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Richard Feynman - Certainly you are joking. I would suggest the biography of J D R Tata: beyond the blue mountains by R M Lala. He is not a physicist but it a good biography


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I found a useful reference that might help answering this question (this doesn't constitute an answer to the questions in post). The reference is from p. 46-47 of the book "From Riemann to Differential Geometry and Relativity" - in those pages the author cites a 1863 letter from Enrico Betti to another mathematician: What gave Riemann the idea of ...


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There is a nice book called Princeton Companion to Mathematics (written by many authors, edited by Gowers) which explains many parts of modern mathematics, and has a chapter on development of principal concepts.


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In Gert Schubring's translation from 2016, the exact statement is: "Every person believes that he knows what a curve is until he has learned so much mathematics that the countless possible abnormalities confuse him." (p. 199).


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Your question is too broad, I think. One nice scholarly book (which covers only the changes around 1850 to 1910, as mathematics went from a mishmash of physics and psychology, and became an abstract deductive science): Jeremy Gray, Plato's Ghost: The Modernist Transformation of Mathematics, Princeton University Press, 2008


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I will skip the pre-history of solving polynomial equations and factoring polynomials. Let me mention that the analogy between long division of numbers and polynomials goes back to medieval Islamic mathematician al-Samawal, see Who invented short and long division?, and the Euclidean algorithm for polynomials was optimized by Hudde, a younger contemporary of ...


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