And from there, assuming that the alien shares the buried assumptions within the math and understands mathematical relations and principles in ways that are mutually comprehensible with ours, one might build a base of mathematical communication, and that might be interesting.
They live in the same Universe that we do, and they'll likely have to devise the same math for describing it.
They could even be motivated by considering their own thoughts. One thought, two thoughts, three thoughts, ...
But how do you get from there to, "I'm hungry," or "You're standing in my spot," or "That's a beautiful sunrise," or "I believe in truth, justice, and liberty"? The things that make communication tricky between fellow humans are things that can't be boiled down to math, or derived from it.
I shall continue. The next step will be to do the parts of physical science (physics, chemistry, astronomy, geology, ...) that are universally shared or at least can be observed by both us and them.
The message would contain pictures of stars and galaxies, and would explain in cartoon form what a star is and what its composition is. To give further hints, one would send over the Periodic Table of Elements, depicting each element's electron shell filling in cartoon form. One would also show how atoms combine into molecules, by cartoons showing electrons shared between atoms. For even more hints, one would send the frequencies / wavelengths of various spectral lines and the atoms and molecules that generate them. Spectral lines of both stars and the interstellar medium.
Once one gets the idea of chemical elements across, one can look smaller and do that with atomic nuclei and elementary particles. One can also look bigger and get across a lot of macroscopic features:
Temperature, solid, liquid, gas, electrical conduction, magnetic properties, light, ..
Also features of planets. Though their planetary system or known planetary systems won't necessarily be very close to what our Solar System has, they'd likely have a good idea of what a planet is.
This seems like a lot of song and dance, but you won't be able to take a glass of water and point to its contents and say "That is water". Instead, you'd have to say "The stars are mostly composed of the first two entries in this table, and two of the first entry and one of the eighth entry combine to make water."
