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Do triangles exist? Is the definition of triangle "bereft of any definition that can be objectively measured"?Certainly not, the definition of a triangle is quite a rigorously defined mathematical construct.
> Is the Ship of Theseus fundamentally different than a triangle, or is it merely a difference of degree?
It is not a difference of quantity but quality. A triangle is an objectively defined, mathematical concept; identities are as I said “You know it when you see it, and different men see it differently.”. — you will not find many mathematicians disagreeing on what is, and is not a triangle.
Another fundamental difference is that a triangle is an abstract reasoning tool, not an empirical one, and no mathematician would suggest that real life objects are, or are not, triangles.
> Ask any random human to define some extremely complex mathematical entity, and you'll get various fuzzy answers.
A lay human being with no mathematical understanding perhaps, but all of these entities have been defined rigorously in mathematical terms.
> Does that mean complex math doesn't really exist?
You were the one who speaks in such terms as “existence”. I will not have you put words in my mouth.
All I said is that one has an objective, rigorous definition such that you will find that every single educated man in the field agrees on what is, and what is not a triangle, such that a machine proof assistant can be fed this definition, and verify machine-level proofs that reason about them, and that the other lacks this such that you find that nigh any man, expert or not, has a different definition of what constitutes the identity of a physical object, and where a new object starts.