It may be that certain theorems, when proved true, counterintuitively retard
progress in certain domains. Lloyd Trefethen provides two examples:
- Faber's Theorem on polynomial interpolation: Interpreted as saying that polynomial interpolants are useless, but they are quite useful if the function is Lipschitz-continuous.
- Squire's Theorem on hydrodynamic instability: Applies in the limit $t \to \infty$ but (nondimensional) $t$ is rarely more than $100$.
Trefethen, Lloyd N. "Inverse Yogiisms." Notices of the American Mathematical Society 63, no. 11 (2016).
Also: The Best Writing on Mathematics 2017 6 (2017): 28.
Google books link.
In my own experience, I have witnessed the several negative-result theorems
proved in
Marvin Minsky and Seymour A. Papert.
Perceptrons: An Introduction to Computational Geometry, 1969.
MIT Press.
impede progress in neural-net research for more than a decade.1
Q. What are other examples of theorems whose (correct) proofs (possibly temporarily)
suppressed research advancement in mathematical subfields?
1
Olazaran, Mikel. "A sociological study of the official history of the perceptrons controversy." Social Studies of Science 26, no. 3 (1996): 611-659.
Abstract: "[…]I devote particular attention to the proofs and arguments of Minsky and Papert, which were interpreted as showing that further progress in neural nets was not possible, and that this approach to AI had to be abandoned.[…]"
RG link.
Best Answer
I don't know the history, but I've heard it said that the realization that higher homotopy groups are abelian lead to people thinking the notion was useless for some time.