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Newton, who discovered the laws of rotation of celestial bodies, made several revolutions in philosophy at once. First, he showed that it is possible to know the heavenly things at all. Secondly, he showed that it is possible to know the heavenly things without trying to give a closed explanation. Gravity is not explained by anything, it just exists. Third, he showed that the laws of celestial bodies and terrestrial ones are the same.
Enlightenment grows out of this – out of a belief in the human mind and its ability to know everything rationally, and therefore society as well. Just as Galileo and Newton redefined Aristotle's physics, so Locke and Montesquieu redefine his politics – ” if Newton could do it, we can do it too.”
Kant's question “how are synthetic judgments possible a priori” is an attempt to analyze Newton, synthetic judgments a priori are Newton's laws.
Darwin, having discovered evolution, showed a new possible principle for constructing historical reasoning, which later begins to be applied everywhere. Society or the universe does not evolve according to Darwin, but it is likely that without Darwin, no one would try to talk about the evolution of society or the universe, that such complex systems can change and self-complicate over time.
Popper actually builds his own version of scientific methodology on the basis of Einstein.
An important but subtle result of general relativity is the discovery that the theory may coincide with observations, but be incorrect. And despite the infidelity, keep working at a certain scale. From this it follows that Popper replaces the claim to truth with plausibility. The theories are not true, they are only “not yet refuted”, but anything can happen.
And from the scientific methodology, Popper deduces the social one, building on similar principles the “open society” as a new version of liberalism, freed from the utopian ideas of the Enlightenment.
And he, relying on Darwin, deduces evolutionary epistemology as an alternative way to answer the question of how cognition is possible. So far, the answers have been limited to the divine gift (excluding Kant, perhaps, but I just don't know him well).
Einstein and quantum mechanics show the importance of relativity, uncertainty, and the subject in cognition.
However, I would hesitate to try to establish causality here – they gave rise to a fashion for relativity and uncertainty, or their concepts became possible, because there was a general mood of loss of faith in the possibility of a single universal objective truth.
This is all at the top, probably in the 20th century there are many more cases when philosophers were inspired by natural sciences, but the richer I am, the more I share.