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 • Ma^thstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#arithmetization, #c-s-peirce, #godel-numbers, #higher-order-sign-relations, #inquiry-driven-systems, #inquiry-into-inquiry, #logic, #mathematics, #quotation, #recursion, #reflection, #reflective-interpretive-frameworks, #semiotics, #sign-relations, #triadic-relations, #use-and-mention, #visualization

Turning to the form of a simple recursive function the clause we used to define it earns the title of “syntactic recursion” due to the way the function name occurring in the defined phrase re‑occurs in the defining phrase

It needs to be clear there is no circle in the definition — each instance of the type 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 and the gyre up via the modifier function

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 • Ma^thstodon
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#arithmetization, #c-s-peirce, #godel-numbers, #higher-order-sign-relations, #inquiry-driven-systems, #inquiry-into-inquiry, #logic, #mathematics, #quotation, #recursion, #reflection, #reflective-interpretive-frameworks, #semiotics, #sign-relations, #triadic-relations, #use-and-mention, #visualization

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.

For example, the factorial function has a definition in terms of the predecessor function and the multiplier function

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
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations

cc: Academia.edu • Cybernetics • Laws of Form • Ma^thstodon (1) (2) (3)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi

#arithmetization, #c-s-peirce, #godel-numbers, #higher-order-sign-relations, #inquiry-driven-systems, #inquiry-into-inquiry, #logic, #mathematics, #quotation, #recursion, #reflection, #reflective-interpretive-frameworks, #semiotics, #sign-relations, #triadic-relations, #use-and-mention, #visualization

Re: William Waites • The Agent That Doesn’t 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
• Reflective Interpretive Frameworks
• The Phenomenology of Reflection
• Higher Order Sign Relations
• Notes On Categories • (1) • (2)
• Survey of Relation Theory
• Survey of Inquiry Driven Systems
• Survey of Semiotics, Semiosis, Sign Relations

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

#arithmetization, #c-s-peirce, #godel-numbers, #higher-order-sign-relations, #inquiry-driven-systems, #inquiry-into-inquiry, #logic, #mathematics, #quotation, #recursion, #reflection, #reflective-interpretive-frameworks, #semiotics, #sign-relations, #triadic-relations, #use-and-mention, #visualization