Sense 1) “‘Actual’ mathematics does not consist in notational embodiments”
‘Actual’ as in essence rather than appearance, territory rather than map. Implies platonic reality of an eternal ephemeral mathematics. Conceived of as beyond and beneath all the particulars of any given representation.
Sense 2) “‘Actual’ mathematics consists in notational embodiments”
‘Actual’ as in embodied. Material. Implies mathematics is the set of mathematics which has been ‘actually’ done.
You describe the second sense as self evident but with mathematics in particular this is bound to cause issue.
I find it hard to even write the description of the second sense without tying some linguistic knot. If ‘actual’ mathematics is its embodiments, what is the thing being embodied? Not actual. So the inscriptions characterise a non existence. What does it mean to describe the characteristics of a thing which doesn’t exist?
From another angle, what does it mean for something to ‘materially exist’ so that it can be considered ‘actual’. I think this concept so nebulous as to make impossible the patching up of all the holes a proper investigation would reveal.
Of course ‘material existence’ is used and understood perfectly well in ordinary circumstances but when the topic at hand is “what counts as ‘actual’ mathematics?” it seems a bold post to tie yourself to.
Amusingly, the one thing that the word "actual" does not permit is lead the conversation to that "extremely interesting question": the word "actual" forces an interlocutor to consider only those contexts in which something is understood as real and existing.
In this case, that would be modern maths notations. So this is kind of artificial nitpicking: English is not a zero-context language, you are still required and expected to understand the context words are used in based on understanding that they wouldn't make a lot of sense in other -even related- contexts.
Of course, if I'd been glib and said this:
> So... can I write, and see, maths?
Then the insinuating that maths and programming are mutually exclusive concepts should definitely lead to a philosophical discussion.