Objects, Models, Theories

[Jon Awbrey]

Objects, Models, Theories • 1
• https://inquiryintoinquiry.com/2026/09/10/objects-models-theories-1-a/

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

I return once more to a recurring subject.

Re: Artem Kaznatcheev • Three Types of Mathematical Models
• https://egtheory.wordpress.com/2013/09/08/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.

Resources —

Functional Logic • Inquiry and Analogy
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
• https://inquiryintoinquiry.com/2025/05/01/survey-of-abduction-deduction-induction-analogy-inquiry-5/

Regards,

Jon

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

[Jon Awbrey]

Objects, Models, Theories • 2
• https://inquiryintoinquiry.com/2026/09/12/objects-models-theories-2-a/

Re: Gödel's Lost Letter • The Graph Of Math
• https://rjlipton.com/2013/11/15/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 —

Functional Logic • Inquiry and Analogy
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy

Survey of Abduction, Deduction, Induction, Analogy, Inquiry
• https://inquiryintoinquiry.com/2025/05/01/survey-of-abduction-deduction-induction-analogy-inquiry-5/

Regards,

Jon

cc: https://www.academia.edu/community/VvEzOe
cc: https://mathstodon.xyz/@Inquiry/117263878675259203
cc: https://www.researchgate.net/post/Objects_Models_Theories

[Jon Awbrey]

Objects, Models, Theories • 3
• https://inquiryintoinquiry.com/2026/09/14/objects-models-theories-3-a/

Re: Peirce List • Tom Gollier
• https://list.iu.edu/sympa/arc/peirce-l/2013-11/thrd3.html#00067
• https://list.iu.edu/sympa/arc/peirce-l/2013-11/msg00069.html

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.

Regards,

Jon

[Jon Awbrey]

Objects, Models, Theories • 4
• https://inquiryintoinquiry.com/2026/09/18/objects-models-theories-4-a/

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”
• https://inquiryintoinquiry.com/wp-content/uploads/2013/11/aristotles-paradigm.jpg

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
• https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Aristotle's_Paradigm

Regards,

Jon