I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics.
I'm going to focus the post (and modulate my genuine idea) about an aspect that I think can be discussed here from an historical and mathematical point of view, according to the following:
Question. I would like to know what are examples of remarkable achievements (in your research subject or another that you know) that arose against the general view/work of the mathematical community since the year 1900 up to the year 1975. Refer the literature if you need it. Many thanks.
An example is the mention that the author of [2] (as I interpret it) about Lennart Carleson and a conjecture due to Lusin in the second paragraph of page 671 (the article is in Spanish).
Your answer can refer to (for the research of pure or applied mathematics, and mathematical physics) unexpected proofs of old unsolved problems, surprising examples or counterexamples, approaches or mathematical methods that defied the contemporary (ordinary, mainstream) approaches, incredible modulizations solving difficult problems,… all these in the context of the question that is: the proponents/teams of these solutions and ideas swam against the work of the contemporary mathematics that they knew at the time.
*You can refer to the literature for the statements of the theorems, examples, methods,… if you need it. Also from my side it is welcome if you want to add some of your own historical remarks about the mathematical context concerning the answer that you provide us: that's historical remarks (if there is some philosophical issue also) emphasizing why the novelty work of the mathematician that you evoke was swimming against the tide of the contemporary ideas of those years.
References:
[1] Roger Penrose, Fashion, Faith, and Fantasy in the New Physics of the Universe, Princeton University Press (2016).
[2] Javier Duoandikoetxea, 200 años de convergencia de las series de Fourier, La Gaceta de la Real Sociedad Matematica Española, Vol. 10, Nº 3, (2007), pages 651-677.
Best Answer
After mathematicians had been been taught for decades that a consistent theory of the calculus based on infinitesimals was impossible, Abraham Robinson was certainly swimming against the tide when he proved otherwise.
Robinson, A. (1961): Non-standard analysis, Indagationes Mathematicae 23, pp. 432-440.
Robinson, A. (1966): Non-standard Analysis, North-Holland Publishing Company, Amsterdam.