Oftentimes open problems will have some evidence which leads to a prevailing opinion that a certain proposition, $P$, is true. However, more evidence is discovered, which might lead to a consensus that $\neg P$ is true. In both cases the evidence is not simply a "gut" feeling but is grounded in some heuristic justification.


Some examples that come to mind:


* Because many decision problems, such as [graph non-isomorphism](https://en.wikipedia.org/wiki/Graph_isomorphism), have nice *probabilistic* protocols, i.e. they are in $\mathsf{AM}$, but are not known to have certificates in $\mathsf{NP}$, a reasonable conjecture was that $\mathsf{NP}\subset\mathsf{AM}$. However, based on the conjectured existence of strong-enough *pseudorandom* number generators, a reasonable statement nowadays is that $\mathsf{NP}=\mathsf{AM}$, etc.
* I learned from [Andrew Booker](https://www.youtube.com/watch?v=ASoz_NuIvP0) that opinions of the number of solutions of $x^3+y^3+z^3=k$ with $(x,y,z)\in \mathbb{Z}^3$ have varied, especially after some heuristics from Heath-Brown. It is reasonable to state that **most** $k$ have an **infinite** number of solutions.
* Numerical evidence suggests that for all $x$, $y$, we have $\pi(x+y)\leq \pi(x)+\pi(y)$. This is commonly known as the "second Hardy-Littlewood Conjecture". See also [this](https://math.stackexchange.com/questions/1072194/source-of-hardy-littlewoods-2nd-conjecture/1073846) MSF question. However, a 1974 paper [showed](https://projecteuclid.org/euclid.bams/1183535510) that this conjecture is incompatible with the other, more likely first conjecture of Hardy and Littlewood.
* Number theory may also be littered with other such examples.


I'm interested if it has ever happened whether the process has ever *repeated* itself. That is:



> 
> Have there ever been situations wherein it is reasonable to suppose $P$, then, after some heuristic analysis, it is reasonable to supposed $\neg P$, then, after *further* consideration, it is reasonable to suppose $P$?
> 
> 
> 


I have [read](https://www.ias.edu/ideas/2011/kennedy-continuum-hypothesis) that Cantor thought the Continuum Hypothesis is true, then he thought it was false, then he gave up.



Have the tides ever turned twice on any open problem?
I think that originally there was a belief (at least on the part of some mathematicians) that for an elliptic curve $E/\mathbb{Q}$, both the size of the torsion subgroup of $E(\mathbb Q)$ and its rank were bounded independently of $E$. The former is true, and a famous theorem of Mazur. But then as curves of higher and higher rank were constructed (cf. [What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?](https://mathoverflow.net/questions/38356/what-heuristic-evidence-is-there-concerning-the-unboundedness-or-boundedness-of/50868#50868)), the consensus became that there was no bound for the rank. But recently there have been heuristic arguments that have convinced many people that the correct conjecture is that there is a uniform bound for the rank. Indeed, something like: **Conjecture** There are only finitely many $E/\mathbb{Q}$ for which the rank of $E(\mathbb{Q})$ exceed 21. (Although there is one example of an elliptic curve of rank 28 due to Elkies.)


