Medieval mathematics was a highly personal craft, with knowledge of mathematics and methods being treated as a liability (for better and for worse) and having that knowledge was treated as secret knowledge that your livelihood depended on.
The reputation of a mathematician depended on knowing techniques others did not know. Mathematicians frequently challenged other mathematicians to semi-public mathematical contests. The two gave each other a set of problems. Whoever solved the problems the fastest, or the most problems in a given time, won the contest and could secure their funding and social status, while the unsuccessful contestant faced public embarrassment.
This created a strong incentive to keep discoveries secret. For example, Scipione del Ferro is now known for discovering a method for solving depressed cubic equations in ~1510. But he kept this method hidden for 20 years because this knowledge gave him professional security. He disclosed the method to his student only on his deathbed.
So knowing a particular formula, algorithm or technique could give the individual significant societal powers and connections; in a way that simply is not possible today. This attitude and orientation of mathematics continued well into late Medieval Era and later; Leibniz complained to the Royal Society after accusations of taking Newton's framework and simply just changing the notation. The Royal Society appointed a committee to investigate (while at the same time Newton was president of the Royal Society), which concluded Newton was the sole author of calculus. Much of the dispute stemmed from Newton’s decision to delay publication and keep his framework a secret.
Today, results are encouraged to be published rather than delayed or kept in secrecy (albeit there are major issues with how journals operate and paywalled papers). Incentives are rather to write papers and the funding problem has largely transitioned to a system of writing grant applications.
This largely social contract appears to be changing, with the advent of large AI companies going after famous problems for PR. Theorems have become cheap. A lot more people, with enough technical skills, can generate largely correct proofs for many problems. Especially undergraduate mathematics, but also graduate problems and previously unsolved problems. These proofs that LLMs can spit out can even be prompted to give a full Lean formalization of the proofs. Some have even gone so far to call the current situation “the fall of the proof economy”.
Now, there is an entirely separate issue (that I will not touch on here) on what to do with these AI slop Lean formalizations that no human has a hope of understanding. But ignoring the issues with slop proofs and the doubt cast on Lean's own soundness, it seems that anyone with enough money to spend on compute can unleash a swarm of typewriter monkeys in advent of rumors of some known problem being close to a possible solution.
For the recent claimed counterexample of Navier–Stokes, Capital&Compute estimated a cost of ~\$6.5 million on compute. TensorFeed estimated a cost of ~\$10–\$15 million, factoring in other costs than pure tokens, and Business Insider estimated a total cost of \$10–\$40 million.
Regardless of the true cost, it seems that professional mathematicians now need to vary about what they put into a LLM and think hard about how to disclose and publish a result. The ability to spend millions of dollars on compute to just grab the full formal proof before anyone else can react is only something companies with large amounts of capital can do. The technology is also being gatekept; because of the insane costs it takes to set up the infrastructure, it is not feasible for any single individual to do it.
It is now the identification of a promising problem which is the scarce and precious resource. We have now seen that even the rumor of someone working on a problem can trigger a massive amount of AI-powered effort to flatten it before the original research project has time to reach its full potential. The incentives may now be pointing in the direction of no longer sharing any promising research directions with the broader community, which would reverse centuries of traditions of open science and do serious long-term damage to the future of the field.
In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained.
While it may be technically infeasible to completely prohibit the use of automated tools to perform indiscriminate solution extraction, I believe that we can still designate many classes of problems as being desirous of a careful analysis that not only solves the problem, but identifies insights from the solution process, and learn more about the difficulty landscape for nearby problems, and for which raw solutions without such analysis would be of negligible or even negative value for these purposes. This is analogous to how a modern food donation drive no longer accepts arbitrary contributions even when they are verified to be technically edible, but instead maintains explicit and socially accepted standards on what level of contributions are actually sought.
When the Millennium Prizes were set up, no one of course anticipated this future technology. I have my own thoughts on how to militate the problem.
The first one is to require that only a human, or a group of humans, can reasonably be named or credited for a result (especially one with a bounty). If no human or group of people can reasonably be said to behind the result, it ought to be recognized as such. This is similar to how US copyright law states that only works created by human beings can be registered. Photos (or accidental selfies) taken by animals cannot be copyrighted. This places the images into public domain.
The second potential solution is to require the full methodology of finding the results to be published. For a human being and a fully human-derived result this methodology involves reading books, reading papers and using their own brain. For an LLM swarm of typewriter monkeys, this would require at least the full system prompts and the setup of the agent system. Because the exact setup is usually treated as a trade secret, this should de-incentivize mega-AI corporations/conglomerates to sweep up the attribution of a solution to a problem in the imminence of a mere rumor.
I recognize that it is kind of an open-ended question, but I think this is an important issue to bring up. Are there feasible ways to prevent the international and collaborative nature of mathematics from devolving in an era of secrecy and professional mathematicians holding onto partial results in fear of some company paying $40 million on compute for result attribution?