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According to Galileo "the world is written in the language of mathematics," and a natural philosopher must learn to read it. How did this approach differ from Descartes' notion of a mathematical universe?

Here's what I've come across so far-

Galileo

  • Invented the telescope, looked at planets, and for the first time, there was hardcore visual proof that not all astronomical bodies orbited the Earth.
  • Got branded a heretic (saved himself by recanting) for it, so Descartes stood down with his ideas since he feared that he'd have to go through the same process.
  • Was slightly accepted after Descartes laid down that God, being a perfect creature, would never try to deceive us, so we can trust our senses. So Galileo trusted his senses and hence trusted what he saw through the telescope.
  • Galileo was more focused on coming up with the math to solve math and physics problems.

Descartes

  • Invented the Cartesian coordinate system and the analytical geometry that we have right now.
  • Descartes believed that Mathematics was the only certain thing in the universe, hence it could be used to reason things out.
  • Descartes, unlike Galileo, wanted to develop math so that he could reach any truth whatsoever.
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You have the difference in the last bullet points of your two lists. Galileo was an experimental scientist, engineer first - math for him was the most comfortable tool to describe the nature's phenomena he studied. From his works, it seems that "why" was less important than "how" for him. Also, note that Galileo's mathematical methods were not very different from the ones used by his peers. Descartes' position, on the other hand, is better captured in his Wax Argument. The experimental results for him are secondary - the thing that captures the nature of the phenomena is the mind. His focus was on philosophy, not on natural science - and success in applying his works in natural sciences only reaffirmed this focus:

Thus, all Philosophy is like a tree, of which Metaphysics is the root, Physics the trunk, and all the other sciences the branches that grow out of this trunk, which are reduced to three principal, namely, Medicine, Mechanics, and Ethics. By the science of Morals, I understand the highest and most perfect which, presupposing an entire knowledge of the other sciences, is the last degree of wisdom.

Source

And there you have it - the classical example of an experimentator versus a theorist, a natural scientist versus a philosopher.

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The differences were best summed up in a Wikipedia article on France's Descartes:

Descartes laid the foundation for 17th-century continental rationalism, later advocated by Baruch Spinoza and Gottfried Leibniz, and opposed by the empiricist school of thought consisting of Hobbes, Locke, Berkeley, and Hume. Leibniz, Spinoza[16] and Descartes were all well-versed in mathematics as well as philosophy, and Descartes and Leibniz contributed greatly to science as well.

The article might have included Galileo with the English philosophers.

Followers of Descartes were "rational," and tended to have their actions dictated by a chain of reasoning. They saw mathematics as a "unifier" of science, from which the principles of science could be deduced; they approached science from a "top down" direction; theory first, then applications. The "Cartesian" coordinate system was a major step in this direction.

People like Galileo were more "empirical, that is, more likely to react intuitively to what their senses and data told them. They preferred to discover their science by "trial and error," (which is the approach Galileo took to religious issues as well, and drove the Catholic Church crazy). To people like Galileo, math was a tool with which to understand scientific discoveries, not a "framework" by which science was deduced.

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