Distinguishing Mathematical Consistency from Physical Realizability

[someonewholikesstuff]

In many physical models, lower-dimensional manifolds are mathematically self-consistent, but dynamically incomplete unless augmented by additional parameters (for example, time for change, or external structures that allow evolution).


This suggests a distinction between mathematical consistency and physical realizability: a model may be well-defined internally, yet require extra structure to represent dynamics or observable processes. I am interested in whether this distinction is already implicit in standard physical frameworks, or whether articulating it in terms of dimensional dependence offers any conceptual clarification.

 

[Astronuc]

In many physical models, lower-dimensional manifolds are mathematically self-consistent, but dynamically incomplete unless augmented by additional parameters (for example, time for change, or external structures that allow evolution).


This suggests a distinction between mathematical consistency and physical realizability: a model may be well-defined internally, yet require extra structure to represent dynamics or observable processes. I am interested in whether this distinction is already implicit in standard physical frameworks, or whether articulating it in terms of dimensional dependence offers any conceptual clarification.

One poses an interesting problem. Can one provide an example?

In the first paragraph, one mentions "time for change" and "external structures that allow evolution", so that implies a time-dependent problem. Then again, one might have a steady-state time-dependent problem, ih which state variables and boundary conditions change slowly or with some being static, or one might have a transient problem with rapidly changing state variables or boundary conditions, or some combination.

Then there is the question/challenge of how well the system is modeled, or how robust is the model, to capture all possible outcomes. There is a spectrum of empirical vs mechanistic based computation efforts, and it is often a matter of time and economics regarding the solution, as well as accounting for what one does not know.

 

[wrobel]

I must confess I reported the thread as AI generated. But moderators said that I was wrong. Nevertheless I am sure it is:)

Phrases like that

lower-dimensional manifolds are mathematically self-consistent, but dynamically incomplete

sound scientifically considerable but actually they are just senseless sequences of words.
Anybody can ensure that by googling "dynamically incomplete manifold"

 

[Astronuc]

dynamically incomplete manifold

Maybe it's an AI term? I googled on the phrase and it came back with a variety of results, including scientific papers, e.g., Variational manifold learning from incomplete data

or L2-Harmonic Forms on Incomplete Riemannian Manifolds ...

 

[someonewholikesstuff]

One poses an interesting problem. Can one provide an example?

In the first paragraph, one mentions "time for change" and "external structures that allow evolution", so that implies a time-dependent problem. Then again, one might have a steady-state time-dependent problem, ih which state variables and boundary conditions change slowly or with some being static, or one might have a transient problem with rapidly changing state variables or boundary conditions, or some combination.

Then there is the question/challenge of how well the system is modeled, or how robust is the model, to capture all possible outcomes. There is a spectrum of empirical vs mechanistic based computation efforts, and it is often a matter of time and economics regarding the solution, as well as accounting for what one does not know.

Thank you for the clarification — that helps sharpen the question.


A simple example I had in mind is a purely spatial 2-D configuration (e.g., a static geometric manifold or field snapshot) which is mathematically self-consistent but does not, by itself, encode change. Introducing time as an additional parameter allows the same configuration to support dynamics (e.g., evolution of fields, boundary conditions, or state variables). In that sense, time is not required for mathematical consistency but is required for physical evolution.


Similarly, reduced-dimension models in physics (such as 1-D or 2-D effective models) can be internally consistent and predictive within their domain, but their applicability relies on higher-dimensional structures or external parameters that are held fixed or averaged over. The “external structure” is not invoked as a new physical dimension, but as part of the modeling assumptions that enable evolution or interaction.


My question is therefore not about necessity in a logical sense, but about the distinction between:

• self-consistent mathematical description, and
• physical realizability and dynamical completeness within a model.

I’m trying to understand how physicists formally draw that line when constructing or interpreting models, especially when simplifying dimensionality.

 

[someonewholikesstuff]

I must confess I reported the thread as AI generated. But moderators said that I was wrong. Nevertheless I am sure it is:)

Phrases like that

sound scientifically considerable but actually they are just senseless sequences of words.
Anybody can ensure that by googling "dynamically incomplete manifold"

Hey, I actually put real time into writing that, you just probably think any type of advanced language is AI—that idea is honestly ludicrous. I’m genuinely interested in dimensional science, and I don’t just throw words together for fun.

 

[PeroK]

Hey, I actually put real time into writing that. I’m genuinely interested in dimensional science, and I don’t just throw words together for fun.

You do, however, throw lots of words together to say something simple. For example:

A simple example I had in mind is a purely spatial 2-D configuration (e.g., a static geometric manifold or field snapshot) which is mathematically self-consistent but does not, by itself, encode change. Introducing time as an additional parameter allows the same configuration to support dynamics (e.g., evolution of fields, boundary conditions, or state variables). In that sense, time is not required for mathematical consistency but is required for physical evolution.

All that says is that we need a time parameter to model physical processes.

And:

Similarly, reduced-dimension models in physics (such as 1-D or 2-D effective models) can be internally consistent and predictive within their domain, but their applicability relies on higher-dimensional structures or external parameters that are held fixed or averaged over. The “external structure” is not invoked as a new physical dimension, but as part of the modeling assumptions that enable evolution or interaction.

I'm not sure what you mean by that, but it looks like words put together because they sound good.

The technical term that you seem to be missing is degrees of freedom:

https://en.wikipedia.org/wiki/Degrees_of_freedom_(physics_and_chemistry)

 

[wrobel]

Hey, I actually put real time into writing that,

Let's be honest: you do not have regular education in math\phys. And all of that pretentious words can not hide that.
I am sure everybody will be happy to help you if you come here with questions from the textbooks you read but not with your homemade terminologies and theories.