Yes as mentioned in the body, there’s a typo on the arXiv version that’s fixed as it’s going through peer-review.
mathematical ideals would not in general map to non-mathematical (physical) entities, hence why it’s a non-trivial constraint added to the formalism—essentially making a kind of an ‘avatar’ of the physical entity labeled within the formalism. There wouldn’t be any other way of referring to a non-abstract entity within an abstract formalism. In this case, I adopted a ZFC mathematics as it was convenient for the formalism, bringing me to:
No, I am not assuming that. For a start, much of logic would qualify and that wouldn’t necessarily be encompassed by mathematics (as Gödel showed) and many abstractions may have very little extra structure. But in this formalism, I choose a way of casting the problem and adopt a minimal amount of structure to add these constraints, it adopts a ZFC set theory.
To achieve the goal of examining the process of the Ansatz—of matching mathematical ideas to non-mathematical entities (or phenomena)
Trying again, “non mathematical” means “not having mathematical existence”, rather than “mathematically indescribable”..? (Example of the former, electrons; example of the latter qualia, ghosts)
But then , why not just say
“matching mathematical ideas to physical entities”
Yes—fair enough — to clarify that, in that quoted sentence, when I said ‘non-mathematical entities’ I actually meant physical entities, rather than eg non-truths/falsehoods within mathematics (eg contradictory theorems and so forth).
That is a good pickup, and a good fix as I try to describe it in english language terms.
Yes as mentioned in the body, there’s a typo on the arXiv version that’s fixed as it’s going through peer-review.
mathematical ideals would not in general map to non-mathematical (physical) entities, hence why it’s a non-trivial constraint added to the formalism—essentially making a kind of an ‘avatar’ of the physical entity labeled within the formalism. There wouldn’t be any other way of referring to a non-abstract entity within an abstract formalism. In this case, I adopted a ZFC mathematics as it was convenient for the formalism, bringing me to:
No, I am not assuming that. For a start, much of logic would qualify and that wouldn’t necessarily be encompassed by mathematics (as Gödel showed) and many abstractions may have very little extra structure. But in this formalism, I choose a way of casting the problem and adopt a minimal amount of structure to add these constraints, it adopts a ZFC set theory.
Trying again, “non mathematical” means “not having mathematical existence”, rather than “mathematically indescribable”..? (Example of the former, electrons; example of the latter qualia, ghosts)
But then , why not just say
“matching mathematical ideas to physical entities”
Yes—fair enough — to clarify that, in that quoted sentence, when I said ‘non-mathematical entities’ I actually meant physical entities, rather than eg non-truths/falsehoods within mathematics (eg contradictory theorems and so forth).
That is a good pickup, and a good fix as I try to describe it in english language terms.
Your sentence is clearer.