Testing the philosophy of mathematics in the history of mathematics

Recent philosophical accounts of mathematics increasingly focus on the quasi-Empirical rather than the formal aspects of the field, The praxis of how mathematics is done rather than the idealized logical structure and foundations of the theory. The ultimate test of any philosophy of mathematics, However idealized, Is its ability to account adequately for the factual development of the subject in real time. As a text case, The works and views of felix klein (1849-1925) were studied. Major advances in mathematics turn out to be most adequately understood as shifts to new conceptualizations at different levels of idealization and abstraction. The implications of the model are explored with special reference to modern philosophical views of the nature and the warrants of mathematical claims to knowledge, And the methodology of the development of this knowledge