I am interested in some reference or book (better an article or a collection of articles, so I don't feel bad if I don't finish a whole book) explaining the diverse lines of thought in mathematics of what constitutes a proof, for example (I know Lakatos' book and only that one). More concretely, if there is anything explaining how the Bourbaki school influenced the European way of seeing Mathematics (I don't know about other ways right now), that would be awesome. Anything would be very interesting to read!
Thank you very much.
The two examples that immediately come to mind is the Bourbaki school and the Russian school. The former emphasizes formal presentation along strictly logical lines (definition, theorem, proof, corollary, that sort of thing) and tends to reduce motivating discussions to a minimum. The Russian school as I see it (though this is based on purely anecdotal evidence) holds intuitions and ideas to be more important than theorems and proofs; the insight lurking behind the proof is the important thing. In conformity with this description, my teacher Misha Gromov once remarked that the only person to have benefited from the Bourbaki project is Pierre Deligne, who really enjoyed reading it from cover to cover (presumably he was able to fill in the missing intuitions and motivations through sheer brilliance). As far as the constructivist/classicist debate, you may enjoy a recent paper published in Intellectica; see here.