My original wording was confusing. I fixed it.
The same thing applies.
All of this becomes very clear if you learn about A.I., Frank. If you are interested in this issue, and know a little computer science, you should pick up a copy of the book "Artificial Intelligence: A Modern Approach" and read through it. Or you can go to the wikipedia artificial intelligence portal but personally I find it much less accessible than that book.
At a fundamental level the primary task of an A.I. programmer is to find ways for an agent to describe the environment state it cares about. For trivial A.I. this can be just a bunch of static data. But as the complexity of an agent's desired behavior increases, it becomes less and less viable to take a brute force approach and try to account for all possibilities ahead of time. You need to think up ways to allow the agent to learn about not only things in the world but also
relationships between things in the world.
And at this point things really start to get interesting, and an educated observer might start to see parallels between the patterns of information processing and the way humans think.
For instance there are entire branches of A.I. dedicated to the data structures and algorithms of logical inference -- reasoning. Did you think reasoning was something only people do? Very wrong -- we have known how program machines to do it for decades. The key is finding ways to reduce logic -- the
relationships between things in the real world -- to the simplest representations and steps possible, things that a computer can deal with.
http://en.wikipedia.org/wiki/Automated_reasoning
And that is the kind of stuff that answers your question -- how can a formal language itself be formalized, or rather, how can the very concept of language itself be formalized?
You can get a high level understanding of this and many other topics that are both fascinating and directly relevant to this discussion if you take a little time to read up on A.I.
Although intuitively it should be clear that you can formalize the idea of language using language because you can do it in English: it isn't hard to describe to someone what language is, is it? Linguists do it all the time.