Dear Daniel,
I thought I was the only Brazilian here; it is good to know I'm not alone! I wish you "boa sorte"! And thanks for your comments. We have very similar ideas, and your comments really got into the point.
Natural language is our first tool, and it is necessary in every beginning of a whole new theory. First you have to do something handmade, just then you can build a machine to do the same. The same holds for using natural language.
In mathematics you can see it clearly. A new mathematical theory begins using a lot of natural language. At this point you say it is not formal. But then it becomes more and more formal and rigorous until the point it is completely formulated in formal language. Now, natural language is no longer necessary and we say that the theory is completely formal.
A similar idea holds for physics. But now, we are not interested in removing natural language, but giving a formal meaning to it. But the meaning of natural language statement is the corresponding mathematical statement. So, a language with formal meaning is equivalent to mathematical language!
How to do that? Simply using my concept of interpretation: the interpretation of a physical theory is what allow us "reading" its mathematical formalism. Therefore, the interpretation allows the translation from mathematical language to natural language, and vice versa. Then you can say the meaning of a sentence in natural language is the corresponding mathematical statement. Note that, doing this we also give a precise meaning to the used concepts.
The set of statements in natural language we can unambiguously translate into mathematical statements are the ones that have complete precision. They define a subset of natural language with complete precision! What I claim is that, every physical theory can define a subset of natural language which has absolute precision.
Once we find the closed formulation of our theories, all these problems are solved. For quantum mechanics, it will not only provide the right quantum logic, but also the right natural language, and concepts to talk about the theory! This will propose a new logic, and also a new conceptual framework. But then, the theory is proposing the conceptual framework in which it can be understood!
I've been working on these questions, and I've proposed a new formulation and interpretation of QM that follows these guidelines. My approach provides anew logic for QM, but it is different from the original quantum logic school. In my manuscript, each result of the theory is formulated in mathematical language and in natural language, and one is simply the translation of the other!
Our classical worldview is not useless; it is just not good enough for some things. That is, the concepts of our worldview are open concepts, and most of them are not very clear. For example, what is reality? what is truth? And I believe that physics has the potential for giving these fundamental concepts a precise meaning! And this is exactly making our worldview more closed!
I don't know much about Mach's philosophy, but this is exactly my view. For me, the meaning of the statement ''the position of an object is (x,y,z)'' would be the corresponding mathematical equation in the formalism of some physical theory.
That's great; you can see how making our worldview semantically satisfactory can be important for physics. I'm doing the same with quantum mechanics, but actually I'm working on a theory that is the basis for both quantum and classical mechanics. And you are right, in a sense; our classical worldview appears in a particular case of this theory.
I'm not sure I've answered all your points, so fell free to ask anything again...
Best regards,
Frederico