For me, particular syntax of a particular representation and implementation is one thing, the isomorphic implementation-free math describing what it represents is semantics.

I.e. A natural number is a list of successor relations. It directly maps to the reality of collecting a series of units, by actually being a series/list of units.

Whereas, a natural number represented in binary doesn't look at all like what it represents, but is an isomorphic representation. It behaves the same way.

And can be interpreted the same way, when it is being used for that purpose.

So: Series/list of steps = Semantics, for a syntactic implementation of binary natural numbers. Only in the right context, where the latter is mean to represent the former, does the isomorphism relate the syntax to the semantics.

In another context, a series of binary values might just be a random pattern we used to mark a bunch of things we think are related. In that case, the semantics would be completely different, even though the syntax is identical.

But I have no idea how other's think about these things.