Your logic is not correct here

You would be correct if you said:

"Isn't it true that any domain that can be described a formal system is also describable by mathematics."

If I have a friend Joe who is largely predictable then in certain situations, he is describable by a mathematical system (a logical system).

His full set of actions go beyond mathematics and if Joe realized how predictable he was, he might stop being so predictable.

Describable by mathematics does not mean "essentially equivalent" to mathematics.

And when Joe started doing something new, you'd just extend your formal system to include all of Joe's new behaviors. Given enough symbolic content, no matter what Joe comes up with, you can model it.

Just like we extended our formal number system into negatives, imaginaries, quaternions, etc.

Math is a terrifically abstract, self-consistent model of reality. But that's all it is: a model. Sometimes the model tells us things we didn't know before, and sometimes we have to change the model to make it work with what we're observing.

I don't think so. My assumption is that Joe is predictable by a mathematical equation. I find that this is only true of certain people and only true in certain circumstances.

People, in general, operate the opposite of computers. We don't think about what we should do, we think about we shouldn't do. So, it is very hard, if not impossible, to represent human behavior by a mathematical system.

This argument is explored in great detail in the book: Godel, Escher, Bach.

I think that's where the "restricted to domains where everyone agrees that there is a factual answer" comes in. (I'd like to have a system answer me if I really violated any traffic laws when I got a ticket last week, but I don't think that's going to be very easy...)