Proofs apply to mathematics but do not apply to science, i.e. any scientific theory is subject to falsification.
In Life Itself Rosen uses the formality of mathematics both in proving why the Newtonian model is inadequate for biology and then in constructing a relational model which is adequate for biology.
Because the book like the author has one foot on each side, I put the word proof in quotes.
Does Rosen explicitly and formally posit the existence of life, specify exactly what differentiates living from non-living, or address the origin of the substance of the organisms he's investigating?