It seems intuitive that a solved problem is unquestionably better than an unsolved problem: a magic button which, when pressed, turns the latter into the former at no cost would thus seem to be a "free lunch" that there would be no hesitation in pressing.
But even such a benign-seeming operation as answering a question can have irreversible costs. One familiar example is that of spoilers to a movie one is looking forward to watching: knowing key plot points in advance may permanently decrease your ability to enjoy the movie for the first time. Similarly, leaking information about a competition before it starts can irretrievably damage the fairness or value of that competition.
In the field of AI, problems whose solutions are already publicly available are deemed "contaminated" and of permanently reduced utility for evaluating such models, as the black-box nature of modern AI makes it extremely difficult to distinguish an AI-generated solution that came from genuinely novel reasoning from one that was somehow already present in training data. Analogously to pre-atomic steel, open problems in mathematics generated before the AI era have, astonishingly, have become something resembling a non-renewable resource. (1/3)
Another example comes from the SAIR/LMFDB inverse Galois challenge that I am co-organizing, which recently completed its first stage at https://competition.sair.foundation/competitions/igp24/overview. The objective of this challenge is to locate polynomials with specified Galois groups. The first stage of this challenge was deliberately posed as a competition, in which contestants did not share the polynomials they used. This of course inhibited the speed of solving the challenge; but it allowed us to create a valuable map https://competition.sair.foundation/competitions/igp24/api of the difficulty landscape of the problem, as one could now measure the difficulty of each potential Galois group by the number of contestants who had managed to locate a polynomial for that group. We are now releasing all the polynomials in order to facilitate the next, collaborative, stage of the competition to make more progress. However, by doing so, we have permanently degraded the ability to crowdsource this type of difficulty map in the future. My fellow co-organizers and I viewed this tradeoff as one worth making, but the decision was not taken lightly. (2/3)
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As Hugo Duminil-Copin wrote recently at https://proofsandprompts.com/2026/08/30/care-for-a-little-more-ai/ , the indiscriminate automated strip-mining of open problems for solutions may destroy the ecosystem from which the next generation of mathematical techniques, problems, and practitioners would have developed, similarly to how using excavators to dig out treasures from an archeological site destroys the rich historical context and data that gives such treasures so much of their meaning and value.
It may become necessary to declare certain classes of mathematical problems off-limits to automated solvers, in order to preserve their broader value to the mathematical ecosystem (for instance, through the training of future mathematicians). Admittedly this can be hard to enforce when such tools are both powerful and widely available. But then again, society has been somewhat successful at using social pressure to discourage, say, widespread spoiling of upcoming movies. Perhaps it can become viable to maintain such protected mathematical sites if one simultaneously opens up other classes of problems as suitable for directing this new AI capability towards. (3/3)
@tao Rich Schwartz has envisioned a future where this doesn’t happen. https://www.math.brown.edu/reschwar/Stories/AI.pdf
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@tao Despite how sad this case study is (and how much I resonate with the message), I think this is quite silly. At the very least, the next few years will be a difficult turbulent time for math academia.
As much as I wish society would be able to achieve this, I think it is impossible. Spoiling or not spoiling a movie is low stakes, but spoiling math problems is high stakes. There are tremendous rewards in terms of careers, or stock valuations for the companies, etc. So it will be done.
@tao , something very similar has already happened in cybersecurity.
Twenty years ago, I developed a 0-day exploit (https://doi.org/10.13140/RG.2.2.14474.11202) that very few people could understand or reproduce. It took six months of research and helped shape techniques that later became foundational to modern exploitation.
A frontier AI could probably reproduce that process today in minutes.
But speed is only part of the story. I have also seen AI develop 0-day exploits using execution paths outside typical human reasoning. Sometimes these techniques are hard to follow, not because humans cannot understand them, but because the cognitive effort becomes exhausting.
Humans build abstractions, assembly language, frameworks, reusable techniques, to make complexity manageable. AI does not need them. It only needs to move a processor and memory from state A to state B, even through paths no human would realistically choose to reason through manually.
The result is uncomfortable: AI can now produce working 0-days through techniques that may be cognitively alien, and people who have never developed one manually can still obtain them.
I understand the concern about preserving the learning ecosystem and the value of discovery. One could make the same argument in cybersecurity. Yet I see another possibility: AI may free future experts to work on deeper problems that have remained unsolved for decades.
With deep respect, and acknowledging my ignorance of mathematics professional ecosystem, I wonder whether a similar perspective applies here. Perhaps some open problems are gateways to deeper ones, and AI may let us reach those frontiers sooner without necessarily diminishing the rigor, meaning, or beauty of discovery.
"Perhaps some open problems are gateways to deeper ones, and AI may let us reach those frontiers sooner without necessarily diminishing the rigor, meaning, or beauty of discovery."
For me at least, I think the point isn't worrying about sacrificing the beauty of discovery or meaning. The point is that there are real utilitarian consequences to hyper optimization. This isn't only the case with AI, it is just one of the more egregious examples of this sort of hyper optimization.
There is a major difference between going towards a local optimum, and going towards a global optimum. Aggressively optimizing some metric (such as solving problems) can lead us to getting stuck in local optimums. In order to avoid getting stuck, and approach a global optimum, you need to sometimes take paths that seem less efficient, and even seem worse until the better optimum is reached, which, to be a bit dramatic, can happen sometimes on the scale of hundreds or even thousands of years. So there is no good way to tell ahead of time.
It might seem like AI can help with this, but there is a problem: In pure mathematics, famous open problems are not goals, they are benchmarks/navigational landmarks. They are there mostly so that while this journey to these long term optimums are playing out (which takes a very long time) we have some reward structures in the short term, and some landmarks that help us navigate so we don't get too lost in the fog.
So, in some cases, instead of helping us, AI rips out the reward structures and the navigational landmarks. Which is more akin to ripping out a dock in a busy harbor or a light house on a foggy night, than it is to being genuinely helpful.
@tao It may be hard to enforce that certain problems are off-limit. We should instead shape the incentive structure so that the archeological site is valued as much as the treasure. (Archeologists have not figured out a way to do that; but perhaps mathematicians can be better. The math is always there, not destroyed.)
“It may become necessary to declare certain classes of mathematical problems off-limits to automated solvers”
How could this mandate be policed?