Are mathematicians just not trying hard enough?
The Riemann hypothesis is one of the most important open problems in math. There's a $1 million prize from the Clay mathematics institute for a proof or disproof of the Riemann hypothesis. At the time of writing, it remains unsolved. From this, we may conclude that one cannot simply buy a solution to difficult mathematical problems.
Or could we? How do we know that buying a difficult maths proof is impossible? Perhaps the Clay mathematics institute is somehow not asking the question in the right way. And it's true that the value of a million dollars has been eroded over time by inflation. One might guess that a Riemann proof would be worth at least 100 million. Would that be enough to conjure it from the collective intelligence of humanity?
Simply directly declaring a $100 million reward for a solution would probably not work. For one thing, there's the issue of corollary-sniping where the prize wouldn't give anyone an incentive to publish solutions to hard intermediate steps of the problem, since the prize as a whole only goes to the one who solves the entire problem as a whole. For another, even the million dollar prize on its own would be plenty of reason for a money-motivated person to solve the problem if a solution were within their grasp. The issue is not merely one of funding, we humans are somehow failing to combine our efforts properly.
Prediction markets are pretty cool
One of the standard ways to buy knowledge is prediction markets. Can we try that here? John Wentworth describes here a scheme for using markets to get mathematical proofs. Here's a scheme that's very similar in the overall idea, though the exact rules are slightly different:
* Shares on mathematical propositions are traded on the market. Propositions should be the kind of things that might be theorems, i.e. they are syntactically meaningful, and contain no free variables, though it's fine if they are false or unprovable.
* Shares on pr