There also seems to be confusion about what Music Theory actually is - it's (usually!) not an attempt to axiomize music, rather it's an attempt to take a musical corpus and explain how pieces in it tend to work (which is why one sees different music theory text books for Jazz and Classical, for example).
I think they try to address this in 1.3 but it comes out more as a complaining and misunderstanding. People get way too hung up on the term "music theory". I have friends who are professional musicians who refuse to learn music theory. Not because they don't like learning it, or they think it stifles their creativity (which are fine reasons to not learn it, you don't have to). But they refuse because they "don't believe in it" or "my music doesn't fit the definition of music theory". Which is totally incorrect, because like you said, music theory is an attempt to explain existing pieces and how they tie together. Not describe some magical pre-existing rules created by our ears and the universe.
It's like learning the grammatical rules behind a natural language. English isn't carved into a mountain somewhere, or programmed into our DNA, it's entirely made up by people, and you're free to break all the rules. But then you're unlikely to be understood. And if you are understood, we'll probably end up giving a name to whatever you did!
English is not, but our perception (the automatic processing of cognitive input to build context) which is essential to language learning/use, is programmed into our DNA.
This is fine as far as it goes, but music has moved on a lot since then and no one - not even film composers - write in that style any more.
So you have the idea of a music theory - which is the set of elements that defines a style - and then you have one particular music theory, which isn't used any more, although some of the ideas are still creatively useful.
Not many people understand that hip hop, dubstep, deep house, death metal, jazz, and shoegaze all have unique theories of their own.
They're defined by elements - some of which are simple cliches, and some of which are more creative - that make the style recognisable.
If you know the elements and how to put them together, you know the style. Simple as.
Classical book theory gives you a small start towards understanding how this works, if you want to take it. But you have to do the rest yourself.
A lot of producers don't bother. They learn one style intuitively, and that's all they know. Other producers want more. Neither has a monopoly on great music.
Does axiomising music help? It may give you a few hints about harmony. But that's all.
It's not even close to being a complete description of classical theory, and it completely fails to say anything useful about the theories of more modern pop styles.
The math part is simply irrelevant to most musicians. Musicians use chords and lines as expressive tools, not as theorems. Knowing where frequency ratios come from is kind of interesting, but there's a lot more going on in music. Trying to mathify it won't get you closer to that.
Nevertheless, I think math can be very useful to describe the various music styles, elements that make up these styles, and the ways to compose these elements. That would be using math in a descriptive way. I like for example the 2 book series "Musimathics" (http://www.musimathics.com/). In the end, nobody knows how much of music can be understood by viewing it through the mathematical lens. I suspect quite a lot, if not all.
Fine, but regardless, there don't seem to be any music theory books that start with the very beginnings (as this submission is trying to). I share the author's deep frustration with existing music theory books: they all assume some magical vocabulary as though you're just supposed to know what they mean already and how dare you question what 'diatonic' means! Fuck that shit.
I teach a class on sound, of which music theory plays a part, and I start with raw sound and human perception: consonance and dissonance. This is the only way you can ever understand WHY there are music theories at all (and don't get me started on how most music theory textbooks don't even MENTION the history of tuning systems. equal temperament ONLY makes sense once you consider the historic alternatives, like Pythagorean tuning).
So I absolutely applaud this effort. This will be a huge help to music students of all ages who aren't comfortable with just accepting a bunch of mystical (and exclusively western) 'wisdom' from on high.
Music theory is as easy (and as difficult) as trig. Learn the language. Learn the axioms. Understand the assumptions, please.
And also understand that those assumptions can change at any time, depending on the musical period you're looking at!