Inquiry Into Inquiry

Riffs and Rotes, Finite Partial Functions • Overview

Posted on November 14, 2026 by Jon Awbrey

Riffs and Rotes are two species of graph-theoretic structure in a mathematical framework which establishes a bijective Gödel numbering between the positive integers $(\mathbb{N}^{+})$ and finite partial functions mapping $\mathbb{N}^{+}$ to $\mathbb{N}^{+}$ . This framework strips away the standard arithmetic properties of numbers to reveal their “purely combinatorial” structure through doubly recursive prime factorization.

The Core Concept: From Primes to Functions

Every positive integer has a unique factorization as a product of primes (The Fundamental Theorem of Arithmetic). Because every prime number has a unique index, e.g. $p_1 = 2,$ $p_2 = 3,$ $p_3 = 5,$ we can recursively decompose both the bases and the exponents of any number into their respective prime indices.

By mapping these recursive index structures, a natural number is transformed into a set of ordered pairs — effectively a finite partial function $(\mathbb{N}^+ \to \mathbb{N}^+).$

Riffs vs. Rotes

The finite partial functions may be visualized using two distinct graph-theoretic structures:

Riffs (Rooted Index-Functional Forests): Represented as directed graphs (digraphs) where nodes reflect the exact recursive assignment of a prime index to its exponent.
Rotes (Rooted Odd Trees with only Exponent Symmetries): Represented as undirected, rooted trees that embed the same structural information but map it into a simplified graphic form.

How the Mapping Works (Examples)

Using the basic rule where a prime factor $p_{n}^{k}$ translates to the functional assignment of input $n$ to output $k,$ expressed as the ordered pair $\langle n, k \rangle,$ numbers map directly to finite partial functions. The first ten cases are shown in the following Table.

Because this construction is a strict bijection, every single positive integer maps to exactly one unique finite partial function, and every finite partial function has a unique integer address (Gödel number). Detailed combinatorial properties and sequence data for these graphs are documented on the OEIS Wiki “Riffs and Rotes” page.

Reference

The above text is adapted from the Google AI Overview for “Riffs and Rotes” “Finite Partial Functions” (Nov 8, 2026).

Resource

OEIS Wiki • Riffs and Rotes

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Posted in Algebra, Arithmetic, Combinatorics, Computation, Graph Theory, Group Theory, Logic, Mathematics, Number Theory, Recursion, Representation, Riffs and Rotes, Semiotics, Visualization | Tagged Algebra, Arithmetic, Combinatorics, Computation, Graph Theory, Group Theory, Logic, Mathematics, Number Theory, Recursion, Representation, Riffs and Rotes, Semiotics, Visualization | Leave a comment

Riffs and Rotes • Overview

Posted on November 14, 2026 by Jon Awbrey

Riffs and Rotes are two species of graph-theoretic structure giving visual representation to the primes factorizations of integers, specifically looking at “doubly recursive factorizations”.

The Core Concept

When a number is broken down into its prime factors, those primes can themselves be broken down based on their index (e.g., the 2nd prime, the 3rd prime, etc.). By recursively expanding these prime indices, an integer can be completely represented as a nested tree or string of prime structural indicators.

These mathematical structures may be mapped into two types of visual diagrams:

Riffs: A family of directed graphs (digraphs) that map out the recursive step-by-step structural relationships.
Rotes: A corresponding family of undirected graphs that visualize the overall underlying network.

Example: The Number 2025

An example will serve to demonstrate how numbers translate to these forms:

2025 is factored traditionally as $81 \times 25 = 3^4 \times 5^2.$
Because $3$ is the $2^{\text{nd}}$ prime $(p_2)$ and $5$ is the $3^{\text{rd}}$ prime $(p_3),$ this is rewritten using prime indices as ${p_2}^4 \times {p_3}^2.$
The indices $2$ and $3$ are then recursively broken down into their own prime structures: $2 = p_1$ and $3 = p_2 = p_{p_1}.$
By dropping terminal 1s, the number collapses into a pure, loss‑less recursive form:

This exact string configuration generates a completely unique geometry — its corresponding Riff and Rote — giving every positive integer its own complex, visual “fingerprint”.

Reference

The above text is adapted from the Google AI Overview for “Riffs and Rotes” (Nov 8, 2026).

Resources

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Objects, Models, Theories • 4

Posted on September 18, 2026 by Jon Awbrey

Re: Objects, Models, Theories • (1) • (2) • (3)

What are objects, models, theories, and how do they relate to one another?

Recurring questions about the relationship between objects, models, and theories take on a different aspect when viewed from the perspective of Peirce’s pragmatic semiotic, in other words, when cast within a framework of triadic sign relations.

In contemplating the array of questions which come to mind I always find it helpful to ruminate on the diagram shown below — I might even call it a mandala for its wealth of symbolic features and its aid in organizing the pro‑&‑con‑fusion of mental impressions.

Aristotle’s “Paradigm”

Here is the corresponding text from Aristotle and the context that leads on to Peirce’s viewpoint.

Aristotle’s Paradigm • Reasoning by Analogy or Example

Resources

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Posted in Adaptive Systems, Analogy, Biological Systems, C.S. Peirce, Information, Inquiry, Inquiry Driven Systems, Learning Theory, Logic, Logic of Science, Mathematical Models, Mathematics, Mental Models, Model Theory, Pragmata, Semiotics, Sign Relations, Triadic Relations, Visualization | Tagged Adaptive Systems, Analogy, Biological Systems, C.S. Peirce, Information, Inquiry, Inquiry Driven Systems, Learning Theory, Logic, Logic of Science, Mathematical Models, Mathematics, Mental Models, Model Theory, Pragmata, Semiotics, Sign Relations, Triadic Relations, Visualization | 1 Comment

Objects, Models, Theories • 3

Posted on September 14, 2026 by Jon Awbrey

Re: Objects, Models, Theories • (1) • (2)
Re: Peirce List • Tom Gollier

Here my task is to build bridges between several different classical and contemporary uses of the word model, so I don’t have the luxury of complete control over the words in play but have to start from the customary senses in the various communities of interpretation. Of course I’m slyly working from a sign‑relational backdrop, but I have to be sleight‑handed about that and not hit people over the head with it.

You can probably guess I’m using object to cover sign‑relational objects, and theories are clearly syntacked together from complexes of sign‑relational signs, so all we have left to pin down is where the various kinds of model sit at the table set with the labels of Object, Sign, Interpretant.

In its theoretical sense, a model of a theory is anything the theory is true of, anything that satisfies the theory. In that sense, a model is very like an object. It is whatever the theory is talking about. In the order of nature, indeed, models come before theories. But there is another order, the order of art, and one may construct artificial models out of almost any stuff, even the stuff of signs. So you see the kind of wiggle room we have to work with.

Things are easier outside of logic, in applied mathematics and the special sciences, where models are just things like analogues, icons, simulations, and similar representations of objects. But that makes them objects serving as signs of other objects, and so you may find some semiotic subtlety lurking there.

Resources

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Objects, Models, Theories • 2

Posted on September 12, 2026 by Jon Awbrey

Re: Gödel’s Lost Letter • The Graph Of Math

GLL:: Kurt Gödel is said to have been a latecomer to appreciating the power of Model Theory. He was of course the greatest architect of Proof Theory, which stands in contrast to Model Theory. Model Theory concerns itself with what could be true, while Proof Theory deals with what can be proved. The latter sounds more definite, but they are supplementary: a statement is capable of being true somewhere precisely when its negation cannot be proved. The question is, where is that somewhere? And when?

What — if anything — is the common sense that connects the different senses of the word model, as it has been used over the years in logic, mathematics, and the special sciences? It’s a problem I’ve been running into for several decades now and I think I can trace the roots of it going back as far as Aristotle’s treatment of analogy.

Resources

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Objects, Models, Theories • 1

Posted on September 10, 2026 by Jon Awbrey

Happy Birthday, Charles Sanders Peirce❢ — September 10, 1839

I return once more to a recurring subject.

Re: Artem Kaznatcheev • Three Types of Mathematical Models

In speaking of models one tends to find denizens of different disciplines talking at cross purposes to one another. Logicians use the word to describe what may be distinguished as logical models, saying a model is whatever satisfies a theory — anything a theory holds true of — and that is the sense used in the logical subject of model theory.

Almost everyone else uses the word to describe what may be called analogical models, analogues being things holding enough properties in common with other things that learning about Thing 2 (the analogue system) can teach us about Thing 1 (the object system).

It is actually quite easy to integrate those two senses of the word model into a coherent picture of the whole situation, namely, the triadic relationship among objects, analogues, and theories. We’ll get into that further as the discussion proceeds.

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Reflective Interpretive Frameworks • Incident 2

Posted on August 27, 2026 by Jon Awbrey

Re: Terence Tao • Modular Arithmetic Challenge

Can a neural network learn to do modular multiplication efficiently?

Incidental Reflection 1

There are alternative models of neural networks which do not depend on threshold neurons and endlessly fiddling with weights.

Incidental Reflection 2

The series of three blog posts linked below present a case study comparing two ways of handling a classic example from the Parallel Distributed Processing paradigm, namely, the “Jets and Sharks” database problem, first taking up the original treatment by McClelland and Rumelhart and then proceeding according to a program I developed for propositional logic modeling. The latter method makes use of ideas from Grossberg’s competition‑cooperation and winner‑take‑all dynamics, but is purely propositional‑logic based, involving no extraneous weights.

Theme One Program • Jets and Sharks • (1) • (2) • (3)

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Posted in Arithmetization, C.S. Peirce, Gödel 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 | Tagged Arithmetization, C.S. Peirce, Gödel 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 | 1 Comment

Information = Comprehension × Extension • Comment 7

Posted on August 20, 2026 by Jon Awbrey

Let’s stay with Peirce’s example of inductive inference a little longer and try to clear up the more troublesome confusions tending to arise.

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

$\text{Figure 2. Disjunctive Term}~ u, \text{Taken as Subject}$

Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign.

$\text{Figure 4. Disjunctive Subject}~ u, \text{Induction of Rule}~ v \Rightarrow w$

One final point needs to be stressed. It is important to recognize the disjunctive term itself — the syntactic formula “neat, swine, sheep, deer” or any logically equivalent formula — is not an index but a symbol.‡ It has the character of an artificial symbol which is constructed to fill a place in a formal system of symbols, for example, a propositional calculus. In that setting it would normally be interpreted as a logical disjunction of four elementary propositions, denoting anything in the universe of discourse which has any of the four corresponding properties.

The artificial symbol “neat, swine, sheep, deer” denotes objects which serve as indices of the genus herbivore by virtue of their belonging to one of the four named species of herbivore. But there is in addition a natural symbol which serves to unify the manifold of given species, namely, the concept of a cloven‑hoofed animal.

As a symbol or general representation, the concept of a cloven‑hoofed animal connotes an attribute and connotes it in such a way as to determine what it denotes. Thus we observe a natural expansion in the connotation of the symbol, amounting to what Peirce calls the “superfluous comprehension”, the information added by an “ampliative” or synthetic inference.

In sum we have sufficient information to motivate an inductive inference, from the Fact $u \Rightarrow w$ and the Case $u \Rightarrow v$ to the Rule $v \Rightarrow w.$

Remark

Here, once again, I have departed from using symbol in the precise technical sense Peirce introduced at the beginning of this discussion, reverting to the more ordinary sense all of us, Peirce included, tend to use on other occasions. Perhaps the best way to smooth the wrinkle in usage is to mark a distinction among symbols, singling out the natural, normal, canonical, or simple symbols within the more general run of artificial, compound, or complex types. Taking that tack has the beneficial side‑effect of aligning the work of abduction and induction, at least, in a pre‑established universe of discourse, with the mainstream of work in computation which takes us from dubious terms to clear signs for the objects of our interest.

References

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432. Archive. Online.

Resources

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Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information = Comprehension × Extension, Inquiry, Intension, Logic, Peirce's Categories, Pragmatic Semiotic Information, Pragmatism, Scientific Method, Semiotics, Sign Relations | Tagged Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information = Comprehension × Extension, Inquiry, Intension, Logic, Peirce's Categories, Pragmatic Semiotic Information, Pragmatism, Scientific Method, Semiotics, Sign Relations | 2 Comments

Information = Comprehension × Extension • Comment 6

Posted on August 16, 2026 by Jon Awbrey

Returning to Peirce’s example of inductive inference in Comment 2, let’s try to get a clearer picture of why he connects it with disjunctive terms and indicial signs.

At this point in time I can’t say I’m entirely satisfied with my understanding of the relationship between disjunctive terms, indicial signs, and inductive inferences as presented by Peirce in his early accounts. What follows is just one of the simplest and least question‑begging attempts at rational reconstruction I’ve been able to devise.

Figure 2 shows the implication ordering of logical terms in the form of a lattice diagram.

$\text{Figure 2. Disjunctive Term}~ u, \text{Taken as Subject}$

Figure 4 shows an inductive step of inquiry, as taken on the cue of an indicial sign.

$\text{Figure 4. Disjunctive Subject}~ u, \text{Induction of Rule}~ v \Rightarrow w$

If there is any distinguishing feature shared by all the instances under the disjunctive description “neat, swine, sheep, deer” then sign users may take that feature as a predictor of being herbivorous, precisely because all the things under the disjunctive description are herbivorous. But everything under the disjunctive description is cloven‑hoofed, so the cases under the disjunctive description serve to indicate, support, or witness the utility of the induction from cloven‑hoofed to herbivorous.

References

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432. Archive. Online.

Resources

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Information = Comprehension × Extension • Comment 5

Posted on August 10, 2026 by Jon Awbrey

Let’s stay with Peirce’s example of abductive inference a little longer and try to clear up the more troublesome confusions tending to arise.

Figure 1 shows the implication ordering of logical terms in the form of a lattice diagram.

$\text{Figure 1. Conjunctive Term}~ z, \text{Taken as Predicate}$

Figure 3 shows an abductive step of inquiry, as taken on the cue of an iconic sign.

$\text{Figure 3. Conjunctive Predicate}~ z, \text{Abduction of Case}~ x \Rightarrow y$

One thing needs to be stressed at this point. It is important to recognize the conjunctive term itself — namely, the syntactic string “spherical bright fragrant juicy tropical fruit” — is not an icon but a symbol.‡ It has its place in a formal system of symbols, for example, a propositional calculus, where it would normally be interpreted as a logical conjunction of six elementary propositions, denoting anything in the universe of discourse with all six of the corresponding properties.

The symbol “spherical bright fragrant juicy tropical fruit” denotes objects which may be taken as icons of oranges by virtue of their bearing those six properties in common with oranges. But there are no objects denoted by the symbol which aren’t already oranges themselves. Thus we observe a natural reduction in the denotation of the symbol, consisting in the absence of cases outside of oranges which have all the properties indicated.

The above analysis provides another way to understand the abductive inference from the Fact $x \Rightarrow z$ and the Rule $y \Rightarrow z$ to the Case $x \Rightarrow y.$ The lack of any cases which are $z$ and not $y$ is expressed by the implication $z \Rightarrow y.$ Taking that in conjunction with the Rule $y \Rightarrow z$ gives the logical equivalence $y = z.$ But that reduces the Case $x \Rightarrow y$ to the Fact $x \Rightarrow z$ and so the Case is justified.

Viewed in the light of the above analysis, Peirce’s example of abductive reasoning exhibits an especially strong form of inference, almost deductive in character. Do all abductive arguments take that form, or may there be weaker styles of abductive reasoning which enjoy their own levels of plausibility? That must remain an open question at this point.

Remark

Readers will notice I have slipped at this point from using symbol in the precise technical sense Peirce introduced at the beginning of this discussion to the more ordinary sense all of us, Peirce included, tend to use on other occasions. Should it become a big problem we can always find a way to mark the distinction but so far it seems context has usually sufficed to resolve any likely confusion.

References

Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
Peirce, C.S. (1867), “Upon Logical Comprehension and Extension”, Proceedings of the American Academy of Arts and Sciences, Vol. 7, pp. 416–432. Archive. Online.

Resources

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The Core Concept: From Primes to Functions

Riffs vs. Rotes

How the Mapping Works (Examples)

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The Core Concept

Example: The Number 2025

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What are objects, models, theories, and how do they relate to one another?

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Incidental Reflection 2

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