In the sense that “every mathematical structure exists”, the concept of “existence” is trivial, as from it follows every “structure”, which is after a fashion a definition of inconsistency (and so seems to be fair game for informal use of the term). Of course, “existence” often refers to much more meaningful “existence in the same world”, with reasonably constrained senses of “world”.
An ensemble-type definition of existence seems more like an attempt to generalize the term than it does an empirical statement of fact. What would it even mean for a mathematical structure to not exist?
Wha?
In the sense that “every mathematical structure exists”, the concept of “existence” is trivial, as from it follows every “structure”, which is after a fashion a definition of inconsistency (and so seems to be fair game for informal use of the term). Of course, “existence” often refers to much more meaningful “existence in the same world”, with reasonably constrained senses of “world”.
How do you know that?
An ensemble-type definition of existence seems more like an attempt to generalize the term than it does an empirical statement of fact. What would it even mean for a mathematical structure to not exist?