Voted up because this is a great topic that I'd like us to try and begin to tackle.

But this post really frustrating to try to respond to. Not because it is especially wrong-headed or poorly written but just because it is a little hard for me to find my way around your theory. It is difficult to find a point of traction. In general, I suspect it just isn't really solving problems but eliding distinctions and ignoring problems (just based on what I do know and the relative shortness of this compared to most other work in philosophy of math). This is pretty much the way I feel about what Eliezer has said on the subject and just about every single thought I've ever had on the subject. I'm also not sure I'm familiar enough with the subject area to be able to examine this post in the way it requires.

So I suspect I'll end up prodding you in a couple places but to begin with: what exactly do you take the Platonist thesis to be? If there is an analogical relationship between a particular expression in our system of inscriptions and our rules for manipulating them (i.e. a written equation) and a physical system (i.e. a system that equation describes) that seems to suggest an underlying structure which is instantiated in both the mathematical expression and the physical system. That such structures exist independently of the mind strikes me as a platonist position. What exactly is wrong with that position? Or what even did you say to contradict it?

Perhaps we need to have a discussion about abstract objects in general before tackling the math.

I do think you're right about the map-territory confusions here. They definitely abound.