There's a paper that stress how this transformation occurred well before in Germany, specially in Göttingen, named "Klein, Hilbert, and the Göttingen Mathematic

There's a paper that stress how this transformation occurred well before in Germany, specially in Göttingen, named "Klein, Hilbert, and the Göttingen Mathematical Tradition" by David Rowe. It makes a good point in how the changes initiated by Weierstrass and Cantor, by throwing away Geometry as a requirement to think in mathematics, ultimately lead the way to us to think in axioms and algorithms that's basically is how we do mathematics today. (The part about Cantor and Weierstrass is not stressed in the paper though).