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, 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 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 MSF question. However, a 1974 paper showed 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

furtherconsideration, it is reasonable to suppose $P$?

I have read that Cantor thought the Continuum Hypothesis is true, then he thought it was false, then he gave up.

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