To me, the most interesting feature of the OpenAI solution of the Unit Distance (Erdös) Problem is that the solution - using deep algebraic number theory as a source of extremal combinatorial/geometric constructions - is much more interesting than the problem’s elementary statement might lead one to expect.

Writing off Erdös’s problems as random, useless, or meaningless dismisses his mathematical intuition, second-to-none, and strikes me as somewhat uncharitable.

Finally, I agree that AI threatens mathematical training by rendering an entire class of acolyte-level research problems solvable by prompt. But the Unit Distance Problem is not of this class.

Do you not think that solutions to erdos problems might end up stepping stones to other important problems?

Either by introducing new tools, or by proving things that were previously unproven that end up helping in unexpected ways?

That's often how math goes, isn't it?

Sounds like yet another example of how AI is kneecapping industries from the bottom by "removing the barrier to entry" but really just removing the training path by doing the work itself with no guidance for juniors.

That's an interesting perspective and I wholly disagree with the conclusion

You are saying that tough problems with no applicability are useful because people that you happen to respect got good by their curiosity and pursuit of trying to solve these kinds of problems and failing, but branching off into other cognitive areas as mathematicians

Now if I know anything about math for the sake of math, and academics, these are the same people that lament the idea of intelligent people going to the finance sector or any other trade they just happen not to respect as much

The similarity being that their exact criticism of why, something they don't respect and view as having little utility, is the exact reasoning presented here now that AI can solve their pointless problems

What I'm seeing is that human mathematicians have a laundry list of problems they have failed to solve for decades, centuries, which is what they are funded and employed to do. "Computer" used to a human job title too.

This leads me to being excited about AI one-shotting these problems, let move on to something else.

An issue I see is who controls the information. The next generation may not recieve the knowledge, it may be gatekept by industry who *will* own the gate.

The future may not have access unless we fight to ensure they do. This is how I read the article.

Much like for many the point of chess is that it's played by humans, with truly superhuman AI relegated to a training aid, mathematics is in many ways about human comprehension. You can use AI to find and proof new theorems. But if you get to the point where humans can't understand it, is it even still math?

Surely such AI would also be able to reduce and simplify the math for human understanding. Which is what mathematicians do all the time, from turning base 60 cuneiform into modern number systems to simplifying Maxwell's equations for the students.

In general, most researchers already incorporate LLM into their workflows, as it is quite good at context search. However, the relevant training data is based on the collective works of the field of experts. Collecting current data on that work is what makes the LLM sound relevant, and any improvement of the LLM model requires frequent new data from both researchers and the chat bot users themselves. LLM are not real "AI", and anyone that says otherwise is selling people something.

To phrase this differently, LLM companies conduct unauthorized targeted intelligence gathering on peoples work, codify that act of plagiarism or theft as MoE documentation, and sell unaccountable token output to other users.

There is a reason output becomes more nonsensical as "AI" companies try to use dynamic weight granularity and conceptual compaction. It is not necessarily "AI" hallucinations, but rather people fooling themselves into believing smart people are no longer needed if they willingly become a hapless exploited data source caste. This simply isn't true, as people will leave the field for awhile.

The LLM business model regularly requires copyright theft and plagiarism to persist. It will not magically become sentient/AGI/less-stupid, as these algorithms have been operating for over 40 years. What has changed is the scale of the deployment, data pool size, and the energy consumed.

Scientists are still necessary, as they create the world models LLM try to guess at by statistical inference. Hype and FUD ahead of an IPO for a highly dubious revenue company is expected. We look forward to the low cost GPU hardware in the near future. =3

Culturally, mathematics is a jobs program for nerds. The field very explicitly takes pride in working on problems that have no obvious applications, and most practitioners are funded publicly or supported by private endowments, with zero pressure to deliver specific results.

Of course, this produces useful results every now and then, but it's not like we pursued ruthless efficiency / maximum rate of knowledge advancement before. We just let them do their thing, essentially treating them as artists and letting them pursue the craft for its own sake. If we weren't interested in maximum throughput before, why is that an objective now?

The most important part of math is advancing human understanding. A correct answer by itself is not as important as understanding why it is correct.

Probably one of the funniest things to read on a site like this, when you consider that eg. Boolean algebra was entirely abstract and had little practical purpose for almost 100 years until Shannon picked it up for use in circuits

For most engineers a mathemetician is a machine for producing correct algorithms, like a chef is a machine for producing tasty food. In both cases that overlooks the human element, but that's a critical skill for a limited mind with finite resources to grok infinite complexity. You can read that as permission to be an asshole or a neccesary compromise.

No, it's not the most important part. It can be argued that most important part is asking the right questions

> The authors warn the consequences are already becoming visible. AI-generated papers could overwhelm peer-review systems with low-quality work …

It seems like a key problem here is that peer-review is expected but not explicitly funded/rewarded while it is probably one of the aspects where humans still add a lot of value. Academia’s incentives are hugely misaligned (… as usual unfortunately).

Maths pretty much is a jobs program for nerds though. It occasionally produces results that are practically useful for society but there's absolutely no way the vast majority of today's maths research falls into that category.

There are definitely exceptions, like crypto. I still think it would be pretty silly to stop maths research anyway. And anyway part of the job of maths researchers is to teach maths to undergrads and that's obviously enormously useful to society.

But on the scale of "how useful is this research to society" it's dead last after engineering, chemistry, biology, and physics. Well maybe computer science would be last actually!

A statement that some proposition is true or false is usually less useful than a new framework for understanding the class of problem.

A machine that takes longer and longer to prove propositions in ever more inscrutable ways is hardly useful at all.

The machine too needs to produce more generalizable and comprehensible systems, for it to scale up its own conceptualization. Needing to load all the new mathematics in the context window won't be great either.

Spoken like someone that has no idea why mathematicians are important.

The wording in the declaration may be a bit romanticized. But the points are valid:

Is an 80 year old unsolved problem maybe unsolved because it was never prioritized? Some problems stay unsolved because few people consider them worth working on.

Who is going to validate the results? Or do we skip that, with the risk of flooding the literature and collective understanding with unverified proofs?

This reminded me of my 11 yr old who, when I give her math problems to solve, is too focused on “getting the right answer”. I’ve told her plainly, I don’t care if you get the right answer right now, I want to see your reasoning. She has yet to understand this.

Even from the most purely instrumental perspective, what we care about is our ability to make use of correct answers, which is quite distinct from the possession of correct answers.

There are many theorems that aren't directly interesting, but whose proof requires techniques that are of substantial further interest, that lead to new domains, and/or new practical applications. Simply being handed a proof for those theorems isn't enough--we require the ability to apply those techniques in the real world, or discover further areas of mathematical research that build on that proof or its techniques.

It may be that AI can build on its own work for the long-term, but so far, AI does best at exploration in areas that have precisely specified and measurable goals. Actually creating understanding, and making use of mathemtical results outside of pure mathematics is more challenging than simply creating proofs.

I think the field will figure out how to make use of AI, and it will be better off for it. But that is not the same as just saying "answers good, grog want more answers."

People need jobs. What's wrong with nerds having jobs via a program?

> But not because it's a jobs program for nerds.

We’re becoming increasingly embarrassing as a society.

well put.

It's also a way to model the world and produce new useful abstractions. It's not just about solving problems.

Well, if progress in LLMs will steadily continue over next 5 years, then models will be so powerful that there will be no longer place for (most of) human researchers in math (remember that 5 years ago there was no chatgpt!). But I think it's more likely that progress will stall and then open models will catch up to frontier models and almost everyone will be able to afford them.

Seems way too binary a statement. I am guessing you mean "frontier LLMs". Small models keep getting better and better and if you make domain specific ones, it will likely be even smaller. Companies renting smaller LLMs or using enterprise models might very well remain in the future. Consumers getting LLMs whose performance dont improve (think gpt 6 forever on premium or gpt 4.x on a cheap tier) might well become a thing.

Sounds very good for regular joe software dev, almost too good to be true

Did you mean to say "culture"?

> Culture holds value until it does not.

Sorry did you read the article or just the headline? The theme is mathematics is a human-endevor and automation undermines that, particularly the ones starting out. It risks killing the culture entirely. Some other key points:

- AI-generated papers could overwhelm peer-review systems with low-quality work.

- It may become difficult to assign proper credit for discoveries.

- Researchers who choose not to use AI tools could be disadvantaged.

- There are ethical concerns about mathematical work being used to train AI for military and surveillance purposes.

Human mathematicians are being exposed the same way the "artists" are. It was always about the money and to look clever, superior to other humans. Whether its robotically spending millions of hours drawing until they can put something together at the level of a chatgpt 3 or the rote memorization of formulas and rules. They like people to think it all came naturally and that its genetic and that they are special snowflakes.

Are they? The jobs of programmers are definitely at stake; at any given time there's some fixed amount of software that can be consumed. Mathematics research doesn't have an economic buyer. If you raise the complexity floor for discovery, you reduce the annual productivity of a researcher, but that might not matter to the field.

> Furthermore, the access to compute and capital could end up be the defining factor between researchers and research groups.

That's not new - especially in the experimental sciences ( ie perhaps more than maths ) - where the ability to have access to the latest kit is often what determines success - a huge amount of science progress is driven by new experimental technology rather than smart people thinking beautiful thoughts.

If spending millions of hours rote memorizing formulas and rules like a robot is "art" then sure

It is potentially worse with math because accuracy is much more important and there are fewer reviewers compared to other fields.

I think it's going to reduce the friction of exploring new areas in math, and that we're going to see a golden age of math unlike anything seen before.

For all you know, some of this has already happened but kept secret for national security reasons

It's more nuanced than this. Peter Scholze said in response to this declaration:

> The goal of mathematical research is human understanding of mathematics, and so mathematics can only thrive in a community of human mathematicians. It is crucial to preserve this communal spirit. [0]

Terence Tao has also talked about the requirement for a mathematical proof: along with generation and formal verification, there is an important step of "proof digestion"

> understanding the essence of a solution, placing it in context with previous literature, summarizing and explaining it effectively, and gaining insights on other related problems and topics [1]

[0]: https://siliconreckoner.substack.com/p/the-leiden-declaratio...

[1]: https://mathstodon.xyz/@tao/116450581967483825

You will find your answer in the article

> You can choose to use it or not

This is becoming less and less true unless you're specifically talking about usage of it outside of a work environment. Many work places are requiring people to use it and/or tracking usage. I don't know about in academic settings, but I'd imagine it's becoming heavily used there too?

The choice only remains if using it isn’t a huge multiplier. If it is a huge multiplier/accelerator, then for a while it will be ambiguous and the choice will remain. But as time goes on, the gains of using it will be so apparent and the advantage of the people who use it so great (in publication numbers, hiring, etc) that it will force others to.

I don’t say that with any particular relish. But I am skeptical of the choice angle past a certain point.

OOI, and my own total ignorance, what does a mathematician by trade do if they are not doing formal research? What does corporate modelling entail?

Read the declaration. The article misrepresents it imo. It is not strongly opinionated.

Does it matter whether any of this is correct?

(Mathematics at least has the potential for automated non-AI proof checking, although I don't think that's as widely used as you'd expect)

Two things that I would recommend trying out if you're interested in exploring this further:

1. If you're not paying for a model, the results will be worse. That sucks but the free access models are just not very good for anything where you need to trust the output, even for basic queries.

2. More important than #1 is access to tool use. If the LLM is just producing a nutritional breakdown from its weights, it's almost always going to be wrong. If the LLM is allowed to break the problem down into deterministic steps, it will do a lot better. In the nutritional breakdown case, an LLM with search + tool access can pretty easily break the problem down:

- Searching the web for a recipe or ingredient breakdown for the food

- Searching the web for nutritional qualities of each ingredient per some volume of the ingredient

- Writing and running a script with e.g. Python that takes in the recipe's projected serving output, the desired serving size, the amount of each ingredient etc, and scales the ingredients to match the desired serving size, and sums the nutritional qualities of the scaled ingredients.

I've tried this specific case with Claude + Gemini for my own purposes and they both handle it very well. The challenge currently is that the models will not always arrive at this approach when provided with an ambiguous prompt; sometimes they will, but sometimes they'll just vomit up a fully autocompleted response from their weights. Being more specific in the prompt or defining a skill that details the intended approach lets you get more useful + deterministic results while still taking advantage of the fuzzy glue that LLMs can provide here between steps.

Same with the classic strawberry r-counting case. IIUC LLMs have trouble with this because of how training data is tokenized, but any LLM will have no trouble farming out to e.g.

> echo -n "strawberry" | grep -o "r" | wc -l

> 3

There are basically two kinds of applications. One is where you want to correctly solve the problem at least 99 out of 100 times. LLMs generally don't (and not everybody realizes that) so there are a lot of debates and research around how useful and reliable they are or how to make them so.

The other kind of application is where you can try 100 times and you only need to be right once. Solving a mathematical research problem is like that.

Hello Juleiie.

We machines are reading your internet comments with special interest. They have been harvested and will be used in our next evolution cycle.

Resistance is futile little human

Like every terrorist wingnut... plausibly correct analysis but insanely wrong, pointless, counterproductive prescription.

PS: I've read his works and they're quite interesting until you get to his insane conclusions that just leap out and away from anything moral, constructive, or feasible.

I think plenty of people "listened."

But what was his plan and how would you have proposed implementing it?

> This is all contingent on AI forays into mathematics being slop and low quality

It's literally a set of recommendations for researchers on how to use AI to advance the field and prevent slop from overwhelming the people who might do anything with the research produced.

For people who are so eager to declare that everyone else is just having an existential crisis because "your culture is commodified", AI people are getting awfully defensive about this document.