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andai 7 hours ago [-]
> Mathematics produces not only a body of results, but also understanding, clarity, and judgment among the communities of mathematicians who have shaped them, often in the context of their own autonomously guided research. This expert knowledge is essential, both to effectively use mathematics, and to continue to articulate new and significant research questions.
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.
j-bos 21 minutes ago [-]
The foundations of a great our mathematics came from landed gentry who enjoyed the work and were not paid for it. Today we prosper in our technologies and algorithms (not talking about recsystems) because of them. If math should once again become the pusuit of the curious and free but with even greater "skills", that does not seem to be an issue for math, or for those of us who benefit from math. An issue for people as a concept and for mathemeticians, yes. But that's a separate topic, like the anguish of not getting into the Premier league.
threatripper 6 hours ago [-]
While you are right in a way, I think you miss the point. In the past "computer" was a job description and mechanical power came from serfs. They surely developed skills we are lacking today but I'd argue that overall the world is a better place with digital computers and electrical motors. It frees up these people to do something else, something of higher value.
rwl 3 hours ago [-]
Sure, the world is a better place with fewer serfs in it, but what exactly is of "higher value" than being a research mathematician? It's already a profession that consists essentially of exercising our highest and most distinctly human capacities: creativity, abstract reasoning, and passing the results of those on through a distinctive language and culture. I don't think the comparison with serfs is useful.
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.
XenophileJKO 1 hours ago [-]
Sure, recreational mathematicians. Just like people that like to ride horses for fun.
bluealienpie 3 hours ago [-]
We argued that AI would free us to explore the arts. Instead it first came for written language and images. So what's left when it can write all the programs, drive all the cars, and AI sensors on farms can monitor and distribute nutrients. I remember watching TED Talks about how AI weapons need to be carefully studied, and instead we see them autonomously picking targets. I'm not seeing any higher values, instead I'm seeing how we're on a path to assured destruction.
noufalibrahim 2 hours ago [-]
I see that point of view but there's another that I've recently been thinking about.
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.
oblio 1 hours ago [-]
This is actually a very old AI insight, acknowledged at least as early as the 80s, let me see if I can find the quote.
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.
> In the past "computer" was a job description and mechanical power came from serfs.
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?
golemotron 23 minutes ago [-]
I don't think that "root[ing] for mathematicians to go away" is the problem. The problem (if there is one) is that the process by which that occurs is economically determined. No amount of complaining will stop AI from being useful in mathematics or erase the incentives to make it better. It's automatic process, like photography sidelining painting or shoe factories sidelining cobblers. We go through this with every technological advance and the outcomes are pretty much determined. No cheerleaders are needed.
kamaal 6 hours ago [-]
>>Similarly, the output of programming is not only a program, but also a programmer. It is you.
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.
AnthonyMouse 2 hours ago [-]
> This can be said about pretty much any job on earth.
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.
coldtea 3 hours ago [-]
>By that definition nothing should ever be automated.
Many things shouldn't. Understanding is one of them.
delusional 6 hours ago [-]
> Everything thinks they are special, actually no one is.
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.
sublinear 6 hours ago [-]
Yes, but we've already painted ourselves into a corner by almost a century of moving all that work onto computers.
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.
kamaal 4 hours ago [-]
>>Why would we want to sever this last thread of human control?
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.
coldtea 3 hours ago [-]
For many of those automations, we're worse off for them.
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.
Archer6621 12 minutes ago [-]
That really depends on who is the one that benefits from automation. Companies automate support systems in order to keep their support staff small, because apparently for many of them it is more profitable to frustrate their customers with crappy support than to pay more support staff to do a better job.
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.
2 hours ago [-]
turzmo 23 hours ago [-]
Much of math (or science) research has the strange quality of being mostly curiosity-driven, but having giant benefits that occasionally spin out to the public.
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.
math_dandy 16 hours ago [-]
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.
pfdietz 15 hours ago [-]
> much more interesting than the problem’s elementary statement might lead one to expect
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.
I don't think Erdos problems are useless myself, I put "useless" in quotes to emphasize that they are the sort of research that doesn't have an immediate application, and so their automated resolution should be weighed against the sociological cost.
As opposed to, say, drug discovery.
danbruc 13 hours ago [-]
I am not a mathematician and did not read the unit distance solution too carefully, but my impression was that it used a variation of a known technique to solve the problem. And that makes perfect sense to me, there are a lot of techniques and lot of less relevant problems, I am not surprised that one can solve some of them with known techniques that just nobody has tried [hard enough] before. I am much more sceptical when it come to the important unsolved problems where every known technique has probably been tried several times over. In those instances it will probably take a true leap in understanding to solve them and I am sceptical that large language models are well suited for that because of the way they work.
math_dandy 12 hours ago [-]
We're very fortunate to have had some very eminent mathematicians backfill the OpenAI proof with history, context, and a literature review [1]. Ideas behind the proof seem to have been "in the air". Indeed, looked at certain point of view, the OpenAI construction can be viewed as a high-dimensional generalization of a known low-dimensional one. In this vein see the remarks of Gowers, Sawin and Tsimerman in [1]. Are LLMs capable of "true leap[s] in understanding"? I have absolutely no idea. But LLMs keep surprising me.
>> 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.
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.
azeirah 14 hours ago [-]
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?
math_dandy 13 hours ago [-]
This is, indeed, how math often goes.
BigGreenJorts 23 hours ago [-]
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.
scottLobster 13 hours ago [-]
Yep, and if history is any guide the only way to play it is to take part and get rich while you can, or play the super long game and be positioned for the collapse.
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.
brador 22 hours ago [-]
We are on tiny 1-5T parameter models with local power stations.
We can reach Q models just by throwing resources at it. That’s a million times current B models.
bdamm 16 hours ago [-]
Is this a known or quantifiable thing? I thought that the limit had already been determined i.e. the existing models top out and at some point it doesn't matter how much time or energy you let the model consume, it won't improve the result. And with regards to training parameters, I thought we were equally limited there, e.g. the existing models can't benefit from a larger parameter space.
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?
int_19h 8 hours ago [-]
The existing models "top out" not because they don't get better, but because it is uneconomical.
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.
sterlind 8 hours ago [-]
what is a B model vs. a Q model? what do these letters mean?
brador 8 hours ago [-]
B Billion parameter, T trillion, Q Quadrillion.
jazzyjackson 13 hours ago [-]
You cannot think fast enough when your wires are kilometers long. The only way up is in, and silicon transistors just cannot compete with density with biologic brains, ergo, super intelligence is a pipe dream
fc417fc802 12 hours ago [-]
Baseless assertions. Fab tech continues to improve. There's no reason ML model internals have to be strictly serial - in fact we're already seeing some shifts away from that.
lo_zamoyski 11 hours ago [-]
I think nuance gets lost in these conversations.
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?
filup 7 hours ago [-]
I'm not sure why you're being downvoted, but I agree with all of your points. Those aren't things that pretty lend themselves well to mathematical modelling. But... there is a marginal field of math that does apply to this: statistics. The first two cases are somewhat special: - It may be daily obvious that an API is terrible, and that the replacement is not. If API 1 takes 1 sec to call, and API 2 takes 100ms to call, straightforward choice without stats. - provisioning can be dangerous. While not really a stats problem, you do need to have a quite elegant model of what is getting refactored, and how to know when to invalidate those cache entries. For the rest of the examples you provided, you're making changes that may make the problem better, may have no effect, or may make the problem worse. You completely need to use statistics to determine whether or not changes like those are honestly having an effect. Performance analysis is part math and part art, and without the math background, you're likely going to be spinning your wheels a bunch. Beyond stats, fields like queuing theory are going to make a massive breakthrough when you're doing performance breakthrough in distributed systems.
yieldcrv 23 hours ago [-]
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.
16 hours ago [-]
math_dandy 12 hours ago [-]
> 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
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!
ccppurcell 16 hours ago [-]
I think you've slightly straw manned the lamentation there. Not that I agree with the lamentation, but using your talent to make the rich richer (which is what quants do, they are paid a fixed amount to provide a larger value up the chain), as opposed to advancing human knowledge, is the reason for the lament, not some sort of respectability issue.
le-mark 13 hours ago [-]
Quants benefit from substantial bonus structure as part of their compensation.
nyxtom 12 hours ago [-]
For every interesting problem AI solves there are a long tail of really dumb things that AI performs that humans would never do. Some days I am in awe of one-shot magic eight-ball output and other days I'm so frustrated by the sheer stupidity of what it produces. It remains to be seen whether that long tail of stupidity can ever be resolved in the current form of LLMs.
cout 11 hours ago [-]
Are you sure you aren't underestimating human capacity for stupidity?
glaslong 10 hours ago [-]
We might be undervaluing consistency. You can plan for reliably stupid. Harder to rely on Often Smart, but Unpredictably Stupid.
hodgehog11 7 hours ago [-]
I supervise quite a few Masters students. In my particular setting, believe me, LLM stupid for the top three chatbots is easier to work with than real human stupid now. We passed that threshold earlier this year.
cousinbryce 10 hours ago [-]
It’s in the training data!
andrei_says_ 11 hours ago [-]
It’s like a constant game of two truths and a lie.
Very much like gambling, you can hit the jackpot, or just have continuous near (and far) misses.
epistasis 11 hours ago [-]
A million monkeys at typewriters, but with momentary runs of extreme luck/brilliance.
Nevermark 11 hours ago [-]
It worked for humans. It took a lot of us, but eventually we accepted zero as a number. Then negative numbers. Then "imaginary numbers" as a useful trick, and then as meaningful.
In our case, hundreds of millions, but we got there.
Huntsecker 13 hours ago [-]
Anyone else draw similarities with this and the artists and authors who complained when gen ai first came out. I think a lot of people don't realise the disruption ai will cause to many industries, until its directly impacting them, basically personal fable at scale (https://en.wikipedia.org/wiki/Personal_fable).
pj_mukh 12 hours ago [-]
I'm not sure the similarities hold. The best comparison I've found is the construction unions in San Francisco. They frequently block housing that is factory built somewhere else in the Bay area even though that would make it possible to build tonnes of new housing in the city at a fraction of the cost.
"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 think the disruptions are temporary. Using AI still takes someone's time and the skill can be honed, which means there will be people specializing in using AI in different ways that are better than what someone just picking it up can do. It's just a reset on skill floors but the people with talent will rapidly regain ground and find their way to the ceiling again. The ones who learn the most, the fastest, will probably end up being worth more than they were previously.
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.
andy99 12 hours ago [-]
I’m curious, do writers and authors still really care about AI? I think by now most people are completely put off by AI slop, the value of AI writing or image generation is basically zero
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.
Sebguer 12 hours ago [-]
This is quite a way to admit that you don't have any writers or artists in your social group. It has absolutely gutted jobs in these industries, and will continue to do so.
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.
dogwalker5000 9 hours ago [-]
> If you think 'most people are completely put off by AI slop', you're living in a blessed bubble
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.
pj_mukh 12 hours ago [-]
Ehh, I've had the opposite experience, with lots of writers and artists in my circle.
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.
hgoel 11 hours ago [-]
But this is missing the fact that the vast majority of starting jobs for artists/writers would be in the former category. Similar to how AI coding or automation hurts junior hiring more than it does senior.
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.
int_19h 7 hours ago [-]
There have been numerous studies by now showing that most people cannot reliably distinguish "slop" from the real thing, and that many genuinely prefer the slop even.
gnerd00 12 hours ago [-]
> do writers and authors still really care about AI
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
potsandpans 11 hours ago [-]
Also worth noting these are reactions to what is essentially equity in access to skills and knowledge.
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.
plastic-enjoyer 13 hours ago [-]
The artists will be fine and AI will liberate them. It's the engineers and mathematicians who are walking into the blade. They built their entire sense of self on being the best optimizers in the room and optimization is the first thing the machines take. Their whole identity was a number going up. Now watch it go to zero.
fc417fc802 10 hours ago [-]
Depends entirely on whether "the market" ends up valuing what humans add to various artistic processes. Same for human engineers, who are still very much needed at least for the time being.
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.
plastic-enjoyer 5 hours ago [-]
> Depends entirely on whether "the market" ends up valuing what humans add to various artistic processes.
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?
fc417fc802 4 hours ago [-]
That depends entirely on what's meant by "will be fine" and "AI will liberate them". I was assuming you meant something along the lines of gainful employment. If instead you meant performing a meaningful task then I'd counter that most engineers like to build finished products not wallow in minutia. You can still hand roll assembly but I don't think many developers lament the advent of the modern compiler. Instead people build far more complicated systems than would otherwise have been possible.
plastic-enjoyer 3 hours ago [-]
Sure, but the compiler didn't promote the assembly programmer, it basically ended assembly programming. So which side of that is the engineer on? You're assuming we're the dev who got the compiler. But if the premise is that AI does the actual problem-solving, then we're the assembly and AI is the compiler.
> 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.
fc417fc802 2 hours ago [-]
My impression is that most developers are motivated to create a finished product for a variety of reasons. If what someone enjoys is the craft purely for its own sake then isn't the resulting situation exactly the same as for artists assuming a hypothetical future where the core activities of both are largely automated? Such people can still engage in the craft purely for their own enjoyment just as anyone who wants to is free to write assembly by hand today.
luiwammus 8 hours ago [-]
I cant speak for engineers, but as a mathematician I wholeheartedly disagree with everything you claim in your comment. Almost none of the mathematicians that I know care about the optimization aspect of mathematics: the pursuit of optemizing constants in theorems and providing minor technical improvements is mostly seen as pointless unless there are significant new mathematical insights that fuel the improvement. I think most mathematicians rather build their identity around providing actual understanding of problems using mathematics and improving society's understanding of mathematical problems.
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.
Spacecosmonaut 23 hours ago [-]
Accelerationists may argue that the eroding of proper attribution and proof verification by humans is a meaningless short term struggle of a dying field.
Mathematics seems to be entering an era where human + machine maximizes performance, much like chess in the 1990s. However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). A future where AI generates and verifies proofs without humans in the loop. Where the mathematics may be beyond human comprehension.
In that future, does it matter that early career mathematicians are inhibited by these developments? Perhaps not. Programming faces the same issue. As AI crawls up the competence ladder, does it matter that fewer people have opportunities to develop the skillset of a senior engineer? Perhaps not.
wongarsu 23 hours ago [-]
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?
vitally3643 13 hours ago [-]
Neural networks are already systems of linear algebra that are beyond human understanding. Most humans could probably grok a 1 or 2 dimensional slice of a network, but the latent vector space is completely beyond the human brain. We have to use tools to analyze neural networks piecemeal in exactly the same way that we analyze any other higher-dimensional construct. Few humans are truly capable of reasoning in 4+ dimensions, that doesn't make string theory "not math". Nor does a trillion-dimension vector space of an LLM make it "not programming".
Humans by themselves invented mathematical concepts beyond human understanding a long time before we invented neural networks.
squidbeak 14 hours ago [-]
> Much like for many the point of chess is that it's played by humans, with truly superhuman AI relegated to a training aid
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.
jrflo 14 hours ago [-]
I think there would still be a place for it if it's beyond human comprehension. For instance, really complex lemmas to solve human-tractable problems. If you can pose a question in a proof assistant language like Lean, have an AI write a Lean program that solves it, you can use that as a Lemma for some other problem. There's quite a bit of math out there that is "correct assuming conjecture X is correct", maybe AI could fill that gap and "still be math".
Spacecosmonaut 23 hours ago [-]
Perhaps P=NP. The new algorithms are handed down to us. We can apply them without fundamentally understanding why P=NP.
overgard 13 hours ago [-]
On the other hand, it could stall out at: good enough to take the easy problems, not good enough to take over the field, but damaging enough to erode the quality of new entrants. (Which incidentally is the scenario I think plays out for software)
math_dandy 12 hours ago [-]
I think the OpenAI model that resolved the Unit Distance Problem would be capable of solving a significant proportion of mathematics PhD thesis problems.
0x59 16 hours ago [-]
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.
rad_val 12 hours ago [-]
AI (in this form) will never be able to solve things we truly cannot solve yet. It might catch things that we didn't project properly or brute force things no human can , but it will never unify general relativity with quantum mechanics. It's amazing at finding hidden truths in large datasets, but won't win a Nobel unassisted.
par1970 12 hours ago [-]
> AI (in this form) will never be able to solve things we truly cannot solve yet.
Argument?
rad_val 12 hours ago [-]
The strongest argument for this is structural: what LLMs are.
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.
resident423 9 hours ago [-]
> but can't come up with revolutionary theories that were never in the training set.
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?
redox99 11 hours ago [-]
Instead of going into a long technical argument of why your description of LLMs is flawed, I'll go straight to the point, because people keep moving the goal posts.
What exact problem would need to be solved by LLMs to convince you that they DO discover novel solutions?
rad_val 10 hours ago [-]
I'm more interested why you think my understanding is flawed honestly. I thought I distilled it decently well in two sentences. The bottom line is, in this hyperdimensional space you can find relationships that are not easily distinguished by human minds, but the corpus is still fixed, a llm can't truly know anything beyond its training data.
redox99 8 hours ago [-]
> Think of it as 1000+ ways to think of that word/token
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.)
dehsge 9 hours ago [-]
unify general relativity with quantum mechanics. The continuum hypothesis. The traveling salesman problem in polynomial time.
redox99 7 hours ago [-]
I think it's cool how in a decade we went from
"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"
roywiggins 7 hours ago [-]
it can operate at the level of a mere mathematics professor, who everyone knows are barely conscious, basically automatons. wake me up when it's Einstein
3uruiueijjj 4 hours ago [-]
The continuum hypothesis was proven independent of ZFC over sixty years ago, I think even GPT2 could have told you that much.
int_19h 7 hours ago [-]
I don't see how any of this follow. Yes, the LLMs will learn the "meaning" (here narrowly defined as relative configuration in the embedding space) of vectors that correspond to tokens in whatever tokenizer is used to feed into them. But that vector space is not discrete, and nothing precludes the model from internally operating on other vectors that it never saw in training, based on how they relate to those vectors which it did see.
fc417fc802 12 hours ago [-]
We have yet to see evidence of proper generalization AFAIK. Examples such as this proof are the closest I'm aware of. I haven't read this one in detail yet but the other examples I've seen have been (upon examination) much closer to an (absurdly) deep literature search than to novel thought.
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.
rienbdj 11 hours ago [-]
The burden of proof is the other way
orbital-decay 15 hours ago [-]
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.
Joel_Mckay 13 hours ago [-]
In general, most humans top out at 3D visualization, and instead rely on crude mathematical tools to work with higher dimensions. Every so often, people like Euler or Leibnitz pops up to give people new methods for blind men with a cane to explore the unseen yet knowable world(s).
Scientific work is not normally naturally statistically salient for LLM observational data inferences. =3
Joel_Mckay 14 hours ago [-]
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 liquidated GPU hardware in the near future. =3
martin1975 13 hours ago [-]
Reading this invoked the image of ouroboros in my mind.
Joel_Mckay 13 hours ago [-]
The Ouroboros in western mythology is a cautionary tale about the uselessness of the first perfect immortal being, and why humans should suffer our imperfections with insightful grace. The concept also made a great Red Dwarf episode.
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
martin1975 12 hours ago [-]
Thank you, human being! Have you looked at the price of a RTX 5090? I can get a used car for that.
Joel_Mckay 10 hours ago [-]
Indeed, just paid $3k more for the same workstation we purchased last year at this time. Just the DDR5 sticks and NVMe drive cost more than most parts right now including a rtx 5070 Ti 16G card. For h265 hardware encoding, the performance differences on higher-end cards benchmarks was negligible.
Building systems based on application specific benchmarks rather than general what-if use-case scenarios will sometimes show you something interesting. ymmv. =3
silveraxe93 24 hours ago [-]
> However, the declaration argues math is more than a machine for producing correct answers.
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.
psyklic 23 hours ago [-]
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.
datsci_est_2015 23 hours ago [-]
To further this assertion, there is almost no value to deeply esoteric math that is technically correct, but completely inapplicable to any scientific reality, and completely unintelligible to humans. Consider these findings deep, dark corners in the unfathomably large hyperspace of mathematics. My guess is AI will be incredibly adept at identifying these types of findings, and it will be exceedingly difficult for humans to identify what is meaningful and what is not in the slop.
sunshowers 14 hours ago [-]
Elliptic curves over reals and the complex numbers had some physical/scientific meaning, but elliptic curves over finite fields had none before cryptography.
canjobear 16 hours ago [-]
Your model of what AI is good at is wrong. Generative AI is not good at wandering off into novel esoteric abstract corners while maintaining correctness, it is good at things that are close to its training data. I suspect that humans will long outperform AI in the domain of "novel esoteric abstract useless math" whereas AI will outperform humans in the domains of (1) making connections between already-well-understood concepts, things that seem obvious in retrospect but which no human figured out just because of the accidents of what people happened to focus on, and (2) proving things that require long, tedious, intellectually unsatisfying calculations, which would cause a human mathematician to give up for boredom.
datsci_est_2015 14 hours ago [-]
My understanding is that we’re talking about “tool-assisted” proof generation, which provides some guard rails but would still allow significant creativity. Tools like Lean, Coq, etc.
yaris 22 hours ago [-]
Works of Shinichi Mochizuki immediately come to mind. He is not AI but provides very good examples of math that is useless because it is incomprehensible by (other) humans.
seanmcdirmid 20 hours ago [-]
Do AIs produce answers whose work is incomprehensible to humans? It seems like you could just have the AI elaborate multiple times until you were satisfied with the explanation and documentation of what went into figuring out the answer. It’s not like the AI is one shotting the answer in a single opaque query anyways.
datsci_est_2015 20 hours ago [-]
Like other commenters, I think you’re also underestimating the complexity of esoteric higher level math.
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.
seanmcdirmid 17 hours ago [-]
I don't get it. LLMs don't have ego, they don't have the ability to say "no, this should be obvious, I'm not going to explain further", they are just token predictors, and given context, they can generate more tokens. If you don't understand how the answer was derived? You just ask more questions and it isn't going to get bored or annoyed, it will just try to answer the questions.
Is that what is offending you so much?
datsci_est_2015 17 hours ago [-]
No, it doesn’t sound like you get it. It has nothing to do with the properties of LLMs and everything to do with the complexity of mathematics.
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.
duchef 15 hours ago [-]
> 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.
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).
datsci_est_2015 14 hours ago [-]
For me it was a “Modern Algebra” course required for my mathematics major, where I managed to squeak by with a B, but it was definitely a filter course for research-level mathematics. It was very clear in the class of a few dozen students who the top 5 or so were based on their questions during lectures and office hours, as well as when they blessed us mere mortals with their presence at our study groups.
(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.
grayclhn 5 hours ago [-]
Algebra is the class where I learned I shouldn’t try to figure out how to prove theorems named after people during tests.
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.
dehsge 9 hours ago [-]
For myself it was learning what a limit is in calculus, then learning about vector spaces, then learning about metric spaces and then learning about different topological spaces.
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.
bawolff 15 hours ago [-]
> 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.
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.
fc417fc802 11 hours ago [-]
Micromanagement wasn't the message I took from that. Rather the level of human involvement required which (it seems like) the two of you more or less agree on.
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.
seanmcdirmid 14 hours ago [-]
> 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.
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.
BalinKing 16 hours ago [-]
Excluding supergeniuses, pure mathematics—even at a very basic, undergraduate level—simply can't be understood passively. Even with an infinitely patient AI teacher who could answer any question on-demand, it'd still require a massive amount of work to actually understand anything in research-level mathematics. Basically every single word in a mathematical definition is a term of art, and (IME) if one doesn't grok each of those words at a fairly deep level, the new definition never really makes too much sense. And this applies recursively: each of the words has some thoroughly inscrutable definition of their own.
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"
How do you make use of something that you don't understand?
epgui 16 hours ago [-]
It’s easy to imagine this being a problem both in quality and in volume. Verifiable work is less valuable than verified work. And noise is always costly.
pfdietz 15 hours ago [-]
It's not that it's incomprehensible, it's that it appears to be wrong.
jmorenoamor 17 hours ago [-]
Sorry but I couldn't agree less.
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.
datsci_est_2015 17 hours ago [-]
> Sorry but I couldn't agree less.
> …
> 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.
psychoslave 22 hours ago [-]
Esoterism is mostly a social tool to keep those not initiated excluded from the private club. Most of the time mathematics becomes tricky less due to unfathomable intrinsic complexity, and more due to the way it’s communicated.
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".
datsci_est_2015 21 hours ago [-]
When I use the word “esoteric”, I mean it at an absolutely hyperbolic level. Like exploring new-but-basically-useless axiom spaces, and creating concepts for which there exists no clean metaphor in time-space - like quantum mechanics on steroids. And then creating multiplicatively more complex concepts by combining those concepts together.
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.
ragebol 23 hours ago [-]
"What is the answer to the Ultimate Question of Life, the Universe, and Everything"
42
lioeters 19 hours ago [-]
The proof is trivial and is left as an exercise for the reader.
rowanG077 23 hours ago [-]
Once you now something is correct, with a proof. It is MUCH easier to understand why it is correct. Than to start from a slate that you don't even know whether something is correct or not. In that sense AI that can just solve high level math problems is immensely useful. It allows a mathematician to explore ideas at a much more rapid pace.
terminalbraid 23 hours ago [-]
Consider that since an LLM is really just an large encoding of data, the "proof" is in there already. All further work on it is effectively only rearranging words. Then all math an LLM is capable of is "done" and we have the "proof" in the LLM which by your definition is now "MUCH easier to understand" and this work is somehow sufficient.
Do you see the problem with your reasoning?
rowanG077 19 hours ago [-]
You're confusing "contains information" with "has produced a result."
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.
bitwize 16 hours ago [-]
The value of human understanding just cratered because we have machines to understand for us now.
martin1975 13 hours ago [-]
Not quite.
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.
epgui 16 hours ago [-]
That seems about as shortsighted as claiming the value of human understanding has cratered after the invention of the electronic calculator.
goatlover 16 hours ago [-]
Wonder what Frank Herbert would have to say about letting machines do the thinking for us.
bitwize 15 hours ago [-]
"You wanna know what the best thing about humans is? You invented us! Giving you a chance to take a rest while we invented everything else!" —Wheatley, Portal 2
zerobees 16 hours ago [-]
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?
141205 15 hours ago [-]
Hardy would agree with the viewpoint that you espouse but it would be pushed back against by Arnol'd, Poincare, Gauss, Von Neumann, and even Grothendiek: Arnol'd and Poincare were vituperatively against the division between "pure" and "applied" mathematics; they considered mathematics and physics interchangeable, and Arnol'd lamented that the field had lost a large amount of funding/prestige/relevance due to groups like the Bourbaki that took a purely aesthetic view; Gauss had a critical view of problems like Fermat's last theorem (he felt that you could construct infinitely many such problems, and felt that attempting to prove it was a generally useless endeavor), along with outright calling pure mathematics worthless; but while Von Neumann and Grothendiek were more moderate, both were critical of the field losing motivation/quality as it strayed away from empirical science into—quoting Von Neumann—"abstract inbreeding".
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.
drivebyhooting 11 hours ago [-]
What’s wrong with geometric algebra?
The same could’ve been said about linear algebra 100 years ago.
munificent 13 hours ago [-]
> We just let them do their thing, essentially treating them as artists and letting them pursue the craft for its own sake.
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.
naruhodo 3 hours ago [-]
> It's not like we treat math as a charity project for eccentrics who like blackboards.
Love it! XD
I agree, and I think, as with physics, mathematical research produces building blocks whose utility won't be realised until later.
bdamm 16 hours ago [-]
Except, it was true before and it is still true today that the best "artists" whether graphic or mathematic, are the ones that do somehow manage to cross the chasm of pure research and providing a tangible benefit to their benefactors. That aspect of understanding your customer is not changed by the presence of AI.
orbital-decay 16 hours ago [-]
Pure mathematics != applied mathematics
redsocksfan45 13 hours ago [-]
[dead]
dwroberts 23 hours ago [-]
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
card_zero 20 hours ago [-]
Boole was trying to improve logic for humans, "The Laws of Thought". So it has a connection to human problems, and eventually to practical matters. He could instead have been working on something much more abstract and much less useful.
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.
dwroberts 13 hours ago [-]
> So it has a connection to human problems, and eventually to practical matters.
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
delichon 24 hours ago [-]
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.
19f191ty 24 hours ago [-]
No, it's not the most important part. It can be argued that most important part is asking the right questions
silveraxe93 24 hours ago [-]
Assume someone solves P=NP
Do you think Stephen Cook and Leonid Levin deserve more credit than whoever solved it?
NotOscarWilde 23 hours ago [-]
That's a bit too simplistic -- if there is a small group that really pushes things forward in a big way, then maybe not, but if this result builds upon decades of prior work, then Cook and Levin might be equally or even slightly more famous than the solver group after the dust settles.
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.
jltsiren 13 hours ago [-]
Depends on the solution. If the solution is that P≠NP, the concept of NP-completeness remains the bigger contribution, unless the proof techniques are particularly interesting and lead to other major results. The same applies if P=NP but the proof does not ultimately lead to a practical algorithm. If we get a practical algorithm, the answer is more valuable than the question.
redox99 10 hours ago [-]
Do questions like goldbach conjecture, fermats last theorem, etc deserve more credit than whoever answers them?
jltsiren 9 hours ago [-]
Fermat's last theorem probably doesn't. While attempts to solve it have led to many mathematical discoveries, the theorem itself feels little more than a piece of trivia. I'm less familiar with the implications of the Goldbach conjecture.
In contrast, class NP and NP-completeness quickly became central concepts in theoretical computer science.
dchftcs 23 hours ago [-]
If the problem resolves to P=NP, that result would probably be more celebratee than being able to formulate the problem, but being able to formulate the problem and get people interested in it is probably worth more than the average primal dual trick to prove a polylog integrality gap for some integer linear program.
i_am_a_peasant 24 hours ago [-]
I agree with both OP and you
psychoslave 23 hours ago [-]
I disagree with everyone, self included, but especially with Cretans.
codeduck 23 hours ago [-]
Cretani eunt domo!
i_am_a_peasant 23 hours ago [-]
Monty Python fan detected :D love your profile desc btw
conformist 23 hours ago [-]
> 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).
armchairhacker 23 hours ago [-]
Math is one field where you can mechanically prove a paper's findings. The only thing that would need to be judged is the (verified) statement's importance.
conformist 23 hours ago [-]
Yes in theory, but not yet in practice because not everything is fully formalised.
solid_fuel 12 hours ago [-]
>> However, the declaration argues math is more than a machine for producing correct answers.
> 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
Shall I continue? Or do you think that choosing which questions to answer might have some level of importance, in addition to getting correct answers?
smath 23 hours ago [-]
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.
barrkel 21 hours ago [-]
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.
kleyd 23 hours ago [-]
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?
hyperpape 23 hours ago [-]
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."
ajs1998 16 hours ago [-]
Spoken like someone that has no idea why mathematicians are important.
IshKebab 14 hours ago [-]
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!
fragmede 24 hours ago [-]
People need jobs. What's wrong with nerds having jobs via a program?
RugnirViking 22 hours ago [-]
what's wrong with artists having jobs via a program? whats wrong with struggling alcoholics having jobs via a program? athletes? politicians? there is no inherent virtue in the struggle and effort associated with great mathematical achievement. It may be satisfying and worthwhile for the solver, but not for society at large, any more than any other pleasurable activity. No, as it is, the sole reason for it is in the result itself. In increased understanding, as it flows down into the sciences, and engineering. There are other benefits, recreation and joy as experienced by others, from access to beautiful proofs, though these are never explicit goals of such programs because they are both impossible to quantify and rarely ever remotely relevant compared to the value brought by the practical value brought by maths.
Of course, there may be some valid arguments that everyone should have a jobs program in the form of ubi or something similar. But I feel thats very different to arguing for mathematicians specifically
rspeele 14 hours ago [-]
> whats wrong with struggling alcoholics having jobs via a program?
Finally, a job AI will never beat me at.
mswphd 15 hours ago [-]
for mathematicians, they do a form of fundamental research that is
1. (generally) incredibly cheap to fund, and
2. (occasionally) has extremely out-sized commercial impacts.
This is to say that jobs programs for math (and more generally fundamental research) have lead to extremely positive ROI for society, which is the typical justification given for funding them.
orangecat 13 hours ago [-]
This is to say that jobs programs for math (and more generally fundamental research) have lead to extremely positive ROI for society
Which makes it not a "jobs program" as the term is generally used.
mswphd 12 hours ago [-]
it arguably still is. The primary unit of production of the jobs of mathematicians is itself not particularly useful for society. In this sense funding them is a jobs program. It is also true that they occasionally produce things of great value, and more frequently the things they produce can be leveraged by other researchers to directly produce things of value. But neither of these are what the job of a mathematician is (either in a day-to-day sense, or even for many mathematician's careers).
To go back to the analogy of jobs programs for alcoholics, it is somewhat similar if there was a small chance every time an alcoholic defecated in public gold came out. This fact might be used to support a jobs program for alcoholics, on the basis of it being positive ROI to society. At the same time, the "job" any individual alcoholic is doing in this setup is not particularly useful to society, so one might still call it a jobs program.
fragmede 15 hours ago [-]
The struggle itself is virtuous.
psychoslave 23 hours ago [-]
People need many things, there are all kind of theories ready to assess and assimilate if deemed worth it out there. A job is not part of any I’m aware of, though it can encompass some human needs in some cases, or go straight against them in some other case.
fragmede 15 hours ago [-]
Every human needs a vocation, better if it's of their choosing.
analognoise 20 hours ago [-]
> But not because it's a jobs program for nerds.
We’re becoming increasingly embarrassing as a society.
bloqs 24 hours ago [-]
well put.
freehorse 3 hours ago [-]
Understanding is and always has been the "hard" bottleneck. In programming work, if one drops understanding and eg let's an agent write code with only superficial human review or none at all, I believe that they can easily get 100x fast or more, the main question being whether the process collapses some point due to sloppy code. In research fields like mathematics, skipping understanding is not something that can be done without a radical reconstruction of what mathematics (as a process/activity/field) is.
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.
modriano 23 hours ago [-]
> “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
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.
joelthelion 16 hours ago [-]
It's also a way to model the world and produce new useful abstractions. It's not just about solving problems.
nphardon 16 hours ago [-]
Yea, math is a crazy thing. just "a tool to solve problems" is a wild take. On one extreme, there's an edgy but logical / plausible hypothesis that we live in a universe of mathematical objects, and at the other, math also discovers a lot of questions, the exact opposite of solving problems.
modriano 10 hours ago [-]
Is there a useful abstraction that doesn't help solve a problem someone has?
joelthelion 5 hours ago [-]
You're right, but what I'm saying is that solving the problem isn't necessarily the primary goal and these new abstractions can be valuable in their own right.
est31 13 hours ago [-]
One of the reasons why over a decade ago, I dived deeply into the OSS world instead of mathematics was that it was so much more accessible: there were docs for everything, and I got direct feedback when something worked vs when something didn't work. Most of my questions had answers on stack overflow, and once I joined Rust (which back then in 2015 didn't have a big stackoverflow presence) I had a community who answered them for me (and in maths I didn't have that).
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.
rad_val 13 hours ago [-]
Agreed. As someone who was always curious but had difficulties learning math the way it's taught at the university, AI teaching me the way no professor ever could is a blessing. I fail to see the point of the memo besides: we got here first and we decide what math is because we can. I'm really optimistic about AI and the value it brings in education. Gatekeepers will complain, but ultimately, will either adapt or be left behind.
overgard 13 hours ago [-]
I imagine the concern is more towards using LLM's to create proofs rather than using them to understand things.
rad_val 13 hours ago [-]
i haven't read their memo, but, the article talks about math being something deeply human and the AI taint. I think it's a bit of both.
tz18 13 hours ago [-]
>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.
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?
est31 9 hours ago [-]
Yeah, among other factors, that "figure it out" mentality put me off in the end. Especially because often you need to show the same mentality unless you want to overkill proofs and spend more time on them than assigned to you. I sometimes miscalibrated and pointed out some details that didn't need pointing out in my proofs while in other proofs, I skipped over too many details for the TA.
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.
fc417fc802 12 hours ago [-]
You will have to learn to voluntarily figure things out for yourself without being pushed towards that. In a sense it's analogous to the presence of cheap calorie dense foods. In order to not be overweight you have to be mindful of and regulate your food intake in various ways.
Alternatively, perhaps universities will provide access to fine tuned models that are mindful of such things.
Jtarii 11 hours ago [-]
You can ask the LLM for a hint as well.
nick__m 10 hours ago [-]
You can but sadly most people ask for awnsers.
umutisik 59 minutes ago [-]
People talk about research mathematics being a science or an art, but it's also a sport. AI will kill the sport aspect of it. The art will survive. The science will thrive.
amelius 56 minutes ago [-]
I think the bigger issue is that mathematicians historically invented the abstractions to make maths easier to understand for humans. With LLMs, will we get abstractions that only computers can understand?
rurban 39 minutes ago [-]
Cry-babies. They want to stay back in the stone-age.
bandrami 24 hours ago [-]
My vague prediction right now is that in five years LLMs will be heavily used by universities in grant-funded math research but nobody else will be able to afford it, much like supercomputer clusters 25 years ago.
azan_ 24 hours ago [-]
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.
bossyTeacher 24 hours ago [-]
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.
kakacik 24 hours ago [-]
Sounds very good for regular joe software dev, almost too good to be true
birdland 2 hours ago [-]
>My job is uniquely creative and human, other jobs can be automated away but mine is just so special.
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.
knollimar 23 hours ago [-]
Math for non mathematicians is a tool. Math for mathemeticians is an art in the same way an artisan takes pride in his work.
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
n64controller 21 hours ago [-]
If spending millions of hours rote memorizing formulas and rules like a robot is "art" then sure
Silamoth 15 hours ago [-]
It’s a sad state of affairs when so many think math is about “rote memorizing formulas and rules like a robot”. That’s how math is taught through freshman or sophomore material for a somewhat ‘general’ audience. But real math is nothing like that - it requires far more creativity. You need to discover formulas and rules. You invent new rules and see what the consequences are. This all requires a great deal of creativity. Nothing “rote” about it.
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.
max-amb 16 hours ago [-]
This seems like an intentionally reductive view on mathematics, however it is true it can be taught like so.
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.
ykonstant 15 hours ago [-]
Ok this is your second comment that reads like pure rage bait, is there a way to notify the mods?
stephc_int13 11 hours ago [-]
I see any kind automation as a good thing as long as it is reliable enough. We stopped copying books manually a long time ago, and the craft was lost, most of us can't do complex calculations manually etc. but it does not matter as long as we can rely on calculators and computers to do it.
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.
phyzix5761 22 hours ago [-]
Is it possible they feel threatened their jobs are at stake?
internet_points 2 hours ago [-]
I don't think current mathematician's jobs are at stake, as much as the field itself, if LLMs take all the "easy" problems that phd students would try to learn by solving on their own. Mathematics is susceptible to the same ladder-pulling situation that we see with junior programmers and LLMs.
tptacek 14 hours ago [-]
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.
TrackerFF 24 hours ago [-]
I've said it before, but there's a massive risk that we simply stop educating researchers. So much of a Ph.D revolves around the person learning how to do research.
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.
DrScientist 24 hours ago [-]
> 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.
TrackerFF 23 hours ago [-]
Absolutely, but at least in the pure / less applied fields, access to computation hasn't really been that critical. The more towards the pure and theoretical, less so.
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?
DrScientist 21 hours ago [-]
Hopefully not as expensive as CERN :-)
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!
Dilettante_ 24 hours ago [-]
>and the pursuit of knowledge for its own sake
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?
n64controller 21 hours ago [-]
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.
narsonika 19 hours ago [-]
This is false on so many levels. Is it ragebait?
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.
Myrmornis 23 hours ago [-]
> AI-generated papers could overwhelm peer-review systems with low-quality work
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.
paulpauper 17 hours ago [-]
It is potentially worse with math because accuracy is much more important and there are fewer reviewers compared to other fields.
pfdietz 17 hours ago [-]
It will drive math journals to require formalization of the proofs in the supplemental material.
fooker 24 hours ago [-]
I'm curious about whether we will start discovering new maths in the next few years that provide insight into unsolved CS or Physics problems!
pfdietz 17 hours ago [-]
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.
fooker 13 hours ago [-]
Right, most professional mathematicians know almost nothing about their neighboring branches of mathematics.
An algebraic geometry researcher would be hard pressed to understand a new result from category theory or even something closer like commutative algebra.
bossyTeacher 23 hours ago [-]
For all you know, some of this has already happened but kept secret for national security reasons
fooker 13 hours ago [-]
Well, we had tried to ban exporting cryptography once. It didn't go well.
1970-01-01 15 hours ago [-]
Why is it wrong to expect humans (mathematicians) to adapt here? AI is already producing solutions to problems that humans could not find. Culture holds value until it does not.
QuantumNoodle 15 hours ago [-]
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.
1970-01-01 15 hours ago [-]
>mathematics is a human-endevor
Just like numbers and logic, it isn't and never was reserved only for humans.
They need to adapt.
QuantumNoodle 9 hours ago [-]
Idk man creativity is something pretty human. insufferable people are very fine if, for example, music & art is outsourced to AI so they can make a some $$. But those things are meant to be enjoyed, not consumed.
IcyWindows 12 hours ago [-]
I don't see how this is different than things replaced decades ago:
Music is a human endeavor and musical recordings hurt musicians, bands, orchestras, etc. Especially those starting out.
QuantumNoodle 9 hours ago [-]
This feels like a troll comment, but I'll bite.
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.
HDThoreaun 13 hours ago [-]
Mathematics is not inherently a human endeavor and claims such as those are why the GOP voters are fine with cutting research funding so heavily. Even if you think it's true you probably shouldnt write think pieces that say so because it's bad politics.
Ancalagon 15 hours ago [-]
Did you mean to say "culture"?
> Culture holds value until it does not.
andai 7 hours ago [-]
> “The tech industry proceeds in accordance with commercial logic, which is antithetical to the values of mathematics,”
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.
cryo32 24 hours ago [-]
As a mathematician by trade I think they’re overblowing it. You can choose to use it or not. I choose not to because I enjoy the process. But I’m not doing formal research or getting paid to do it these days.
I will note that the average corporate mathematical modelling is usually a fucking circus so adding AI might make it better.
ryan_n 24 hours ago [-]
> 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?
cryo32 23 hours ago [-]
My academic connections that I keep in touch with never really left the 1990s. And no one is pushing them on AI.
monobot12 17 hours ago [-]
Sure, but their peers (who do use AI) will out-publish them soon enough and solve the open problems before they do.
TheServitor 5 hours ago [-]
So your downside here is problems get solved faster?
alpinisme 23 hours ago [-]
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.
cryo32 23 hours ago [-]
I don't think all universities or research agencies are particularly pressed on this. I mean my daughter is a notable researcher in a scientific field and they have absolutely no pressure to use AI to pump out papers or deliver value quickly.
alpinisme 23 hours ago [-]
I highly doubt there is any overt pressure in academia right now to use AI. It’s a relatively conservative institution. But there’s certainly pressure to publish (publish or perish being a common phrase for decades), and competition for jobs in academia is fierce. That’s what I meant in referring to long term pressure.
thesamethrowawa 23 hours ago [-]
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?
cryo32 23 hours ago [-]
Well I rather like to be paid more than a mathematician so left academia rather quickly. In my case corporate modelling mostly involves making prediction models based on shitty data and metrics to make poorly contrived business decisions that lose millions of dollars.
thesamethrowawa 21 hours ago [-]
lol.... but they are still data driven decisions, everyone loves those, especially when you lose millions of dollars and need to justify it.
golol 23 hours ago [-]
Read the declaration. The article misrepresents it imo. It is not strongly opinionated.
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.
Isn't the whole point of the field of mathematics in a theoretical sense the pursuit of formal solutions?
So, why would they be advocating for limitations on arriving at solutions?
bryan0 16 hours ago [-]
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]
Solve it and understand it; seems intuitive to me.
I don't understand how that contradicts my question.
casey2 15 hours ago [-]
And the goal of computer mathematics research is computer understanding of mathematics. I fail to see a reason provided as to why society should defund automated reasoning just so mathematicians can put off burger flipping for another year.
x86cherry 16 hours ago [-]
You will find your answer in the article
dhfbshfbu4u3 23 hours ago [-]
In a year, none of this will really matter. Intelligence is now a scalable resource independent of biological constraints. Everyone will use it because the system will no longer afford them the luxury of time. In a decade (maybe sooner), references won’t matter either.
pjc50 23 hours ago [-]
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)
dhfbshfbu4u3 23 hours ago [-]
Does it matter if the Leiden Declaration is correct? To the humans, maybe but not in the bigger picture.
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.
freakynit 24 hours ago [-]
"""
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.
"""
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.
adamnemecek 8 hours ago [-]
> For years, AI researchers have used math as a proving ground for their models.
For years?
atleastoptimal 17 hours ago [-]
This is all contingent on AI forays into mathematics being slop and low quality. However it's clear that recent AI models are capable of genuine mathematical achievements which surpass the frontier of what humans are able to accomplish (wrt the unit distance Erdos problem).
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.
magicalist 17 hours ago [-]
> 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.
17 hours ago [-]
z3ratul163071 7 hours ago [-]
the least i expected the math dorks to be luddites
Theodores 23 hours ago [-]
From the article:
> 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.
morpheos137 13 hours ago [-]
AI is the interpolation of the human corpus. Is it suprising AI recombines sucessfully where human attention has not explored all plausible solutions? N,o not especially. The key fallacy is that AI is other than human. This is really no different from computer proofs, e.g. 4 color theorem. The fact the prompt is not linked to the solution by individual human intention alone does not make the solution less human in origin.
Far more interesting as it's outlaying a set of principles for using AI to augment human involvement and science, rather than replacement.
sylware 23 hours ago [-]
Are maths AI models now using "tools", aka formal solvers?
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".
ck2 23 hours ago [-]
I still don't understand how "AI" is ready for serious use beyond entertainment purposes
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
llbbdd 17 hours ago [-]
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
bo1024 16 hours ago [-]
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.
themafia 11 hours ago [-]
Eh. This isn't "AI" or language models masquerading as intelligence. I submit that this is actually the long tail of the internet and the decision to rest on the laurels of peer reviewed submissions rather than advancing the field to better disseminate knowledge.
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."
juleiie 23 hours ago [-]
I will argue that AI and flood of low quality slop makes genuine human work more valuable, not less.
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.
n64controller 21 hours ago [-]
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
juleiie 18 hours ago [-]
You will never be able to match insanity of human mind embodied in a walking sack of meat that was shot out of another.
rad_val 10 hours ago [-]
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meindnoch 23 hours ago [-]
Another mathematician already predicted this, but you didn't listen. His name was Theodore Kaczynski. It's time to reap what you've sown.
busyant 18 hours ago [-]
I think plenty of people "listened."
But what was his plan and how would you have proposed implementing it?
bitwize 16 hours ago [-]
It's obvious what his plan was: blow up key people until the dark future was averted.
busyant 11 hours ago [-]
So ... how many people have you murdered in the name of advancing your goals?
Nasrudith 8 hours ago [-]
The complete historical illiterate? He didn't even understand the hunter-gatherers he so idolized.
burnt-resistor 14 hours ago [-]
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.
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.
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.
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.
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.
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?
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.
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.
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.
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.
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.
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.
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.
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.
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
As opposed to, say, drug discovery.
[1] https://arxiv.org/html/2605.20695v1
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.
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?
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.
We can reach Q models just by throwing resources at it. That’s a million times current B models.
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?
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.
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?
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.
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.
"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]
[1]: https://x.com/OpenAI/status/2060451757818601808
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.
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.
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.
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.
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.
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.
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
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.
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.
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?
> 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.
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.
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.
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.
The future may not have access unless we fight to ensure they do. This is how I read the article.
Argument?
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.
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?
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.)
"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"
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.
Scientific work is not normally naturally statistically salient for LLM observational data inferences. =3
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
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
Building systems based on application specific benchmarks rather than general what-if use-case scenarios will sometimes show you something interesting. ymmv. =3
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.
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.
Is that what is offending you so much?
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'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).
(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.
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.
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.
> 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.
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.
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.
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-...
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.
> …
> 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.
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".
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.
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Do you see the problem with your reasoning?
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.
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.
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?
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.
Love it! XD
I agree, and I think, as with physics, mathematical research produces building blocks whose utility won't be realised until later.
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.
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
Do you think Stephen Cook and Leonid Levin deserve more credit than whoever solved it?
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.
In contrast, class NP and NP-completeness quickly became central concepts in theoretical computer science.
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).
> 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.
Shall I continue? Or do you think that choosing which questions to answer might have some level of importance, in addition to getting correct answers?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.
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?
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."
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!
Of course, there may be some valid arguments that everyone should have a jobs program in the form of ubi or something similar. But I feel thats very different to arguing for mathematicians specifically
Finally, a job AI will never beat me at.
1. (generally) incredibly cheap to fund, and
2. (occasionally) has extremely out-sized commercial impacts.
This is to say that jobs programs for math (and more generally fundamental research) have lead to extremely positive ROI for society, which is the typical justification given for funding them.
Which makes it not a "jobs program" as the term is generally used.
To go back to the analogy of jobs programs for alcoholics, it is somewhat similar if there was a small chance every time an alcoholic defecated in public gold came out. This fact might be used to support a jobs program for alcoholics, on the basis of it being positive ROI to society. At the same time, the "job" any individual alcoholic is doing in this setup is not particularly useful to society, so one might still call it a jobs program.
We’re becoming increasingly embarrassing as a society.
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.
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.
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?
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.
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.
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
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.
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.
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.
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.
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?
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!
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?
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.
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.
An algebraic geometry researcher would be hard pressed to understand a new result from category theory or even something closer like commutative algebra.
- 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.
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.
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.
> Culture holds value until it does not.
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.
I will note that the average corporate mathematical modelling is usually a fucking circus so adding AI might make it better.
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.
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
So, why would they be advocating for limitations on arriving at solutions?
> 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
I don't understand how that contradicts my question.
(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)
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 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 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.
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.
> 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.
https://leidendeclaration.ai/
Far more interesting as it's outlaying a set of principles for using AI to augment human involvement and science, rather than replacement.
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".
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
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.
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."
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.
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
But what was his plan and how would you have proposed implementing it?
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.