Q:

Can we rigorously define certainties of famous conjectures?

Is there some precise probabilistic framework to define levels of certainty of mathematical conjectures given current mathematical knowledge? If so, what would be the certainties attached to statements like the Riemann Hypothesis and $P\neq NP$?

A:

Very recently this problem was addressed with the notion of a logical inductor. The context for this work is artificial intelligence, decision theory, and economoics. Basically, Homo economicus is too rational even ignoring the usual critiques. If I'm a rational agent, and I was going to bet on the provability of the Riemann Hypothesis1, I would bet all my money for it or against it based on whether it was, in fact, provable which I would know because I'm perfectly rational. This is not only an unrealistic portrayal of people, but it is completely infeasible, even impossible, via any mechanism. It takes time and effort to actually deduce the logical consequences of some statements. So a slightly more realistic rational agent would have the worst-case of analysis paralysis possible.
Since for any statement that is provable or refutable in a fixed formal theory we can eventually deduce that, there needs to be some "computational limit" to get probabilities other than $0$ or $1$ for such statements. A logical inductor gives a sequence of probabilities for statements in some fixed formal theory. The logical inductor is defined with respect to a deductive process which is a monotonically growing sequence of (finite) sets of theorems. Conceptually, the deductive process would be an enumeration of all theorems deducible from the theory. For example, all theorems deducible with one step of inference, with two steps, ..., with $n$ steps. From the logical inductor criterion (see the paper), we can show that given an efficiently computable sequence of formulas, $\{\varphi_n\}_{n\in\mathbb N}$, that are all eventually provable, then given $\varepsilon\in\mathbb Q$ there is some $N$ such that for all $n > N$, $1-P_n(\varphi_n) < \varepsilon$ where $P_n$ is the probability assignment at step $n$. More generally, if $p_n$ is the fraction of formulas $\varphi_m$ for $m<n$ that are provable, a logical inductor will eventually converge on assigning probability $p_n$ to $\varphi_n$ at step $n$. Notice that, if $\varphi_n$ is only deduced at step $2^n$, say, then for large enough $n$, the logical inductor will believe $\varphi_n$ long before it is proven. Essentially what's happening is that a logical inductor will start to believe any statement that fits a(n efficiently computable) pattern of provable statements. When that pattern is legitimate, it will thus believe things far in advance of them being proven. If that pattern is spurious, it will eventually notice and correct its beliefs.
The good news is there is a computable implementation of a logical inductor and it's not trivial insofar as it can give you accurate probabilities well before a theorem has been deduced. Unfortunately, the algorithm provided is not practically efficient. More fundamentally, there is no way to calculate $N$ in general. On the one hand, this is no different from the fact that new information can cause one to dramatically alter their beliefs. On the other hand, it is only "in the limit" that the probability assignments are even coherent. For example, $P_n(\varphi\lor\psi)=1$ does not mean $P_n(\varphi)+P_n(\psi)=1$. It will, however, eventually recognize these patterns.
So you can run the algorithm or some other implementation of a logical inductor adding all theorems we've proven (in the chosen formal logic) as axioms so they don't need to be rediscovered. Run it until you get bored, then take the probability assignment at that time. There's no direct way to have it focus on the Riemann Hypothesis, though you can ask for the probability it assigns to that formula at any point.
1 Not to be confused with the statement that anyone will find a proof.

