analysis of Sequential Observation Data
http://web.archive.org/web/20180828161616/http://intersci.ss.uci.edu/wiki/index.php/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data

Differential Analytic Turing Automata
https://oeis.org/wiki/Differential_Analytic_Turing_Automata

Survey of Theme One Program
https://inquiryintoinquiry.com/2020/08/28/survey-of-theme-one-program-3/

Blog Dialogs
============

Animated Logical Graphs
https://inquiryintoinquiry.com/2015/01/08/animated-logical-graphs-1/
https://inquiryintoinquiry.com/2015/01/14/animated-logical-graphs-2/
https://inquiryintoinquiry.com/2015/01/26/animated-logical-graphs-3/
https://inquiryintoinquiry.com/2015/01/27/animated-logical-graphs-4/
https://inquiryintoinquiry.com/2015/01/28/animated-logical-graphs-5/
∙∙∙
https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-30/
∙∙∙
https://inquiryintoinquiry.com/2020/11/12/animated-logical-graphs-45/
https://inquiryintoinquiry.com/2020/11/15/animated-logical-graphs-46/
https://inquiryintoinquiry.com/2020/11/27/animated-logical-graphs-47/
https://inquiryintoinquiry.com/2020/11/30/animated-logical-graphs-48/
https://inquiryintoinquiry.com/2020/12/03/animated-logical-graphs-49/
https://inquiryintoinquiry.com/2020/12/05/animated-logical-graphs-50/
https://inquiryintoinquiry.com/2020/12/20/animated-logical-graphs-51/
https://inquiryintoinquiry.com/2020/12/21/animated-logical-graphs-52/

Regards,

Jon





Dirk Baecker <baecker@...>
 

Dear Jon,

You don't have, by any means, one or two papers which put your ideas together?

best

Dirk

Am 19.01.2021 um 15:24 schrieb Jon Awbrey <jawbrey@att.net>:

Cf: Survey of Animated Logical Graphs • 3
http://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/

Dear Laws of Form Group,

A lot of the work I've done on the C.S. Peirce/Spencer Brown approach
to the “mathematical hypostases underlying logic” currently goes under
the heading of “Animated Logical Graphs”. There's a Survey Page on my
blog which I update from time to time. See the link above and I'll put
a partial transcript below.



I just updated my Survey of blog and wiki posts relating to
Animated Logical Graphs. A great many links went missing
when my old worksite, the InterSciWiki, went offline so
I've been repairing those as I run across them.


 

Dear Dirk,

There's a few published papers from the Peirce's Eye View
of everything on my OEIS Wiki user page —

https://oeis.org/wiki/User:Jon_Awbrey#Presentations_and_Publications

Some are also accessible at Academia.edu —

https://independent.academia.edu/JonAwbrey

Those are all very general, broad scope pieces but may be of interest.

The first two blog posts I listed might be good to break ground —

Logical Graphs : Introduction
https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/

Logical Graphs : Formal Development
https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/

In the last many posts on the Animated Logical Graphs thread
I've been focused on drawing out more clearly than I've ever
managed to do before the duality between the “entitative” and
the “existential” interpretations of (my extension) of logical
graphs. As all aficionados know, Spencer Brown emphasized the
former while Peirce began with it but then shifted to the latter.
Nonetheless, both were clearly aware of the Very Abstract Calculus
underlying the dual logical interpretations.

Regards,

Jon

On 1/19/2021 9:28 AM, Dirk Baecker via groups.io wrote:
Dear Jon,
You don't have, by any means, one or two papers which put your ideas together?
best
Dirk
Am 19.01.2021 um 15:24 schrieb Jon Awbrey <jawbrey@att.net>:

Cf: Survey of Animated Logical Graphs • 3
http://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/

Dear Laws of Form Group,

A lot of the work I've done on the C.S. Peirce/Spencer Brown approach
to the “mathematical hypostases underlying logic” currently goes under
the heading of “Animated Logical Graphs”. There's a Survey Page on my
blog which I update from time to time. See the link above and I'll put
a partial transcript below.


Dirk Baecker <baecker@...>
 

Dear Jon,

Thank you. That helps a lot. And I see with fascination that you wrote about Integrative Universities. I am about to leave from teaching at a private university here in Germany which always tried to, but never quite managed to, get integrative.

very best

Dirk

Am 19.01.2021 um 16:20 schrieb Jon Awbrey <jawbrey@att.net>:

Dear Dirk,

There's a few published papers from the Peirce's Eye View
of everything on my OEIS Wiki user page —

https://oeis.org/wiki/User:Jon_Awbrey#Presentations_and_Publications

Some are also accessible at Academia.edu —

https://independent.academia.edu/JonAwbrey

Those are all very general, broad scope pieces but may be of interest.

The first two blog posts I listed might be good to break ground —

Logical Graphs : Introduction
https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/

Logical Graphs : Formal Development
https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/

In the last many posts on the Animated Logical Graphs thread
I've been focused on drawing out more clearly than I've ever
managed to do before the duality between the “entitative” and
the “existential” interpretations of (my extension) of logical
graphs. As all aficionados know, Spencer Brown emphasized the
former while Peirce began with it but then shifted to the latter.
Nonetheless, both were clearly aware of the Very Abstract Calculus
underlying the dual logical interpretations.

Regards,

Jon

On 1/19/2021 9:28 AM, Dirk Baecker via groups.io wrote:
Dear Jon,
You don't have, by any means, one or two papers which put your ideas together?
best
Dirk
Am 19.01.2021 um 15:24 schrieb Jon Awbrey <jawbrey@att.net>:

Cf: Survey of Animated Logical Graphs • 3
http://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/

Dear Laws of Form Group,

A lot of the work I've done on the C.S. Peirce/Spencer Brown approach
to the “mathematical hypostases underlying logic” currently goes under
the heading of “Animated Logical Graphs”. There's a Survey Page on my
blog which I update from time to time. See the link above and I'll put
a partial transcript below.




 

Cf: Animated Logical Graphs • 53
http://inquiryintoinquiry.com/2021/01/22/animated-logical-graphs-53/

Praeclarum Theorema Proof Animation
https://inquiryintoinquiry.files.wordpress.com/2012/01/praeclarum-theorema-2-0-animation.gif

All,

A good deal of the work I've been doing on the C.S. Peirce/Spencer Brown
approach to “the mathematical hypostases laying the grounds for logic”
currently flies under the banner of Animated Logical Graphs. There's
a Survey of related resources I update from time to time at the
following location.

• Survey of Animated Logical Graphs
https://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/

Resources
=========

• Logic Syllabus ( https://oeis.org/wiki/Logic_Syllabus )
• Ampheck ( https://oeis.org/wiki/Ampheck )

• Logical Graphs ( https://oeis.org/wiki/Logical_Graphs )
• One ( https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/ )
• Two ( https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ )

• Propositions As Types Analogy
( https://oeis.org/wiki/Propositions_As_Types_Analogy )

• Propositional Equation Reasoning Systems
( https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems )

• Survey of Abduction, Deduction, Induction, Analogy, Inquiry
( https://inquiryintoinquiry.com/2020/12/16/survey-of-abduction-deduction-induction-analogy-inquiry-2/ )

Regards,

Jon


 

Cf: Animated Logical Graphs • 54
http://inquiryintoinquiry.com/2021/01/27/animated-logical-graphs-54/

Re: Peter Cameron
https://cameroncounts.wordpress.com/about/
::: Doing Research
https://cameroncounts.wordpress.com/2009/11/11/doing-research/

Re: Gil Kalai
https://gilkalai.wordpress.com/about/
::: Chomskian Linguistics
https://gilkalai.wordpress.com/2009/09/29/chomskian-linguistics/

Speaking of dreams, the night before last I had a dream where
I was listening to a lecturer and something he said made me think
of a logical formula having the form “if if if a, b, c, d”, which
I visualized as the Peircean logical graph shown below.

If If If
https://inquiryintoinquiry.files.wordpress.com/2021/01/if-if-if.jpg

I knew I had seen something the day before prompting that fragment
and a search through my browser history turned up Gil Kalai's post
on Chomskian Linguistics where I'd read the phrase “anti anti anti
missile missile missile missile”.

Regards,

Jon


 


 

Cf: Animated Logical Graphs • 55
http://inquiryintoinquiry.com/2021/01/30/animated-logical-graphs-55/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/sociological_reading_of_lof/79753680
::: William Bricken ( https://groups.io/g/lawsofform/message/19 )

<QUOTE WB:>

Kauffman’s 2001 piece on Peirce (title is “The Mathematics of Charles Sanders Peirce
( http://homepages.math.uic.edu/~kauffman/Form.html )”) is IMO fundamental to this
discussion.

Here’s a brief excerpt from a piece I did in 2005:
“Boundary Logic and Alpha Existential Graphs (AEG)”

4.3 LoF and Alpha Graphs Compared

AEG applies the diagrammatic structure of enclosure specifically to logic.
The representations of LoF and AEG are isomorphic, while the systems of
transformation rules are remarkably close to being the same. ...

</QUOTE>

Dear William,

Many thanks for your excerpt. It highlights many of the most critical points
in comparing the systems of Peirce and Spencer Brown so I'll take up the topic
of duality first. Over the years I've always found that to be one of the stickier
wickets in the whole field. I'll discuss it in my Animated Logical Graphs series
as that's where I've recently redoubled my efforts to explain the issue and why
it's important.

Regards,

Jon


william bricken
 

Hi Jon,

Weird how we’ve been doing this for so many years! I look forward to what you have to say. Dunno if you’ve seen this, may be of interest.

We built some stuff similar to logic graphs, we called distinction networks (d-nets), in the deep past. Here’s some implementation details (1995) for asynchronous d-net computation. Ran it first on an Intel Hypercube with 16 nodes (ugh, course-grain parallelism — a technical abstract (1987) at http://wbricken.com/pdfs/01bm/05arch/01dnets/03para-eng-ijcai.pdf), and eventually migrated to a distributed network architecture in which each node was an independent operating system, more for the convenience of doing VR than for the elegance of fine-grain logic parallelism.


take care
wm

On Jan 30, 2021, at 9:45 AM, Jon Awbrey <jawbrey@...> wrote:

Cf: Animated Logical Graphs • 55
http://inquiryintoinquiry.com/2021/01/30/animated-logical-graphs-55/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/sociological_reading_of_lof/79753680
::: William Bricken ( https://groups.io/g/lawsofform/message/19 )

<QUOTE WB:>

Kauffman’s 2001 piece on Peirce (title is “The Mathematics of Charles Sanders Peirce
( http://homepages.math.uic.edu/~kauffman/Form.html )”) is IMO fundamental to this
discussion.

Here’s a brief excerpt from a piece I did in 2005:
“Boundary Logic and Alpha Existential Graphs (AEG)”

4.3  LoF and Alpha Graphs Compared

AEG applies the diagrammatic structure of enclosure specifically to logic.
The representations of LoF and AEG are isomorphic, while the systems of
transformation rules are remarkably close to being the same.  ...

</QUOTE>

Dear William,

Many thanks for your excerpt.  It highlights many of the most critical points
in comparing the systems of Peirce and Spencer Brown so I'll take up the topic
of duality first.  Over the years I've always found that to be one of the stickier
wickets in the whole field.  I'll discuss it in my Animated Logical Graphs series
as that's where I've recently redoubled my efforts to explain the issue and why
it's important.

Regards,

Jon







 

Cf: Animated Logical Graphs • 56
http://inquiryintoinquiry.com/2021/02/06/animated-logical-graphs-56/

Re: Re: Animated Logical Graphs • 55
https://inquiryintoinquiry.com/2021/01/30/animated-logical-graphs-55/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/animated_logical_graphs/79952098
::: William Bricken
https://groups.io/g/lawsofform/message/78

<QUOTE WB:>

Weird how we’ve been doing this for so many years! I look forward to what you have to say. Dunno if you’ve seen this, may be of interest.

We built some stuff similar to logic graphs, we called distinction networks (d-nets), in the deep past. Here’s some implementation details (1995) for asynchronous d-net computation. Ran it first on an Intel Hypercube with 16 nodes (ugh, course-grain parallelism — a technical abstract (1987) at “The Losp Parallel Deduction Engine” (PDF) ( http://wbricken.com/pdfs/01bm/05arch/01dnets/03para-eng-ijcai.pdf ) ) and eventually migrated to a distributed network architecture in which each node was an independent operating system, more for the convenience of doing VR than for the elegance of fine-grain logic parallelism.

Distinction Networks
====================

Abstract. Intelligent systems can be modeled by organizationally closed networks of interacting agents. An interesting step in the evolution from agents to systems of agents is to approach logic itself as a system of autonomous elementary processes called distinctions. Distinction networks are directed acyclic graphs in which links represent logical implication and nodes are autonomous agents which act in response to changes in their local environment of connectivity. Asynchronous communication of local decisions produces global computational results without global coordination. Biological/environmental programming uses environmental semantics, spatial syntax, and boundary transformation to produce strongly parallel logical deduction.

Reference
=========

• Bricken, W. (July 1995), “Distinction Networks” (PDF)
( http://wbricken.com/pdfs/01bm/05arch/01dnets/04distinction-networks.pdf ) .

</QUOTE>

Dear William,

Thanks for the readings. Maybe I've just got McCulloch on the brain right now
but the things I'm reading in several groups lately keep flashing me back to
themes from his work. What you wrote on distinction networks took me back
to the beginnings of my interest in AI, especially as approached from
logical directions. There's a couple of posts on my blog where I made
an effort to point up what I regard as critical issues. I'll reshare
those next and see if I can throw more light on what's at stake.

Regards,

Jon


 

Cf: Animated Logical Graphs • 30
https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-30/

All,

This upgrades an earlier post where I began to focus more
even-handedly on the dual interpretations of Peirce's and
Spencer Brown's graphical calculi for propositional logic.

The duality between Entitative and Existential interpretations
of logical graphs is a good example of a mathematical symmetry,
in this case a symmetry of order two. Symmetries of this and
higher orders give us conceptual handles on excess complexity
in the manifold of sensuous impressions, making it well worth
the effort to seek them out and grasp them where we find them.

In that vein, here’s a Rosetta Stone to give us a grounding in
the relationship between boolean functions and our two readings
of logical graphs.

Boolean Functions on Two Variables (see also attached image)
https://inquiryintoinquiry.files.wordpress.com/2020/11/boolean-functions-on-two-variables.png

Regards,

Jon


 

Cf: Animated Logical Graphs • 57
http://inquiryintoinquiry.com/2021/02/11/animated-logical-graphs-57/

| All other sciences without exception depend upon
| the principles of mathematics; and mathematics
| borrows nothing from them but hints.
|
| C.S. Peirce • “Logic of Number”

| A principal intention of this essay is to separate
| what are known as algebras of logic from the subject
| of logic, and to re-align them with mathematics.
|
| G. Spencer Brown • Laws of Form

The duality between entitative and existential interpretations
of logical graphs tells us something important about the relation
between logic and mathematics. It tells us that the mathematical
forms giving structure to reasoning are deeper and more abstract
at once than their logical interpretations.

A formal duality points to a more encompassing unity, founding
a calculus of forms whose expressions can be read in alternate
ways by switching the meanings assigned to a pair of primitive
terms. Spencer Brown’s mathematical approach to Laws of Form
and the whole of Peirce’s work on the mathematics of logic
shows both thinkers were deeply aware of this principle.

Peirce explored a variety of dualities in logic which he treated on
analogy with the dualities in projective geometry. This gave rise to
formal systems where the initial constants, and thus their geometric and
graph-theoretic representations, had no uniquely fixed meanings but could be
given dual interpretations in logic.

It was in this context that Peirce’s systems of logical graphs developed,
issuing in dual interpretations of the same formal axioms which Peirce
referred to as “entitative graphs” and “existential graphs”, respectively.
He developed only the existential interpretation to any great extent, since
the extension from propositional to relational calculus appeared more natural
in that case, but whether there is any logical or mathematical reason for the
symmetry to break at that point is a good question for further research.

Resources
=========

• Duality Indicating Unity
https://inquiryintoinquiry.com/2013/01/31/duality-indicating-unity-1/

• C.S. Peirce • Logic of Number
https://inquiryintoinquiry.com/2012/09/01/c-s-peirce-logic-of-number-ms-229/

• C.S. Peirce • Syllabus • Selection 1
https://inquiryintoinquiry.com/2014/08/24/c-s-peirce-syllabus-selection-1/

References
==========

• Peirce, C.S., [Logic of Number — Le Fevre] (MS 229),
in Carolyn Eisele (ed., 1976), The New Elements of
Mathematics by Charles S. Peirce, vol. 2, 592–595.

• Spencer Brown, G. (1969), Laws of Form,
George Allen and Unwin, London, UK.

Regards,

Jon


Lyle Anderson
 


 

Cf: Animated Logical Graphs • 58
http://inquiryintoinquiry.com/2021/02/11/animated-logical-graphs-58/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/animated_logical_graphs/79952098
:: 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.). Online
https://plato.stanford.edu/archives/win2017/entries/symmetry-breaking/

Dear Lyle,

Thanks for the link to the article on symmetry and its breaking. I did once
take a Master's in Mathematics, specializing in combinatorics, graph theory,
and group theory. As far as the applications to logical graphs and the
calculus of indications goes, it will take careful attention to the details
of the relationship between the two interpretations recognized 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
taste or facility 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 will
shortly share more of that work.

Regards,

Jon


 

Cf: Animated Logical Graphs • 59
http://inquiryintoinquiry.com/2021/02/21/animated-logical-graphs-59/

Re: Richard J. Lipton
https://rjlipton.wordpress.com/about-me/
::: The Art Of Math
https://rjlipton.wordpress.com/2020/11/12/the-art-of-math/
Re: Animated Logical Graphs
https://inquiryintoinquiry.com/2021/02/11/animated-logical-graphs-57/
https://inquiryintoinquiry.com/2021/02/11/animated-logical-graphs-58/

All,

Returning to the theme of duality and more general group-theoretic
symmetries in logical graphs, here's an improved version of the
introduction I gave two years ago.

Cf: Animated Logical Graphs • 30
https://inquiryintoinquiry.com/2019/08/25/animated-logical-graphs-30/

The duality between Entitative and Existential interpretations
of logical graphs is a good example of a mathematical symmetry,
in this case a symmetry of order two. Symmetries of this and
higher orders give us conceptual handles on excess complexity
in the manifold of sensuous impressions, making it well worth
the effort to seek them out and grasp them where we find them.

In that vein, here's a Rosetta Stone to give us a grounding in
the relationship between boolean functions and our two readings
of logical graphs.

Table. Boolean Functions on Two Variables (see also attached)
https://inquiryintoinquiry.files.wordpress.com/2020/11/boolean-functions-on-two-variables.png

Resources
=========

• Logic Syllabus
( https://oeis.org/wiki/Logic_Syllabus )

• Logical Graphs
( https://oeis.org/wiki/Logical_Graphs )

• Duality Indicating Unity
( https://inquiryintoinquiry.com/2013/01/31/duality-indicating-unity-1/ )

Regards,

Jon


Lyle Anderson
 

Jon,

Definition 1: A group (G, ∗) is a set G together with a binary operation ∗ : G×G → G satisfying the following three conditions: 1. Associativity - that is, for any x, y, z ∈ G, we have (x ∗ y) ∗ z = x ∗ (y ∗ z). 2. There is an identity element e ∈ G such that ∀g ∈ G, we have e ∗ g = g ∗ e = g. 3. Each element has an inverse - that is, for each g ∈ G, there is some h ∈ G such that g ∗ h = h ∗ g = e.

From the Pierce/Brown diagrams, we can derive arithmetic, algebra, and group theory. We also get computer and information theory.   We can do this because the Creator of the Universe intended us to do this.  He gave us a clue by starting the Torah with the letter Bet which is also the number 2.  Bet started out as the pictograph for a "house" which was an early distinction between inside and outside.

It is also interesting to note that one of the possible English interpretations of Genesis 1:1 is: "In the Beginning () created Elohim."  The Kabbalists call () Ein Sof.

Jewish Kabbalah is a set of esoteric teachings meant to explain the relationship between the unchanging, eternal God–the mysterious Ein Sof (אֵין סוֹף‎, "The Infinite")–[5][6] and the mortal, finite universe (God's creation).[3][5] It forms the foundation of mystical religious interpretations within Judaism.[3][7]
https://en.wikipedia.org/wiki/Kabbalah
7
3,
Lyle


Lyle Anderson
 

All,
I got an email outside the group asking the source of this particular translation of Genesis 1:1. It comes from my friend Stan Tenen's book "The Alphabet That Changed the World", "How Genesis Preserves a Science of Consciousness in Geometry and Gesture." It is available on Amazon in Kindle and Paperback.  https://smile.amazon.com/gp/product/B01GOHTLOI/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1

meanings of this string of 28 Hebrew letters:
• In the beginning creates Elokim the essence of the Heavens and the essence of the Earth.
• [He] creates a 6-thorn that is the essence of the Heavens and the essence of the Earth.
• [He] creates a woven network that is the essence of the Heavens and the essence of the Earth.
• By means of (“ in the”) fire-thorn God creates…
• By means of a “tadpole” (a newborn “extended head”), a source of fertilization
 
Tenen, Stan. The Alphabet That Changed the World: How Genesis Preserves a Science of Consciousness in Geometry and Gesture (Kindle Locations 1222-1227). North Atlantic Books. Kindle Edition. 

This is where I first learned about Laws of Form and George Spencer-Brown:

The idea that distinction constitutes the very first principle in terms of which things come into manifestation has fairly recently become a topic within mathematical logic. As we mentioned in the introduction, in the 1960s, the British logician/ mathematician G. Spencer-Brown developed a formal language for representing every possible “universe” as being brought about through a First Distinction. 7 Spencer-Brown posited that every world comes into being by a primordial space being severed by an act that distinguishes something within it from everything else. Imagine a sheet of paper with a circle drawn on it. The circle severs the space into an inside and an outside. Any distinction whatsoever is like that.
 
Tenen, Stan. The Alphabet That Changed the World: How Genesis Preserves a Science of Consciousness in Geometry and Gesture (Kindle Locations 1186-1192). North Atlantic Books. Kindle Edition. 


Fabian Strobel
 

Sorry Lyle,
 
as someone who studied Hebrew language history I want to point out that the claims made in this and in Lyle's previous email are Stan Tenen's own ideas. They are contradicting everything that is known historically about the Hebrew language and alphabet. Tenen claims to continue the history of Kabbalah (and Kabbalah in itself is desinterested in historic evidence) but he does not even stand on the ground of kabbalistic traditions in his claims. I do not want to start a discussion about his claims here, everyone interested can check them out for herself. I just want to contradict his veneer of scientific judaism.
 
All the best, keep the logic coming,
 
Fabian
 
 
Gesendet: Sonntag, 21. Februar 2021 um 21:31 Uhr
Von: "Lyle Anderson" <LylePhone@...>
An: lawsofform@groups.io
Betreff: Re: [lawsofform] Animated Logical Graphs
All,
I got an email outside the group asking the source of this particular translation of Genesis 1:1. It comes from my friend Stan Tenen's book "The Alphabet That Changed the World", "How Genesis Preserves a Science of Consciousness in Geometry and Gesture." It is available on Amazon in Kindle and Paperback.  https://smile.amazon.com/gp/product/B01GOHTLOI/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
 
meanings of this string of 28 Hebrew letters:
• In the beginning creates Elokim the essence of the Heavens and the essence of the Earth.
• [He] creates a 6-thorn that is the essence of the Heavens and the essence of the Earth.
• [He] creates a woven network that is the essence of the Heavens and the essence of the Earth.
• By means of (“ in the”) fire-thorn God creates…
• By means of a “tadpole” (a newborn “extended head”), a source of fertilization
 
Tenen, Stan. The Alphabet That Changed the World: How Genesis Preserves a Science of Consciousness in Geometry and Gesture (Kindle Locations 1222-1227). North Atlantic Books. Kindle Edition. 

This is where I first learned about Laws of Form and George Spencer-Brown:
 
The idea that distinction constitutes the very first principle in terms of which things come into manifestation has fairly recently become a topic within mathematical logic. As we mentioned in the introduction, in the 1960s, the British logician/ mathematician G. Spencer-Brown developed a formal language for representing every possible “universe” as being brought about through a First Distinction. 7 Spencer-Brown posited that every world comes into being by a primordial space being severed by an act that distinguishes something within it from everything else. Imagine a sheet of paper with a circle drawn on it. The circle severs the space into an inside and an outside. Any distinction whatsoever is like that.
 
Tenen, Stan. The Alphabet That Changed the World: How Genesis Preserves a Science of Consciousness in Geometry and Gesture (Kindle Locations 1186-1192). North Atlantic Books. Kindle Edition. 


 

Cf: Animated Logical Graphs • 60
http://inquiryintoinquiry.com/2021/02/21/animated-logical-graphs-60/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/animated_logical_graphs/79952098
::: 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
this 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.

Regards,

Jon