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

• Functional Logic • Inquiry and Analogy
• Survey of Abduction, Deduction, Induction, Analogy, Inquiry

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

#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

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

• Functional Logic • Inquiry and Analogy
• Survey of Abduction, Deduction, Induction, Analogy, Inquiry

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

#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

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.

Resources

• Functional Logic • Inquiry and Analogy
• Survey of Abduction, Deduction, Induction, Analogy, Inquiry

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

#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