Not to be too picky (am a mathematician too) but you ought to read about the history of the theory of "limit cycles" of planar vector fields and Dulac's "Theorem". Quite a lesson.

Edit: sorry, could not help writing this later (was on my iPhone before). The thing is that Dulac 'solved' an important part of one of Hilbert's problems. Was a 'proof' for about 60 years.