Recently I've become interested in the history of the way people think about mathematics. What I mean by this is for example how Godels proof basically put an end to the whole school of thought proposed by Bertrand Russel.
Now I wonder if there are any similar 'schools of thought' in current mathematics. And if there are any good sources on the complete history of how all those views and schools of thought came to be and how they ended.
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It sounds like you are interested in what is often called philosophy of mathematics. While there are many books and articles on this subject out there, I can personally recommend Shapiro's Thinking About Mathematics, which takes the reader through an introduction to both the history of the philosophy of mathematics, as well as more recent ideas and schools of thought.
For the "foundational" debate of the '30s, see :
For some impact on mathematics of one of the above philosophical "schools" (e.g. Intuitionism and related : Intuitionistic Logic, The Development of Intuitionistic Logic, Luitzen Egbertus Jan Brouwer, Set Theory: Constructive and Intuitionistic ZF) see Constructive Mathematics and :
It is also worth noting a "connection" between Intuitionism/Constructivism, Alonzo Church's Lambda Calculus and N.G. de Bruijn's Automath (and see here), a formal language "precursor" of proof assistants.