How do we prevent mathemathics from devolving into the Medieval Era of secrecy?

Recorded: Sept. 17, 2026, 9:09 p.m.

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How do we prevent mathematics from devolving into the Medieval Era of secrecy?

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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 wary 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.
Terence Tao, on Mastodon, has also raised this issue:

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 mitigate 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?

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5 revs, 2 users 93%Markus Klyver

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Is "vary" a typo for "worry"?
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Steven Landsburg
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Steven Landsburg

2026-09-15 17:48:05 +00:00

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Is "militate" a typo for "mitigate"?
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Qfwfq
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Qfwfq

2026-09-15 17:51:02 +00:00

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2 days ago

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I like the phrase "AI slop Lean formalizations that no human has a hope of understanding." I agree that this is part of the problem.
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David White
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David White

2026-09-15 18:20:03 +00:00

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2 days ago

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Some of this secrecy is inherent to the game, rather than "medieval" (actually Renaissance is more to the point than medieval here, but OK). Abel once accused Gauss of "covering his tracks," with G. replying that it wouldn't do to leave the scaffolding standing once the building was finished. The pursuit of mathematics is an inherently competitive enterprise. And not only that, but inherently tied up with intelligence (crypto), the defense industry, etc. All in all, many reasons for secrecy already in place, nothing to prevent that from happening. Probably happening already.
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R.P.
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R.P.

2026-09-15 18:51:26 +00:00

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I also do not see how even with the advent of prompting AI for proofs we go all the way to a guild-like secrecy. For sure we might be more tight lipped about ongoing work except with a tight circle of collaborators. But by talking with some colleague about ongoing work, it helps me gauge potential interest, and I am also hoping to get a bit of pushback to be later more clear about necessary hypothesis or weed out silly counterexamples. But there is also an implicit level of trust, that they will not try to scoop me (now by using AI).
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D. Corro
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D. Corro

2026-09-15 21:40:59 +00:00

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I will try to answer the question in a way that does not frame AI as the principal issue. I think AI is indeed a major issue, one that exposed a lot of problems, but these problems have been evolving for some time and I think are now unavoidable because of the advent of new technology.
Let's start with your historical anecdote concerning the cubic equation in 1500's Italy. What happened after that? Well, later in that century, the world would soon see the rise of the modern age of science, with the birth and rise of Galileo. This was enormously transformative, and would inspire later works including that of Newton. Science, at this time, was not restricted to 'scientists'; intellectuals and elites from all over Europe were fascinated. I was stunned to find out recently that Voltaire, not a name one would normally associate with science, had apparently read and completely understood Newton's work, and played a huge role in popularizing it. This is almost unthinkable in today's environment.
Along those lines, the explosion of interest in mathematics was intricately linked to mathematics' ability to explain natural phenomena. One case that I learned of recently is Kepler's Law: Kepler, who wrote down the law before the advent of calculus, had used an iterative method to apply it. Newton, over 60 years later, refined Kepler's laws with the new mathematics he developed that we now call calculus. However, the actual computation method remains iterative; it is what led to what we call the Newton-Raphson method. Lagrange gave a series solution that doesn't seem to always work, and the precise issue of convergence was only settled by Cauchy in the 19th century, which gave rise to complex analysis.
All of the above I learned only recently, despite being a professional mathematician for over 10 years. Complex analysis had always been presented as a 'pure mathematics' subject, with a sprinkling of applications.
My point is, the entire subject of mathematics seemed to have been drifting for a while. We now extol the 'virtues' of the intellectual 'purity' of mathematics, that it appears untethered to any other subject. This is certainly an attractive narrative for some, like myself and I believe many other current mathematicians, but it fundamentally disconnects from the general population, even other intellectuals.
One thing I have noticed is that mathematicians tend to pontificate, without realizing it: outreach attempts are almost always about telling people 'why the stuff I do is interesting/important', and the question doesn't seem to be 'what questions coming from natural phenomena or from lived experiences can be addressed mathematically'. This has led to, over time, to a situation where the vast majority of people have no interest nor ability to read a mathematics paper.
Going back to the question at hand and the anecdote, during that time it was necessary to keep methods a secret because the demand for mathematical knowledge itself was very limited. What mathematicians produced was seen as of little value, so whatever value one finds must be closely guarded. A century later, the value produced by mathematics was so manifest that everyone wanted more, eventually leading to the notion of professional mathematicians. But somehow over time we either ran out of easy mathematical approaches to resolve questions from natural phenomena, or we simply drifted away from such questions having formed our own identity, but either way the jig is up: there is a realization that the collective value produced by mathematics as a profession seemingly does not justify the number of workers.
In conclusion, the work that we need to do is to work hard on truly making mathematics accessible, so that even lay people can manifestly find the value of not only mathematics as a subject (this part I think is pretty clear), but of continuing to do mathematics research. If the public decides that because LLM's can seemingly crack long-standing open problems more or less autonomously and reach the conclusion that mathematicians are now worthless, I think that reflects a major failure on our part, spread over several generations.

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Stanley Yao Xiao

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@SamHopkins I think you are conflating two very different issues. To pure mathematicians anything that's not math is 'applied', but to an actual applied scientist, that means something completely different. An 'applied mathematician' working in mathematical biology would not be considered an applied scientist. Your concerns seem relevant to applied science, but not to 'applied mathematics', broadly interpreted
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Stanley Yao Xiao
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Stanley Yao Xiao
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2026-09-15 21:41:19 +00:00

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@D.Corro indeed, you have identified a very pertinent issue to mathematics. We are so siloed that even two slightly different sub-disciplines can't talk to each other. There are plenty of number theorists who do not understand the work of other number theorists; I know this for a fact.
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Stanley Yao Xiao
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Stanley Yao Xiao
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2026-09-15 21:56:52 +00:00

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I suspect there are plenty of poets who don't understand the work of other poets.
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Gerry Myerson
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Gerry Myerson

2026-09-16 01:05:29 +00:00

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I think the problem is that the production of ideas is not valued by society as much as we would like. If I had $100 million, I would be prioritizing funding classics and various minor humanities subjects well before mathematics, not because they are more important, but because the need is greater.
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Alexander Woo
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Alexander Woo

2026-09-17 01:42:04 +00:00

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19 hours ago

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@GerryMyerson Considering the very frequent popular opinion that poetry is irrelevant elitist nonsense, this isn't a great comparison for mathematics. Many leading poets put a lot of effort into outreach to try to explain why it's still relevant, and a poet who doesn't try to make their work understandable is intentionally devaluing their entire field. Even physicists have the same issue too, and they're working with stuff that actually does things. It's not unreasonable to expect the same in maths.
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Graham
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Graham

2026-09-17 10:34:13 +00:00

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10 hours ago

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Your historical example suggests that the funding structure in the past disincentivized sharing mathematical knowledge. In view of this, a long run solution presents itself: we can shift incentive systems away from a system which often provides (provided?) the bulk of the mathematical credit to the last person to solve an explicitly stated conjecture.
Instead, we can attempt to re-orient our incentive systems to better reward those who recognize that these conjectures should be made in the first place (at least while artificial intelligence doesn't surpass humans in this regard) as well as to reward those whose work contributes to human mathematical understanding, regardless of whether this work was the "first" to approach a mathematical idea or is the first to provide the solution to some well-posed problem. (A similar ideal is discussed in this recent blog post by Daniel Litt.)
Of course, this does not answer the question of how to prevent mathematics in the short term from becoming more secretive but it suggests that so long as we actually do change our incentive systems in the long term this won't be an issue.

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Tom Gannon

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Putting millions into solving pure maths problems is just to show off. It is about being the first and presenting what you can do. Being the second is boring. Anyone remember Apollo 12?
So the companies will soon move on and work on some other tasks. Your adversaries are now the rich universities that maybe can pour 10,000€ or even 100,000€ into solving a certain question. They will be much faster than other universities, similar to experimental physics or biology where some universities and institutes own the fancy machines.
So if you work at such a rich university, you can try to do bleeding edge AI supported maths research and find all the counterexamples that others have searched for decades. But if you are at a poor university, it does not make sense to chase the same results with pen and paper, and trying to keep your partial progress secret. You need to find something orthogonal to the things that AIs are good at, otherwise your research becomes meaningless.

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JF Meier

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Medieval mathematics was characterized by a highly personal craft where knowledge and methods were treated as liabilities, creating strong incentives for secrecy among mathematicians. Reputation was tied to possessing specialized techniques, often secured through contests where speed and the secrecy of discovery conferred social status and funding. This drive for secrecy, exemplified by figures like Scipione del Ferro, meant that knowledge was guarded to ensure professional security, a dynamic that persisted into later eras, as seen in disputes over frameworks like calculus where delays in publication were employed.

The dynamic shifted in response to the rise of the modern scientific age, moving from secrecy to a system where results were encouraged to be published, and funding mechanisms evolved toward grant applications. However, this shift is currently facing disruption due to the advent of large artificial intelligence companies, which can generate proofs autonomously, leading to concerns about the "fall of the proof economy." The current threat to open mathematics stems from the potential for entities with substantial computational resources to extract solutions quickly, which could flatten promising research directions before the broader community can react.

This situation raises a critical question regarding the ecosystem of mathematical progress. Terence Tao suggests that the identification of promising problems is now the scarce resource, and the risk is that the pursuit of immediate solutions incentivizes withholding research findings from the community, which would damage the long-term future of the field. To counteract this, the authors propose reorienting incentive systems rather than attempting to prohibit the use of automated tools.

One proposed mitigation strategy involves establishing clear attribution standards: only human or groups of humans should be credited for results, similar to copyright law, to prevent corporations from claiming ownership of AI-derived solutions. Furthermore, requiring the publication of the full methodology used to reach a result would act as a deterrent against large entities claiming credit for findings derived from computational swarm processes. This approach attempts to ensure that value is attributed to human intellectual contribution and understanding.

Beyond these specific measures, there is an underlying philosophical need to make mathematics more accessible. The authors argue that the current intellectual focus on the purity of mathematics has led to a disconnection from the general public, as the value of mathematical research is not widely perceived. To sustain progress, the collective value created by mathematics as a profession must be adequately recognized, which requires making the work manifestly accessible so that the public understands the value of continuing mathematical research. Ultimately, the long-term solution lies in fundamentally changing incentive structures to reward the recognition of promising conjectures and contributions to human mathematical understanding, irrespective of who first achieves a purely computational solution.