Leiden Declaration on Artificial Intelligence and Mathematics

https://leidendeclaration.ai/

by zvr • 2 months ago

143 83 2 months ago

leidendeclaration.ai

Freak_NL• 2 months ago

(Leiden being the town in the Netherlands where Leiden University is.)

cactusplant73743• 2 months ago

Beautiful area. I lived there for a time during the pandemic and I really enjoyed walking the canals.

Semkas2• 2 months ago

Arguably the most beautiful town in the Netherlands

rho138• 2 months ago

With a gorgeous botanical garden and great cafes along the canals

curt15• 2 months ago

It's worth remembering Thurston's essay on mathoverflow (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."

pfdietz4• 2 months ago

That'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• 2 months ago

There 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.

magicalist3• 2 months ago

That's the product of math from the point of view of mathematicians. But is it the point of view of those funding math?

Yes, and your examples are exactly examples of what the GP quote is talking about.

Of course people paying money want applications, which includes "power" in your kind of reductive framing (applications to war being only one of many types of applications, or we could redefine any gradient provided by expanded understanding as "power", in which case the choice of word just seems melodramatic).

What we've also learned over the centuries, a lot more clearly in the last few, is that seemingly pointless or applicationless understanding can very quickly become useful. This is why it's clearly worth still funding pure math.

enugu• 2 months ago

Power depends on understanding - Seeing a larger scale view of what is happening as opposed to an arbitrary sequence of manipulations.

The foundations of the WW2 technologies you cite were dependent on previous theoretical efforts (ex:relativity) to develop a good understanding.

Without understanding, you get brittle demos which fail as the environment or problem description changes.

joe_the_user• 2 months ago

The debate on the funds of scholarship and education has been between democratic vision of "an informed citizenry" versus the technocratic vision of education for jobs and technical progress. Both of those might be "power" (Foucault said "everything is power" and if so, it tells us nothing). Thus I don't think your particular framing is useful.

For many who pursue mathematics, it is a refuge from ordinary life (or normal "power"). Within mathematics, there has historically been a tension between the "Babylonian" model of pure calculation and the "Greek" model of advancement through understanding. If current AI models subsume mathematics entirely, the Babylonian model will have "won" and the possibility of an informed citizenry will be in doubt.

palmotea• 2 months ago

That's the product of math from the point of view of mathematicians. But is it the point of view of those funding math?

That's AI in a nutshell: the only point of view that matters is the point of view of people with a lot of money, and we've finally developed a technology that will allow all those other points of view to be squashed and discarded. The powerful won't need to be bothered with them anymore.

For them, math is an instrument. Disagree? Fuck you, you don't matter anymore. Be excited about the future!

Supermancho2• 2 months ago

The 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.

applfanboysbgon4• 2 months ago

This 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

Supermancho2• 2 months ago

I'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.

Lerc3• 2 months ago

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.

But that is the nature of establishment, when something is a sufficiently firmly established tradition, people see it as a truism.

Crediting people is a social convention. Plagiarism is a social construct. It can be useful, in many areas of science, to reference to support your arguments. This is less important in proofs, because a proof is a proof, but references aid in understanding.

These are all reasons to reference and attribute that benefit the writer, and could be done voluntarily. The notion of a duty to reference or attribute has no impact on the validity of the claims being made. It is a collective decision to proportion prestige.

Turning the duty to do so into an unquestioned truism means it has to be done regardless of whether it accurately represents any property of merit.

There are many instances where prestige delivered grossly mismatches what an impartial observer would consider a fair balance of effort and ability.

We should at least recognise that this is so because we have chosen to let it be this way.

cactusplant73743• 2 months ago

Is there a connection to Leiden, NL?

jauco• 2 months ago

yes, 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• 2 months ago

These type of questions are the reason why LLMs will take your job :-)

cactusplant73743• 2 months ago

Questions 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.

lioeters• 2 months 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• 2 months ago

John 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• 2 months ago

1. 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

Semkas2• 2 months ago

4. then you write empty cliches

gilleain• 2 months ago

add too numbers

Did you do this on purpose to anger both Mathematicians and keen spellers?

moi2388• 2 months ago

Probably. I do it 2.

ShinyLeftPad2• 2 months ago

Who 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?

applfanboysbgon4• 2 months ago

Let's not sell them short, they're closer to $xxxx billion.

briandw3• 2 months ago

Most 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• 2 months 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? 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.

seanhunter2• 2 months 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_simpl...

Lerc3• 2 months ago

Generally, yes, but once broken down you end up with a large number of items that individually each is obviously true, so you know the combined statement Is true, but you don't find out if it is saying what you think it is saying.

In combining the parts you have the correct answer to a question, but is it that question you want to know?

Consider a proof that in the future all people will be happy.

You can methodically show this to be true but at the same time inadvertently include a proof that the number of people in the future will be zero.

It doesn't make the claim wrong, it stays undoubtedly true. It's just not what you assume it means.

seanhunter2• 2 months ago

It is, but it is somewhat worse for machine-generated proofs, especially when the proof is very long and was done by brute force (eg the 4 colour map theorem[1] is the famous example), or depends on a lot of niche results in disparate areas (which LLMs are wont to sometimes do).

Even when the proof is produced by the llm in a formal system like Lean4 it may not be “honest”[2] and it can be hard to tell if the proof is very long and complex and especially if it includes highly specialized results from lots of different areas of maths. Llms can (and do) do this just fine, but for a human proof that would require a team each of which was specialized in a particular area. Those people are more likely to be able to cross-check each other.

[1] https://pubs.ams.org/ebooks/conm/098/ and https://en.wikipedia.org/wiki/Four_color_theorem

[2] An “honest” proof may contain bugs or errors but it does not constitute a deliberate attack on the proof system or the math libraries it uses. Systems like Lean aim to not incorrectly validate an honest proof with mistakes but don’t guarantee anything in the case of a proof being dishonest. This is the sense used here https://lean-lang.org/doc/reference/latest/ValidatingProofs/ .

athrowaway3z• 2 months ago

https://lean-lang.org/doc/reference/latest/ValidatingProofs/...

magicalist3• 2 months ago

Your list is cherry picked from the list of "potential threats" to the values of the mathematical research community identified by this document. They aren't criticisms or absolute statements, they're literally a list of potential new problems for the future of mathematical research, and they all seem reasonable to me, if not all at the same levels of magnitude or plausibility.

Notably you don't seem to be looking at either the list of identified values or their recommendations to researchers in their use of LLMs, which would seem much more important to engage with in any non-shallow dismissal of the document as "feel[ing] like gate keeping and resistance to change".

It's also kind of a bad look (and actively harmful for discourse) for people working on AI to be so dismissive of fields actively engaging with how their field is changing due to AI. I haven't seen any other field engaging this actively with its possible futures, have you? Usually we seem to only get some extreme of over-hyped utopia, doomerism, or dismissal of everything as slop.

dwaltrip• 2 months ago

You aren’t really engaging with the substance or heart of the post, and your reading feels a bit knee-jerky and bad-faith to me.

briandw3• 1 month ago

So what was the substance that I missed?

onetimeusename2• 2 months ago

There is some of that but I wouldn't call it gatekeeping. Universities lately promote citations and publications so there's a sense that results are all that matters. Results matter, yes, but there's a human side too where we're kind of asking about human creativity and ability. To me an appropriate analogy is in climbing Mt. Everest. Proving something, or even writing a thesis, is like climbing Mt. Everest. A lot of the value is actually in the effort you put into it. You could take a helicopter ride up to the top and then climb a few steps and claim "You climbed to the peak of Everest". That's like using AI. But if you asked them about what it was like, how they prepared, etc. their answer would not be helpful. So I think there is a lot of value in the journey itself and outsourcing all this to AI would destroy the human part of it.

bcjdjsndon4• 2 months ago

It's completely antithetical to the whole enterprise to hide anything these researchers produce behind a paywall. Id be glad to see that go.

or even writing a thesis, is like climbing Mt. Everest. A lot of the value is actually in the effort you put into it.

As an analogy, in the music industry, if you need a jingle written, you wouldn't care if someone spent five minutes or five years writing it. AI is now filling that formulaic space very well. It won't replace the top end of humans output but it completely outdoes all the boilerplate stuff humans take an age creating

magimas• 2 months ago

I don't know, I don't think this "effort for effort's sake" is a very convincing argument. In particular, I think it's very much affected by recency bias in a way?

What we perceive as "effort worth taking" instead of "dull occupational therapy" is very prone to change with technology.

If you would argue that modern photographers need to take the time to physically develop their photos and use chemicals to get their effects rather than applying photoshop filters, you'd not be taken very seriously - in the 80s and 90s it would have been a very different discussion where people saw photoshop as "taking the helicopter to the summit of Mt Everest".

Same even with paper writing. I still had old school teachers in the 90s and early 2000s who insisted that writing anything on a computer was a "shortcut" that would encourage worse writing because you could undo stuff etc. They did all their handouts and worksheets on their old typewriters.

There is a discussion to be had on AI in maths, but I don't think it's this one. I think mathematicians should be talking about what the future of their field is supposed to look like in a time where AI will be able to find the proofs. Maybe maths will turn into a more "experimental" science, where you already know the proof of a theorem, but you want to find a particularly elegant way that helps humans understand it or find other ways to apply the knowledge. Or rewrite old theories from different angles based on all the new proofs generated by AI. I don't know, but I think there's a lot of mathematics to do out there for humans even in a time with AI.

scarmig• 2 months ago

Isn’t this a problem with human proofs as well?

Human proofs are themselves a kind of a proof of work. They certainly write flawed proofs, but you can expect a human author of a paper to have put in more effort--substantially more--than the human reader needs to verify it. Arguably, this asymmetry disappears for generated proofs.

Automated theorem provers help a bit here, but they don't eliminate the human verification cost.

derpism999• 2 months ago

That doesn’t mean it isn’t a problem for human proofs as well. Mathematicians use affiliations and c.v.’s as their primary means of gatekeeping for review, not the effort that went into a proof or its superficial appearances. Especially in hard and big problems which have had any number of flawed proofs.

Even Grigori Perelman took years to have his proof of Poincaré conjecture accepted, and he had a Ph.D. and a Berkeley fellowship.

Expecting a human author to put substantially more effort into a proof than needed to verify it is oversimplifying. It is more a matter of credentialing and collegiality in mathematics whereby someone’s reputation and work-product demonstrates that a purported proof put forward for review is likely to be true or at least a valuable or interesting contribution even if imperfect.

AI makes this a much bigger challenge.

1970-01-01• 2 months ago

It is said that every discipline, every task, will eventually be challenged by AI. I see it simply as an adapt-or-die situation. There is no rule in Principia Mathematica stating this field of philosophy is rightly reserved only to humans. Hopefully, this is just evidence of the old-guard culture starting to break.

https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Pri...

SilverElfin• 2 months 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• 2 months ago

A 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.

Mattasher2• 2 months ago

Interesting 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_b3• 2 months ago

I 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.

Mattasher2• 2 months 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.

bcjdjsndon4• 2 months ago

Ever since I found out the french chopped their kings head off, I've had a quiet respect for those crazy frogs

magicalist3• 2 months 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• 2 months ago

So they are recommendations. At global level.

max0077• 2 months ago

1. 'incentives' hmm

"Now, here, you see, it takes all the running you can do, to keep in the same place."

brcmthrowaway• 2 months ago

In a word, they're cooked

Spacecosmonaut2• 2 months ago

Accelerationists 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.

bcjdjsndon4• 2 months 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

Spacecosmonaut2• 2 months ago

Sure, but if we continue that analogy it does mean that there will be no human contributions to frontier mathematics.

josh-sematic• 2 months ago

Chess is a game where we play and observe for the sheer joy of mastery. Mathematics has some of that “done for its own sake” aspect but not as cleanly. There’s always the chance that some applied mathematics might help something in science or engineering. Whether we care more about process vs outcome is now more important than ever before.

wslh• 2 months ago

imagine a future where even talented mathematicians are nothing but noise

That would be AGI. My conjecture is that LLMs alone are not enough for that future. They are incredible, but AGI needs other breakthroughs.

In that sense, I think math is very different from chess or Go. Chess and Go are complete-information games with fixed rules and a fixed board. Math is open-ended.

gobdovan• 2 months ago

Chess has set rules and is a closed world with a set objective. In maths, you make up the rules. For every clearly defined problem that everybody cares about and is not yet proven, there's someone who first who recognised the importance of that given problem and managed to define it clearly enough for it to be recognised as such.

There's also the separate, less glamorous issue that people don't want to talk about, which is proof reliability. [0] If you have systems to help you formalise the problems and leave an algorithm or AI or whatever solve it in a verifiable way, that's a win for both the mathematicians and the rest of the world.

The deeper question is whether AI can replace the human role in deciding what mathematics should be done and what concepts matter. If that's automated, then yeah, we're screwed.

[0] https://lamport.azurewebsites.net/tla/proof-statistics.html

newsicanuse• 2 months ago

Keep dreaming

potsandpans• 2 months ago

Let'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.

FergusArgyll2• 2 months ago

Recommendations for policymakers in government and elsewhere

  > 2. Don’t believe the hype
  > 3. Regulate the artificial intelligence industry

mimd• 2 months ago

So, 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.

Leiden Declaration on Artificial Intelligence and Mathematics

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