Just like if you look at things closely enough you see that stuff is really mostly nothing - e.g. [link|http://www.lbl.gov/Science-Articles/Archive/scanning-tunneling-microscopes.html|I'm not touching this keyboard], the electron wave functions that are part of "me" are repelling the electron wave functions of this keyboard thingy. Rutherford showed that atoms are [link|http://www.chemsoc.org/timeline/pages/1911.html|mostly empty space].
You're right that it's hard, and perhaps impossible, to construct a logical framework that doesn't involve some basic assumptions. An important one in our understanding of the universe is that the same physical laws apply everywhere. We have no way to prove that, but assume it to construct a framework to understand our surroundings. So far, it has been a pretty good assumption.
As you say, it's important to realize the limitations of our understanding, but to argue (as some - not here - do) that because we can't prove everything we can't prove anything is sophistry.