I previously asked whether abstract objects can be split into categories, groups or sets of their component parts, and was told, definitely, and later another question occurred to me, take an expression like '2x', I asked myself, what is an expression? Is it simply a set and arrangement of symbols or is it a 'meaning' that can be understood from seeing the set of symbols? Is it in a way both?
For example for the set of symbols we can 'associate' a meaning, and maybe both can be 'associated' through a third entity that links them, perhaps this is an 'expression'? In this case it's not a wild jump to suggest the 'expression' is actually a set containing two abstract objects, the visual information from the symbols, and the mathematical 'meaning' itself? I found that simply 'associating' the expression with it's meaning and set of symbols gave me what felt like an exactly similar structure to before without any language like 'set' or 'collection'.
This led me to ask, what is the difference with abstract objects, with being a member of a set or collection and being simply 'associated' with the object, and the other members?
After all, unlike a real collection, I can't group these all together as one, or glue them together and make a single object.