Brite - One explanation in favour of Mathematics as the medium of communication could be - The Universe has a common origin, the Big Bang, the laws of physics are the same everywhere. If the extraterrestrials are sufficiently intelligent, they would have discovered the same laws governing our world and Universe, as their own. Mathematical equations would have to be the same. No translation required. If you show the alien, Newton's universal gravitation equation, the alien would immediately recognize it -- instant communication. Nature is best described by calculus and differential equations. Everything in nature from the atom to galaxies can be described by the appropriate set of differential equations. Newton's second law, F = ma is one example of a differential equation. The space-time fabric of the Universe itself, is made of number.