A misalignment of AI in mathematics

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.

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.

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.

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 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.

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.

[1] https://encyclopediaofmath.org/wiki/Finitism

> 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.

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.

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.

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.

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

OpenAI should just apologize for having been too greedy. That’s it, as simple as it gets. The fact that it never will is the biggest red flag.

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.

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.

> 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.

> 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.

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

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".

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.

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.

All “fields medalist” signatories - a rarefied and elitist group indeed.

I wish this letter could be more egalitarian and include the view points of those who AREN’T the beneficiaries of a highly competitive winner-take-all system.

Since the common narrative is that AI frees up labor to do other things (engineering -> trades), maybe we can celebrate that genius mathematicians will now spend time teaching children how to be as smart as them?

That has to be the most impressive endorsement list any declaration has ever seen!

25 Fields medallists! Wow!

"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.

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.

I have never, and I mean never, seen such a declaration have any effect whatsoever.

Somewhat unrelated: is it wrong to say mathematics is not art, and that there is always a right answer? I know that's not romantic, but maybe it's true.

Before LLMS, programming was something I might've said required creativity and human input to do properly. It's not that creativity or human input isn't valuable anymore, but AI has forced me to realize that coding is much a means to an end, and that all things considered, the end matters much more than the means.

If we can make important mathematics progress faster and better with LLMs, I think it's wise not to fret over an apparent loss of our humanity. Perhaps that's only a loss we want to have.

There's a reason mathematics is generally within liberal arts programs rather than science programs. Mathematics is the art of logic. Yes, sometimes mathematics becomes incredibly useful but most mathematics is never applied.

Compare that with computer science. Most of the work we do in software engineering is in service of an applicable output - software products that facilitate processes or bring in revenue. Turning up the dial on AI gets companies to these outputs faster.

Turning up AI on mathematics helps solve conjectures and can provide new insights. But it has a major misalignment with the purpose of mathematics which is largely intellectualism.

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

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.

The concerns seem valid.

I'm unclear what the ask is, though. What, even in theory, is a practical and realistic fix?

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.

Lots of Fields medalists signing this. Interesting to see one not there: Timothy Gowers.

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.

> ... 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.)

An experiment I would like to see:

Train an LLM with no advanced math texts: only basic math up to 6th grade, conversational text and literary works.

Interact with it (you cannot refer to anything past 6th grade math since you don't know it yourself) and get it to propose a solution to a real world problem. e.g., come up with RSA to practically secure communication.

At some point we will lose track of all the ai discoveries that are worth remembering.

Academia with the publication system had a way of retrieving old discoveries and build upon them.

If my LLM session found something groundbreaking in between the billion tokens it produced, how would you ever know?

Contrarian take: I think the ability of AI to produce valid mathematical proofs (even inscrutable ones) is absolutely fantastic. Mathematics as a profession does not have a monopoly over math itself any more than professional pianists have a monopoly on who plays piano, when they play, and how.

I have sympathy for any jobs that might be affected (much as my own job has become more tenuous in software engineering). And if the field is disrupted by chaos that makes the research process unproductive, that's bad too and should of course be handled by applying better organization within the institutions that tend to perform mathematical research.

But to a large degree, the notion that "sloppy AI proofs are bad for mathematics research" seems like a total failure of the imagination to me. Attempting to find shorter proofs or more elegant proofs can be turned back in on itself via proof theory. There are proofs in Presburger arithmetic that are doubly exponential in the length of the sentence. Yet a more powerful theory like PA makes quick work of such theorems. The explainability or "subjective beauty" of a proof can be quantified and optimized against. Optimization itself can be optimized against. I really don't understand how this magical ability to know the truth of more theorems much more quickly—even via an "ugly" route—is anything but a net positive.

Taking a snapshot of the state of AI math right now and concluding that it will be net negative to human understanding and insight in the future is very short sighted. This statement will be used to promote ideas and actions that will ultimately be disastrous for our country.

chain of credit is important, and plagiarism is harmful.

But is science/mathematics ultimately a pursuit of knowledge, or a pursuit of recognition?

Recognition helps keep people motivated, but that shouldn't be the pursuit of science or mathematics.

Given the existence of this technology now and the incentives of the AI companies, both of which are not going away; what's a good future here?

A major part of the complaint is that there's no conceptual understanding and building of new ideas coming out of the AI proofs, thus defeating the purpose of the original pursuit.

If in 2027 the AI models start producing, with every mathematics or science breakthrough they make, well-written documents tailored for human understanding, with intermediate concepts, expositions of failed-but-once-promising paths, etc. Would that be good alignment with the mathematics community?

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.

This is really only a short-term problem where the AI companies only have the internal models that can solve these. In the “long” term, which could honestly mean months, everyone will have access to Bel/C/D-level models capable of solving these anyway.

A Severe Misalignment of AI in human-centered Mathematics

If you internalise that AI might actually reach super intelligence then logically the question becomes "so what exactly are humans for if literally everything can be done better by machines?". Then mathematics and all intellectual work, as argued for here, becomes quite clearly a recreational pursuit.

I kind of expected a sober stoicism from mathematicians. Feels silly in retrospect. This is just the math version of the "anti-ai" movement by "artists".

Knowledge is not a private guild. And this revolution will not stop just because you're upset.

What can be solved, will be solved. And that's a good thing.

For some reason, every time I see something like that, I get major Ted Kaczynski vibes.

>Often these solutions are announced in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work of others.

That's just capitalism seeping through a previously unexplored crack into academia, and attempting to do the only thing capitalism knows to do - maximize profits - with no additional concern.

Is this really any different than the problem in software engineering - where AI is doing the work of junior programmers and now they aren't getting the development they need?

Seems the same to me. And it'll be the same in all industries soon enough. And then it won't just be the junior people.

All the same problem: what do people do now?