Tag Archives: Mathematics

Unknown's avatar Jon Awbrey 5:48 am on September 22, 2026
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Objects, Models, Theories • 4

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

Aristotle’s “Paradigm”

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

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Unknown's avatar Jon Awbrey 8:48 am on September 15, 2026
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Objects, Models, Theories • 3

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.

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Unknown's avatar Jon Awbrey 8:24 am on September 13, 2026
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Objects, Models, Theories • 2

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.

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Unknown's avatar Jon Awbrey 12:32 pm on September 10, 2026
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Objects, Models, Theories • 1

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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Unknown's avatar Jon Awbrey 10:52 am on August 28, 2026
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Reflective Interpretive Frameworks • Incident 2

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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Unknown's avatar Jon Awbrey 7:00 am on June 14, 2026
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Icon, Likeness, Likely Story, Likelihood, Probability • 4

Re: Icon, Likeness, Likely Story, Likelihood, Probability • 3
Re: Laws of Form • Lyle Anderson

Lyle,

We are here engaged in the wider context of which Peirce’s systems of graphs for propositional logic and Spencer Brown’s calculus of indications constitute a prominent corner, one might even say a “cantonical field”, but still just one corner of the larger picture, abstractly syntactic and formally deductive in character.

Over and above that niche the overarching edifice of Peirce’s Logic of Science, supported by the theory of signs and the theory of inquiry, must cover all three forms of inference — abductive, inductive, deductive — plus the bridge from qualitative logic to quantitative statistics.  That is the architecture of inquiry with which we’ll be occupied for quite some time.

Continuing from where I left off last time —

What intrigues me about the recently cited passages from Aristotle is the way he uses what we now regard as semiotic terms — icon, index, sign — to describe the elements and structures of logical syllogisms, including the modes of non‑demonstrative inference.

The roles of signs informing sign relations and the rules of inference guiding inquiries are subjects Peirce explored in depth.  Especially in the early years the subjects of signs and inquiry are so entwined in Peirce’s relevant lectures and papers that he passes from one to the other with little sense of discontinuity between the two.

Over the years, both in Peirce’s work and the community of researchers following after, there develops such an intense focus on the problem of classifying signs that the theory of signs takes on the character of a separate subject, detached from its natural connection to the theory of inquiry.

One of our tasks is to heal that rift and regain a sense of the original common root.

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Unknown's avatar Jon Awbrey 11:34 am on May 23, 2026
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Icon, Likeness, Likely Story, Likelihood, Probability • 3

Re: Peirce List • Phyllis Chiasson

A more complete excerpt and the translator’s notes are very helpful here.

A probability (εικος) is not the same as a sign (σηµειον).  The former is a generally accepted premiss ;  for that which people know to happen or not to happen, or to be or not to be, usually in a particular way, is a probability :  e.g., that the envious are malevolent or that those who are loved are affectionate.  A sign, however, means a demonstrative premiss which is necessary or generally accepted.1  That which coexists with something else, or before or after whose happening something else has happened, is a sign of that something’s having happened or being.

An enthymeme is a syllogism from probabilities or signs ;  and a sign can be taken in three ways — in just as many ways as there are of taking the middle term in the several figures :  either as in the first figure or as in the second or as in the third.

  • E.g., the proof that a woman is pregnant because she has milk is by the first figure ;  for the middle term is ‘having milk’.  A stands for ‘pregnant’, B for ‘having milk’, and C for ‘woman’.
  • The proof that the wise are good because Pittacus was good is by the third figure.  A stands for ‘good’, B for ‘the wise’, and C for Pittacus.  Then it is true to predicate both A and B of C ;  only we do not state the latter, because we know it, whereas we formally assume the former.
  • The proof that a woman is pregnant because she is sallow is intended to be by the middle figure ;  for since sallowness is a characteristic of woman in pregnancy, and is associated with this particular woman, they suppose that she is proved to be pregnant.  A stands for ‘sallowness’, B for ‘being pregnant’, C for ‘woman’.

If only one premiss is stated, we get only a sign ;  but if the other premiss is assumed as well, we get a syllogism,2 e.g., that Pittacus is high-minded, because those who love honour are high-minded, and Pittacus loves honour ;  or again that the wise are good, because Pittacus is good and also wise.

In this way syllogisms can be effected ;  but whereas a syllogism in the first figure cannot be refuted if it is true, since it is universal, a syllogism in the last figure can be refuted even if the conclusion is true, because the syllogism is neither universal nor relevant to our purpose.3  For if Pittacus is good, it is not necessary for this reason that all other wise men are good.  A syllogism in the middle figure is always and in every way refutable, since we never get a syllogism with the terms in this relation4 ;  for it does not necessarily follow, if a pregnant woman is sallow, and this woman is sallow, that she is pregnant.  Thus truth can be found in all signs, but they differ in the ways which have been described.

We must either classify signs in this way, and regard their middle term as an index (τεκµηριον)5 (for the name ‘index’ is given to that which causes us to know, and the middle term is especially of this nature), or describe the arguments drawn from the extremes6 as ‘signs’, and that which is drawn from the middle as an ‘index’.  For the conclusion which is reached through the first figure is most generally accepted and most true.  (Aristotle, Prior Analytics 2.27, 70a3–70b6).

Translator’s Notes

  1. If referable to one phenomenon only, a sign has objective necessity ;  if to more than one, its value is a matter of opinion.
  2. Strictly an enthymeme.
  3. If the signs of an enthymeme in the first figure are true, the conclusion is inevitable.  Aristotle does not mean that the conclusion is universal, but that the universality of the major premiss implies the validity of the minor and conclusion.  The example (<all> those who have honour, etc.) quoted for the third figure contains no universal premiss or sign, and fails to establish a universal conclusion.
  4. i.e. when both premisses are affirmative.
  5. Signs may be classified as irrefutable (1st figure) and refutable (2nd and 3rd figures), and the name ‘index’ may be attached to their middle terms, either in all figures or (more probably) only in the first, where the middle is distinctively middle.
  6. Alternatively the name ‘sign’ may be restricted to the 2nd and 3rd figures, and may be replaced by ‘index’ in the first.

Reference

  • Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

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Unknown's avatar Jon Awbrey 8:45 am on May 20, 2026
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Icon, Likeness, Likely Story, Likelihood, Probability • 2

Re: Peirce List • Phyllis Chiasson

I’m still a bit fuzzy on how Aristotle’s account relates to Peirce’s usage, though I’m pretty sure Peirce must have taken Aristotle’s usage into account, but it does seem that Aristotle drew some sort of distinction here, using a term “tekmerion” which gets translated as “index” to make the following remark later on in that chapter.

We must either classify signs in this way, and regard their middle term as an index [τεκµηριον] (for the name ‘index’ is given to that which causes us to know, and the middle term is especially of this nature), or describe the arguments drawn from the extremes as ‘signs’, and that which is drawn from the middle as an ‘index’.  For the conclusion which is reached through the first figure is most generally accepted and most true.  (Aristotle, Prior Analytics, 2.27.70b1–6).

Reference

  • Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

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Unknown's avatar Jon Awbrey 4:48 pm on May 18, 2026
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Icon, Likeness, Likely Story, Likelihood, Probability • 1

Re: Peirce List • Benjamin Udell • Michael Shapiro

Here’s a likely locus classicus for “icon” in its logical sense —

A probability (εικος) is not the same as a sign (σηµειον).  The former is a generally accepted premiss;  for that which people know to happen or not to happen, or to be or not to be, usually in a particular way, is a probability:  e.g., that the envious are malevolent or that those who are loved are affectionate.  A sign, however, means a demonstrative premiss which is necessary or generally accepted.  That which coexists with something else, or before or after whose happening something else has happened, is a sign of that something’s having happened or being.  (Aristotle, Prior Analytics, 2.27.70a3–10).

Reference

  • Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Resource

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Unknown's avatar Jon Awbrey 9:44 am on May 3, 2026
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Animated Logical Graphs • 2

Re: Peirce List • Jim Willgoose

It’s almost 50 years now since I first encountered the volumes of Peirce’s Collected Papers in the math library at Michigan State, and shortly afterwards a friend called my attention to the entry for Spencer Brown’s Laws of Form in the Whole Earth Catalog and I sent off for it right away.  I would spend the next decade just beginning to figure out what either one of them was talking about in the matter of logical graphs and I would spend another decade after that developing a program, first in Lisp and then in Pascal, that turned graph‑theoretic data structures formed on their ideas to good purpose as the basis of its reasoning engine.

I thought it might contribute to a number of long‑running and ongoing discussions if I could articulate what I think I learned from that experience.

So I’ll try to keep focused on that.

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Unknown's avatar Jon Awbrey 10:10 am on April 30, 2026
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Animated Logical Graphs • 1

For Your Musement …

Here are some animations I made up to illustrate several different styles of proof in an extended topological variant of Peirce’s Alpha Graphs for propositional logic.

  • Proof Animations
    • Double Negation

      • Double Negation

    • Peirce’s Law

      • Peirce's Law

    • Praeclarum Theorema

      • Praeclarum Theorema

    • Two‑Thirds Majority Function

      • Two‑Thirds Majority Function

A full discussion of logical graphs can be found in the following article.

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Unknown's avatar Jon Awbrey 4:36 pm on April 21, 2026
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Reflection On Recursion • Discussion 1

Re: Reflection On Recursion • 1
Re: Laws of Form • John Mingers

JM:
This is a very important and interesting topic.  I think you should consider the relationship to self‑reference, indeed are they really the same thing?

Also the work of Maturana and Varela on autopoiesis and the neurophysiology of cognition which also has recursion at its heart.

Thanks, John.  Yes, we certainly find the whole array of self concepts coming into play here — selfhood, autopoiesis or self creation, self reference and self transformation, just to name a few.  But one thing I need to emphasize from the start is how radically different such concepts appear when viewed in the x‑ray vision of Peirce’s pragmatic semiotics.

I forget where I first heard it, but it’s fairly common observation that the persistence of a recurring problem is a symptom of how unlikely it is to be solved in the paradigm where it keeps occurring.

After a while, it simply becomes time to change the paradigm …

Just by way of a first example, take the very idea of “self‑reference”.  The moment we place it in the medium of triadic sign relations we realize signs do not refer to anything at all except insofar as an interpreter refers them.

And when we ask, “What is this, that we call an interpreter?”, the pragmatic theory of signs tells us we cannot tell when we turn out the light but under the x‑ray of the pragmatic maxim the sum of its effects is effectively modeled by an extended triadic sign relation.

Everything I’ll be working at here will be done within a framework like that.

Regards,
Jon

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Unknown's avatar Jon Awbrey 10:10 pm on April 18, 2026
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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

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Unknown's avatar Jon Awbrey 8:30 am on April 14, 2026
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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

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 : \mathbb{N} \to \mathbb{N} \times \mathbb{N} \times \mathbb{N} such that F(n) = ( p(n), n, f(n) ).
  • The three arrows at the penultimate node represent a function m : \mathbb{N} \times \mathbb{N} \to \mathbb{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.

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Unknown's avatar Jon Awbrey 2:40 pm on April 9, 2026
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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

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.

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