Gromov just posted on his page the paper In a Search for a Structure, Part 1: On Entropy. June 19, 2012. With much interest I started to read it and my first impression is that I have seen something like this before (but I might be wrong, please excuse me if so) in the HUGE last paper (book)
by Jean-Marie Souriau, the inventor of symplectic geometry and geometric quantization, among others.
Specifically, I refer to the way Souriau treats probabilities in “CLE 8: Calcul des Hasards”, p. 209.
The book is a must-read! It is a summum of Souriau mathematical view of Nature, specifically concerning symmetry, entropy, relativity and quantum mechanics.
Gromov, with his enormous geometrical knowledge (different than Souriau’ though) points to sofic groups, this I need a lot of time to understand.
UPDATE: I am starting to understand the sofic group notion of Gromov and learning to appreciate it, it’s related to constructions with approximate groups, apparently.