Reflection On Recursion

[Jon Awbrey]

Reflection On Recursion • 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.

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.

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.

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

Regards,

Jon

cc: https://www.academia.edu/community/L24rvm

[Jon Awbrey]

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.

Regards,

Jon

cc: https://www.academia.edu/community/L24rvm

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

[Jon Awbrey]

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.

Regards,

Jon

cc: https://www.academia.edu/community/l74XM7
cc: https://mathstodon.xyz/@Inquiry/116402930549503424
cc: https://www.researchgate.net/post/Reflection_On_Recursion

[Jon Awbrey]

Reflection On Recursion • 4
• https://inquiryintoinquiry.com/2026/04/18/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 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

Regards,

Jon

cc: https://www.researchgate.net/post/Reflection_On_Recursion

[Jon Awbrey]

Reflection On Recursion • 4
• https://inquiryintoinquiry.com/2026/04/18/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 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

Regards,

Jon

cc: https://www.researchgate.net/post/Reflection_On_Recursion