First Contact and Language as a Way of Knowing

I caught the tail end of today's episode of This American Life, in which Ira Glass speaks to Paul Davies, who is the go-to guy for figuring out how to communicate with aliens. And his answer (you guessed it) is that we should communicate mathematics (quantum mechanics) via binary code.

So when we are thinking about knowledge, there is a question about the link between knowledge and language - if we can't express it, do we know it? Put it on an intergalactic scale, wherein the best minds figure the only way to communicate in a first-contact situation is a way I personally cannot communicate. In that sense, what do I as an individual human really know? Nothing?

So perhaps the link between language and knowledge is overblown. Learning new languages, we may find ourselves provided with better nuances we can use to express with precision the knowledge we have, but does this make language an actual way of knowing? Is language nothing more than a way of sharing context? A system of symbols for putting us on the same page?

Is the feeling of being unable to put your finger on an idea a lack of knowledge or a lack of language for expressing knowledge - thereby blocking the knowledge from actualization and utilization? Is it one and the same? Is intuition knowledge without language?

(In reading over this post, my dependence on metaphor and idiom to express my ideas strikes me as pertinent, insofar as metaphor and idiom are so language- and even culture-specific, and so difficult to translate.)