Reflection On Recursion • 1.3
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/

Comment 5 —

Recursion is rife in mathematics and computation, typically sporting its recursive character on its sleeve in the fashion of syntax sketched above.

But mathematics and computation are overlearned subjects and practices, enjoying long histories of being gone over with an eye to articulating every last detail of any way they might be conceived and conducted.

So it's fair to ask whether all that artifice truly tutors nature or only creates a rationalized reconstruction of it. Then again, even if that's all it does, is there anything of use to be learned from it?

Comment 6 —

The prevalence of recursion in mathematics arises from the architecture of mathematical systems.

Mathematical systems grow from a fourfold root.

• “Primitives” are taken as initial terms.

• “Definitions” expound ever more complex terms in relation to the primitives.

• “Axioms” are taken as initial truths.

• “Theorems” follow from the axioms by way of inference rules.

Recursive definitions of mathematical objects and inductive proofs of the corresponding theorems follow closely parallel patterns. And again, in computation, recursive programs follow the same patterns in action.

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Reflection On Recursion • 1.2
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/

Comment 3 —

If we discard from the idea of recursion what is not of its essence, we find recursion occurs when our understanding of one situation has recourse to our understanding of other situations.

Very typically, the object situation presents itself as complex, difficult, or unfamiliar while the resource situations are regarded as being better understood.

It must be appreciated, however, that any ranking of situations by level of understanding is contingent on the circumstances in view and may vary radically in alternate settings.

Comment 4 —

Recursion occurs more markedly in “syntactic recursion”, where the recursive process shows its character as such in the symbols of its syntactic expression.

A sense of the difference can be gained by looking at a case of “ostensible syntactic recursion”. (How much substance backs the ostentation is a subject we'll take up, maybe at length, but later …)

Consider the following diagram for the computation of a simple recursive function.

Simple Recursion
• https://inquiryintoinquiry.com/wp-content/uploads/2026/03/simple-recursion-fn-mn-fpn.png

For example, the factorial function f(n) = n! has a definition in terms of the predecessor function p(n) = n-1 and the multiplier function m(j, k) = j∙k.

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Reflection On Recursion • 1.1
• https://inquiryintoinquiry.com/2026/04/06/reflection-on-recursion-1/

Ongoing conversations with Dan Everett on Facebook have me backtracking to recurring questions about the relationship between formal language theory (as I once learned it) and the properties of natural languages as they are found occurring in the field.

A point of particular interest is the role of recursion in formal and natural languages, along with collateral questions about its role in the cognitive sciences at large.

It has taken me quite a while to bring my reflections up to the threshold of minimal coherence — and the inquiry remains ongoing — but it may catalyze the thinking process if I simply share what I've thought so far …

Comment 1 —

Recursion is where you find it — so, myself not being a natural language researcher, when someone who is says they don't find it in a given corpus I just take them at their word …

Comment 2 —

The question to which I keep returning has to do with the relationship between two ways we find recursion occurring.

One way I'd call “pragmatic recursion” — if I wanted to be precise and cover its full scope — since so many of its operations occur without conscious direction, but for now I'll defer to more familiar language, calling it “cognitive” or “conceptual” recursion.

Resources —

Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

Reflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#RIF_1

The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_Reflection

Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

TCO in both Python and Rust? What a time to be alive. As a FP advocate, I am in awe.

And if you don't know what TCO is, let me share with you the best conference ever https://www.youtube.com/watch?v=-PX0BV9hGZY&t=1s

Python: https://blog.reverberate.org/2025/02/10/tail-call-updates.html

Rust: https://www.mattkeeter.com/blog/2026-04-05-tailcall/

#python #rust #programming #tco #functionalProgramming #recursion

You know that you are a nerd, when you spent half an hour to get the page number of the index entry for recursion into the displayed pages of the #recursion entry.

#texlive

Reflective Interpretive Frameworks • Incident 1
• https://inquiryintoinquiry.com/2026/03/26/reflective-interpretive-frameworks-incident-1/

Re: William Waites • The Agent That Doesn't Know Itself
• https://johncarlosbaez.wordpress.com/2026/03/20/the-agent-that-doesnt-know-itself/

WW: ❝Why Has Nobody Done This?❞

People who study C.S. Peirce would say reflective reasoning requires triadic relations at core and there is work being done on that. One of the challenges is clarifying the role of triadic relations in category theory and raising them into higher relief as fundamental operations.

Note. I was looking for a word to describe a random encounter with something that jogs one's memory of a recurring theme — “incident” plays into the “reflection” theme and looked worth trying for now.

Resources —

Inquiry Driven Systems • Inquiry Into Inquiry
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview

Reflective Interpretive Frameworks
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_10#Reflective_Interpretive_Frameworks

The Phenomenology of Reflection
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_11#The_Phenomenology_of_Reflection

Higher Order Sign Relations
• https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

Notes On Categories
• https://inquiryintoinquiry.com/2013/02/22/notes-on-categories-1/
• https://inquiryintoinquiry.com/2021/07/31/notes-on-categories-2/

#Peirce #HigherOrderSignRelations #Inquiry #InquiryIntoInquiry #Logic #Mathematics
#Recursion #Reflection #RelationTheory #Semiotics #SignRelations #TriadicRelations

Use Warnsdorf's rule to find a closed knight's tour in Python

https://rodstephensbooks.com/knights_tour_warnsdorf.html

#Python #Algorithms #KnightsTour #Recursion #WarnsdorfsRule

​In about 30 years, it’ll be released—featuring an Indiana Jones fully reconstructed from 40 terabytes of archival footage.

​The plot is peak recursion: an AI generates code for Harrison Ford's digital twin to battle a rogue algorithm in virtual pixelated jungles. Pure digital cyberpunk wrapped in a "cassette futurism" and Cold War aesthetic:
​"Soviet scientists in a secret bunker under Magadan accidentally awakened an ancient Sumerian algorithm trapped within copper circuits. This proto-AI doesn't just want to conquer the world, it wants to rewrite history by purging everything chaotic and illogical... namely, the USSR."

​In the end, Indiana defeats the virus by simply pulling the plug. A digital silence falls over the world, while the 2056 audience pays for their tickets in crypto-yuan, fully aware that the film itself was created by the very AI the hero was fighting on screen.

#futureofcinema #GenerativeAI #SciFi #Cyberpunk #CassetteFuturism #RetroFuturism #DigitalTwin #DeadInternetTheory #Recursion #Dystopia #TechNoir #ColdWarAesthetic