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