alt Hacker NewsA misalignment of AI in mathematics
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I never expected this many people (on this thread) arguing semantics and what not. I know that not everyone has morality and ethics, but I didn't realize it was this bad.
I'm afraid of the ripple effect of the agenda pushed by AI companies will have. In future and even now, they say AI has significantly progressed math and scientific research in general. There is truth to this, but the narrative has done more damage (so far) to the students, researchers, and the culture of knowledge transfer in academia. Many graduate students (I know) are having a crisis if any of their research worth it? If AI can (or will) do everything, what's the point of doing experiments and all? This will eventually deter a whole generation of curious minded students from research.
I guess, only time will whether this is for the good or bad. And how good AI models get without new data from research and experiments.
You can couch it different words but the basic shape of all these is the same, whether it was voice artists earlier, IT outsourcing, or now disciplines like mathematics. A small set of people (relatively) who were the primary source of getting something done, suddenly find that technology has made it accessible for others to do what they specialized in. It's a tough pill to swallow and it is natural to not be comfortable with this for most people.
However, as it has happened in the past, once there is a technological wedge, technological advances will move forward, whether some community likes it or not, and human ingenuity will find ways such that the benefit is greater than the risk.
To me it doesn't seem like what AI has destroyed is the ability for mathematicians to develop understanding and share it with each other, but rather it's destroyed the yardstick (solving open problems) that has traditionally been used to measure how much they have contributed to that understanding.
I do see how this is a problem in terms of assigning credit, but I think the cat is already out of the bag in terms of these models being capable. Even without AI labs spending millions of dollars to solve millennium prize problems, there are plenty of other people who will use them to pick low hanging fruit. I don't think any social solution is going to make things go back to the way they were, where you could share your progress towards a famous open problem without risking someone "scooping" you within a couple of days.
I think that the most likely outcomes are either mathematics becomes more secretive, or there is a more deliberative approach to assigning credit than who was "first" to solve some problem. In the former case, this may slow down progress, and in the latter case, this could mean that credit would become more subjective, and be a continual source of controversy.
Tao's critique of AI in the field of mathematics reminds me of what French art critic Charles Baudelaire said in the 19th century about photography [0].
Baudelaire argued that photography became a haven for failed painters, the sorts of hacks that could not finish proper training. Photography, as a mechanical rendering of the world, could only record what already existed; it couldn't transform reality the way a painting could.
He also criticized the public's craze for "rushing" into it, and complained that this technical "progress" was weakening the arts.
Do you see some parallels as well?
[0] https://fr.wikisource.org/wiki/Curiosit%C3%A9s_esth%C3%A9tiq...
This is a PR problem, not a mathematical problem. It's possibly the worst PR problem mathematics has faced since the execution of Hippasus for whistleblowing on the cover-up of the regular dodecahedron. It's still a PR problem.
So what went wrong? Mathematics education. Math below grad school is all about solving stated problems. Credit is given for solving puzzles successfully. Homework is problem sets. Everybody below a very advanced level is taught math that way. Even at the higher levels, puzzles remain important. Awards in mathematics are often tied to solving puzzle-like problems. That's still the criterion for becoming Senior Wrangler at Cambridge, "the greatest intellectual achievement attainable in Britain". This despite Polya's attempt at reform a century ago. Puzzle solving gets good grades and class rank. So it's the status indicator mathematics presents to the outside world.
Then reasonably good AI comes along. AI has become rather good at solving puzzles. So people aim powerful AIs at known hard puzzles, with some success. That blows up the status indicator system. Mathematics itself is fine. It's the status symbols that have a problem.
Maybe the Fields Medalists need to hire a crisis management team to reframe what success means in mathematics. That's what they're trying to do with that letter, but they're mathematicians, not PR people, and they don't know how.
This sounds a lot to me like people in the 90's complaining that computers were destroying chess. Thirty years later, chess is more popular than it ever was, and chess players are better than they ever have been. I wouldn't be surprised if there are now more chess books now than there ever have been. Furthermore, it turns out that a lot of chess books written before computers were just wrong about a lot of things. It turns out having an oracle for the "right" answer in chess, even without an explanation, used properly, allows humans to develop broader, more accurate insights.
The argument here sounds similar. The fear, as I understand this statement to be saying, is that by being given the correct answer, in the form of a 100-page Lean proof, humans will be robbed of the chance to from insights about the structure of mathematics itself. I don't see any reason that humans can't continue to develop insights as they try to digest the 100-page Lean proof into something more manageable; but with more certainty and fewer false starts.
> We are witnessing a general threat to intellectual work
This is the crux of it and goes far beyond Mathematics or Computer Science. To get a bunch of humans to do anything, you have to motivate them. Kleos and Timē; renown and stuff. These AI companies threaten to rip this away from everyone but themselves, and this recent millennium prize is the perfect example.
Solving this problem as a human would have led to tremendous Kleos; my name would be written in the annals of mathematics, lecture tours of praise were mine to be had for the rest of my days. This one victory would have earned my recognition throughout history. The greatest a mortal may hope for. Ripped away.
It would also have given me great Timē. The prize money, the professorships, the book deals. Gone.
If all hope of “renown and stuff” in the intellectual realm is now taken by the AI companies, they will remove all human motivation to pursue these endeavours.
Perhaps the glory will come from slaying these fell beasts.
> solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight.
This is the effect of AI on most intellectual disciplines, and it’s a real worry.
I understand this stance and where they are coming from, but I can't help but think this sounds very analogous to engineers' arguments against AI-assisted and vibe-coding, especially with regard to cognitive debt. Yet the software industry is plowing ahead, reportedly pushing mountains of unreviewed code to Prod, and the world hasn't ended.
Of course, nobody's really comfortable with it, so this is also a forcing function for the industry to adapt and figure out new techniques to manage complexity and trust. I think the same will happen with Mathematics.
But it is also possible we will end up with three forms of Mathematics: the one we understand, the one we don't, and the one we don't understand but can prove to work. Kind of like magic -- with all the positive and negative connotations of the word.
It is pretty evident that these models will soon exceed our cognitive capabilities. Is it right to hold them back just because we can't keep up? Many of those discoveries will be so beyond us that we can't do anything with them, but that also means they can't hurt us. On the other hand, there could be many discoveries that we can parlay into practically useful applications, even if we don't understand them.
Just like LLMs.
To be honest I didn’t get that much outrage here: https://news.ycombinator.com/item?id=49662116
It seems like a lot of the issue here is that these problems aren’t interesting in and of themselves, but they lead down interesting roads. It defeats the purpose if you solve them without getting any real understanding.
It’s akin to saying you’ve solved “pancake flipping” problems with a waffle maker, or “travelling salesman” problems with a zoom meeting.
There's an unpopular branch of mathematics which does not have infinities - finiteism.[1] The constructive version of finitism takes the position that there is no such thing as infinity, just arbitrarily large upper bounds. You can have theorems about arbitrarily large numbers, but you never get
1 + 1/2 + 1/4 + 1/8 ... = 2
The big objection to finiteism is that it's a lot more work. Infinity swallows many special cases. Proofs get longer without infinity, and most of the special cases are uninteresting. That's not a problem for AIs.
Someone may start up an AI and make it grind through Hilbert's program for putting mathematics on a fully consistent foundation, starting from a finiteism base. This is a huge, unrewarding job. Great for machine work.
This letter is complaining that human understanding has been crucial to advancing of mathematics, and AI companies are not bothering with it. But the promise (and horror) of AI mathematics is that, if it succeeds, human understanding becomes irrelevant. That's the goal. So this letter's message will fall on deaf ears.
Keep in mind employees at AI companies are publicly stating that they believe they're risking a >10% chance of human extinction. They're knowingly risking the lives of every man, woman, and child to continue the work. The lives of their own sons and daughters. A person already rationalizing that isn't going to shed a tear for the careers of mathematicians. Just a bug on the windshield.
OpenAI: "Our mission is to ensure that artificial general intelligence benefits all of humanity."
- Except the mathematicians who we'll scoop and cause existential dread among their entire field.
- Except the software developers. They'll need to become plumbers or live on UBI.
- Except the people in countries that can't afford the cost of AI tokens to keep up with the rest of the world.
Just keep picking off groups of humans for the "benefits of all humanity"... while building larger and larger disparities been the have a lots and the just have enoughs.
We're going to build humans a utopia but along the way we'll leave a trail of destruction because that's not our problem.
Job protectionism for elite mathematicians under guise of caring about student development. The glory of the super smart math person will need to shift to more creative modes, just like art had to handle photography. Attribution is legitimate issue but should not stall progress as it is easy to address via the same research mechanisms that agents already do.
> We are witnessing a general threat to intellectual work, with misalignment between the outcome of the use of AI and its initial purpose. In many fields and activities, years of training have traditionally served not only to produce a final answer or product, but also to develop understanding and the ability to formulate new questions and ideas. However, building on a vast body of previous human work, AI systems are becoming increasingly capable of producing the results of such work directly, and these goals cease to align.
This is about how good taste in both research direction and in design are essential to steering AI, but we have no plan at all for instilling that taste in students or practitioners in a post-AI world.
> The issues the mathematical community faces now are similar to issues that other scientific and creative professions are facing, and indicate issues that all of humanity might face: how to make sure that, as AI changes the way work is done, we do not lose sight of what that work was meant to achieve in the first place.
Besides eroding taste and taste-building, this is about just how useful friction is as signal.
Everyone coding with AI knows it routes around difficulties like a river around a stone, which is not necessarily a good thing. It will do it tirelessly 1000 times instead of learning anything from it. AND if the AI does not fail in this, the human driver will get no signal, and never know it happened. This seems to be getting worse, not better.. my theory is that more models are cross-trained on cybersecurity stuff where the goal is success and the method doesn't matter. Fine for pen-testing, ultimately pretty bad for coherent code or math or physics.
Discrete tasks where we don't want to be bothered is a real use-case, but optimizing for it everywhere is terrible for the future of durable abstractions that we can build on and ratchet up our understanding with. Bad for the models too eventually! They can maintain a codebase with millions of special cases or juggle tons of free variables in equations, but that just encourages bad abstractions.. they have a ceiling for this too, even if it's higher than humans.
Looks like mathematicians (like people in many other professions) have to redefine what their work means and how to define success. Hard to agree that a tool that can find a proof is detrimental by itself, rather it voids some assumptions people relied on previously
In internet culture there’s this phrase “Hydrogen Bomb vs Coughing Baby”, meant to highlight the absurd power difference between two combatants.
In almost any scenario even tangentially involving mathematics, twenty-five Fields medallists uniting to denounce something would be a veritable Tsar Bomba.
It should give you pause that here they feel like the ailing infant.
Sounds like the field needs to adapt. The world is different now, better get used to it. Trying to artificially hold back progress just so people can continue to flex on their peers is cowardice. Every single industry throughout history has had moments like this, and looking back, we'd have changed none of it.
> I am proud to be among the list of 25 initial signatories — all Fields Medallists — to the declaration below
I wonder if there’s a Fields Medalist group chat.
The rise of AI is going to lead to a lot of similar issues in many fields as it grows and develops further. This can be seen form 2 perspectives. The death of intelligence as we no longer need to think for ourselves or understand anything since AI can do it.
Alternatively, and this is what I choose to believe, it will lead to further intellectual enlightenment and advancement for use as a species as we start to discover new problems and areas of research that we had never conceived of before.
If we let AI take over all of our thinking then we are heading in the wrong direction. If we continue to ise it as the tool it is it will help us grow and advance as a species.
Everyones outraged all the time. It doesn't mean anything anymore. It's that meme from years ago about the red ants and the black ants living in a box peacefully until someone shakes the box and they start trying to kill each other. They go after each other and not the one shaking the box.
OpenAI/Anthropic are shaking the box.
Puts the onus on the AI companies to provide a specific replacement mechanism, no? Unless I'm unfamiliar with something else he's written that proposes something more specific and constructive
To Tao’s credit he obviously identified the problem very clearly and admits understandably "we did not have the time to have a more consultative process, as with Leiden; but we decided that the urgency of the situation was such that we needed to release a statement sooner rather than later".
Mathematics is about discovering and understanding the logical implications of assumed axioms under various inference rules.
Alternatively, some claim that mathematics is about understanding these implications.
Under the first definition, AI is already, and forevermore will be faster and better at proving theorems. Just like it is better at checkers, chess, and now go.
The author asserts that AI proofs are incomprehensible to humans, and so under the second definition AI is merely a tool to overcome one hurdle on the way to understanding.
So which is it? The author seems to claim the second definition, but bemoan the end of mathematics under the first.
Im surprised about the sentiment in this discussion.
I totally see the problem Terence is describing. We are loosing a lot in understanding and focus if it continues like that. The solution found for Navier Stokes doesn’t have much „real value“ - but what almost always happened in the past when people worked on the difficult problems, these sparked new ideas / new theorems that broadened our knowledge. Think back at your grad studies, figuring out a proof as homework was hard, sometimes incredibly hard, but while doing it we gained a lot of understanding how things work. Now asking AI for the solution and „just“ getting it, risks our understanding, our creativity and our ability to connect the dots with other territories. I see it in students nowadays, there is much less understanding, much less creativity in finding solutions. I truly think this „short-path“ solution with the „death of struggle is one of the biggest risks with AI already for human development
Tao et al. are effectively calling for diseases like childhood cancer to remain persistent for longer.
Physics and Biology will see major breakthroughs that WILL fundamentally alter our world. That is one key thing missing from alot of discussion here is the narrow focus on math (or parallels with software engineering). Doing well in math is key to doing well in physics and other sciences.
This reads like people lamenting a bygone era and making a desperate attempt to bring it back. I'm sorry. Outside of the good ol' boys club, no one cares about some process they've romanticized simply because "that's how its always been done". Absolute nonsense.
We are moving forward and if that means no human wins a fields medal because they didnt spend three decades working on a problem that could be solved in three days, the world will be better for it.
My interpretation is they don't care about scooping mathematicians they were just trying to scoop a competitor. They had a short window in which to complicate Anthropic's priority, when Anthropic announced be able to say "ok nice but we did that too". Upon realizing they'd misunderstood, incredibly they said to this academic (who did not resolve NS) okay well just go ahead and claim the prize, so long as you're not Anthropic let's make this a good story.
If you spend $20M working out a Millennium Prize problem, in what universe would you offer that an unrelated effort should take credit? This is a branding game rather over whether software engineers are going to use codex or claude. In that light $20M (or whatever it was) might be worth it to squash even the rumor that claude code is more capable. Engineers look up to mathematics, while at the same time business and probably most engineers think the problem was to solve the problem. GPTs solved one the hardest known problems so they can solve my company's problem.
Some go further looking at these people, very on-the-nosely likened by one commenter here to ants, talking about education and responsibility and "core values" etc and just don't care. There was a major problem at the beginning of the week that is not a problem now and that is uncomplicated progress.
It's not wrong for OpenAI/Anthropic to do math for product development or even just branding but seemingly at no cost now they could work in an arena real mathematicians aren't interested in, versus just mowing the field. Everyone involved on their side should admit the purpose of these demonstrations is not to engage mathematical ideas it's about Claude/Codex. there's no shame in that. Which is better at solving random hard Diophantine systems? That would seem to tell me as much as I need to know insofar as a model's value is represented by raw mathematical power - then, take my money just as well!
Assuming the worst accusations are not true I think there are ways forward going to be acceptable for all. The labs themselves do not represent Terrance Tao as some kind of gate-keeping dinosaur in this. They're not interested, not the kind of entity that can care about theoretical mathematics. These dudes are paid 7-8 figure salaries ultimately for the product, they solve a Millennium Prize problem then pretty quickly seem to move past it.
I don’t think they make a coherent argument here? This seems to hinge on some argument that because AI doesn’t properly explain its breakthroughs, therefore the breakthroughs are less fertile for human understanding? This makes no sense. Why wouldn’t these under-explained breakthroughs be extremely fertile soil for explanations?
Imagine time traveling back in time and offering Leibniz a packet of proofs from the intervening years, but with the caveat that there would be no explanations. Would he say no?
"how to make sure that, as AI changes the way work is done, we do not lose sight of what that work was meant to achieve in the first place."
That sounds like an "us problem", not an AI or OpenAI/Anthropic problem.
And I am outraged by their dinosaur mindset and the gatekeeping mentality that force every student to follow their same archaic system that no longer makes sense 20 years ago, let alone now.
It sounds like we are not happy with getting answers like 42 to our questions about life, the universe, and everything
The average programmer is outraged by OpenAI's methods, too.
I don't care how good Astra or any subsequent models they may release might be... I am never going back to those token reset shenanigans.
I'm so bored by HackerNews commenters on decelerationism stuff. Model doesn't care; the ability for math problems is emerging, not trained.
These mathematicians are not suggesting anything interesting, and the announcement more like desperate crying stuff
> We are witnessing a general threat to intellectual work, with misalignment between the outcome of the use of AI and its initial purpose. In many fields and activities, years of training have traditionally served not only to produce a final answer or product, but also to develop understanding and the ability to formulate new questions and ideas. However, building on a vast body of previous human work, AI systems are becoming increasingly capable of producing the results of such work directly, and these goals cease to align.
This is Terry Tao talking about AI's impact on Math, but this could just as well be a software engineer talking about AI's impact on software development.
Do mathematicians deserve more job security than software engineers?
The concerns seem valid.
I'm unclear what the ask is, though. What, even in theory, is a practical and realistic fix?
> But solving problems is only a tool and proxy for achieving the primary goal of conceptual understanding and insight. Forgetting this in the world of AI may turn the tool against the primary goal.
I will quote Richard Feynman - “the prize is in the pleasure of finding the thing out, the kick in the discovery”.
I am not saying that should be the case for everyone in every discipline. But if there is one subject that is mostly pure curiosity driven (instead of worldly impact), it is math. Robbing them the primary motivation is brutal.
If all problems are solved by a machine, what do we have left to satisfy our curiosity, our desire to explore, and where can we find the pleasure of “figuring the thing out”.
Any proof for or against a mathematical conjecture, bruteforced by AI can be the spark for new insights. I'll concede that to the AI companies.
But I agree with the sentiment that the marketing behind these "discoveries" is disingenious. They pretend they solved the problem, but it still takes a bunch of humans to reduce the solution to a simplified and sensible explanation.
In the Economist article Tao links, Hugo Duminil-Copin, draws a comparison: airdropping someone on the summit of Mount Everest is very different from climbing it.
The fundamental issue with AI solving any perceived difficult problem is that we have lost the journey. The sight atop Mount Everest looks much different when you have climbed compared to being dropped from above.
Lots of Fields medalists signing this. Interesting to see one not there: Timothy Gowers.
I agree with many of the sentiments here. But an open letter signed exclusively by Fields Medalists that purport to define precisely what the "mathematical community" (who is inside and outside) and what their goals are raises my hackles for some reason.
It is true that the manufacturing of "true/false" statements is not the same as gaining understanding of a problem. However, for many mathematicians, true/false statements are already manufactured by others. Think of a student who is given a conjecture to investigate, with their advisor describing it as "it must be true". Most exercises in a textbook are stated such that the outcome is known before you begin. That's not really a problem -- investigating the conjecture/exercise yields its own dividends, whether or not the outcome is known. It is also the case that defining new directions involve understanding and synthesizing related problems, asking the right questions, and deciding upon the right directions, and it's not clear that AI can do that at all.
The real risk, I think, its the public's (and funding agencies') perception of the importance of "human" mathematics, but that's already a struggle. For example, it's tough to explain to the lay person why it's still important to research group theory -- the main example people cite is RSA encryption, which was invented almost 50 years ago.
> ... 25 initial signatories — all Fields Medallists —
As a non-native English speaker, I initially understood this to mean that all living Fields Medallists had signed. I later realized that it meant only that all the signatories were Fields Medallists.
(Apparently, there are 47 living Fields Medallists today.)
Back in the day you could think of a cool idea. I don't know maybe a plane that could fly without drag. To even see if this was feasable you had to understand physics, engineering, and then from there you had to have a math person see if it was actually possible.
Now, I can ask ChatGPT about this and get back a proof that shows "a passive airframe cannot sustain zero-drag motion through still, viscous air"
So, I think if anything now, Maths has changed for the better. More ideas can be proven false or true from a get go instead of wasting so much to see if its even feasible to find out it isn't.
Progress if anything is about to leap frog anything we have ever known.
To me, the "meaning" of proof is twofold. First is the understanding which is completely absent from a proof which depends on exhaustive iterations of instances or is asserted by fiat from myriad individually contestable paths. That's what I think makes AI proofs risky: they absence of understanding.
The second is the utility. We get to do navigation because the maths about angles and spheres checks out. The social utility downstream of AI proofs may be huge.
I don't like this disjunction.
The root of all these is the culture in mathematics (and science in general) to only reward those who “get there first”. This creates a perverse incentive to compete. When no one can out compete a tireless swarm of AI, no one gets rewarded any more.
But nothing’s stopping anyone to still work out an alternative proof, or a more elegant proof, or just trying to prove for the sake of understanding, just like doing homework without looking at the solution. It’s just that you can’t get paid doing that anymore.
Playing a devil's advocate. Why do we need understanding ? To take an example i would say ~99% of the population do not understand how combustion engines or how semiconductors work, what say another 1% ? Is the fear post-apocalyptic in nature ? We need some human priesthood to carry on tradition ? why ?
Let's assume in the next decade GPT-7 PRO Ultra is cheaply ubiquitous, inspectable, reproducible, transferable, reasonably un-constrained by any institutional interests.
What say the 1% ?
Lifted up my comment for addition visibility
The women who made up the workforce of telephone switch operators would like to have a word.
Meaning - every new technology has both been perceived as a threa and often forced change in society. Agree maybe “it feels different” this time, but don’t you think everybody before us just said the same thing?
Also not clear if this is an actual called action.
Academics should never leak their research to ClosedAI lest their work be stolen. Universities and corporations will have to build their own compute to not have their data stolen.
Nothing AI companies will change the value of math, as William Thurston said: The product of math is clarity and understanding, not theorems by themselves. What they seek for the IPO is devilish and misleading, and doesn't serve the true purpose of the math.
This is really well written and exposes a core tension between science and something akin to engineering. The "engineering" of proofs has become "easy" (a compute and $) problem, rather than hard (a time and conception problem).
Without the ability to do things the "hard" way it is difficult to figure out if doing things the "easy" way will help us advance the frontier of math and science.
I may be wrong but historically we had this version of science discovery for a long while (empirical observation and brute force application) rather than first principles leading to applications (tools, the wheel, mills etc). Then somewhere along the way it flipped after Newton and the enlightenment period and started understanding first principles before they become engineering applications.
Perhaps it is not required, and we can just keep doing things the "easy" way like we used to, or we might find ourselves out of the ability to brute force things and then we go back to needing to do this the hard way, at which point this period of AI brute forcing would be seen as a detriment.
As a mathematician maybe I am a little more optimistic than this declaration.
I am thinking of Mochizuki's abc conjecture: He worked in relative isolation, and dumped a huge incomprehensible proof on the community (to oversimplify a bit). That's not totally unlike what might happen if AI generates a huge, incomprehensible proof of let's say RH.
Well, what is the result? In the Mochizuki case, it was a lot of skepticism, but it also generated conferences, papers, talks in the hallway, discussions with students, and so on--a flurry of exactly that kind of community process that the declaration says is the main driver of mathematics.
Ultimately we think a fatal flaw was found in Mochizuki's proof, so it didn't lead anywhere in particular. But in our hypothetical "AI lean-verified proof of RH" situation, it would presumably generate substantially more of that community activity we saw in the Mochizuki situation. And if it's correct, that community activity would be productive (expository talks, students given problems to flesh out or generalize, etc).
Maybe mathematics just becomes a little more like other fields--relying on labs with lots of money for compute, digging through a corpus of AI-generated proofs, etc.