How to understand Freeman Dyson's Saying:
After quantum mechanics, nature itself suddenly became linear.
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Until quantum mechanics, linear differential equations arising in physics were the result of approximation (linearization) of non-linear ones. So it was believed that the fundamental laws of nature are described by non-linear differential equations, while linear ones are only used to approximate them.
With the invention of quantum mechanics we know that the most fundamental equation of physics (Schrodinger equation) is a linear differential equation. And that the space of states of a physical system is a vector space (in classical mechanics it is a manifold, again a non-linear object). As QM is the most fundamental theory that we currently possess about the real world, the statement of Dyson is well justified.
QM is indeed the most fundamental theory: most physical theories are thought to be approximate. This is the only one that seems to be exact, at the present state of our knowledge.