Date
1  20 of 29
Animated Logical Graphs
Cf: Survey of Animated Logical Graphs • 3
http://inquiryintoinquiry.com/2020/08/23/surveyofanimatedlogicalgraphs3/ 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/logicalgraphs1/ Logical Graphs : Formal Development https://inquiryintoinquiry.com/2008/09/19/logicalgraphs2/ 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/peirceslaw/ • OEIS Wiki https://oeis.org/wiki/Peirce%27s_law Praeclarum Theorema • This Blog https://inquiryintoinquiry.com/2008/10/05/praeclarumtheorema/ • 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/surveyofthemeoneprogram3/ Blog Dialogs ============ Animated Logical Graphs https://inquiryintoinquiry.com/2015/01/08/animatedlogicalgraphs1/ https://inquiryintoinquiry.com/2015/01/14/animatedlogicalgraphs2/ https://inquiryintoinquiry.com/2015/01/26/animatedlogicalgraphs3/ https://inquiryintoinquiry.com/2015/01/27/animatedlogicalgraphs4/ https://inquiryintoinquiry.com/2015/01/28/animatedlogicalgraphs5/ ∙∙∙ https://inquiryintoinquiry.com/2019/08/25/animatedlogicalgraphs30/ ∙∙∙ https://inquiryintoinquiry.com/2020/11/12/animatedlogicalgraphs45/ https://inquiryintoinquiry.com/2020/11/15/animatedlogicalgraphs46/ https://inquiryintoinquiry.com/2020/11/27/animatedlogicalgraphs47/ https://inquiryintoinquiry.com/2020/11/30/animatedlogicalgraphs48/ https://inquiryintoinquiry.com/2020/12/03/animatedlogicalgraphs49/ https://inquiryintoinquiry.com/2020/12/05/animatedlogicalgraphs50/ https://inquiryintoinquiry.com/2020/12/20/animatedlogicalgraphs51/ https://inquiryintoinquiry.com/2020/12/21/animatedlogicalgraphs52/ Regards, Jon


Dirk Baecker <baecker@...>
Dear Jon,
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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>:


Dirk Baecker <baecker@...>
Dear Jon,
toggle quoted messageShow quoted text
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>:


Dear Dirk,
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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/logicalgraphs1/ Logical Graphs : Formal Development https://inquiryintoinquiry.com/2008/09/19/logicalgraphs2/ 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, Am 19.01.2021 um 15:24 schrieb Jon Awbrey <jawbrey@att.net>:


Dirk Baecker <baecker@...>
Dear Jon,
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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>:


Cf: Animated Logical Graphs • 53
http://inquiryintoinquiry.com/2021/01/22/animatedlogicalgraphs53/ Praeclarum Theorema Proof Animation https://inquiryintoinquiry.files.wordpress.com/2012/01/praeclarumtheorema20animation.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/surveyofanimatedlogicalgraphs3/ 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/logicalgraphs1/ ) • Two ( https://inquiryintoinquiry.com/2008/09/19/logicalgraphs2/ ) • 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/surveyofabductiondeductioninductionanalogyinquiry2/ ) Regards, Jon


Cf: Animated Logical Graphs • 54
http://inquiryintoinquiry.com/2021/01/27/animatedlogicalgraphs54/ Re: Peter Cameron https://cameroncounts.wordpress.com/about/ ::: Doing Research https://cameroncounts.wordpress.com/2009/11/11/doingresearch/ Re: Gil Kalai https://gilkalai.wordpress.com/about/ ::: Chomskian Linguistics https://gilkalai.wordpress.com/2009/09/29/chomskianlinguistics/ 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/ififif.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: Survey of Animated Logical Graphs • 3
https://inquiryintoinquiry.com/2020/08/23/surveyofanimatedlogicalgraphs3/ All, I updated my last Survey page on Animated Logical Graphs and added links to the series of posts on CSP, GSB, & Me. https://inquiryintoinquiry.com/2017/07/19/charlessanderspeircegeorgespencerbrownandme/ https://inquiryintoinquiry.com/2017/07/20/charlessanderspeircegeorgespencerbrownandme1/ https://inquiryintoinquiry.com/2017/07/21/charlessanderspeircegeorgespencerbrownandme2/ https://inquiryintoinquiry.com/2017/07/31/charlessanderspeircegeorgespencerbrownandme3/ ••• https://inquiryintoinquiry.com/2017/08/25/charlessanderspeircegeorgespencerbrownandme10/ https://inquiryintoinquiry.com/2021/01/18/charlessanderspeircegeorgespencerbrownandme11/ https://inquiryintoinquiry.com/2021/01/25/charlessanderspeircegeorgespencerbrownandme12/ https://inquiryintoinquiry.com/2021/01/26/charlessanderspeircegeorgespencerbrownandme13/ Regards, Jon


Cf: Animated Logical Graphs • 55
http://inquiryintoinquiry.com/2021/01/30/animatedlogicalgraphs55/ 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,
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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 (dnets), in the deep past. Here’s some implementation details (1995) for asynchronous dnet computation. Ran it first on an Intel Hypercube with 16 nodes (ugh, coursegrain parallelism — a technical abstract (1987) at http://wbricken.com/pdfs/01bm/05arch/01dnets/03paraengijcai.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 finegrain logic parallelism. take care wm


Cf: Animated Logical Graphs • 56
http://inquiryintoinquiry.com/2021/02/06/animatedlogicalgraphs56/ Re: Re: Animated Logical Graphs • 55 https://inquiryintoinquiry.com/2021/01/30/animatedlogicalgraphs55/ 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 (dnets), in the deep past. Here’s some implementation details (1995) for asynchronous dnet computation. Ran it first on an Intel Hypercube with 16 nodes (ugh, coursegrain parallelism — a technical abstract (1987) at “The Losp Parallel Deduction Engine” (PDF) ( http://wbricken.com/pdfs/01bm/05arch/01dnets/03paraengijcai.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 finegrain 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/04distinctionnetworks.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/animatedlogicalgraphs30/ All, This upgrades an earlier post where I began to focus more evenhandedly 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/booleanfunctionsontwovariables.png Regards, Jon


Cf: Animated Logical Graphs • 57
http://inquiryintoinquiry.com/2021/02/11/animatedlogicalgraphs57/  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 realign 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 graphtheoretic 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/dualityindicatingunity1/ • C.S. Peirce • Logic of Number https://inquiryintoinquiry.com/2012/09/01/cspeircelogicofnumberms229/ • C.S. Peirce • Syllabus • Selection 1 https://inquiryintoinquiry.com/2014/08/24/cspeircesyllabusselection1/ 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
There has been a lot of work on symmetry. In fact "Group Theory" is really the "Theory of Symmetry Groups."
Here is a very good article on symmetry and symmetry breaking in Physics and Philosophy. Here is the outline:
https://plato.stanford.edu/entries/symmetrybreaking/


Cf: Animated Logical Graphs • 58
http://inquiryintoinquiry.com/2021/02/11/animatedlogicalgraphs58/ 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/symmetrybreaking/ 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/animatedlogicalgraphs59/ Re: Richard J. Lipton https://rjlipton.wordpress.com/aboutme/ ::: The Art Of Math https://rjlipton.wordpress.com/2020/11/12/theartofmath/ Re: Animated Logical Graphs https://inquiryintoinquiry.com/2021/02/11/animatedlogicalgraphs57/ https://inquiryintoinquiry.com/2021/02/11/animatedlogicalgraphs58/ All, Returning to the theme of duality and more general grouptheoretic 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/animatedlogicalgraphs30/ 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/booleanfunctionsontwovariables.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/dualityindicatingunity1/ ) 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 6thorn 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”) firethorn 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 12221227). North Atlantic Books. Kindle Edition.
This is where I first learned about Laws of Form and George SpencerBrown: 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. SpencerBrown developed a formal language for representing every possible “universe” as being brought about through a First Distinction. 7 SpencerBrown 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 11861192). 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 6thorn 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”) firethorn 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 12221227). North Atlantic Books. Kindle Edition.
This is where I first learned about Laws of Form and George SpencerBrown: 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. SpencerBrown developed a formal language for representing every possible “universe” as being brought about through a First Distinction. 7 SpencerBrown 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 11861192). North Atlantic Books. Kindle Edition.


Cf: Animated Logical Graphs • 60
http://inquiryintoinquiry.com/2021/02/21/animatedlogicalgraphs60/ 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

