> However, the declaration argues math is more than a machine for producing correct answers. The discipline, its authors believe, is a deeply human endeavor built on creativity, understanding, collaboration, and the pursuit of knowledge for its own sake.
Generation X was the last generation that had 'general knowledge', as in an abundance of fairly useful information stored in 'grey matter' that could be recalled quickly. When search engines came along there really wasn't much need to know anything since most things could be looked up. However, you still had to think.
With LLMs, thinking is kind-of optional. This really is an existential threat to our intelligence since 'use it or lose it applies'. I am glad these mathematicians are doing their duty as canary in the coal mine.
I understand that the "language interface" of a "maths AI" could be some specialized trained LLM (Large Language Model) that to convey, with human language, "high level" mathematical mental contructs and intuition.
But then, you would need some models which does the reasoning using formal mathematical solvers (and probably a ton of "scratch" memory, it would be interesting to see how those models end up storing "mathematical" lema data). I guess you can have ML (Machine Learning) for those models on 'general maths', but also we can think about more mathematically focused ML for a specific problem, area, etc. And in the end, ML for maths, would it be mostly permutations of truth statements fed to a neural net?
When we were talking about "AI", one decade ago, that was what most had in mind (it may help a bit in physics, but it seems less likely, because reality/experiments are hard to teach to "AI"s).
If that becomes a reality (aka easy hardware access, and some "working" models), mathematicians will have to be as good in maths than in maths ML. And this is were there is an issue: training honestely good mathematical human brains may become very hard with some broad availability of good general maths reasoning "AIs".
Far more interesting as it's outlaying a set of principles for using AI to augment human involvement and science, rather than replacement.
I mean, what field doesn't? Everyone works to make money.
Slightly unrelated, but, their website "https://leidendeclaration.ai/" itself gives an eerie feeling of being built by Sonnet. That color scheme and the layout is what Sonnet chooses by default most of the times.
For years?
The barrier to entry just got lowered. This has happened many times before in history. We just end up with fewer of what David Graeber would call "bullshit jobs."
I briefly studied at a pure math department. We were learning linear algebra and I found the symbol heavy, proof oriented approach very difficult and unintuitive. But when I squinted at the diagrams I realized, oh wait, this actually has dozens of practical applications! Across dozens of different fields! How fantastic!
And the textbook, for some reason, chose to mention precisely none of them. Which I found quite disappointing, because it made the whole thing seem quite abstract (which it actually wasn't), and made it harder to understand.
I mentioned this to my colleagues, who became extremely upset, and informed me that I was in the wrong department.
In a word, the job of the mathematics department is not only to produce mathematics, but mathematicians.
Similarly, the output of programming is not only a program, but also a programmer. It is you.
Outsourcing the work deprives you of who you become by writing it.
Some questions are more urgent and practical. My feeling is that the more directly practical a question is, the more likely the research community is to support AI usage in that question.
The annoying thing about recent AI advances is that they target questions on the wrong end of the spectrum: Erdos problems are exactly the sort of "useless" questions that people might answer purely for the love of the game. The sort of questions that a young person might cut their teeth on and gain confidence.
Solving questions like these automatically, I think, is not good for the long-term health of research. At least for the foreseeable future you still would like people to become interested and develop skills in these fields. These developments, and especially how they are presented, directly discourage that.
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.
There might be more to maths than that, but that is definitely the most important part. I love science funding. But not because it's a jobs program for nerds.
As a former physicist and current data scientist/engineer, I know for a fact that commercial utility drives math research and researchers.
Math is a tool to solve problems. Some mathematicians might only love the process of using the tool, but commercial logic absolutely drives mathematician attention to develop commercially useful tools.
AI makes the math world more accessible than before. If you have a question about a proof in the lecture, you can just ask it. Of course, one can't trust it blindly, but fundamentally it's amazing.
I think that's a good thing, but of course this means that a lot has to change in culture and behaviors, also in the research world.
The software engineering world is more or less in the same situation, it's also changing. But for now I think it still holds true that someone who knows maths plus an LLM is better than someone who doesn't know maths plus LLM. At least in software it does.
He states that he struggled to come up with problems which would be challenging for AI to solve (at the below site) and thus forced to accept that mathematicians have to rethink their profession.
FrontierMath: Benchmarking AI against advanced mathematical research by Epoch AI - https://epoch.ai/frontiermath
As a follow up to the above, see "First Proof: Mathematicians Putting AI to the Test" featuring eminent mathematicians - https://www.youtube.com/watch?v=AaICCTpkI7Q
That's why there's a disconnect when you go from math for engineers to the stuff above it. It feels less useful and very different
That's not a problem unique to math, or even to academia. It's a problem in every context in human life where people communicate via written documents.
They learn how to read papers and literature rigorously. They get low-hanging fruits to practice on, which can take months. Their funding doesn't come from thin air either.
So what happens when the group leaders would rather spend money on compute, and get models to solve the low-hanging fruit? Which the models could very well do in mere hours, compared to months.
Nor does it help that publishing is the number 1 measure in academia. Furthermore, the access to compute and capital could end up be the defining factor between researchers and research groups.
It is basically the "junior problem", but even more severe.
Except when someone hands you a magic button that just gives you knowledge?[at least in the framing of this "warning"] Then it's about peoples' livelihoods, about "culture", etc?
"Computer" used to be a job. Did science on the whole lose or gain by making these clerks obsolete?
At this stage, the current wave of AI is not reliable enough that it would be safe to lose the abilities it can replace.
The failures modes are often turned into memes and jokes, but they are the thing we should really pay attention to, IMO.
I will note that the average corporate mathematical modelling is usually a fucking circus so adding AI might make it better.
Every time I ask ChatGPT to make a table for a subject I know well, I will find an error in one of the results and it is very confident about it until I question it in detail
Every time I ask ChatGPT for nutritional breakdown of some dense food source and give it a quantity like 8 ounces and ask for the weight of each ingredient, the weights will be wrong and add up to more than the original weight of 8 ounces
These are variations of the old "how many Rs in strawberry" problem, it's still not solved, "AI" cannot reassemble a complex problem properly
A lot of what it tells me in detail about some subjects sounds suspiciously like Reddit posts reassembled out of order
The ability to clearly outmatch trillion dollar machines is a very unique satisfaction. I even write ordinary internet comments with an intention to make them clearly better and more fun to read than boring Claude output.
So, why would they be advocating for limitations on arriving at solutions?
The issue is, how is a group of intellectuals, whose identity derives from their ability to do something rare, useful, and requires many years to get good at, react when a machine can produce all of their useful output nearly automatically, can verify its own outputs, and is getting better exponentially? It is the complete annihilation of one's sense of value and purpose when the binding element to your culture is commodified.
I think there will be a lot of arguments trying to claim that the point of mathematics is curiosity, or that there is always some ineffable human element that AI can't replicate, but I fail to see how somehow these wishy-washy human centered values somehow mean anything compared to the amoral pursuit of mathematical truth, which has nothing to do with humans.
It's just that we humans happened to be the only beings in the universe good at math until ~2025. Now there is another species which can do many of the things we do, and it is not bound by the size of the human brain, our short term memories, or the architectural limits of biological computation. To imagine that humans would retain supremacy in this very un-human like discipline seems like wishful thinking.
This assumption may well turn out to be correct, but it is not self-evident.
Nearly everyone who has ever got interested in mathematics got discouraged at some point and they left the field. Mathematics is very hard. Those very few that remained certainly have talent, but they also have characteristics that are necessary for success in a competitive field, which are perhaps less valuable per se. Such characteristics as may be over-represented in males for instance. This is not a point about gender differences, but about the intrinsic merit of different success factors.
It seems equally possible that the above assumption will turn out to be diametrically incorrect. People that would have been discouraged before LLMs will now retain their curiosity longer. Democratisation is surely a possible outcome.
Arguably, chess has never been as popular and accessible. And that discipline fell to AI three decades ago.
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?
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.
I agree that automation is good when it frees us up for higher pursuits, but what is the end goal? All human labor is replaced? No human can produce anything of value relative to a machine? What does our life look like then?
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?
I don’t say that with any particular relish. But I am skeptical of the choice angle past a certain point.
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
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.
- 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.
> Culture holds value until it does not.
(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)
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
> 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...
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.
This can be said about pretty much any job on earth.
By that definition nothing should ever be automated.
Everything thinks they are special, actually no one is. You become special by being rare. Find something that can be done by no one or only a scant few.
Just because AI can do something that resembles work should not mean outsourcing work to it. Mathematicians should not outsource their work to AI just like programmers should not outsource programming to AI.
Humans working with AIs in a tight loop means intellectual work becomes more high-level and creative, but a human should always own the work, validate it and stake their reputation to it. Simply ban any humans who produce low quality work using AI.
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.
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?
That isn't really true. After push button elevators with floor-logic relays eliminated the need for "elevator operator" to be a job, nobody needed to be an elevator operator anymore. The equipment could do 100% of the job and if the equipment was out of order then you call a repair technician or install a new elevator rather than needing to find an elevator operator to pull out of retirement, since knowing how to repair or install elevators was never part of their job to begin with.
The trouble with AI-generated code is that it can't do 100% of the job, so you still need a programmer to do the parts that it can't, but then you need the programmer to understand how to do the parts that it can't, which in turn requires them to also understand how to do the parts that it can.
Many things shouldn't. Understanding is one of them.
That sounds very nice, but isn't true. Most of the people I know, myself included, don't consider themselves special broadly. They're special in their own community, but not globally.
As opposed to, say, drug discovery.
This is reinforced by the immediate (human) use of the idea to resolve in the negative another significant problem, the sum-product conjecture on reals.
Explanation of what was involved: https://www.erdosproblems.com/forum/thread/blog:6
Businesses will not adapt until they are incentivized to do so, and very few businesses have a multi-decade outlook. Even before AI, the senior 10x employee who retired and took all his domain knowledge with him because there was never any funding to train his replacement was a problem.
The future may not have access unless we fight to ensure they do. This is how I read the article.
"But Jobs" they scream and hijack the council to block new housing. I'm sorry folks, the point of housing isn't the jobs it creates, the point of housing is housing! New jobs ARE actually created, they are just higher leverage ones in the house factory.
Mathematicians are now facing the same... calculation (pun intended). And I think they are empowered to create a lot more leverage, and they shouldn't be afraid of it. A lot of them are catching on to this [1]
I don't think any of these AI layoffs are actually because AI replaced a human. I think a lot of the layoffs are actually just due to a faltering economy or greedy companies trying desperately to get a piece of the pie, so they're sacrificing their long game for short term gambles. I don't think that's going to pay off for them.
They don't seem to realize that 100X the resources have been dumped into building coding/design tools as other industries. As soon as that hose gets turned towards their industry they are going to be in a similar or worse boat.
I personally do wonder (worry) about where all of this pans out and what society looks like post generative llms. But at the same time there is a particular flavor of amusement that I can't help feeling watching folks simultaneously balance, "llms produce nothing of value" and "llms are so harmful and dangerous to our culture that we need to start policing use within our community"
Where that harm essentially stems from devaluing hard earned skills within the community. And while I do not take joy in the displacement of labor, never in my wildest dreams could I have anticipated how harsh and irrational of a reaction to the equity of these skills could be. Which, I would like to point out, though hard earned were earned under the tremendous privilege to pursue these goals in the first place.
Llms are an amplifier of an individuals intuition and taste. That these supposed pillars of the community are not bravely exploring how to push and wrangle these bounds, and instead are retracting into conservative stances under the guise of human centric morality is (IMHO) demonstrative of lack of confidence and creativity within these fields more generally.
I believe that this lack of creativity and imagination is how we find ourselves in the personal fable you're noting: the experts are so myopic that they can't even imagine how they're field can be disrupted until it's disrupted outside of their control, and feel the need to control rather than explore.
> There might be more to maths than that, but that is definitely the most important part. I love science funding. But not because it's a jobs program for nerds.
I can produce an infinite number of verifiably correct papers, if that's all that matters.
1 + 1 = 2
1 + 2 = 3
1 + 3 = 4
1 + 4 = 5
1 + 5 = 6
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.
If you don’t believe me, crack open a text on something like graph theory (that’s pretty accessible, and if you’re a programmer, you’re familiar with graphs) and read through some proofs. Or better yet, try to prove some theorems yourself. No amount of rote memorization of formulas or rules will replace the creativity needed to write these proofs. Doubly so for discovering the facts in the first place.
If you are interested, perhaps check out 3Blue1Brown on youtube, they manage to show some of the (very) real beauty in mathematics!
Edit: Also, theoretical computer science is a subset of mathematics, and considering where we are on the internet, I get the feeling you like computer science.
Both mathematics and art are comprised of two phases, the first, technical one, where the novice grinds the skill and the second, the creative one which can only be achieved if you have the means (skill) to express yourself. What you described is the technical phase, not the creative one. There is intrinsic value to it that has nothing to do with money or cleverness, something that if you ever experienced it yourself even once, wouldn't need to be explained to you. Only people who never reach phase two have your stance. Artists and mathematicians who pick academia didn't exactly have great commercial prospects before AI was a thing, yet they still chose those paths because that's what having a real passion looks like.
>They like people to think it all came naturally and that its genetic and that they are special snowflakes.
No, they don't. Most of them are the humble people that know the value of cultivating a skill and when they do pride themselves it's precisely because they know the staggering amount of hard work and commitment they invested. Most of them are worried for unemployment and don't want all their work to be reduced to training data and on top of that not be given well-deserved credit for it.
The only thing being exposed here, is how much AI in its current form was being underestimated and constantly labeled as "not real/good enough intelligence". This was and still is a shared sentiment even among tech people. Can't really blame them for going through a bargaining or acceptance stage.
And since you also sound like the kind of person who thinks prompting can replace the "robotically spending millions of hours" of practice, I've got news for you: it cannot. You are about to learn the hard way the value of skill and human understanding because as much as capitalism rewards "impact" and "results", the market never values easy things.
But now you have people like Gowers and Tao, pure mathematicians, hyping up what the SOTA models can do - and I figure they both are getting access and tokens us mortals can't afford.
So I guess the question is - will everything be as expensive as applied fields?
An algebraic geometry researcher would be hard pressed to understand a new result from category theory or even something closer like commutative algebra.
Just like numbers and logic, it isn't and never was reserved only for humans.
They need to adapt.
Music is a human endeavor and musical recordings hurt musicians, bands, orchestras, etc. Especially those starting out.
At scale, correctness and reward are becoming increasingly disconnected. Example: capital continues to compound regardless of whether it reflects underlying human welfare, just as information can spread regardless of whether it is true. Reality still matters, of course. If you want airplanes to stay in the air, somebody eventually has to be correct. The problem is that our economic and social systems are becoming less effective at distinguishing between what is true and what is merely rewarded.
I don't understand how that contradicts my question.
Why would we want to sever this last thread of human control? What is there to gain from it? I don't think I have to convince anyone how much there is to lose.
The situation being created with an overdependence on AI is looking much more like the burning of Alexandria, and less like a utopian dream or even the oft-warned-about authoritarian hellscape. The AI hype is over and revealed to be delusional and politically motivated.
I'm sure most research mathematicians would like more freedom from some of the drudgery of their work (grading, admin, etc.), just like the rest of us. But we should be aiming for a world that allows more people to become mathematicians, not fewer.
Serfs, all right, but in what world do you live where "computers", people who did manual computing (i.e. mechanical additions/multiplications/... with very large numbers) are the same as actual research mathematicians, who are basically pure logicians?
The only perspective where it makes sense to root for mathematicians to go away is if you're a misandrist that thinks humanity should be replaced by robots (for reasons...). Or isn't logic something that's a defining human trait, and one of the main reasons we became the dominant species on the planet?
5 guys were on that truck. 1 driver and 4 guys that actually lifted up various shaped trash cans and dumped them into the truck.
Today I live in an apartment complex. 100 families take their trash to the compactor. 1 guy in a garbage truck comes once a week to collect the compacted refuse.
I wonder what happened to the other 4 guys. 80% of the garbage collecting labor… freed up to do something of higher value.
Maybe they cured cancer.
There is a huge difference between the two. Mathematicians work on discovering fundamental truths of the universe that go into the corpus of human knowledge forever. Programmers create utilities.
The idea that we just "trust everyone to carefully check and learn from AI output" as our barrier to human skillsets eroding is never going to work.
There is an Anthropic engineering post on HM front page that addresses this exact issue:
"... supervise the agent’s behavior via a human-in-the-loop. Claude Code previously protected against agents taking unintended actions by asking users for permission at each turn. Theoretically that works, but we’ve found the approach to be fallible. Our telemetry showed users approved roughly 93% of permission prompts. The more approvals a user sees, the less attention they pay to each, becoming over time much less diligent in their supervision. "
We can reach Q models just by throwing resources at it. That’s a million times current B models.
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.
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 liquidated GPU hardware in the near future. =3
So I suspect that the cloud will pass on math too, initial demos get extrapolated and people get worried but in the end slop is slop and serious people aren’t getting replaced or even threatened.
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?
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).
I think that is great, really! but does anyone remember asking a TA or teacher or prof or parent and getting told you can work it out for yourself, or maybe just given a hint? What if that is an essential part of learning, having to work through things you don't understand, but that you have the tools, the foundation, to figure out.
A calculator can't teach you math. A forklift can't build your strength. This is really a double edged sword, as far as education or accessibility goes.
You have to constantly ask... what do I lose by not figuring it out myself?
Your distinction between the practical and the theoretical is important. Practicality is important - everything we do is a matter of practicality of means or method, even how we pursue theoretical ends - but two points.
First, there is more to life than the practical. Some truths are known for their own sake, even if they also tell us about still more profound truths (also known for their own sake) or may have incidental practical relevance and consequences in some other context.
Second, while the theoretical terminus is the truth for its own sake, the practical terminus is always something other than itself. Well, what is that "something else"? You can't have an infinite regress of practicality. The meaning of a proximate, practical end is always other than itself. The practical requires an end beyond itself to justify it.
I agree that most people don't seem to inquire much about such ultimate ends. Their thoughts are confined to the proximate. Of course, how have they determined what the proximate should be? Something for people to contemplate.
Where science is concerned, it depends. On the one hand, there are fields that are certainly more theoretically oriented. It's not "the game" that motivates theory - that would make it mere recreation, with the truth taking a backseat - but the truth. (For this reason, I hesitate to call Erdos theoretically motivated. AFAICT, he was motivated by the challenge of problem solving and not the truth, insight, and understanding to be gained which would have been merely incidental and instrumental for him.)
However, I would also say a good chunk of science is motivated by a background motivation of technology production and the mastery of nature. Think Francis Bacon who viewed science as an instrument of power and showed a preference for the "how" over the "what" (τόδε τι) or the "why" (τὸ διότι). This set the tone for a great deal of modern science. A great deal does less explaining and more predictive modeling, because predictive modeling can be sufficient for control. Indeed, a truly theoretical causal account and understanding of a thing's nature can be less useful as a practical instrument than a merely predictive model.
Now, AI is a practical tool. I think they can be enormously useful as research aids, even in theoretical contexts, provided that one
1. understands their nature;
2. understands the purpose of the theoretical activity undertaken.
What is their nature? Well, they're statistical models that can unearth interesting and useful correlations and patterns. But they are not reasoning and knowing things. Their results are generated mechanically and mindlessly. Knowing this means taking their results with a healthy skepticism and a critical eye.
What about the purpose of theory? By analogy, think of a student in school who uses AI to complete all his assignments. Has he satisfied the purpose of those assignments? No, because the purpose of the assignments isn't to produce the effect - the solutions - per se, but to learn something. Theoretical work is like that; it's purpose is to understand and to grasp some truth. An AI can be used to assist this process, just as a calculator or a search engine can, but if you use it in a manner that circumvents that purpose instead of supporting it, then you're not achieve that purpose and wasting your time. What's the point?
It sounds plausible that LLMs help generate insights that humans have missed. But there are many open questions, eg the rate of generating insightful vs uninsightful but plausible statements, which can affect how useful they will be, and of course "open"ai has no incentive to share how much effort/cost (tokens and/or human-review) had been put into investigating erdos problems before coming up with this solution.
AI is simply not able to innovate, only combine.
This excessively pro-AI article brought to you by private equity.
If you love mathematics so much, and it's not the prestige and accolades that drive you, then what stops you from just solving problems on your free time even if they are already solved by AI?
Why does your field have to remain economically viable for you, why does this not apply to textile manufacturing or something? Someone's positions in society is owed to textile manufacturing too, and it has a culture that some people would lament the loss of and so on.(See guild system, craftsmanship in Europe).
I can't predict whether this will be a good thing in the long run, but this is literally the same complaint that every industry affected by automation ever had, and many who are now complaining would dismiss it if it were about something they personally do not care about or isn't sufficiently "noble" or intellectual.
I know it hurts, but the core complaint is just economic displacement, many have had to deal with that before. Most people who have something they love have to do that on their free time because it's not economically viable as a job, tough luck.
Mathematicians of all people should be free from such emotion-driven thinking. I guess people’s self interest in continuing to make an income trumps all.
Pick and place robots, or humanoid robots that can fold laundry, are still a lot tougher than automating knowledge workers and a lot more expensive to the point it's questionable if they're worth it.
We may not be on a path to assured destruction, we may be on a path to becoming livestock.
Yeah. UBI. We'll probably be seeing that in the next 15 years.
(In Europe, at least. In America the culture may not be able to stomach it, and I see even odds of a massive fake jobs program instead.)
Yeah I'm seeing this with the attitude towards AI. Especially as the economic benefits increase, we will justify increasingly reckless approaches. (Probably until some major catastrophe. That seems to be how these things go.)
While minimizing energy spent worked well in historic periods where survival was hard, in this era of abundance and a complex, interconnected and fragile civilization, the same instinct becomes harmful.
IME a vastly more common sentiment among mathematicians regarding mathematical talent leaving the nest to apply their skills in other fields is that those other fields are lucky to get them!
In our case, hundreds of millions, but we got there.
Humans by themselves invented mathematical concepts beyond human understanding a long time before we invented neural networks.
It's much more than just training. Humans use the engines to prepare openings and find promising novelties. Over time these novelties unearthed by engines fill out theory. It's easy to fine elite games where neither player is out of book for dozens of moves. Modern players are full hybrids in that sense. Looking back at chess, it seems natural that Mathematics will go the same way.
Scientific work is not normally naturally statistically salient for LLM observational data inferences. =3
Argument?
Trust me a fair bit of boomers and the generation before lost jobs to computer automation in the 1990s through the 2000s. And they used pretty much the same justification, every bit of work, take for example designing something like a machine spare that was earlier done through painstaking process of bringing the thing to life from the meticulous work on the drafting board till machining was now in the domain of computers.
In India alone, banking jobs were considered those commanding tremendous prestige and income potential, got automated through computers. Tax consultants, accountants, postal services etc etc. The list is endless.
AI is some what like that for us in this generation.
Many of the fields that were traditionally considered for "smart" people (STEM etc.) are the ones that are being really hammered by AI. Whereas, things which people considered lightweight often involving social relationships and interpersonal skills are still beyond the scope of AI (much of it even theoretically beyond the scope although perhaps robots might have an effect there).
There used to be a sysad T-shirt from the BOFH days "Go away or I'll replace you with a very small shell script" which pushed the idea that whatever could be replaced by a computer was something trivial. Now we find that the things which we thought were only for "smart people" are the very things being replaced by computer programs which is telling. Perhaps what we considered tough and smart really wasn't.
I was under the impression that improvements are arriving via how the models are trained and how model prompting context is constructed, rather than just by how much data or how much energy is spent searching over the model space for a particular prompt.
Is there some evidence that we have not reached a pleateau with just resource consumption on existing models?
I don't want to call anyone out, but I emailed one fairly famous mathemetician, and he literally said: "This is very interesting, I thought about it for a while, couldn't figure it out, but I thought ChatGPT had an interesting response..." and he linked me to his chatgpt transcript... (which, was actually helpful, because he asked it a better question than I was asking).
I have a suspicion that math will quite soon be exactly like programming and fall to the same machinery that coding is.
One thing that I noticed is that a common workflow I had was isolating hard subquestions in a self contained way and then "surveying" multiple different LLMs in a totally clean context. They would often say: "Oh, this is a obvious example of such-and-such" and immediately clear the barrier.
In a brutal simplistic way: each token is represented in a high dimensional vector. LLMs operate on them. They are the true, underlying meaning of the token for the LLM. Think of it as 1000+ ways to think of that word/token. Those meanings are baked in at training time. So, LLMs might be able to cross-reference them and solve a class of problems that flew under our radar, but can't come up with revolutionary theories that were never in the training set.
Of course, they will help winning a Nobel in the years to come, no doubt, but can't speak mathematics we can't understand (beyond simple obfuscation) and won't discover anything substantial on their own.
Obviously that doesn't mean we won't eventually achieve novel thought, or even that the current form is fundamentally incapable of it, merely that we've yet to see evidence of it and thus the default assumption is that we aren't there yet.
it is demographics.. there is no single answer, you are talking about millions of people with varying amounts of this JOB description
> most people are completely put off by AI slop
this is almost pathological.. most people consume media not produce it. Those in the business of media have been eliminating people for thirty years, and this AI tooling has multiplied that effect
> the value of XXXX writing or image generation generation is basically zero
yes - bingo.. the average capable person now can expect to be paid ZERO for their ability to personally produce writing or image generation.. and, if you don't start somewhere, you will never get to ascend the ladder of success in those fields, by definition
> I suspect that the cloud will pass on math too
consistent with the other statements here, this is 180 degrees false.. substantiation? the content of the letter signed by world class mathematicians, who are visibly quite concerned
Of course AI threatens this too, but the threat is of a much lesser degree. One could even argue that AI is helpful here with getting mathematicians to the 'frontier of knowledge' as AI is usually good in combining ideas from different fields.
Of course I agree that if the student just asks LLM to do their homework, they have not learned anything. But it's sad if one can't ask questions about a proof or such. Having the LLM around to review the homework submission is also useful, to make sure that the arguments are solid.
Alternatively, perhaps universities will provide access to fine tuned models that are mindful of such things.
Though having said that - the ~5 billion for the LHC now seems cheap ( even inflation adjusted ) in the context of Google investing 180 billion in infrastructure just this year!
What we do know is that a model "tops out" wrt training data - that is, for a model of a given size, there's only so much training data you can squeeze into the set before you stop seeing gains. But conversely it means that if you already have a model of say 1 Ttok that is "trained to capacity", then a model of 2 TTok needs roughly twice as much training data to fully utilize all those weights. Which means that the cost of training it is not 2x but 4x (twice as many params x twice as many tokens). And then of course serving it is 2x more expensive, but even with optimal training the gains aren't 2x. So it very quickly becomes uneconomical.
A good example of that kind of model is (was) GPT-4.5. The prices and the consequent lack of demand show why companies don't really do that sort of thing anymore.
But no, there's no evidence of a plateau as such. I'm not sure what "evidence that we have not reached a plateau" would even look like.
Can you elaborate? I don't think the solution to the unit distance problem was in the training set, but I'm guessing you mean there's some higher bar for revolutionary theories LLMs cant reach? If so where do you expect the limit will be?
What exact problem would need to be solved by LLMs to convince you that they DO discover novel solutions?
> In 2025 he left academia to become "Founding Mathematician" of Axiom Math in Palo Alto, California to research the application of artificial intelligence to fundamental mathematics.
If you think 'most people are completely put off by AI slop', you're living in a blessed bubble because: most people cannot even tell that the slop is slop, and are happy to engorge themselves on it.
Certainly the scenario where a human touch isn't valued by the market raises lots of very difficult philosophical and economic questions but that's a separate issue.
Arnold's polemics are perhaps the most infamous and easily found online (see "On Teaching Mathematics"), but the written opinions of Poincare et seq. are also easy to find. Even today the vast majority of research funding for mathematics, at least in the United States, is dolled out for highly applied fields like partial differential equations. The field does not even close to unanimously (contemporarily or historically) "explicitly take pride" in working on problems that have no obvious application, or being a "jobs program for nerds": the notion of such "pure" or "nonapplied" mathematics is at the very least a highly fractious and controversial subject, with a number of big names taking opposing viewpoints (often vehemently).
I think your picture of the field is over-represented on the internet, much like the fixation on certain niche fields: Category Theory, Homotopy Type Theory or, worst of all, outright dubious fields like Geometric Algebra; fields with a large number of online promoters, but with much less funding and relevance in the actual academic space. Of course there are reputable people with PHDs that feel this way,—but I can only imagine that there's a legion of tyros, pop math consumers, and undergraduate students who disproportionately promote this viewpoint.
I think we generally did that because that seemed to be the best known process for maximizing the quantity of useful mathematics that they occasionally stumble upon.
It's not like we treat math as a charity project for eccentrics who like blackboards. What we want is new mathematical discoveries that have a huge positive impact on other areas of the world. It's just that math and/or human brains are such that seemingly the best way to find those discoveries was to let mathematicians wander around randomly in mindspace.
If a more guided structured process produced more results, we'd probably do that. But it doesn't seem to, so we don't. I don't think anyone knows yet what the best process for producing useful mathematics with humans + AIs looks like.
42
By which I'm trying to make an abstract point about the inevitability of staying somewhat down to earth. I mean "pure" curiosity is great, except it isn't ever really pure, and abstract mathematics isn't ever totally abstract, it's just sort of meta in relation to practical things that humans care about.
Do you think Stephen Cook and Leonid Levin deserve more credit than whoever solved it?
Mathematics requires substantial creativity at every level. There is problem selection, conjecture formation, proof strategies, definitions, models, and explanations. Yes, it's constrained and guided by logic and rigor but having logic won't give you creativity.
> Music is a human endeavor and musical recordings hurt musicians, bands, orchestras, etc. Especially those starting out.
The medium it is recorded on has no bearing on what composed the music. If people don't get rewarded for composing they won't. Same with mathematics. If people don't get paid for being creative they just won't be creative.
I am not saying I agree with everything in the article. OP of this thread just made a low effort comment that was addressed in lengths during the article.
Like not being able to get some actual human when you call support, and talking to some fucking automatic system.
This includes many of the " 1990s through the 2000s" ones, and earlier ones too. Sometimes what was lost was an added layer of attention and quality that was previously required, but it was sacrificed away for efficiency.
Found it:
> Rodney Brooks explains that, according to early AI research, intelligence was "best characterized as the things that highly educated male scientists found challenging", such as chess, symbolic integration, proving mathematical theorems and solving complicated word algebra problems. "The things that children of four or five years could do effortlessly, such as visually distinguishing between a coffee cup and a chair, or walking around on two legs, or finding their way from their bedroom to the living room were not thought of as activities requiring intelligence. Nor were any aesthetic judgments included in the repertoire of intelligence-based skills.
I studied math at MIT and have several friends who are professors now and they deal with cranks all the time and since they're very kind and conflict averse people they tend to respond with perfunctory emails when they get inbounds like that.
So just be wary. Your external validation may not be as strong as you think it is, though kudos to you for at least trying to step out of the AI vortex to attempt to ground yourself.
In reality people would be thrilled to have such response even with a finished preprint on arXiv. Anyway if you really hit the jackpot hope it will be smooth working out the details and get it published!
I think most knowledge workers don’t like AI because most of them are aware that AI was created to replace them.
Just about every CEO that has given a speech about AI at universities have gotten booed by the students which isn’t surprising as those CEOs are effectively promoting technology that will take their future from them.
You have machines that can aid and expand your (human) understanding greatly, that wasn't possible without machines.
Machines don't think. They aren't human. They have no soul, agency/free will, self-reflection/awareness, moral imperatives or ethics.
You've been watching too much Terminator, son.
In addition to your last paragraph: lots of things that we used to do the less efficient way had side-benefits that were not immediately obvious, probably because they compounded over time. Now that we're not doing them anymore, we notice all kinds of widespread societal problems (in particular among young people) that come up that were never there before.
What you propose makes fine rhetorical sense, and I can assure you it did reach me, it's just that a (very) cursory search yielded me no significant employment rate changes or drastic layoffs in the related sector over the decades. Instead, it suggested that people have been reshuffled to do waste sorting and other related activities rather, and that the field actually grew, directly contradicting your smug, sarcasm-laden, willfully demagogue framing. Traits that are not exactly the hallmark of epistemic rigor to begin with, nor do they further it, even if the given narrative did hold up.
It's so easy to be asinine and make up a story, especially when one feels morally justified in doing so, and considers the base facts & analogy to be "obviously correct". I don't think that setting people up for failure by feeding them correct sounding lies - or sending related discourse into a nosedive in quality just to get in some cheap zingers - would help the cause a whole lot though. Do you?
Put differently, it helps if the example provided actually holds up as an example for the topic discussed. Especially if that example is as dramatic as 80% of a given job disappearing, and the people involved just plain losing their livelihood supposedly.
LLM are more like the Mechanical Turk trick, but the persons inside the machine running the con is unaware of how their actions affect the confounded observers.
Have a wonderful day =3
No, it does not, at least for arts. There is a chance, that AI will free art from it's utility or the necessary to create value. Maybe graphic designers and illustrators will lose their jobs due automatization, but most paid art (concept work, corporate design) was instrumental compromise anyway. Take away the commercial floor and what's left is the part they cared about most. Engineers on the other hand identify mostly by being optimizators and creating value. They may keep their jobs, but at what cost?
The markets that have replaced writers and artists with slop never valued them in the first place, and the markets that do will never replace them with AI, and I say this as an AI engineer.
Writing movies, writing theater, creating clearly original illustrations for various purposes, these are all tasks AI will never threaten, because there is just no point. And also, the market sizes for this kind of thing are a rounding error compared to say coding or back office automation which is incidentally the bulk of the token spend right now, confirming all this.
Do you see the problem with your reasoning?
LLMs don’t give a shit about social side effects, leave alone on unconscious level, because they are void of any intention. At most they are tuned on their thin edge layer to lean toward this or that kind of output, but that’s it.
Now the landscape shift as it’s sold (I guess) is that anyone can take a postdoc gibberish infused with the hard gained academic winks and subtle references and turn it into a ELI5 "does it have any applicability for my concrete issue at stake, prove it through Lean, good let’s deploy".
Deep esoteric research and trivial looking boring research can be as useful as state of the art trending areas.
"Jobs for nerds" as has been stated, has given surprising and unexpected advances, or leveraged incredible advancements.
An standard and boring bacteria in a specific Spanish biome, gave us CRISPR-Cas. There ar hundreds of examples.
True knowledge is, and will be, a human endeavor, deiven by human curiosity. Promoting curiosity is the sign of a developed society.
One way to make money is to create a great product that solves a problem people have and market it effectively. That's a sort of idealised situation, an aspiration, but it's what I would call socially valuable. Another way is to help people who are already rich move their money around so that they become richer, in return for a fraction of the increase in wealth. I personally have no problem with that, everybody has to make a living, but that is not socially valuable. It is debatable how socially valuable pure mathematics research is. But you take a fixed fee and all your best work is public domain.
As artificial intelligence (AI) muscles its way into field after field, researchers have wrestled with what it means for the future of their disciplines. Few communities have felt that pressure more acutely than mathematicians, who have haplessly watched AI get frighteningly smart, frighteningly fast.
Today, 16 math specialists have turned that unease into a public cry for help—and call to action. Part warning, part manifesto, the 11-page Leiden Declaration on Artificial Intelligence and Mathematics cautions that unchecked automation threatens not only how math is practiced, but what the discipline stands for. It also lays out principles for using AI in ways that support, rather than erode, the field.
For years, AI researchers have used math as a proving ground for their models. By their nature, math problems are easily gradable and offer an effectively unlimited training pool, and building AI models that can prove theorems is widely seen as a steppingstone toward higher performing reasoning systems. But AI is no longer confined to just straightforward tricks. Last month, OpenAI reported one of its newest general-purpose reasoning models had independently solved a famous, 80-year-old conjecture—only the latest feat in a series of breakthroughs pushing AI to the frontiers of the subject.
However, the declaration argues math is more than a machine for producing correct answers. The discipline, its authors believe, is a deeply human endeavor built on creativity, understanding, collaboration, and the pursuit of knowledge for its own sake. Those values often clash with the incentives driving AI development. “The tech industry proceeds in accordance with commercial logic, which is antithetical to the values of mathematics,” declaration co-author Michael Harris of Columbia University told The New York Times.
The authors warn the consequences are already becoming visible. AI-generated papers could overwhelm peer-review systems with low-quality work, make it difficult to assign proper credit for discoveries, and disadvantage researchers who choose not to rely on AI tools. The authors also raise ethical concerns about mathematicians’ work being used to train AI systems for military and surveillance purposes.
The declaration, which is endorsed by the International Mathematical Union, the discipline’s leading global body, is now open for signatures from individuals and organizations worldwide. It will also be discussed at next month’s International Congress of Mathematicians in Philadelphia, which comes as governments are also beginning to take a greater interest in the advancement of AI systems. Today, for example, President Donald Trump signed an executive order aimed at giving federal officials advance access to the most capable AI models before their public release, in part so agencies can prepare for any security threats AI might enable.
This story is part of Science’s AI in Science reporting initiative, which is supported by Ray Rothrock & family.
Understand first what Axiom Math does (https://www.youtube.com/watch?v=abYcV5LHMG4). It is only after he realized the possibilities of AI that he started Axiom Math with a student he mentored.
Love it! XD
I agree, and I think, as with physics, mathematical research produces building blocks whose utility won't be realised until later.
I found myself thinking about this issue when I was experimenting with an MCP server to handle tuning some precision parameters for scientific simulations. Claude did a much better job than I used to do when I was a fresh PhD student, yet being given tasks like that was how I learned, so it almost felt like pulling the ladder up after myself.
In the sciences, I think this is less of a problem because the PhD to scientist pipeline is pretty normalized, labs are used to the idea of having to let younger people take longer on problems that experienced people could solve much faster. But this doesn't seem to be as normalized elsewhere.
A proof being latent in an LLM is no more significant than a proof being latent in a book, a theorem prover, or the axioms themselves. Einstein's papers were latent in the genetic code of his parents and the environment of his time. That doesn't mean general relativity was "already done" before Einstein was born.
By your logic, no computation has ever accomplished anything because the output was always implicit in the inputs.
The entire purpose of computation is extracting information from representations where it's difficult to see into representations where it's easy to see.
So no, this isn't a problem with the original reasoning. It's a problem with yours.
There’s no way to “ELI5” this type of complexity. I’m talking about concepts exponentially more esoteric than quantum mechanics, and even within quantum mechanics there is nothing to ELI5 for a concept like “spin”. The best you can do is say that it’s a property of a particle. But imagine the words “property” and “particle” are also completely meaningless to you because they’re built on even more layers of conceptual mathematical abstraction.
> …
> True knowledge is, and will be, a human endeavor, deiven by human curiosity. Promoting curiosity is the sign of a developed society.
Unless I misunderstand, it sounds like you do agree? My point is that without human mathematicians LLM output is meaningless, and without human mathematicians holding the reins, LLMs would probably quickly devolve into “proving” things that are not only completely unintelligible by humans, but have no utility.
Your examples of esoteric mathematical concepts are anecdata. The vast majority of esoteric mathematics does not have utility. Mathematics is an incredibly large space of concepts. Consider the number of provable theorems in number theory alone, perhaps even related to specific subsets and sequences of numbers. The vast majority of the findings in that domain will not be isomorphic to some real world problem, they will be trivia.
We will need mathematicians to separate the signal from the noise.
But in relation to parent post I was replying to, it did not provide an answer or solution to anything. It has much closer relation to philosophy than anything.
Focusing on only ‘solutions’ in any field is shortsighted because you can’t know how the dots will connect. Someone’s seemingly pointless curiosity or experiment can unlock something unexpected, just like Boole
But it is a moot point anyway. Cook and Levin are very well known already in TCS, and credit is not directly enumerable like money, so "more than a lot of credit" doesn't make too much sense.
For this problem in particular, asking the right kind of question was really important for the field and led to a lot of discoveries even before it will be answered.
---
it's not a great discovery, it's a pretty minor question, that I thought would be easy and it's not -- i've just been poking off and on at it for weeks, and I'm relying on lean to verify everything. It's actually a quite specific CS-adjacent problem that I came up with trying to write code, that just is hard to solve, and nobody in the literature that I could find has looked at directly. The end result of it will have exactly zero consequences other than proving an interesting lower bound for a question that as far as i can tell, nobody has bothered even looking at but me. The reason it touches on multiple fields is that it's sort of both an algebra problem and a CS problem, so i keep having to flip between them to understand what I'm looking at, and there are a lot of sub-fields that span both that have different tools, and it took me a while to find the right one.
I assume you used 1000 because that's in the ballpark of the vector size. But these are not independent scalars, like each might store a certain property. Just like in 2D you can have 4 quadrants (or subdivide further), with a vector of size 1000 you can encode an insane amount of meaning.
> Those meanings are baked in at training time. So, LLMs might be able to cross-reference them and solve a class of problems that flew under our radar, but can't come up with revolutionary theories that were never in the training set.
There's a lot of jumping to conclusions here, but I'll try to answer more generally.
This idea of how LLMs work is mostly to build an intuition, like with a CNN you'd say imagine a layer does edge detection, and so on. And to some degree you can detect those kinds of behavior, but a NN is a VERY general architecture. It needn't work like you say, it can calculate any function and running under a loop and a scratchpad (basically an agent) is turing complete.
Even ignoring that, this part is misleading
> Those meanings are baked in at training time.
Being baked in at training time does not mean it didn't build novel meanings at training time.
This is even more significant when you take into account post training RL.
A simple proof that transformers can generate novel, superhuman solutions, is that you can build a transformer based chess bot, feed it 0 human games, and train it with RL until it can beat any human, completely novel and unconstrained by human gameplay (because it would've never seen it).
You can do that with any task that's verifiable, like coding or math.
(Also as a separate fact, as long as a task is easier to verify than solve (basically always), you have somewhat of a million monkeys with a typewriter, and with temperature sampling the model might eventually stumble it's way onto a solution.)
> "it is comparatively easy to make computers exhibit adult level performance on intelligence tests or playing checkers, and difficult or impossible to give them the skills of a one-year-old when it comes to perception and mobility."
Way to miss the mark (and also shift the discussion to woke conversation points on a comment from 4+ decades ago).
The point of his entire comment is that it seems like the "hard things" (aka abstract science) will be a lot harder for AI than "easy things" (a 5 year old or a dog understanding their environment in great detail, from depth perception to smells, sounds, etc, etc).
Your comment looks like it was written by exactly the kind of man Brooks was mocking.
Building systems based on application specific benchmarks rather than general what-if use-case scenarios will sometimes show you something interesting. ymmv. =3
My point is to say many things have been replaced by technology over the years, but people do them as hobbies.
Liquidity is extremely socially valuable from my perspective
I view the concept of "the market" as a construction project, that isn't finished. Not finished until every device around you can be traded instantly, with fractional shares even, with high liquidity and a capability for to the second price discovery, and there is a liquid options market on top of that.
The price of anything is not really resolved, its ability to be collateral and access a more liquid form of exchange at any time is not resolved.
Time is valuable, all of this reduces the time. There are still people waiting 90 days to access cash tied up in their home's equity, when another part of the market has split second collateralized lending. All of the market for anything should be that way.
The ability to exchange is valuable and wealthy people have liquidity issues. Poor people have liquidity issues. Everyone has a liquidity issue and doesn't know it.
Anything that slows things down slows down the whole construction project. A market with five 8 hour trading sessions a week with settlement the next day moves far slower than a market with three times as many trading sessions during the same time frame, where the trade is the settlement. The opportunities become endless for people aiming to accumulate more, the liquidity of traders to do actual business and negotiation and acquire goods and services and raise capital becomes vastly greater and far faster. Proving a new venture all the way to an exit becomes far faster, and results in the wealth distribution to the employees, vendors, and everyone else far faster and far greater.
That's what I see and look forward to. That has extremely high social value. Promoting liquidity and promoting velocity of transactions helps solve the actual reservations people have about the market at all. More paths for people on the poorer side of the bell curve to afford things.
Also, who do you think were the vast majority of AI researcher in the 50s, 60s, 70s?
The things like proving complicated theorems are things that are acquired by education within a lifetime, and that's why they're easy for AI.
The things a child can do are acquired through millions of years of evolution. While they don't require much explicit education, that doesn't mean they're easier.
"Neural networks will never be able to understand this sentence that's obvious to humans"
to
"LLMs must be able to solve problems that humanity hasn't been able to after almost a century, and that might even be unsolvable"
> Engineers like to build finished products, not wallow in minutia
This only works if what you loved was having built the thing. If what you loved was the building itself, the solving, then "here's a way bigger system, the AI figured it out" isn't a win. It's just a promotion from maker to manager. And a lot of engineers specifically tried to avoid this promotion in their career.
Consider the “Magnus Carlsen” of mathematics, who is more capable of understanding mathematics than any other human. But then also realize that that individual has probably devoted their entire career into a specific subdomain of mathematics. Within other deep recesses of mathematics, this Magnus equivalent will be less capable than their peers without years of rewiring their brain to understand the esoteric concepts and properties within that other subdomain.
LLMs will be able to dig deeper and broader than any human mathematician, and find results that are completely useless to humans because it would take more than an entire lifetime to “speak the language” of the concepts the LLMs have produced. The only way those results can become useful to humans is if then the LLM itself finds a way for it to be practical to humans once again.
So, no, I don’t think this represents the “democratization” of mathematics where mathematicians are no longer necessary because anyone can just prompt the LLM to explain it. The bar for entry level mathematics is lower, for sure, but research level mathematics will continue to be unapproachable for anyone who hasn’t devoted their career to it.
How about python3:
>>> input() + input()
2
2
'22'
>>> .2 + .2 + .2 == .6
FalseAs an example, the kid who can solve Math problems has less of an edge over AI than the kid who automatically becomes the captain of the neighbourhood football team but older human beings often assume that the former is smarter.
How are you going to get observational data for a different universe? Does such a thing even exist? What is its nature? You're operating well outside the bounds of human knowledge.
What you are actually saying there is that you can't imagine 2+2 being anything other than 4. That's perfectly reasonable but it's not the same thing.
Is that what is offending you so much?
In contrast, class NP and NP-completeness quickly became central concepts in theoretical computer science.
Of course it'd be super helpful to have, say, a teacher who could tailor explanations to anyone's precise background (e.g. where possible, using examples that come from the student's field of study when explaining some abstract concept). Or, if some definition comes with some precondition that has no obvious purpose, perhaps an omniscient teacher could explain why it's there with concrete counterexamples.[0] But even granting all this, I think that mathematical intuition is necessarily based on a lot of hard work actually exploring definitions on one's own, with pencil-and-paper and a lot of thought. That is to say, even though the process could probably be sped up a lot with a nigh-omniscient teacher[1], I doubt that a student wouldn't still need years of training to even have a clue what's going on.
(I'm saying all this, by the way, as someone who is terrible at all this and has very little mathematical maturity[2]—I'm speaking from my own frustrating experience....)
[0] c.f. Lakatos' excellent book Proofs and Refutations
[1] without the "curse of knowledge," or else we're back to square one of "answers that are correct but useless"
[2] e.g. the "post-rigorous stage" described in https://terrytao.wordpress.com/career-advice/theres-more-to-...
At the most fundamental level, you can only have a discreet or a non-discreet universe. If it’s discreet, there are countable things and 2+2 = 4 is true. In a non-discreet universe there are no countable things, but the universe itself is countable. If the universe were non-discreet and infinite, you could still count the infinities so it’s still true.
Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly.
Without a human holding the reins, consider an LLM a rudderless superboat speeding erratically towards the horizon, finding and proving meaningless theorems that not even your most talented classmate could ever begin to understand.
My point is the human is a critical piece to the puzzle, but not just any human, a career mathematician.
I do have a PhD so I kind of know how that feels. I watched my entire field (PL) get eaten up by AI though, the problems that I thought were huge 10 years ago are just silly footnotes now.
> Without a human holding the reins, consider an LLM a rudderless superboat speeding erratically towards the horizon, finding and proving meaningless theorems that not even your most talented classmate could ever begin to understand.
I don't disagree with that. LLMs are a tool, a super fast pattern matcher, research, token predictor. I don't expect it to go out and define its own esoteric (or useful) problems to pursue without human interaction. That's for the humans to do.
I don't understand what that has to do with my original comment though. I wasn't addressing what problems the LLMs were answering, just how to review and dissect the answers that they would come up with.
Why something in unsolvable or undecidable can be as important as the output of a theorem.
Questions like these, fields medal level problems or Karp’s 21 NP-complete problem are problems working mathematicians are interested in.
Will LLMs help as an human assistant in the future? Probably.
Will LLMs answer these questions themselves, provide insights and bounds to these new mathematics and teach other mathematicians why this new math they create is true?
Will these models have phds and take candidates teaching them how to apply and think about the maths problems they are interested in?
LLMs don't produce concepts, they just predict next tokens; they can't invent new concepts, only synthesize what is in their probability distribution already. They can mix/fuse vast areas of math together that are inaccessible to individual mathematicians, but can't create new concepts not present in their probability distribution.
You literally can't prove that you aren't a brain in a vat so I have no idea how you expect to make sweeping claims about the fundamental nature of reality. It is certainly convenient and practical to take certain basic assumptions as fact in order to go about higher level tasks but that does not make them so.
I'm really interested in this anecdote. I have never experienced this but have a reasonable academic background (BSc, MSc, MD) - and I am certainly not the person you're describing. Could you elaborate? Is this something more exclusive to pure mathematics (my bsc/msc are CS).
> Without a human holding the reins, consider an LLM a rudderless superboat speeding erratically towards the horizon, finding and proving meaningless theorems that not even your most talented classmate could ever begin to understand.
This feels like a little bit of a jump to me. AIs arent actually alive so of course someone is going to have to pose the question. They arent going to just do stuff on their own. And of course mathmaticians are going to need to interpret the results if we are to glean anything beyong if the conjecture is true or false.
But you seem to be suggesting that mathematicians will have to micromanage every step. That seems like a bit of a jump which i dont see much evidence for.
Then I had to relearn how a limit worked.
From a proof with epsilon delta inequalities. To a proof with showing for some n dimensional metric spaces that has all the properties needed to converge does in-fact converge. Finally to a proof that for any space that is metric there is an isometric function into that metric space that also converges.
And that does touch measure theory, functional analysis or set theory. So there’s still so so much more for me to learn.
The meaning I took was how far it's possible to travel from the shore - ie the scope of the state space. The mathematics we're exposed to is all quite shallow compared to what will (presumably) be possible between digital formalization and massive ML models. But the latter probably can't ever be understood by regular biological humans.
The toilet brush is a much better tool for unclogging the average toilet.
The plunger is actually meant to unclog sinks as far as I can tell, since it can attach much better to the sink and through its action can create pressure to unclog the much smaller sink drain pipe.
(Aside, this was one of the only undergrad courses where I felt I needed to attend study groups in order to not fail.)
The first exam was easy to pass based on intuition alone, as the topics were isomorphic to concepts I was familiar with like geometry or algebra. The midterm was a wake up call when it was made clear that just understanding the homework wasn’t sufficient, you were going to be asked to prove things that were much more difficult than what I’d ever encountered, and under time pressure (I had been doing math proofs since age 13 in geometry, and I was 22 at that point).
Maybe if you did discrete math, combinatorics, or linear algebra I would say it was 5x to 10x more abstract and difficult. Probably 2x more difficult and abstract than Theory of Calculus, if you had taken that or a similar course.
Edit: I also do endurance running and play soccer into my 30s. Seeing people run literally twice as fast as me (world record pace), and playing against former college athletes is equally as humbling. The time has passed for me to have anything near their ability haha.
If you are so certain of your claim then why are you seemingly incapable of defending it using logic and reason?
> What does philosophy have to do with anything.
If you took the time to look up the field of analytic philosophy to see what it's about, particularly with regards to metaphysics, that would presumably answer your question. There are literally treatise on the underlying nature of numbers and mathematical concepts (among other things) and you will find that there are multiple competing views on the matter.
When someone says "hey it seems like you're unaware of thing" and you think "WTF even is that" it is at that point generally a good idea to think to yourself "hey maybe there's something important that I don't know here" and then at least perform a topical check of the thing.
And I think you’re underestimating the jump from discrete math and linear algebra to abstract algebra… I think I attended each of those classes and opened their textbooks a total of 3 times each and did fine - once for each exam. But fml abstract algebra and measure theory were rough.