As the other commenters have indicated, I think that your distinction is really just the distinction between physics and mathematics.
I agree that mathematical assertions have different meanings in different contexts, though. Here’s my attempt at a definition of mathematics:
Mathematics is the study of very precise concepts, especially of how they behave under very precise operations.
I prefer to say that mathematics is about concepts, not terms. There seems to me to be a gap between, on the one hand, having a precise concept in one’s mind and, on the other hand, having the ability to articulate that concept in words.
It is more difficult to say what mathematical statements mean.
Often, a mathematical statement will just be about the mathematical concepts it mentions. A statement of geometry, for example, is often just about the concepts of triangle, circle, or whatever geometrical concepts it mentions.
However, in a mathematician’s working practice, these statements often stop being about the original mathematical concepts. Rather, they might become about isomorphism classes of concepts (which are themselves concepts), where “isomorphism” is taken in the sense of some mathematical theory. Or the statements might become mathematical concepts in their own right. When the mathematician thinks of them this way, they no longer refer to anything outside of themselves. They are themselves the precise concepts being studied mathematically.
Thus, different utterances of a mathematical statement can mean different things. That is, the things that it refers to, and what it asserts about those things, can change with different utterances (while never being about actual physical things, which is the domain of physics). Usually, mathematicians try to arrange it so that a given statement has (provably) the same truth value in all cases, but not always.
As the other commenters have indicated, I think that your distinction is really just the distinction between physics and mathematics.
I agree that mathematical assertions have different meanings in different contexts, though. Here’s my attempt at a definition of mathematics:
Mathematics is the study of very precise concepts, especially of how they behave under very precise operations.
I prefer to say that mathematics is about concepts, not terms. There seems to me to be a gap between, on the one hand, having a precise concept in one’s mind and, on the other hand, having the ability to articulate that concept in words.
It is more difficult to say what mathematical statements mean.
Often, a mathematical statement will just be about the mathematical concepts it mentions. A statement of geometry, for example, is often just about the concepts of triangle, circle, or whatever geometrical concepts it mentions.
However, in a mathematician’s working practice, these statements often stop being about the original mathematical concepts. Rather, they might become about isomorphism classes of concepts (which are themselves concepts), where “isomorphism” is taken in the sense of some mathematical theory. Or the statements might become mathematical concepts in their own right. When the mathematician thinks of them this way, they no longer refer to anything outside of themselves. They are themselves the precise concepts being studied mathematically.
Thus, different utterances of a mathematical statement can mean different things. That is, the things that it refers to, and what it asserts about those things, can change with different utterances (while never being about actual physical things, which is the domain of physics). Usually, mathematicians try to arrange it so that a given statement has (provably) the same truth value in all cases, but not always.
I think the following says something similar quite nicely:
— Edward Teller