What does it mean that our language is linear and finite?

Question

In our everyday language, and in formal logic, our sentences and proofs are linear and finite. (Even if it has a tree structure, we usually write it down linearly on paper.) This seems to give a limit on what our logic can express. I can't say cleanly, but for example, the compactness theorem of first-order logic seems to be come from this kind of limitation. Do you have any ideas or reference for this?