For Galileo, mathematics was very important because nature itself was mathematically ordered (nature understood as a great book written in the language of mathematics). As for the scientific method, in Newton’s use of mathematics is the same as in Galileo’s. The philosophical view is partially different: Newton thinks that mathematics is a tool that can be used by man to understand reality.
He makes a comparison: mathematics is like a dress made by a tailor for someone; so mathematics is a tool we use to dress up nature: the scientist uses the right mathematics to understand nature.
Galileo thinks that it is God who created the mathematical world, while Newton thinks that nature is like this, it is what it is, we do not know why it is so but, regardless of whether God created the world, we adapt mathematical tools to nature , it is we who must adapt to situations.
Newton is also very important in mathematics because he fixed the infinitesimal calculus (discovered in the 17th century by various authors, but fixed by Newton and Leibniz) to study physical phenomena.
In Principia Newton considers the 3 principles of mechanics (principle of inertia, F = m.a, principle of action and reaction) as axioms, then gives definitions and from these he deduces all the physical laws (including that of gravitation). The procedure is similar to that of Euclid and Spinoza, but this analogy must not mislead us, there is a basic difference: Euclid’s axioms are evident (the whole is greater than the part), instead the 3 principles of mechanics are not not at all obvious, but the result of experimental research. So Newton uses the axiomatic-deductive method, the 3 principles are the result of continuous study and continuous research (for this reason it can be said that he practiced, like Galileo, the interaction between sensible experiences and necessary demonstrations). Despite the different method of exposition, Newton is more like Galileo than Descartes. The same applies to Newton as to Galileo, namely that science presupposes both sensible experiences and the necessary demonstrations.
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