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Abstract: Galileo isn't really remembered for his mathematics.
There is nothing called "Galileo's Theorem," for instance.
But Galileo did make a fundamental contribution to mathematics,
arguably more important than any new theorem, namely a new (or re-discovered)
conception of what mathematics could mean. In the decades before
Galileo, higher mathematics was an essentially static and obscure
corner of philosophy, barely connected to physical reality.
After Galileo, mathematics became the scaffolding
of physics, and (apparently as a consequence) subject to rapid development.
This revolution in Galileo's thought, and in the philosophy of
mathematics more generally, had to come from outside mathematics:
in Galileo's case it had its roots in literature, the arts, and quite
possibly the theology of the High Middle Ages.