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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.

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  • Is association in the mind? See Wiki's Association.
    – Mauro ALLEGRANZA
    Oct 17, 2022 at 17:02
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    In mathematics the "associating" is done by "(binary) relations". However, the relation of belonging to the same set is only a very special case of general relations that "associate" objects. The verb is more commonly used when there is a map that "associates" images to their pre-images, e.g. the determinant map "associates" numbers to square matrices. Mapping "expressions" to their "meanings" is called "interpretation" in formal logic, your '2x' will be interpreted as a function.
    – Conifold
    Oct 18, 2022 at 7:13

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Definitions map symbols to ideas. The symbol is the symbol, the idea is the idea, and the definition is the definition. None of the three is identical with the other.

Ideas may or may not model processes external to the mind, and the model, if it exists, may be a good one (it generates predictions that comport well with measurements) or not.

You could certainly define a new symbol, such that the definition maps the new symbol to the idea of the set of the old symbol and the old idea. However, this is a new symbol, a new definition, and a new idea.

We could do this in infinite recursion if we wanted: a symbol that represents the idea of [a symbol and an idea of [a symbol and an idea of [a symbol and an idea of...]]]

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[W]hat 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?

It's both. The symbols are generally interpreted as syntax. The meaning they lead to is semantics.

[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'

No need to reinvent the wheel; you can use the triangle of reference which includes the symbol, reference, and referent. Another German idea was that of sense and reference.

[W]hat 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?

Well, an overview of abstract objects can be read here at SEP. However, if you are looking for an explanation for the intuitive notion, it's possible to see a name as representing 1 or more things. In math for instance, x can be understood to be a distinct natural, say x:=5, or as any member of the set of natural numbers written x∈ℕ.

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