Yes. That is helpful. I was going to bring this up in my first comment but decided to focus on one thing at a time. Your view seems very similar to nominalist-structuralism, which I find appealing as well. At least I can read nominalist-structuralism into your post and comments. Afaict, it’s considered the most promising version of nominalism going. They basically take your use of isomorphism and go one step farther. The SEP discusses it some but it’s a pretty poorly written article. You might have to do a lot of googling. Structuralism argues that math does not describe objects of any kind but rather, structures and places within structures (and have no identity or features outside those structures). Any given integer is not an object but a places in the structure of integers. As you can imagine once you think of mathematical truths this way it become obvious how math can be used to describe other physical systems: namely those systems instantiate the same structure (or in your words, are isomorphic?). Nominalist structuralism involves disclaiming the existence of structures (which are abstract objects) as independent of the systems that instantiate them
ETA: One issue here is the work being done by the word “resembles” when we say “the structure of real numbers resembles the structure of space’, or “the structure of macroscopic object motion resembles the structure of simulated object motion in your physics simulator”. Which is the same issue as “what is the work being done by ‘isomorphic’.
Yes. That is helpful. I was going to bring this up in my first comment but decided to focus on one thing at a time. Your view seems very similar to nominalist-structuralism, which I find appealing as well. At least I can read nominalist-structuralism into your post and comments. Afaict, it’s considered the most promising version of nominalism going. They basically take your use of isomorphism and go one step farther. The SEP discusses it some but it’s a pretty poorly written article. You might have to do a lot of googling. Structuralism argues that math does not describe objects of any kind but rather, structures and places within structures (and have no identity or features outside those structures). Any given integer is not an object but a places in the structure of integers. As you can imagine once you think of mathematical truths this way it become obvious how math can be used to describe other physical systems: namely those systems instantiate the same structure (or in your words, are isomorphic?). Nominalist structuralism involves disclaiming the existence of structures (which are abstract objects) as independent of the systems that instantiate them
ETA: One issue here is the work being done by the word “resembles” when we say “the structure of real numbers resembles the structure of space’, or “the structure of macroscopic object motion resembles the structure of simulated object motion in your physics simulator”. Which is the same issue as “what is the work being done by ‘isomorphic’.