there is a certain style of type of proof in mathematics something like a "barrier theorem" but which also relates to widespread mathematical beliefs/ "conventional wisdom".
- an example would be the proof that trisecting an angle is impossible.
- another was the historical belief (but not proof) that the parallel postulate was derivable from other geometric postulates, ie here the "barrier" is the previously widely-held idea that no other consistent geometry is possible without parallel lines.
- another case that comes to mind is Von Neumann's proof of the impossibility of local hidden variables in quantum mechanics, later overturned in its particulars but extended further by Bell.
- yet another old example is the discovery that $\sqrt{2}$ is irrational by Hippasus (supposedly leading to his sentence of drowning from a boat by Pythagoreans according to one legend).
- there was some shock at the time at the proof that the quintic polynomial is not solvable by algebra
- the discovery of the undecidability of Hilberts 10th problem after decades of investigation shattered Hilberts own program at the turn of the 20th century of maximally extending provability.
etc! these barrier proofs or beliefs are sometimes very complex or deep seated in mathematical thinking. however, they make assumptions or impose conditions that sound reasonable but later are understood to be quite subtle and sometimes new discoveries are made that somewhat defy expectation wrt the "known barriers". barrier theorems are also important in TCS where several have been built wrt potential P vs NP proofs. (e.g. RJ Lipton summarizes some here, see also this cstheory question by Kintali.)
What are other key cases in mathematics of barrier theorems or beliefs that were later overturned, sometimes dramatically?
Are there references that study this phenomenon of reversal broadly/ in general?