Reflection On Recursion • 4
• https://inquiryintoinquiry.com/2026/04/18/reflection-on-recursion-4/

A feature worth noting in the recursion diagram is the function traversing the square from one triadic node to the other. It preserves an image of the object n all the while its precedent p(n) 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.

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

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

Reflection On Recursion • 3
• https://inquiryintoinquiry.com/2026/04/13/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.

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

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 F : N → N×N×N such that F(n) = (p(n), n, f(n)).

• The three arrows at the penultimate node represent a function m : N×N → N such that m(j, k) = jk.

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
• 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

Reflection On Recursion • 2
• https://inquiryintoinquiry.com/2026/04/09/reflection-on-recursion-2/

Turning to the form of a simple recursive function f(n) = m(n, f(p(n))), the clause we used to define it earns the title of “syntactic recursion” due to the way the function name “f” occurring in the defined phrase “f(n)” re‑occurs in the defining phrase “m(n, f(p(n)))”.

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

It needs to be clear there is no circle in the definition — each instance of the type f is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat. Instead of a circle then we have two gyres, the gyre down via the precedent function p and the gyre up via the modifier function m.

cc: https://www.academia.edu/community/L24rvm
cc: https://www.academia.edu/community/LE2mrr
cc: https://www.researchgate.net/post/Reflection_On_Recursion

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

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

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

Propositions As Types Analogy • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-analogy-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the Propositions As Types Analogy. And I see hints the 2‑part analogy can be extended to a 3‑part analogy, as follows.

Proof Hint ∶ Proof ∶ Proposition
∷
Untyped Term ∶ Typed Term ∶ Type

or

Proof Hint ∶ Untyped Term
∷
Proof ∶ Typed Term
∷
Proposition ∶ Type

See my working notes on the Propositions As Types Analogy —
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#Mathematics #CategoryTheory #ProofTheory #TypeTheory
#Logic #Analogy #Isomorphism #PropositionalCalculus
#CombinatorCalculus #CombinatoryLogic #LambdaCalculus
#Peirce #LogicalGraphs #GraphTheory #RelationTheory

Sign Relations • Semiotic Equivalence Relations 2.3
• https://inquiryintoinquiry.com/2025/12/31/sign-relations-semiotic-equivalence-relations-2-c/

The semiotic equivalence relation for interpreter A yields the following semiotic equations.

• [“A”]_A = [“i”]_A

• [“B”]_A = [“u”]_A

Display 4
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-4.png

or

• “A” =_A “i”

• “B” =_A “u”

Display 5
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-5.png

In this way the SER for A induces the following semiotic partition.

• {{“A”, “i”}, {“B”, “u”}}.

Display 6
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-6.png

The semiotic equivalence relation for interpreter B yields the following semiotic equations.

• [“A”]_B = [“u”]_B

• [“B”]_B = [“i”]_B

Display 7
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-7.png

or

• “A” =_B “u”

• “B” =_B “i”

Display 8
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-8.png

In this way the SER for B induces the following semiotic partition.

• {{“A”, “u”}, {“B”, “i”}}.

Display 9
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-9.png

Taken all together we have the following picture.

Tables 7a and 7b. Semiotic Partitions for Interpreters A and B
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/semiotic-partitions-for-interpreters-a-b-2.0.png

Resources —

Sign Relation
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/VBAXbj
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2026/01/01/sign-relations-semiotic-equivalence-relations-2/

#Peirce #Inquiry #Logic #Mathematics #RelationTheory
#Semiosis #Semiotics #SignRelations #TriadicRelations

Sign Relations • Semiotic Equivalence Relations 2.2
• https://inquiryintoinquiry.com/2025/12/31/sign-relations-semiotic-equivalence-relations-2-c/

In the application to sign relations it is useful to extend the square bracket notation in the following ways. If L is a sign relation whose connotative component L_SI is an equivalence relation on S = I, let [s]_L be the equivalence class of s under L_SI. In short, [s]_L = [s]_{L_{SI}}.

A statement that the signs x and y belong to the same equivalence class under a semiotic equivalence relation L_SI is called a “semiotic equation” (SEQ) and may be written in either of the following forms.

• [x]_L = [y]_L

• x =_L y

Display 3
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/sign-relation-ser-display-3.png

In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes that can be useful. Namely, when there is known to exist a particular triple (o, s, i) in a sign relation L, it is permissible to let [o]_L be defined as [s]_L. This modifications is designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.

Applying the array of equivalence notations to the sign relations for A and B will serve to illustrate their use and utility.

Tables 6a and 6b. Connotative Components Con(L_A) and Con(L_B)
• https://inquiryintoinquiry.com/wp-content/uploads/2025/12/connotative-components-con-la-con-lb-3.0.png

Resources —

Sign Relation
• https://oeis.org/wiki/Sign_relation
• https://mywikibiz.com/Sign_relation
• https://en.wikiversity.org/wiki/Sign_relation

Survey of Semiotics, Semiosis, Sign Relations
• https://inquiryintoinquiry.com/2025/05/06/survey-of-semiotics-semiosis-sign-relations-6/

cc: https://www.academia.edu/community/VBAXbj
cc: https://www.researchgate.net/post/Sign_Relations_First_Elements
cc: https://stream.syscoi.com/2026/01/01/sign-relations-semiotic-equivalence-relations-2/

#Peirce #Inquiry #Logic #Mathematics #RelationTheory
#Semiosis #Semiotics #SignRelations #TriadicRelations