André Weil's likening his research to the quest to decipher the Rosetta Stone (see this letter to his sister) continues to inspire contemporary mathematicians, such as Edward Frenkel in Gauge Theory and Langlands Duality.
Remember that Weil's three 'languages' were: the 'Riemannian' theory of algebraic functions; the 'Galoisian' theory of algebraic functions over a Galois field; the 'arithmetic' theory of algebraic numbers. His rationale was the desire to bridge the gap between the arithmetic and the Riemannian, using the 'Galoisian' curve-over-finite-field column as the best intermediary, so as to transfer constructions from one side to the other. (See also 'De la métaphysique aux mathématiques' 1960, in volume II of his Collected Works.)
That fitted rather neatly with demotic Egyptian mediating between priestly Egyptian (hieroglyphs) and ordinary Greek on the real Rosetta Stone. But just as one might have expressed that text in a range of other contemporary languages - Sanskrit, Aramaic, Old Latin, why should there not be other columns in Weil's story? Frankel himself adds a fourth column (p. 11) 'Quantum Physics'.
So now the questions:
Are there other candidate languages for Weil's stone? Might there be a further language for which we would need intermediaries back to the arithmetic? Could there be a meta-viewpoint which determines all possible such languages.
Presumably the possession of a zeta function is too weak a condition as that would allow the language of dynamical systems.