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Leiden Declaration on Artificial Intelligence and Mathematics
- cactusplant7374 4mo agoIs there a connection to Leiden, NL?
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- jauco 4mo agoyes, that's where a conference was held that kickstarted the group that drafted this declaration. > In September 2025 the Lorentz Center at Leiden University in the Netherlands hosted a conference entitled Mechanization and Mathematical Research. The around 60 participants from 10 countries comprised mathematicians, computer scientists, philosophers, historians and social scientists, including those with experience in industry and in government.
- root-parent 4mo agoThese type of questions are the reason why LLMs will take your job :-)
- cactusplant7374 4mo agoQuestions are in the spirit of community. It isn't listed on the about page. If you rely on LLMs for everything you will never interact with people.
- Freak_NL 4mo ago(Leiden being the town in the Netherlands where Leiden University is.)
- cactusplant7374 4mo agoBeautiful area. I lived there for a time during the pandemic and I really enjoyed walking the canals.
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- Semkas 4mo agoArguably the most beautiful town in the Netherlands
- rho138 4mo agoWith a gorgeous botanical garden and great cafes along the canals
- curt15 4mo agoIt's worth remembering Thurston's essay on mathoverflow (https://mathoverflow.net/questions/43690/whats-a-mathematician-to-do https://mathoverflow.net/questions/43690/whats-a-mathematici...): "The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding."
- pfdietz 4mo agoThat's the product of math from the point of view of mathematicians. But is it the point of view of those funding math? I suggest if one looks at the history of funding for mathematics and science, the product of these efforts is not understanding, but rather power. Funding went way up after WW2 when the war demonstrated that power flows from them. Math not only contributed to the scientific weapons of the way, but was directly used in operation planning (the birth of the field of Operations Research) as well as in cryptography. The reason this matters is that AI is also a quintessential power-oriented technology. From the point of those providing the monetary lifeblood on which modern mathematical practice depends, the current math-AI discussion presents no issue worthy of concern.
- repelsteeltje 4mo agoThere is more to math, than input (money) and output (power). Sure, there is some relation between applied sciences and how knowlegde can assist effecting world events. But for the most part, math discovery relied more on human curiosity than on resources to "do math". Conversely, if people allocate lots of money to developing AI, that doesn't mean mathematicians have an obligation to take the money provide ROI to investors.
- pixl97 4mo agoI mean, in real life it's a combination of both. Some money is for math as an exploration of our world that will never pay off. Some money is learning things that may pay off long after we're dead (planting trees so our great grandchildren have shade). Some money is for solving problems right now. Getting funding can be quite difficult at times, so you'll see some portion of researchers (or mathematicians in this case) take the dollars they can get.
- Supermancho 4mo agoThe potential threats section reads like panic, rather than a critique of AI. I can see where #2 has some legs, if I thought tradition was sacrosanct. 1. AI proofs might be incorrect and difficult to demonstrate why. This implies they are not like human proofs in these qualities. 2. AI proofs are difficult to attribute correctly, because they don't follow established traditions. Nothing to do with the math, but ok. 3. Mathematicians without AI (for political or practical reasons) will not necessarily be able to participate in AI-assisted research. This history of Mathematics is littered with people having uneven access. 4. People/orgs are publishing that AI found things are fact before they are properly evaluated. Same issue. 5. All these things are bad, because AI might muddy the field with lots of unknowns.
- applfanboysbgon 4mo agoThis appears to be a very bad faith post that intentionally misrepresents what is being said. 1. pertains to the quantity of output adding stress to review processes; LLMs can feasibly produce a million plausible but incorrect 'proofs' in the time that a human can produce one. We already see this effect in software development, with bug bounty programs shutting down and open-source software rejecting AI contributions or closing altogether because LLMs flood review channels with an amount of spam for which there is no sufficient amount of human bandwidth to handle. 2. is nothing about "following established traditions" but rather the general concept of crediting people for their prior work, unless you think that "not plagiarising" is a trifling established tradition. 3. is more or less accurate to the point they made, but "it has historically been this way" isn't a compelling justification for "it should always be this way and also it's okay if it gets worse" 4. An existing issue being made 100x more common is a point worth bringing attention to even if it already existed, actually 5. said nothing that could possibly be interpreted in the vein of "muddying the field with lots of unknowns" at all. Point 5 was actually about economic incentives and the risk of mathematic research becoming beholden to tech monopolies
- Supermancho 4mo agoI'm not sure it's constructive to explain our differences, point by point. eg > 2. is nothing about "following established traditions" > undermine the traditional system of attribution Literally does. Suffice to say, I find your interpretations to be surprising and disconnected and it has not changed my views.
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- lioeters 4mo ago> Terence Tao - Professor, University of California, Los Angeles > This has been the result of months of community input about the fundamental values and goals of the mathematical community. In retrospect, these were questions we should have been systematically discussing years ago, but in any event the exercise was extremely valuable, and the end result is excellent. I wholeheartedly endorse the statements and recommendations in this declaration.
- u1hcw9nx 4mo agoJohn Carlos Baez: >I support this declaration. I have one small comment: the document notes that "Technologies which affect the way in which mathematics is practiced may disturb the current system of incentives." The current system of incentives seriously is flawed in many ways, and I don't think maintaining the status quo should be our goal. However, we should work to improve it, not let it be corrupted by outside forces, as has already been done for decades by university administrators, journal oligopolies, etc.
- dist-epoch 4mo ago1. first they ignore you <<<< GPT-4 can barely add too numbers without making a mistake 2. then they laugh at you <<<< the International Math Olympiad is basically just high school math 3. then they fight you <<<< this declaration 4. then you win
- Semkas 4mo ago4. then you write empty cliches
- gilleain 4mo ago> add too numbers Did you do this on purpose to anger both Mathematicians and keen spellers?
- moi2388 4mo agoProbably. I do it 2.
- ShinyLeftPad 4mo agoWho exactly is the implied underdog "you" that we so desperately want to win here-- do you mean those poor struggling $xx billion companies or current US government apparently beholden to them?
- applfanboysbgon 4mo agoLet's not sell them short, they're closer to $xxxx billion.
- tcp_handshaker 4mo ago[dead]
- briandw 4mo agoMost of the arguments here feel like gate keeping and resistance to change. I didn't see any arguments that were directly about advancing the state of knowledge of math. “Current automated techniques can produce plausible but unreliable (or even incorrect) arguments which are difficult to distinguish from correct mathematical proofs.” That seems like a problem for mathematics with or without AI. Isn’t this a problem with human proofs as well? “Many current models are also built on data obtained by systematically exploiting licenses and access arrangements that were not made with artificial intelligence in mind, or indeed by simply violating copyright protections” Copyright? The copyright arguments have been hard to make in domains where copyright is much stronger, mathematical knowledge isn’t even subject to copyright. “Technologies which affect the way in which mathematics is practiced may disturb the current system of incentives” Resistance to change again. “Proper evaluation is endangered if results are communicated through informal channels” Gatekeeping again.
- chasd00 4mo agoCan't all proofs be eventually broken down into their fundamental pieces and then it's clear as day if it's right or wrong? Seems like a proof would be the best place to determine if an AI is right or not because the output is either right or wrong, there's no subjectivity and the, now common, excuse "well a human would have done the same" won't hold up.
- seanhunter 4mo ago> Can't all proofs be eventually broken down into their fundamental pieces and then it's clear as day if it's right or wrong? You’d think so, but not really. There are mathematical structures which are unimaginably huge but have little if any reducible structure. For example, in algebra, one of the most basic structures is a Group. When trying to understand a group, one of the most important tools is to break a group into chunks using what’s called a “normal subgroup”. However it turns out that there are some absolutely enormous groups that are “simple” (ie have no normal subgroups). So, there is a set of 26 of these known as the “sporadic simple groups” that just don’t fit any kind of pattern. Proving results about these has proved very difficult because they can’t be broken down (they have no normal subgroups) and by definition just don’t fit any kind of other pattern. One of these, the “monster” group has approximately 8x10^53 members. So you have a set that is unimaginably massive and has very little internal structure as it is “simple” and so can’t be broken down further. The proof that there are 26 of these sporadic simple groups is part of the theorem known as the classification of finite simple groups, sometimes known as the “Enormous Theorem”.[1] It took over 100 mathematicians nearly 50 years and resulted in hundreds of papers. Even with that many mathematicians involved, there were still errors and revisions needed to the original proof. Some of the original authors are gradually publishing a somewhat simplified version of the proof but it’s still a massive effort. [1] https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.wikipedia.org/wiki/Classification_of_finite_simpl...
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- SilverElfin 4mo ago> Technologies that draw extensively on the published mathematical commons undermine the traditional system of attribution. This just feels like something that has always been true. Defending attribution in this way feels more like a panicked gatekeeping rather than something valuable and principled. I’m a bit disappointed to see people like Terence Tao endorse this.
- TimorousBestie 4mo agoA career in mathematics is very unlikely to deliver fortune, fame, or even respect. Attribution is in many cases the only way we know that someone else thought our work was valuable. The bar for being cited has never been lower in the history of mathematics: getting an arXiv account is easier than getting a lobste.rs account (at least by my experience, for I have the former but not the latter). Tao knows what he is talking about, he was an early adopter of LLMs for proof-generation.
- adamzwasserman 4mo ago[flagged]
- Mattasher 4mo agoInteresting document. Especially interesting compared to the Bourbaki movement from a century ago, which was much more focused on universality and correctness, and much less focused on process and attribution (in fact, demanded anonymity).
- adrian_b 4mo agoI consider attribution very important, less because it were associated with some merits for the humans quoted as the sources of certain new ideas, but more because it facilitates going backwards to the historical documents where those ideas were first introduced. At least for me, in many cases I have achieved a much better understanding of various things after I studied the historical development of the ideas related to them. Therefore I agree with the point "2." at "Potential Threats". For me a novel mathematical demonstration that is not presented in a way which disentangles its really new elements from the previously known elements, by proper quotation of all relevant older sources, has a value that is many times lower than that of a demonstration with proper attributions.
- Mattasher 4mo ago> At least for me, in many cases I have achieved a much better understanding of various things after I studied the historical development of the ideas related to them. Completely agree with this. Math, and especially related studies like probability/stats, are presented as laws of the universe, when quite often they represent branching paths that could have gone a different way. What we take as correctness is often just one specific metric, or one justifiable epistemological pathway, among many, and it's worth understanding why we ended up where we are, and what those other forking paths might have been, and often still are.
- bcjdjsndon 4mo agoEver since I found out the french chopped their kings head off, I've had a quiet respect for those crazy frogs
- magicalist 4mo ago> Especially interesting compared to the Bourbaki movement from a century ago, which was much more focused on universality and correctness, and much less focused on process and attribution (in fact, demanded anonymity). The Bourbaki group was (is) about documenting existing mathematics rigorously, not doing original research. This document is specifically about the future of mathematical research.
- zkmon 4mo agoSo they are recommendations. At global level.
- max0077 4mo ago1. 'incentives' hmm "Now, here, you see, it takes all the running you can do, to keep in the same place."
- brcmthrowaway 4mo agoIn a word, they're cooked
- Spacecosmonaut 4mo agoAccelerationists may argue that the eroding of proper attribution and proof verification by humans is a meaningless short term struggle of a dying field. Mathematics seems to be entering an era where human + machine maximizes performance, much like chess in the 1990s. However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). A future where AI generates and verifies proofs without humans in the loop. Where the mathematics may be beyond human comprehension. In that future, does it matter that early career mathematicians are inhibited by these developments? Perhaps not. Programming faces the same issue. As AI crawls up the competence ladder, does it matter that fewer people have opportunities to develop the skillset of a senior engineer? Perhaps not.
- bcjdjsndon 4mo ago> However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). Isn't chess more popular than ever? Ai dominating the game didnt seen to matter
- Spacecosmonaut 4mo agoSure, but if we continue that analogy it does mean that there will be no human contributions to frontier mathematics.
- FergusArgyll 4mo ago(This doesn't disprove your point because it seems like it should be trivial to make but interestingly) I don't think there's any computer system which autonomously come up with new directions in openings. As far as I know; a GM looks at stockfish's evaluation of the top x moves and analyzes one that hasn't been played a lot etc.
- bcjdjsndon 4mo agoIf the output is better who cares?
- 4mo ago
- potsandpans 4mo agoLet's assume that there is an advantage to using llms. E.g. the use of an llm lends a competitive advantage in a given field. There is no moral or ethical obligation to disclose tool use. The disclosure in of itself presents an asymmetric disadvantage to the disclosee. Especially in this charged environment where large swathes of people are champing at the bit to discredit or diminish any effort that leverages these tools. This system incentivizes people to hide tool use to gain a competitive advantage. This moralistic grandstanding will be seen as a reactionary movement of people trying to cope with transformative technology. Lie about tool use, don't admit it. Use it as you see fit and rely on your taste, expertise and best judgement.
- FergusArgyll 4mo ago> Recommendations for policymakers in government and elsewhere > 2. Don’t believe the hype > 3. Regulate the artificial intelligence industry
- mimd 4mo agoSo, no one's buying our textbooks anymore, it's getting costly to do our work, stop over claiming results for your marketing and insulting our work? I would have stuck with that. Everything else seems to be organizational or standard politics. I would have avoided the political sections as it just sets those signatories up for getting into those fights, having not realized what they cosigned. The issue of attribution is a hard problem that has festered for a while, that has mostly been obscured by the inability to discern an authors entire body of ingested information. I would be careful as, the proposed opt-out restrictions will be eventually imposed on people, which sorta blows up the whole math enterprise of universities/etc. Ie. the proposed leads to a book author revoking a person's ability to reference their arguments in a mathematic paper for money, spite, etc.