#arithmetization — Public Fediverse posts

Reflection On Recursion • Discussion 1

Re: Reflection On Recursion • 1
Re: Laws of Form • John Mingers

JM: This is a very important and interesting topic.  I think you should consider the relationship to self‑reference, indeed are they really the same thing?

Also the work of Maturana and Varela on autopoiesis and the neurophysiology of cognition which also has recursion at its heart.

Thanks, John.  Yes, we certainly find the whole array of self concepts coming into play here — selfhood, autopoiesis or self creation, self reference and self transformation, just to name a few.  But one thing I need to emphasize from the start is how radically different such concepts appear when viewed under x‑rays of Peirce’s pragmatic semiotics.

I forget where I first heard it, but it’s fairly common observation that the persistence of a recurring problem is a symptom of how unlikely it is to be solved in the paradigm where it keeps occurring.

After a while, it simply becomes time to change the paradigm …

Just by way of a first example, take the very idea of “self‑reference”.  The moment we place it in the medium of triadic sign relations we realize signs do not refer to anything at all except insofar as an interpreter refers them.  And when we think to ask, “What is this that we call an interpreter?”, the pragmatic theory of signs tells us we do not know when we turn out the light but under the x‑ray of the pragmatic maxim the sum of its effects is effectively modeled by an extended triadic sign relation.

Everything I’ll be working at here will be done within a framework like that.

Regards,
Jon

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#Arithmetization #CSPeirce #GödelNumbers #HigherOrderSignRelations #InquiryDrivenSystems #InquiryIntoInquiry #Logic #Mathematics #Quotation #Recursion #Reflection #ReflectiveInterpretiveFrameworks #Semiotics #SignRelations #TriadicRelations #UseAndMention #Visualization

Reflection On Recursion • 4

A feature of special note in the recursion diagram is the function traversing the square from one triadic node to the other.  It preserves an image of the object all the while its precedent is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#Arithmetization #CSPeirce #GödelNumbers #HigherOrderSignRelations #InquiryDrivenSystems #InquiryIntoInquiry #Logic #Mathematics #Quotation #Recursion #Reflection #ReflectiveInterpretiveFrameworks #Semiotics #SignRelations #TriadicRelations #UseAndMention #Visualization

Reflection On Recursion • 3

One other feature of syntactic recursion deserves to be brought into higher relief.  Evidence of it can be found in the recursion diagram by examining the places where three paths meet.  On the descending side there is the point where three paths diverge.  On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.

The arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.

• The three arrows from the initial node represent a function such that
• The three arrows at the penultimate node represent a function such that

For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.

Resources

• Inquiry Driven Systems • Inquiry Into Inquiry
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#Arithmetization #CSPeirce #GödelNumbers #HigherOrderSignRelations #InquiryDrivenSystems #InquiryIntoInquiry #Logic #Mathematics #Quotation #Recursion #Reflection #ReflectiveInterpretiveFrameworks #Semiotics #SignRelations #TriadicRelations #UseAndMention #Visualization