As a mathematician, I will never understand how my colleagues had never foreseen this. Okay, maybe you can believe in a plateau but at least accept as a possibility that there won't be one. This applies to every other human endeavour.
I am also sad, but my sadness is, in a way, priced in.
It was really quick. I stumbled across a comment from KConrad on MathOverflow on how useless AI was for research. That comment was from 2024. When it disproved the unit distance conjecture in May, the solution involved a breath of knowledge of the literature that it made sense it took a computer to make the connection. It only took 5 months for it to advance to the point that it was better than humans at almost everything. Even if you expected it, the mind boggles at how fast it was.
Also, it's impossible to really know how you're going to react to something until it happens to you. When ChatGPT came out, I already thought "in the long run they're going to automate almost everything humans love to do". Despite having 4 years to brood about it, I did not really know what it would feel like until it happened to me directly.
Programmers, artists, anyone who used to create any form of documentation, translators, there are literally so many.
Either these mathematicians thought it would never come for their jobs, because all of these other jobs were simpler or they've been living under a rock.
Either way, I don't feel for them any more than I do for all of humanity.
I happen to be (re)reading Strogatz' book "Nonlinear Dynamics and Chaos" right now. It's beautiful thing, and nothing in the "age of AI" threatens to equal it or in any way diminish my enjoyment of it.
I think the point is that in the "age of AI" virtually nobody will have the opportunity to follow a career and produce works like Strogatz was able to do.
I'm not disagreeing with the post. My personal opinion is that LLM "thinking" is qualitatively different to human thinking, that the "convex hull of ideas" idea is probably right in some sense, that the social structures we have built to allow and encourage human intellectual exploration are not obsolete and that there is a high risk of catastrophic damage being done to them by those who fail to see this.
It's really a fascinating book and topic. And i too enjoy appreciating it. I guess it's appreciating the beauty of nature and as you said, AI will not be able to threaten that.
I still think mathematicians are going to be as important as ever. The LLMs are only finding what was technically possible given our current breadth of knowledge after brute force solving problems. Eventually it will run out of things that it can do with its training and the only way to get it to do more will be training it on new human found knowledge.
Finding that new knowledge may be exponentially more difficult, but it will be much more valuable.
> Eventually it will run out of things that it can do with its training and the only way to get it to do more will be training it on new human found knowledge.
I don't see how its output possibilities could be infinite given a finite set of data. Taken to the extreme if you give an LLM only a single piece of information then that is going to be the only thing it can produce. Given it two pieces of information and maybe it can produce every combination of those two things. Given all of the worlds knowledge I'm sure its possible outputs are incomprehensibly large, but still not infinite. Then having its outputs match a set of really specific instructions is going to be a small subset of that.
Sure, a given LLM can only produce so much (though it might be a lot), and then you'll need fresh inputs. But you contend that those have to be human, while I am arguing that they could be
- random,
- from the environment,
- from other AIs.
That increases the space considerably.
In particular, note that your argument also applies to humans: A human, being finite, can only produce finitely many things. But you add other humans, the environment, and pure chance to the mix, and it'll be pretty infinite for all practical purposes.
I would be curious to see the end result of letting it endlessly update its inputs from its own outputs. I was under the impression that eventually this would cause delusions and if you let it do this long enough and it is solving problems that no humans can verify as correct that it isn't useful.
Maybe that is the limits of my imagination, but it seems unwise to take humans completely out of the loop. Having mathematicians around that can understand what it is doing so that we can utilize that knowledge in a way that we can have some confidence in seems important.
It seems far more likely to me that current LLMs have learned generic reasoning capabilities which will allow them to solve effectively infinite problems, rather than whatever vague limit on capabilities you seem to think there are.
Typical thought experiment - go back to the early years of humanity, even before we've created language itself. Communication is basic grunts and other such things, cutting edge knowledge is 'poke him with the pointy end'. Compile all of this knowledge into the training set for an LLM, let it run with infinite processing power and infinite time. It'll just continually remix the knowledge of that snapshot of time. You're certainly not getting to the Moon, discovering the secrets of the atom, or even discovering mathematics.
It sounds increasingly dumb to bring this up given how advanced models have become, but it remains true - they are still fundamentally token prediction algorithms. And that imposes hard constraints on their potential. These constraints are relatively academic in our current world because there's just so absurdly much information and no human can hold more than a negligible amount in his mind at once. So they'll certainly be able to connect many 'obvious' dots that we don't see simply for a lack of carrying capacity, but they're not going to discover concepts that do not clearly follow from the comprehensive knowledge of an era.
How is this different from human brains being made up of a finite number of neurons in a finite number of configurations? I know very little about LLMs and nothing about neurology and maybe you can explain to me why one is so much more powerful than the other.
As I understand it, the advantages of AI have not just been speed of processing.
Humans have cognitive limits. Humans hold a specific slice of all knowledge at a time and necessarily limits what correlations can be leveraged. This is similar to "context windows" in modern AI. Human minds are wildly variable in efficiency. AI is has constant inefficiency, largely controlled by the speed of light, physical RAM, and processing power, with superior (no cost) "thought" context switching. The AI also has superior accuracy without the need for "concentration" because the AI is processing on a Turing machine with the limited operations that it's superior at in terms of time and energy - eg find an element in a list of 1 million elements.
> a finite number of neurons in a finite number of configurations
Neurons and synapses change, especially synapses (the connections between neurons). Some neurons have a few connections to other neurons. Others have hundreds of thousands of connections to other neurons. These grow and change. "Estimates are between 100 trillion and 1,000 trillion neural connections" Is it finite? Depends on your definition of finite. If anything countable is finite, then it is finite but the number may be practically uncountable. Is it fixed? No.
That's 100G to 1T. Neurons are more complicated than weights, so add a x1000 somewhere, but they have more redundancy because they are noisy, unrelilable and may even die, so divide by some small number.
1. LLMs aren't run deterministically - they use a bit of noise to improve the output. So even with the exact same input you're going to get a whole distribution of outputs.
2. Feedback from the environment. LLMs (and humans!) work best when they can test ideas. That's why coding agents work so well. If you ask an agent 1000 times to just think of a compression algorithm, it'll probably spit out very samey answers just like a human would. If you give it a test corpus and let it try out different things it will absolutely be able to come up with novel ideas.
AIs cannot create things like cubism, surrealism or hip-hop.
Or maybe it can; the early image generators famously did output horrific images sometimes because of their limitations, but these outputs qualify nonetheless creations.
Now the interesting thing is that it is humans who evaluate them as horrific, worthless or interesting.
Certainly AIs could pretend they have that sort of opinions and judgement, but hopefully humans will remain more interested in the opinions of other humans.
> Certainly AIs could pretend they have that sort of opinions and judgement, but hopefully humans will remain more interested in the opinions of other humans.
Online, how would they even know? Offline -- either we give up on the internet, or the offline discussion will be heavily influenced by what AI said online.
Amusingly, LLMs are good at detecting patterns because they are modelled after the human brain, which is very good at detecting patterns (this e.g. is annoying when you are a game developer who wants to reuse assets). People are still able to tell if the source of an artwork (or text) because LLMs tend to output "patterned" text (like the famous "load bearing"). That being said, the false positive/negative rate of humans for this task has yet to be measured.
Yeah, I think that's a great essay and it really helped clarify some of my own ideas.
At the same time, I think it's pretty amazing how this is the fundamental question of AI's trajectory, yet people on both sides seem so sure of their viewpoint with little actual evidence or data to back it up.
That is, it's possible that LLM-based approaches will be forever limited to this "convex hull" of current training data, and thus humans will always have a role to play (and will retain primacy when it comes to overall intelligence), because it appears that humans are not limited to this convex hull based on previous discoveries (e.g. I think basically every scientist would argue Einstein's general relativity was outside the convex hull that existed at that time).
Yet, if LLM-based approaches are eventually able to surpass humans on this "novel discovery" axis, then we as a species are basically done for when it comes to intellectual primacy - and then it becomes a very strange and scary world. And many AI researchers at the top companies clearly believe current approaches will scale and eventually (pretty soon I'll add) lead us to RSI.
I have no idea which side is right, and I'll admit I really hope LLMs eventually hit a wall. But again, I just am surprised that many seem to be putting forth their beliefs, but I would love to know if there is any good evidence one way or the other (or better yet, a more concrete and verifiable definition of the "convex hull" so that researchers could provide some falsifiable examples).
> Think about a world where non-mathematicians leapfrog the field after a few AI prompts. It would be a world with more knowledge. But it might be a world with less understanding.
This seems like a good summary of the article, and I think the answer here is obvious.
So what?
News flash folks: we do not have the knowledge to survive on our own. If a solar flare deletes the electric grid, the first thing that happens is 8 billion people die in the next year.
We survive on abstractions others have built, and we have a global dependency web.
AI abstracting knowledge is neither new or dangerous; if it is dangerous, this certainly isnt the argument to prove it.
I think they might find comfort in knowing it's not just mathematicians - this is coming for all of us. Computers putting us out of work has been coming for a long long time.... and now it's here.
I'm convinced the people who do not see what Mathematicians see simply have not considered that their jobs are just as vulnerable. The day when OpenAI replaces their own AI researchers with GPT will be upon us, and the response from the Mathematicians will be "First time?"
I have little sympathy, since I think it is a result of many decades of complacency when it comes to thinking about why want to people do what they do and instead just enjoying being the ones with the results and money. People who love the process and see the human as inherently valuable won't have a problem. I'm more shocked that there are so many brilliant people who are so results-oriented and don't even have a concept of value outside of that. I'm kind of disgusted by the tears actually
This is exactly my perspective as well. Seeing humans as inherently valuable is the key. It's shocking to me that so many people seem to have been reducing humans to the results they are able to produce and now that the results appear to be drying up, their perception of their own human value is going down with it.
I’m fully STEM: software developer with a physics PhD with a mechanical engineer wife, but this is certainly just deserts for the STEM people, who mocked the humanities and especially philosophy for centuries (literally, this started back with Kant even) for spending time thinking about these questions. STEM people are not at all spiritually prepared for it, and this is the main disappointment for me. I spent many years struggling with publish or perish in physics but all those around me claimed they simply loved doing physics and that that’s what mattered. Now as soon as their exclusive ability to give results is slightly threatened the love of physics/maths/coding is no longer spoken about. I get maybe they weren’t the same people, but I thought more people loved the process than it seems today. If anything these new developments make it even clearer for me what I love about this kind of work. I’m so happy it has made me not have to defend based on results, a razor sharp contrast that makes it so clear really why humans do things
it is a little ironic that after the "creative destruction" of industry after industry all of a sudden now it's unfair and we weep. I say this as a software dev with his neck on the line like everyon else.
It’s not necessarily their fault. Our economic system puts zero intrinsic value in humans, it only puts value on their output. It’s hard not to feel this type of way when it’s the only system you know.
> I have little sympathy, since I think it is a result of many decades of complacency when it comes to thinking about why want to people do what they do and instead just enjoying being the ones with the results and money.
It's not clear from your comment: Are you talking about math professors?
If so, what money? They make a living, but it's not that great. I just checked the salary of two associate professors at my university: $80-85K.
And keep in mind they often start their careers quite late. Often multiple postdocs after PhD. Starting a tenure track position in your mid 30's is not common.
> I'm more shocked that there are so many brilliant people who are so results-oriented and don't even have a concept of value outside of that. I'm kind of disgusted by the tears actually
The whole point of the article is that the person crying is because he loves the process and is worried the human connection will be gone.
As a software engineer it is hard for me to feel sorry for these mathematicians. Software engineers are getting laid off because of advancements made by AI. Math researchers are employed at universities, and universities aren't laying off mathematicians now that OpenAI has solved some of their puzzles. Mathematicians can still feed their families, software engineers displaced by AI can't. Yet I see far more of these blogs about the plight of mathematicians, not enough about the plight of software engineers.
Software engineers are getting laid off because of the end of zero interest rates, wars started by Putin and Trump, and a bit because of pandemic over hiring. The part that is due to AI is less hiring of juniors. Know your enemy before starting a fight.
There are orders of magnitude fewer mathematicians with full time academic positions than there are software engineers. The economics are different and consequently academia tends to be more slow moving in reacting to market trends. It doesn't mean it isn't coming, but it will look somewhat different
I don't personally grieve for the likes of Strogatz or Tao, it seems likely both of them will be fine however this pans out. But many more mathematicians have been toiling away in precarity and obscurity and without tenure for years and years. And of course many mathematicians went into private sector roles as software engineers, actuaries, analysts, ML engineers and similar, and have been displaced there.
Visible blog posts written by famous mathematicians closely aligned with the news and hype cycles getting frontpaged on HN doesn't mean that mathematicians are disproportionately whiney, or that software engineers aren't writing about it or ending up on the front page
The idea that a junior software engineer who can't get a software job _and also can't get any other kind of job or move somewhere cheaper to live_ to the point that they can't afford to feed their families is what's unrealistic.
It's not like the options are write software or starve. Billions of people find jobs and can feed their families without writing software.
I also don't often meet junior software engineers who actually have families to feed. Most juniors work in the industry for a few years before deciding to get married and have kids.
Unfortunately this often means moving away from family. What are you going to say to the kid whose parents toiled to get him through college on the expectation that he would be able to take care of them? "Sorry Mom, I have to move to North Dakota to work on the oilfields"?
> Software engineers are getting laid off because of advancements made by AI.
This is said by a lot of CEOs trying to make layoffs look good, but I haven’t seen anything that suggests that it’s true.
> Yet I see far more of these blogs about the plight of mathematicians, not enough about the plight of software engineers.
Really?!? We’ve been living through 3 years of blog posts, LinkedIn hot takes, and podcasts talking about the plight of software engineers in the era of AI. There isn’t a week that goes by without an HN post about it.
I can feel empathy for the practitioners in a field that may have been profoundly changed, the same way I can feel empathy toward software engineers who are struggling with the changes in their own field.
In the end, we’re all trying to figure out where this leads. And how this will affect our lives and our work.
When software engineers spend the last 50 years eliminating the work of clerical, secretarial, and dozens of other back-office functions, were they concerned about how those people would feed their families? Do you think anybody but other software engineers feels any sorrow for them today, now that they are facing the same fate?
While I mostly agree with you, there's an argument to be made that classical programming tended to eliminate the mechanical parts of work that people hate doing, while AI is removing the intellectually stimulating part.
It's not fun to think of a future where knowledge workers become irrelevant in society and we're back to just physical work and managerial elites.
I'm not really one of them, but there are many developers for whom the intellectually stimulating part is designing the system. Writing the code was boring drudgery that they hated doing. They tend to call themselves "architects" or something similar, not "programmers."
That wasn't software engineers eliminating that work. And neither is it today AI eliminating software jobs. Let's name the straw man in the room: it's humans displacing humans, and extracting the value for themselves.
Just like in past history.
Yes, it sucks, for software engineers, those in the past, and those in the future. As long as you and others fall for the straw man perhaps you and others will get more and more tricked and hurt, misplacing your aggression and hatred on each other rather than those receiving most of the extracted value.
Yes in some cases, but the reduced costs also allow lower prices which benefits all consumers of whatever product is being produced. At least for those products where there is a competitive market.
What an asinine comment. Do you think I, as a software engineer, am sitting around twirling my mustache, cackling as I plot to destroy livelihoods? I'm trying to make a living like anybody else. I've had my own opportunities lost to outsourcing and H-1B's since day 1.
The economist John Keynes predicted almost 100 years ago that technological advances would lead to a 15 hour workweek. If AI now makes you 5x as productive, then why aren't you working 1 day a week for the same pay? Why didn't Keynes' vision come true.
It is not for the same pay. Real wage increased dramatically during that time period. And despite all obsoleted jobs, society is still more or less at full employment.
You may not have sat around twirling your moustache but you have actively been involved in an industry where the goal is to automate stuff that people would previously do manually. That is the entire point. Which, if you believe the message on many threads/posts here, puts people out of jobs.
I don't believe it is that black and white. Yes, automation eliminates some jobs, but overall moves everyone (on average) further up the productivity line, which improves lives, again, on average.
That is why we live like kings now compared to people of the past.
But circling back to the point, if AI does the same thing, eliminates expensive manual coding and other similar work, that is essentially the same thing we've been doing to other people for decades.
As a software engineer? The career whose very reason, largely, for existing in this market is to automate jobs, with few notable exceptions? That one is somehow deserving of sympathy and that of mathematicians is not? Come off it. The notion is undignified.
Are phones an "automation of existing jobs"? Maybe couriers (??), but most of what people use them for goes far beyond that.
What about the software used to drive trains or autopilot planes? We still have conductors and pilots, but now the outcomes are safer on average and more efficient.
Or something like Google Maps -- all that replaced was the paper booklet people used to keep in their cars, it's not like we used to employ 'navigators' to go out on a road trip.
Or something like Intuit QuickBooks: there's still an accountant in the loop, and if anything more accountants today than there were a few decades ago.
Certainly some software is essentially automatory in nature. But to characterize 'nearly all' as being such just betrays a lack of awareness for the rest of the field.
> That one is somehow deserving of sympathy and that of mathematicians is not
He didn't say that mathematicians aren't deserving, he was griping that people said this about SWEs. Do you think mathematicians are deserving but SWEs aren't?
How did you manage describe a litany of automation while being fully unaware that is what you were doing? I mean you even said the word autopilot lmao. Yes, not all tasks in a job are fully automatable. iPhones, automating communication. Quickbooks and turbotax, automating business workflow steps.
The words and excels of the world, automating documentation and bookkeepery. It all makes some kind of existing thing we already do faster and more convenient, and often more efficient. Most software simply does nothing new. The thing you didnt manage to list which is the actually good example is games.
You seem to be unaware of the existence switchboard operators (people redirecting calls to the correct person) were hundreds of thousands in the US alone at some point.
The difference is it takes great intelligence to be a mathematician, whereas anyone can be trained to write code. It’s like the difference between an automotive engineer and a mechanic. I would argue writing code takes less skill than carpentry.
Do you write much code? Do you design software? It sounds like you don't. The challenge with software is managing complexity. This is unique to this field. Mathematicians are humans.
What's your source for "Software engineers are getting laid off because of advancements made by AI". I haven't experienced that in my field and can't find any data to support that notion.
It’s only been a few months since LLMs started solving significant open problems in math. I wouldn’t jump to the conclusion that they have it better than software engineers.
In many ways it will likely be the opposite. Tech firms still need engineers to talk to customers and run experiments. What will the mathematicians be needed for?
Unless this is satire, have you not considered how many proles have been laid off because they were replaced by technology or told that a spreadsheet model had decided their job was superfluous? Capitalism has been using technology to fond new ways to enrich itself for the whole age of personal computing, and arguably long before. Those Andreesen Horowitz comments about 'software eating the world' may have been exciting to people working at startups, but came off as rather sinister to people who were in the world outside softwareand just wanted to build a modest but comfortable life rather than develop unicorns and cash out.
I still remain hopeful that this is a step change in capability and capacity that we can adopt and adapt towards our own purposes. What was taking decades for us to do has been reduced to days and dollars. So while we’re witnessing a forest fire consuming entire fields of open problems in math, the best upside would be finding new problems that come from learning how to work with the new tools.
This is what it seems like so far for software. The speed limit has changed. It’s interesting to explore what that means. Extending the analogy, long ago you could be a race car driver part time; that would be pretty laughable for an F1 driver. It’s a very specialized skill and the optimal athlete is quite different for driving a car than for running, for example. That feels like what I’m seeing in software.
The most effective folks are much faster and better with these tools than average. It’s a different skill set.
I think that formal verification makes Math much closer to code than many other fields. But I’m eager to see the first mathematicians that are dramatically more proficient with the new tools than average.
I love that we have new ways to solve problems. And it's been hard to accept that my own unique intelligence—which I've spent decades cultivating—just isn't as important as it used to be. I am developing new ways of thinking to better adapt to an AI-enhanced world, but I'm not done with the mourning process. We're all finding our way through this.
It's probably because with the release of each model it is becoming more and more difficult to see how a person's livelihood and way of life continues. Uncertainty stacked on uncertainty.
How will it pan out, no one really knows. The leading frontier labs are continuing to hire aggressively. But who knows for how long or for long or near-term goals.
I found it amusing the one time that I've seen dang recently steer a thread and actively participate was in a complete one-off on Sino cultural standards (and the perception thereof). To me it almost seemed like he was issuing a threat because, amusingly, I thought, Xi's minions are watching.
The "moderation by less", as well as the guidelines, are both farcical in nature. They've given way to ruination of an otherwise decent tech discussion forum (or news aggregator, or YC propaganda machine or whatever you want to call it).
If you want actual discussion around things, lobsters, awful or even 4chan are better than HN.
I guess there really is a first time for everything!
> The "moderation by less", as well as the guidelines, are both farcical in nature. They've given way to ruination of an otherwise decent tech discussion forum
The guidelines have been constant (with minor updates) for most of the two decades HN has existed. Dan’s a moderator for 14 years and I’ve been one for several. Both of us have been active participants almost since the beginning. When did the ruination begin?
Your comment seems a generic vent, largely unrelated to the parent comment but if you have any actionable suggestions you’re welcome to email us.
> Your comment seems a generic vent, largely unrelated to the parent comment
There it is, the "off topic" play. Well done, Tom.
If you'd put anywhere near the amount of attention in moderating the site for over all quality that you put into my "largely unrelated" comment, this entire comment thread would not exist. Think it over.
Please don’t insinuate that I’m “playing”. You opened this subthread with an insinuation that we’re shills for the CCP. Yes, it breaks the guidelines along with basic decency. The only reason this site exists for people to bemoan that it’s not as good as it used to be (which people have been saying since about three months after it started) is that we have guidelines and uphold them in a way that people can trust, often against pressure from many different people with all kinds of agendas. As I said, if you have any actionable suggestions you’re welcome to email them but all I see here is someone complaining about quality whilst having no concern for the quality of their own contributions here. If you don’t like the guidelines you should take your own advice and find a site that’s more to your liking.
What I was aiming for was to point out the oddity of dang selecting the one odd ball thread to tell everyone to play nice, in what is otherwise a site that is daily filled with AI doomer posts, as per the opening comment of this thread to which I responded. And I did mention that I found it amusing - because I doubt either of you are actually shilling for CCP. This wasn't even the main point of what I was after.
That neither of you are at the very least bothering to steer the greater discourse towards some higher ground is a lack of balance that I am highlighting.
To remark my comment as off topic in the context of the thread opener is to dismiss what I am saying using a very lazy tactic that I've seen used countless times before. No, it is not off topic. It is very much on topic and relevant.
And if you have a problem with my comment, feel free to email me.
We are posting comments in threads to uphold the guidelines every day about all kinds of topics, which anyone can see by looking at our comment feeds. There are posts about AI all the time here because it’s the biggest topic in technology and even the world right now. Some topics/discussions are negative and many are very positive, reflecting the broader discourse. We’re curating the content all the time to provide a good balance of topics and perspectives. By statistical trends and what we observe in the quality of the best posts and discussions, HN is the healthiest it’s been in years.
But one of your central complaints was about the guidelines, which have been constant for nearly two decades, so we’re really not having a serious conversation here.
And how! All the “nothing has changed, stop whining posts” are neglecting that they haven’t had to go look for a job. My previous company became quite toxic about LLMs and employee usage. It’s not “move over to the paradigm or get left behind” instead it was “jump on the hype-wagon and sing LLM’s praises all day, every day or you’re out”.
I'm no mathematician, but my impression as someone who at least does a lot of applied work with maths is that the main thing that's changed is that the yardstick by which academics earn recognition is being re-assessed.
But from my perspective proofs have never been the most interesting part of maths. What really interests me is the intuition that leads to new applications, and to new questions. Newton, for example, gets more credit for calculus than Leibniz because he's the one who demonstrated how darned useful it is. Erdos is arguably more famous for asking interesting questions than for answering them.
I still don't see LLMs seriously contributing to that facet of mathematics any time soon. Every so often I ask one to suggest a good binomial confidence interval for a particular application, and it still tends to just suggest one of the subpar options that everyone learns in a first year stats course.
Proving theorems is one of the main ways that mathematicians turn math into money.
The recent barf-out done by LLM companies is useful if what are you are interested in is an unreadable mess and someone giving you the final answer. You might be able to extract some new "aha!" moments out of it, but we will have to wait and see what happens once people go through it. I would hypothesize the LLM output will be good at hinting at some non-obvious relationships and patterns, but I doubt it will yield anything of substance in its own right.
Historically, proving difficult theorems required mathematicians to sometimes invent new fields, new ways of looking at problems, the discovery of which itself yielded incalculable contributions to both mathematics and other fields. It is this that is the greater contribution of and to mathematics.
> Proving theorems is one of the main ways that mathematicians turn math into money.
Mathematicians working at universities are paid to teach, to engage in the politics and business of government contracting, and only in very rare circumstances to play prestige games on famous problems.
It is unlikely that Congress will decide to slash budget for NSF's MPS directorate on account of AI getting good at proving theorems. That's just not how Congress works.
It is also unlikely that LLMs will be able to do, in the near future, do the sort of applied mathematics that happens in the context of occasional NIH or DoD funding that flows into Mathematics departments, for a variety of complex sociological reasons I'm happy to expound upon.
And it's unlikely Engineering degree programs will stop requiring calculus sequences, since being able to do the computation was has not been the actual point of teaching calc since he late 20th century (if ever).
So, the good news is that I doubt any of this will have much of an effect on the job market for Mathematicians who want to prove theorems for a living. The bad news, of course, is that the job market for Mathematicians who want to prove theorems for a living was already nearly non-existent.
> One doomer post per day to keep the valuations up.
Yes, outsiders (investors, journalists) are given a false impression while insiders (software engineers) are supposed to give up. Double win for the agitprop machine.
Mathematics as a field of study is maybe 2500 years old and a lot has been figured out in that time. But look how far AIs have come in just ten years. I wouldn't expect progress to stop now, would you? A decade from now I wouldn't be surprised if machine mathematicians are essentially oracles. They tell us the answer but can't explain it to us any more than you could explain long division to a golden retriever.
The doom posts will continue until profits are up.
Jokes aside, its like working for a company, individually you can lift mountains and someone else will take credit. When you use LLMs for a single bit, all credit goes to LLMs lol
the above, a "sorting" machine is the primary usage of hyperscalers currently, they're just trying to find new uses to boot for extra money/legitimation.
the browser industry
the phone industry
the CAs
JavaScript
it's all paid for by those wanting to sort humanity into whatever bucket the can fear monger the peasantry into for aweek or however long it takes the classification system to sort out.
gentrification works, but we chose forcible displacement now we have to live with the forever consequences.
would you give guns and bombs to the displaced native americans who taste blood over their beads and trinkets?
we've been inside the event horizon of the singularity for years, the tax bill will only get worse unless we recognize it for the tick that it is.
This was the top comments when the submission disappeared from the front page.
Now the thread has been cleaned up and a pro-defeatism comment is the top comment and the submission is on the front page again (I'm not alleging moderator intervention, user mechanisms are sufficient).
This piece circles, without naming it, around the concept of “deontic scorekeeping” central to the work of Robert Brandom on semantics. The idea that our ability to create and sustain conceptual systems depends on the give and take of reasons within a social setting. Brandom has been front of mind for me lately because elements of his theory both explain why LLMs should work in principle, and where they are bound to be radically deficient. An LLM encodes the normative regularities of our linguistic/conceptual games into its weights, but it cannot be a real participant in our games because it is incapable of staking durable commitments or changing its dispositions over time.
Worth reading: Robert Brandom, “Making It Explicit: Reasoning, Representing, and Discursive Commitment”
Mathematicians always define their terms exactly, in order to begin their discussion. Philosophers never define their terms exactly, because then they would have nothing to discuss.
This may not be helpful or strictly accurate, but when I see deontology or variants of that word, I substitute "SHOULD". It is like a duty or obligation.
I think in future, each company will built its own universe of mathematics consisting of prized discoveries. Maybe one Mathematical Universe per Company in the multiverse of mathematics.
I mean, many companies operate within particular domains but don't have much skin in the domain itself in a way that is meaningful to other participants there -- like Oracle builds software for healthcare, ostensibly, but any actual user of their products will tell you they are nightmarish with regard to usability, and research shows that patients are often harmed as a result. So healthcare isn't their game, it's selling healthcare software.
Likewise OpenAI doesn't have "skin" of any shape or value in e.g. the mathematics community or even in mathematics research per se. In my cynical view, they are exploiting the exposure of mathematical research to automation in an effort to create a case study of their product obsoleting a highly visible and well-leveraged and relatively small profession. Regardless, I doubt even they would say they have durable skin in mathematics in any sense, certainly none of the ones I mentioned above. But their output inhabits the same space as mathematicians, who have to contend with the consequences
What a coincidence, I've also picked up inferentialism but never really got into it.
I'm interested in the reversal that Sellars proposes - it seems clear to me that "representation" is not prior to the space of reasons, despite how unintuitive it is.
> An LLM encodes the normative regularities of our linguistic/conceptual games into its weights, but it cannot be a real participant in our games because it is incapable of staking durable commitments or changing its dispositions over time.
I see no reason why this can't be the case in the future, even though I think inferentialist-based theories are generally strong. Can't you force LLM's (or whatever AI architecture will exist in a decade) to stake durable commitments itself, then? They're at least autonomous enough to run computers. I agree that that is a core part to true meaning & autonomy, but nobody just lets them yet. I don't think inferentialism is a consciousness-based argument, either, so that escape hatch isn't available. I also don't think this is a necessary condition to engage in the space of reasons (but maybe then I'm misunderstanding it, or retreating to a more Sellarsian understanding). Could you explain on this more?
Though again, I only have the book sitting on my desk, and I haven't cracked it open...
I actually agree with you. LLMs could become participants in the future, but this would require (1) durability of real awareness over time (an architecture change from the current fixed-weights + transient context architecture) and (2) that humans grant them the status of equal participants (probably not a terribly high bar given how people already treat these things). I'm not aware of any real progress toward (1), and all the stand-ins people have developed live in the context layer or textual artifacts, which are poor stand-ins for durable memory.
And yeah, consciousness based arguments are bunk IMO. It's clear that LLMs are in a basic sense "aware", and even to some extent "sapient" (in Brandom's usage), i.e. they can justify claims and object to things at least locally. I follow Rorty on the question of consciousness, which is to say that consciousness has no deeper meaning in practice beyond "are you awake", and in philosophical discussion it's largely a proxy for "are you one of us" in some amorphous sense.
Well that's too bad, since Rorty completely misses the point on consciousness.
Consciousness, the interesting part anyway, is qualia, or the "it is like something to be me" as Nagel put it.
Intention, as Dennett used it in his book "The Intentional Stance" is the "are you one of us" angle, and is distinct from consciousness because as the book title suggests, it is a stance taken with respect to some other system (person, cat, dog, turtle, LLM) rather than a description of subjective experience.
So I think it is your (and Rorty's) positions that are bunk. It is very, very far from clear that LLM's are in any basic sense "aware" let alone "sapient". They are extraordinarily able to manipulate language in a way that leads to text generation that has traditionally been assumed to indicate consciousness and/or intelligence. But we have no way of knowing if it is like anything to be an LLM, even though most of us are busy taking the Intentional Stance towards them.
I guess I'd say that your qualia aren't a something, but they're not a nothing either. We can't say what they are. They're not publicly describable, and therefore not publicly attributable. As Wittgenstein would say, "They cancel out."
And as I think Dennett himself would say, qualia are just Sellars's Given rearing its ugly head again.
Well, while I love much of Dennett's work, I emailed him after "Consciousness Explained" was published to say that I was disappointed that it appeared to describe what we are conscious of, not how we are conscious. He sent back a short note to say "You're probably right".
Humans change over time due to what they experience. It’s different from waking up from a ten-year coma and having access to a record of what happened to yourself in an alternative timeline where you didn’t fall into the coma.
Brandom wrote a shorter version of the book, called "Articulating Reasons", which is available in full on his faculty page [1], or from booksellers. He's a very wordy writer (kind of exhausting to read, even for someone who enjoys philosophy), but if you get ahold of Making It Explicit, chapters 2 and 3 have the meat of the ideas I'm talking about here.
I think it means if we all have to justify ourselves to each other, we collectively come up with an sustainable ecosystem of ideas that we can understand. So if we stop justifying ourselves to each other, the ecosystem doesn't get built or sustained, and we stop understanding things. Not the person you responded to, but that's my reading.
If a short sentence like that could accurately summarize Robert Brandom’s deontic scorekeeping, he would have wasted a good chunk of his career.
I mean this with zero snark: if you’re interested, you’d be best off asking an LLM or doing a search. But the overall idea is that Brandom described how concepts and meaning are created socially by a kind of game in which people keep track of the claims that other people commit to.
If you accept Brandom’s framework, then it can be argued that LLMs can’t meaningfully participate in this process. This could be considered an explanation of why people have a negative reaction to LLM involvement in activities that are meaningful to humans - even though most people won’t be familiar with the theory, they would instinctively recognize that something important is lacking, such as commitment over time.
When anyone can push a button on a machine and out comes a math result, it still doesn't become theirs or anyone else's until they can internalize it and make it part of their thinking.
"I can’t solve these things, but maybe you people in the future can stand on my shoulders."
Seeing AI being lauded for solving problems like this makes me feel... uncomfortable.
We are feeding all works of the giants that we have all stood on into a huge vat, in the hope to generate something meaningful.
If we can't understand the output, internalize it, or build from the time and experience that a human would have gained from solving the same problem, it feels like a dead end.
When all the problems have been "solved" by AI, what's next? If humans no longer struggle to solve the problems that the previous generation have left behind, how do we progress?
I strongly feel most angst is because Mathematics has become yet another capital intensive game to play. Rather than a noble, purely intellectual practice.
Getting those fields wiped just shows how unnecessary those fields where in the first place. I'm a programmer. I am the adapter between a stakeholder and a machine. I am just needed because the stakeholder lacks the knowledge to tell the machine what he wants. If he can use this new fancy adapter called Agentic AI now I happily quit my job and go on finding something that makes more sense in the first place.
The same goes for math.
Let's find some things that bring value other than just the price of a stock.
Broken take, with out this trail of mathematics that lead us here we'd have never reached this place. You think multi dimensional matrix math comes natively to humans?
At worst its time is past, but even that is unlikely.
I'm wondering if mathematicians are sort of going through what programmers have been going through recently, really questioning whether their craft can continue. But now it feels like, in the software world, at least from my seat, people can handle much larger scope and work on much more complex problems. If anything, my job has gotten a lot _more_ interesting.
So I wonder if mathematics will go through that too? A period of chaos and doubt, but finding new and more effective ways of working.
People need to separate out the alien intelligence vs human intelligence.
Its easy to demonstrate this: The math hysteria passes Lean but no human can make sense of it at present. This is akin to bots telling humans where to dig. Thats fine but lets be clear about whats going on.
Development is more interesting, I agree, but also more burnout inducing, and the not yet understood implications create unreasonable expectations on developers. It will take some time to find the new normal. Right now me and every colleague are working harder than ever and drowning in tasks. Far from being redundant.
Yeah, for sure, but at least to me, it feels like we will figure that out. We're in a transitionary phase. But it also seems to me like, to your point, we're busier than ever, and in a lot of ways, the problems we're working on are harder than ever.
My friend, who is a math professor, told me that they weep because they are about to become just like us developers: prompt writers and AI output reviewers.
I think Thompson has no understanding of endeavors like pure math or the theoretical sciences. In the words of Einstein, "I want to know how God created this world. I am not interested in this or that phenomenon, in the spectrum of this or that element. I want to know his thoughts; the rest [is] just details."
I think mathematicians aren't bemoaning their lost sunk costs. They're now aware that they may never again have the joy of being the first to pull up the edge of the tarp of ignorance and see the raw truth beneath _for the first time_. Their voyages of discovery, which give life meaning, may never leave dry dock again.
> The story of human discovery and the triumph of the human spirit will soon be excised from this discipline. The Dinitz-Garg-Goemans counterexample was the most egregious demonstration. It revealed that the process of prompting novel proofs will be as auraless as ordering doordash. Watch as magic and mystery evaporate. Watch as the sun sets on our heroic age. Is there not something evil in the act of blocking all future generations of mathematicians from the experience of discovery? Forget about accuracy or even attribution. Something fundamental to the experience of mathematics is being taken.
> You are a helpless onlooker. Before you a channel through which humans have accessed the ineffable and sacred for thousands of years is being sealed for eternity.
This is absurd and silly. Discovery doesn't go away because more new things are being discovered. This position only makes sense if you make an arbitrary cut point where some instances "don't count". If you were to go back to scientists or mathematicians several hundred years ago and tell them that one day science would be done by tens of millions of full time researchers around the globe, often working in teams, some of them might also bemoan the loss of the "hero scientist" as any commoner could now participate. But science and math have only benefited from this, and this has increased, not decreased, the beauty and wonder in these fields.
It is not the same because the act of problem solving for discovery is now to be done exclusively by machines, as humans are no longer remotely competitive.
If you read the linked article, the author goes into more detail about how doing mathematics also has been an act of communication with past and present human mathematicians. This is very much true when math and science are done by millions of human researchers, but not so when it is done by massive swarms of agents at a couple labs.
I am sure science and math will benefit in the sense of solving more problems than ever before. But you are not really understanding the agony of the humans caught in the change.
As someone with friends in the mathematics community, they've expressed similar sentiments.
I have to wonder if this is the equivalent of teleporting back in time and finding Marco Polo or any of those early explorers who traveled for the love of the game (and fame and money), and opening up Google Maps on your phone and showing them the zoomed out world. And showing them any island or village they could name.
my analogy is that it is like an explorer being handed access to satellite imagery. and then being told that your experience will still be valuable in telling the satellite where to point and interpreting the photos. folks who got joy from traversing unknown terrain might find it distressing to become an image analyst.
That's such a nice way to put it! I recently stumbled into "The Last First- Winter K2"[1] and saw the same spirit along with a similar misguided effort to win without the will, nor capacity to do it — e.g. influencers taking a stab at K2 is like prompting your way to results.
> They're now aware that they may never again have the joy of being the first to pull up the edge of the tarp of ignorance...
Not "never again", it would be extremely difficult but science and math are also built on random, trial and error, and intuitions that could not arise from AI.
That's the issue - trial and error. A human can try a couple ideas in a day, but realistically, there are not that many sane ideas.
Computer can iterate without frustration on a completely random garbage. Once in a while it stumbles upon a gem. Even if it's one in 10000, that's still hundreds of good ideas in a day (provided enough GPUs).
> they may never again have the joy of being the first to pull up the edge of the tarp of ignorance and see the raw truth beneath _for the first time_
Until we can demonstrate AI consciousness, “seeing the raw truth” will be a human activity. Even now, verifying and understanding these AI proofs is a massive endeavor. Coming up with an intuitive (to humans) distillation of the finding remains an open challenging demanding expertise.
i think to certain types of people (who are probably more likely to be mathematicians), the "self" discovery aspect of it is important, they don't want to read about a new proof for a problem they were grappling with, they want to come about it through their own thoughts.
LLMs are not feeding us things, we are asking it and getting answers. This dialectic I think is super powerful and it still requires the human to ask questions and continue expanding intellectual curiosity to go down the rabbit hole and discover. LLMs are just an incredible rubber duck but the discovery is still there.
I think I disagree, if I had a puzzle on my desk and I told an LLM to explain how to solve it, I think I have robbed myself of the enjoyment of solving the puzzle. Is it really any different than googling a solution?
There are different avenues of "solving" something though. Asking an LLM to one shot something vs iterating on it collaboratively are different ways of achieve potentially different ends.
If the result is black and white, then yeah, its probably not super fruitful but if the end result is subjective, then there really no is objective "solve".
> Until we can demonstrate AI consciousness, “seeing the raw truth” will be a human activity.
I don't think this matters a bit. We individually can't demonstrate "consciousness" in other humans, it's not even a word with a definition. What we can do is miss the feeling of discovering something on one's own rather than having it explained to you over and over again until you understand it.
I mean, dead mathematicians are definitely not "conscious," but their books keep you from discovering things that you may have discovered yourself if left to your own devices.
The joy that was mentioned is the joy of knowing a bit of gossip that no one else knows, of being the first explainer. Interpretation is just being able to pass gossip on; a decidedly lesser joy. And there's no reason that AI shouldn't be able to explain findings in an intuitive way - LLMs at least purely operate based on human intuition as written in human communications and then captured in matrices.
If these proofs are good (doubtful, imo) and can be reduced (intuitively distilled, sorry), there's no reason that an LLM shouldn't be able to do that.
It is strange, because in a sense, when you start learning math, you never explore, you are being taught.
Now mathematicians were able to exercise curiosity. That is going to change. Back to the ranks of eternal learners if you will.
The fun was in the research. Now I think it will leave the door open toward more Applied math. What can people do with all the discoveries. The curiosity will be there.
I think it is better that way.
It will also put a cap on abstractions that overlaps. In essence, AI is less likely to create new theories, coin new terminology, etc. But it is better suited to recognize emerging patterns.
That will probably simplify mathematics.
Unless the domain of abstraction is infinite and with each new abstraction comes its own results but I doubt it. Who knows though...
Most of these questions Mathematicians posed to each other have no practical application, and probably will never have a practical application. (ex: the solution to the Navier Stokes question doesn't change anything at all.)
In many ways, the teleporting machine that teleports to the solution is against the point of the questions. No one wants the solution to be "Run $10,000,000 worth of tokens to cover all possible Lean formulations of the problem".
I think for Mathematicians, I'd say that the human level formulation is still possible and with a new perspective. We as a society (and outside of Mathematics) need to also understand that this AI teleport to the answer thing has less value than we all expected. The value is in the journey.
The open question is if people will still want to take the journey if the teleport machine exists. Or if the existence of the teleport machine halts exploration.
> "Or if the existence of the teleport machine halts exploration."
You're overlooking that the teleport machine costs lots of money to use. The AI providers are not going to waste money by teleporting randomly around.
Someone is going to have to find a new destination that is worth spending the money to teleport to (or walk by themselves) and that someone is mathematicians.
I need to do some more research on this. Because I've always been told that AI, more specifically LLMs, are "glorified autocompletes". Yes they have huge data sets but ultimately they can only provide answers based on data that they were trained on.
So how can they possibly solve problems that humans have worked for decades or centuries on without being able to solve? How did they find something new? Is it that they were able to deduce the solution by examining all current data sets? Or simply brute force it until the produced solution passed all necessary tests?
The following certainly does not describe all mathematics or all LLM operations, but its my mental model for why LLMs are so effective in several fields.
Lots of mathematics entails recognizing that a tool/techniques that works in one area can be adapted to a completely different problem. For a human to do this requires having a robust mental model of the technique and the problem itself, and probably having just the "right" mental model to see how they fit together. And we are talking about extremely hard/sophisticated techniques and extremely hard/sophisticated problems, so the ability to deeply appreciate either can require years of study. And on top of that, even if the mathematician has sufficient understanding to develop the intuition that the technique might work, they will need profound fortitude, skill, and patience to see the solution through.
Unlike human mathematicians, who will develop a deep understanding of only a relative handful of techniques and problems in their lifetime, LLMs have ALL written techniques, problems, solutions, etc stored in their internal representations. So when they ably match a problem to a technique, we can imagine the operation as a supercharged autocomplete, or even an extremely proficient search engine. Their stroke of genius is really a consequence of capacity, moreso than cleverness.
Yes this is it. So many breakthroughs in math (and other sciences) have been like "I went to a conference and happened to mention my work to someone in a completely different area/field and they said: well that sounds like something i just noticed in my research".
Combining spread out ideas is very powerful and LLMs are great at that.
They have access to a much larger corpus of information than any single human can manage in his one brain. No human mathematician reads all the papers and has perfect recall on them. The answers have always been there, but no individual has been able to make all the right connections to pull them together and formalize a new discovery or proof.
Yeah. "Glorified autocomplete" might be all the human brain is too. "Glorified autocomplete" can do an amazing number of things, including perhaps pass the Turing test and perhaps ... "think".
"Glorified autocomplete" is meant to describe the _underlying mechanism_, but unfortunately it also sounds like one of the things LLMs are actually used for, so it gives the mistaken impression that the claim is all they are good for is a component inside an editor application.
485 comments
[ 2.8 ms ] story [ 47.8 ms ] threadI am also sad, but my sadness is, in a way, priced in.
Also, it's impossible to really know how you're going to react to something until it happens to you. When ChatGPT came out, I already thought "in the long run they're going to automate almost everything humans love to do". Despite having 4 years to brood about it, I did not really know what it would feel like until it happened to me directly.
Either these mathematicians thought it would never come for their jobs, because all of these other jobs were simpler or they've been living under a rock.
Either way, I don't feel for them any more than I do for all of humanity.
Finding that new knowledge may be exponentially more difficult, but it will be much more valuable.
How do you know this?
- random,
- from the environment,
- from other AIs.
That increases the space considerably.
In particular, note that your argument also applies to humans: A human, being finite, can only produce finitely many things. But you add other humans, the environment, and pure chance to the mix, and it'll be pretty infinite for all practical purposes.
Maybe that is the limits of my imagination, but it seems unwise to take humans completely out of the loop. Having mathematicians around that can understand what it is doing so that we can utilize that knowledge in a way that we can have some confidence in seems important.
It sounds increasingly dumb to bring this up given how advanced models have become, but it remains true - they are still fundamentally token prediction algorithms. And that imposes hard constraints on their potential. These constraints are relatively academic in our current world because there's just so absurdly much information and no human can hold more than a negligible amount in his mind at once. So they'll certainly be able to connect many 'obvious' dots that we don't see simply for a lack of carrying capacity, but they're not going to discover concepts that do not clearly follow from the comprehensive knowledge of an era.
Humans have cognitive limits. Humans hold a specific slice of all knowledge at a time and necessarily limits what correlations can be leveraged. This is similar to "context windows" in modern AI. Human minds are wildly variable in efficiency. AI is has constant inefficiency, largely controlled by the speed of light, physical RAM, and processing power, with superior (no cost) "thought" context switching. The AI also has superior accuracy without the need for "concentration" because the AI is processing on a Turing machine with the limited operations that it's superior at in terms of time and energy - eg find an element in a list of 1 million elements.
Neurons and synapses change, especially synapses (the connections between neurons). Some neurons have a few connections to other neurons. Others have hundreds of thousands of connections to other neurons. These grow and change. "Estimates are between 100 trillion and 1,000 trillion neural connections" Is it finite? Depends on your definition of finite. If anything countable is finite, then it is finite but the number may be practically uncountable. Is it fixed? No.
1. LLMs aren't run deterministically - they use a bit of noise to improve the output. So even with the exact same input you're going to get a whole distribution of outputs.
2. Feedback from the environment. LLMs (and humans!) work best when they can test ideas. That's why coding agents work so well. If you ask an agent 1000 times to just think of a compression algorithm, it'll probably spit out very samey answers just like a human would. If you give it a test corpus and let it try out different things it will absolutely be able to come up with novel ideas.
Or maybe it can; the early image generators famously did output horrific images sometimes because of their limitations, but these outputs qualify nonetheless creations.
Now the interesting thing is that it is humans who evaluate them as horrific, worthless or interesting.
Certainly AIs could pretend they have that sort of opinions and judgement, but hopefully humans will remain more interested in the opinions of other humans.
Online, how would they even know? Offline -- either we give up on the internet, or the offline discussion will be heavily influenced by what AI said online.
https://birdsnfrogs.github.io/2026/09/12/Felix_qui_potuit_re...
At the same time, I think it's pretty amazing how this is the fundamental question of AI's trajectory, yet people on both sides seem so sure of their viewpoint with little actual evidence or data to back it up.
That is, it's possible that LLM-based approaches will be forever limited to this "convex hull" of current training data, and thus humans will always have a role to play (and will retain primacy when it comes to overall intelligence), because it appears that humans are not limited to this convex hull based on previous discoveries (e.g. I think basically every scientist would argue Einstein's general relativity was outside the convex hull that existed at that time).
Yet, if LLM-based approaches are eventually able to surpass humans on this "novel discovery" axis, then we as a species are basically done for when it comes to intellectual primacy - and then it becomes a very strange and scary world. And many AI researchers at the top companies clearly believe current approaches will scale and eventually (pretty soon I'll add) lead us to RSI.
I have no idea which side is right, and I'll admit I really hope LLMs eventually hit a wall. But again, I just am surprised that many seem to be putting forth their beliefs, but I would love to know if there is any good evidence one way or the other (or better yet, a more concrete and verifiable definition of the "convex hull" so that researchers could provide some falsifiable examples).
This seems like a good summary of the article, and I think the answer here is obvious.
So what?
News flash folks: we do not have the knowledge to survive on our own. If a solar flare deletes the electric grid, the first thing that happens is 8 billion people die in the next year.
We survive on abstractions others have built, and we have a global dependency web.
AI abstracting knowledge is neither new or dangerous; if it is dangerous, this certainly isnt the argument to prove it.
I think they might find comfort in knowing it's not just mathematicians - this is coming for all of us. Computers putting us out of work has been coming for a long long time.... and now it's here.
it is a little ironic that after the "creative destruction" of industry after industry all of a sudden now it's unfair and we weep. I say this as a software dev with his neck on the line like everyon else.
It's not clear from your comment: Are you talking about math professors?
If so, what money? They make a living, but it's not that great. I just checked the salary of two associate professors at my university: $80-85K.
And keep in mind they often start their careers quite late. Often multiple postdocs after PhD. Starting a tenure track position in your mid 30's is not common.
> I'm more shocked that there are so many brilliant people who are so results-oriented and don't even have a concept of value outside of that. I'm kind of disgusted by the tears actually
The whole point of the article is that the person crying is because he loves the process and is worried the human connection will be gone.
I don't personally grieve for the likes of Strogatz or Tao, it seems likely both of them will be fine however this pans out. But many more mathematicians have been toiling away in precarity and obscurity and without tenure for years and years. And of course many mathematicians went into private sector roles as software engineers, actuaries, analysts, ML engineers and similar, and have been displaced there.
Visible blog posts written by famous mathematicians closely aligned with the news and hype cycles getting frontpaged on HN doesn't mean that mathematicians are disproportionately whiney, or that software engineers aren't writing about it or ending up on the front page
It's not like the options are write software or starve. Billions of people find jobs and can feed their families without writing software.
I also don't often meet junior software engineers who actually have families to feed. Most juniors work in the industry for a few years before deciding to get married and have kids.
Unfortunately this often means moving away from family. What are you going to say to the kid whose parents toiled to get him through college on the expectation that he would be able to take care of them? "Sorry Mom, I have to move to North Dakota to work on the oilfields"?
This is said by a lot of CEOs trying to make layoffs look good, but I haven’t seen anything that suggests that it’s true.
> Yet I see far more of these blogs about the plight of mathematicians, not enough about the plight of software engineers.
Really?!? We’ve been living through 3 years of blog posts, LinkedIn hot takes, and podcasts talking about the plight of software engineers in the era of AI. There isn’t a week that goes by without an HN post about it.
I can feel empathy for the practitioners in a field that may have been profoundly changed, the same way I can feel empathy toward software engineers who are struggling with the changes in their own field.
In the end, we’re all trying to figure out where this leads. And how this will affect our lives and our work.
It's not fun to think of a future where knowledge workers become irrelevant in society and we're back to just physical work and managerial elites.
Just like in past history.
Yes, it sucks, for software engineers, those in the past, and those in the future. As long as you and others fall for the straw man perhaps you and others will get more and more tricked and hurt, misplacing your aggression and hatred on each other rather than those receiving most of the extracted value.
Yes in some cases, but the reduced costs also allow lower prices which benefits all consumers of whatever product is being produced. At least for those products where there is a competitive market.
The economist John Keynes predicted almost 100 years ago that technological advances would lead to a 15 hour workweek. If AI now makes you 5x as productive, then why aren't you working 1 day a week for the same pay? Why didn't Keynes' vision come true.
governments implemented (and still implement) keynesian economics, and it has turned out to be a very bad idea and he was very very wrong.
I don't believe it is that black and white. Yes, automation eliminates some jobs, but overall moves everyone (on average) further up the productivity line, which improves lives, again, on average.
That is why we live like kings now compared to people of the past.
But circling back to the point, if AI does the same thing, eliminates expensive manual coding and other similar work, that is essentially the same thing we've been doing to other people for decades.
Are phones an "automation of existing jobs"? Maybe couriers (??), but most of what people use them for goes far beyond that.
What about the software used to drive trains or autopilot planes? We still have conductors and pilots, but now the outcomes are safer on average and more efficient.
Or something like Google Maps -- all that replaced was the paper booklet people used to keep in their cars, it's not like we used to employ 'navigators' to go out on a road trip.
Or something like Intuit QuickBooks: there's still an accountant in the loop, and if anything more accountants today than there were a few decades ago.
Certainly some software is essentially automatory in nature. But to characterize 'nearly all' as being such just betrays a lack of awareness for the rest of the field.
> That one is somehow deserving of sympathy and that of mathematicians is not
He didn't say that mathematicians aren't deserving, he was griping that people said this about SWEs. Do you think mathematicians are deserving but SWEs aren't?
The words and excels of the world, automating documentation and bookkeepery. It all makes some kind of existing thing we already do faster and more convenient, and often more efficient. Most software simply does nothing new. The thing you didnt manage to list which is the actually good example is games.
You seem to be unaware of the existence switchboard operators (people redirecting calls to the correct person) were hundreds of thousands in the US alone at some point.
Say you have a friend who likes making music (not professionally - just a hobby).
If an invention came out tomorrow that took the joy completely out of making music, would you respond with "It's hard for me to feel sorry about you."
Just because things suck for you doesn't mean it can't suck for others (in different ways).
In many ways it will likely be the opposite. Tech firms still need engineers to talk to customers and run experiments. What will the mathematicians be needed for?
What does your being software engineer has anything to do with feeling sorry for Mathematicians' job ?
It's a classic "I have it worse than them...everyone should feel sorry for ME and not them".
This is what it seems like so far for software. The speed limit has changed. It’s interesting to explore what that means. Extending the analogy, long ago you could be a race car driver part time; that would be pretty laughable for an F1 driver. It’s a very specialized skill and the optimal athlete is quite different for driving a car than for running, for example. That feels like what I’m seeing in software.
The most effective folks are much faster and better with these tools than average. It’s a different skill set.
I think that formal verification makes Math much closer to code than many other fields. But I’m eager to see the first mathematicians that are dramatically more proficient with the new tools than average.
Nothing has changed in math. Math is not solved nor has the human been kicked out of the loop. Same with programming.
How will it pan out, no one really knows. The leading frontier labs are continuing to hire aggressively. But who knows for how long or for long or near-term goals.
The "moderation by less", as well as the guidelines, are both farcical in nature. They've given way to ruination of an otherwise decent tech discussion forum (or news aggregator, or YC propaganda machine or whatever you want to call it).
If you want actual discussion around things, lobsters, awful or even 4chan are better than HN.
Never been invited, but I've also never felt smart enough for that community. I'm glad I can read it though!
I guess there really is a first time for everything!
> The "moderation by less", as well as the guidelines, are both farcical in nature. They've given way to ruination of an otherwise decent tech discussion forum
The guidelines have been constant (with minor updates) for most of the two decades HN has existed. Dan’s a moderator for 14 years and I’ve been one for several. Both of us have been active participants almost since the beginning. When did the ruination begin?
Your comment seems a generic vent, largely unrelated to the parent comment but if you have any actionable suggestions you’re welcome to email us.
There it is, the "off topic" play. Well done, Tom.
If you'd put anywhere near the amount of attention in moderating the site for over all quality that you put into my "largely unrelated" comment, this entire comment thread would not exist. Think it over.
That neither of you are at the very least bothering to steer the greater discourse towards some higher ground is a lack of balance that I am highlighting.
To remark my comment as off topic in the context of the thread opener is to dismiss what I am saying using a very lazy tactic that I've seen used countless times before. No, it is not off topic. It is very much on topic and relevant.
And if you have a problem with my comment, feel free to email me.
But one of your central complaints was about the guidelines, which have been constant for nearly two decades, so we’re really not having a serious conversation here.
But many people perceive this very differently, see link below. And perception is reality after all.
https://news.ycombinator.com/item?id=50093180
And this is only the beginning.
But from my perspective proofs have never been the most interesting part of maths. What really interests me is the intuition that leads to new applications, and to new questions. Newton, for example, gets more credit for calculus than Leibniz because he's the one who demonstrated how darned useful it is. Erdos is arguably more famous for asking interesting questions than for answering them.
I still don't see LLMs seriously contributing to that facet of mathematics any time soon. Every so often I ask one to suggest a good binomial confidence interval for a particular application, and it still tends to just suggest one of the subpar options that everyone learns in a first year stats course.
The recent barf-out done by LLM companies is useful if what are you are interested in is an unreadable mess and someone giving you the final answer. You might be able to extract some new "aha!" moments out of it, but we will have to wait and see what happens once people go through it. I would hypothesize the LLM output will be good at hinting at some non-obvious relationships and patterns, but I doubt it will yield anything of substance in its own right.
Historically, proving difficult theorems required mathematicians to sometimes invent new fields, new ways of looking at problems, the discovery of which itself yielded incalculable contributions to both mathematics and other fields. It is this that is the greater contribution of and to mathematics.
Mathematicians working at universities are paid to teach, to engage in the politics and business of government contracting, and only in very rare circumstances to play prestige games on famous problems.
It is unlikely that Congress will decide to slash budget for NSF's MPS directorate on account of AI getting good at proving theorems. That's just not how Congress works.
It is also unlikely that LLMs will be able to do, in the near future, do the sort of applied mathematics that happens in the context of occasional NIH or DoD funding that flows into Mathematics departments, for a variety of complex sociological reasons I'm happy to expound upon.
And it's unlikely Engineering degree programs will stop requiring calculus sequences, since being able to do the computation was has not been the actual point of teaching calc since he late 20th century (if ever).
So, the good news is that I doubt any of this will have much of an effect on the job market for Mathematicians who want to prove theorems for a living. The bad news, of course, is that the job market for Mathematicians who want to prove theorems for a living was already nearly non-existent.
Yes, outsiders (investors, journalists) are given a false impression while insiders (software engineers) are supposed to give up. Double win for the agitprop machine.
Is it like Chess and Go? There are still plenty of people playing for fun, but the machines are better at it than we are.
Jokes aside, its like working for a company, individually you can lift mountains and someone else will take credit. When you use LLMs for a single bit, all credit goes to LLMs lol
https://en.wikipedia.org/wiki/IBM_and_World_War_II
the above, a "sorting" machine is the primary usage of hyperscalers currently, they're just trying to find new uses to boot for extra money/legitimation.
the browser industry the phone industry the CAs JavaScript
it's all paid for by those wanting to sort humanity into whatever bucket the can fear monger the peasantry into for aweek or however long it takes the classification system to sort out.
gentrification works, but we chose forcible displacement now we have to live with the forever consequences.
would you give guns and bombs to the displaced native americans who taste blood over their beads and trinkets?
we've been inside the event horizon of the singularity for years, the tax bill will only get worse unless we recognize it for the tick that it is.
Now the thread has been cleaned up and a pro-defeatism comment is the top comment and the submission is on the front page again (I'm not alleging moderator intervention, user mechanisms are sufficient).
Wow that's very well worded.
Worth reading: Robert Brandom, “Making It Explicit: Reasoning, Representing, and Discursive Commitment”
Deontology - `normative ethical theory that the morality of an action should be based on whether that action itself is right or wrong`
Super roughly:
* Virtue ethics: character and motivations
* Deontological ethics: rules and obligations
* Consequentialist ethics: utility and outcomes
The participant is not an LLM, the participant is a company. The company learns. It adapts. It has a trillion dollars of skin in the game.
Likewise OpenAI doesn't have "skin" of any shape or value in e.g. the mathematics community or even in mathematics research per se. In my cynical view, they are exploiting the exposure of mathematical research to automation in an effort to create a case study of their product obsoleting a highly visible and well-leveraged and relatively small profession. Regardless, I doubt even they would say they have durable skin in mathematics in any sense, certainly none of the ones I mentioned above. But their output inhabits the same space as mathematicians, who have to contend with the consequences
They hire math and physics grads and good command of math is very useful in computer science, machine learning and AI.
And how do you see these qualities changing over, say, the next decade?
And potentially steals [1].
[1] https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_priority...
I'm interested in the reversal that Sellars proposes - it seems clear to me that "representation" is not prior to the space of reasons, despite how unintuitive it is.
> An LLM encodes the normative regularities of our linguistic/conceptual games into its weights, but it cannot be a real participant in our games because it is incapable of staking durable commitments or changing its dispositions over time.
I see no reason why this can't be the case in the future, even though I think inferentialist-based theories are generally strong. Can't you force LLM's (or whatever AI architecture will exist in a decade) to stake durable commitments itself, then? They're at least autonomous enough to run computers. I agree that that is a core part to true meaning & autonomy, but nobody just lets them yet. I don't think inferentialism is a consciousness-based argument, either, so that escape hatch isn't available. I also don't think this is a necessary condition to engage in the space of reasons (but maybe then I'm misunderstanding it, or retreating to a more Sellarsian understanding). Could you explain on this more?
Though again, I only have the book sitting on my desk, and I haven't cracked it open...
And yeah, consciousness based arguments are bunk IMO. It's clear that LLMs are in a basic sense "aware", and even to some extent "sapient" (in Brandom's usage), i.e. they can justify claims and object to things at least locally. I follow Rorty on the question of consciousness, which is to say that consciousness has no deeper meaning in practice beyond "are you awake", and in philosophical discussion it's largely a proxy for "are you one of us" in some amorphous sense.
Consciousness, the interesting part anyway, is qualia, or the "it is like something to be me" as Nagel put it.
Intention, as Dennett used it in his book "The Intentional Stance" is the "are you one of us" angle, and is distinct from consciousness because as the book title suggests, it is a stance taken with respect to some other system (person, cat, dog, turtle, LLM) rather than a description of subjective experience.
So I think it is your (and Rorty's) positions that are bunk. It is very, very far from clear that LLM's are in any basic sense "aware" let alone "sapient". They are extraordinarily able to manipulate language in a way that leads to text generation that has traditionally been assumed to indicate consciousness and/or intelligence. But we have no way of knowing if it is like anything to be an LLM, even though most of us are busy taking the Intentional Stance towards them.
And as I think Dennett himself would say, qualia are just Sellars's Given rearing its ugly head again.
[1] https://sites.pitt.edu/~rbrandom/Courses/2022%20Phil%20of%20...
I read this sentence 5 times and I still cannot parse it.
I mean this with zero snark: if you’re interested, you’d be best off asking an LLM or doing a search. But the overall idea is that Brandom described how concepts and meaning are created socially by a kind of game in which people keep track of the claims that other people commit to.
If you accept Brandom’s framework, then it can be argued that LLMs can’t meaningfully participate in this process. This could be considered an explanation of why people have a negative reaction to LLM involvement in activities that are meaningful to humans - even though most people won’t be familiar with the theory, they would instinctively recognize that something important is lacking, such as commitment over time.
Seeing AI being lauded for solving problems like this makes me feel... uncomfortable.
We are feeding all works of the giants that we have all stood on into a huge vat, in the hope to generate something meaningful.
If we can't understand the output, internalize it, or build from the time and experience that a human would have gained from solving the same problem, it feels like a dead end.
When all the problems have been "solved" by AI, what's next? If humans no longer struggle to solve the problems that the previous generation have left behind, how do we progress?
The same goes for math.
Let's find some things that bring value other than just the price of a stock.
At worst its time is past, but even that is unlikely.
So I wonder if mathematics will go through that too? A period of chaos and doubt, but finding new and more effective ways of working.
Its easy to demonstrate this: The math hysteria passes Lean but no human can make sense of it at present. This is akin to bots telling humans where to dig. Thats fine but lets be clear about whats going on.
I think mathematicians aren't bemoaning their lost sunk costs. They're now aware that they may never again have the joy of being the first to pull up the edge of the tarp of ignorance and see the raw truth beneath _for the first time_. Their voyages of discovery, which give life meaning, may never leave dry dock again.
> You are a helpless onlooker. Before you a channel through which humans have accessed the ineffable and sacred for thousands of years is being sealed for eternity.
https://kirwinhampshire.substack.com/p/the-dark-night-of-mat...
If you read the linked article, the author goes into more detail about how doing mathematics also has been an act of communication with past and present human mathematicians. This is very much true when math and science are done by millions of human researchers, but not so when it is done by massive swarms of agents at a couple labs.
I am sure science and math will benefit in the sense of solving more problems than ever before. But you are not really understanding the agony of the humans caught in the change.
I have to wonder if this is the equivalent of teleporting back in time and finding Marco Polo or any of those early explorers who traveled for the love of the game (and fame and money), and opening up Google Maps on your phone and showing them the zoomed out world. And showing them any island or village they could name.
1: https://en.wikipedia.org/wiki/The_Last_First:_Winter_K2
Not "never again", it would be extremely difficult but science and math are also built on random, trial and error, and intuitions that could not arise from AI.
Computer can iterate without frustration on a completely random garbage. Once in a while it stumbles upon a gem. Even if it's one in 10000, that's still hundreds of good ideas in a day (provided enough GPUs).
> The reaction from the math community has been a mix of indignation, astonishment, and sorrow.
Really? Not an iota of positivity or optimism? I'm not sure I can believe it.
Until we can demonstrate AI consciousness, “seeing the raw truth” will be a human activity. Even now, verifying and understanding these AI proofs is a massive endeavor. Coming up with an intuitive (to humans) distillation of the finding remains an open challenging demanding expertise.
If the result is black and white, then yeah, its probably not super fruitful but if the end result is subjective, then there really no is objective "solve".
I don't think this matters a bit. We individually can't demonstrate "consciousness" in other humans, it's not even a word with a definition. What we can do is miss the feeling of discovering something on one's own rather than having it explained to you over and over again until you understand it.
I mean, dead mathematicians are definitely not "conscious," but their books keep you from discovering things that you may have discovered yourself if left to your own devices.
The joy that was mentioned is the joy of knowing a bit of gossip that no one else knows, of being the first explainer. Interpretation is just being able to pass gossip on; a decidedly lesser joy. And there's no reason that AI shouldn't be able to explain findings in an intuitive way - LLMs at least purely operate based on human intuition as written in human communications and then captured in matrices.
If these proofs are good (doubtful, imo) and can be reduced (intuitively distilled, sorry), there's no reason that an LLM shouldn't be able to do that.
In many ways, the teleporting machine that teleports to the solution is against the point of the questions. No one wants the solution to be "Run $10,000,000 worth of tokens to cover all possible Lean formulations of the problem".
I think for Mathematicians, I'd say that the human level formulation is still possible and with a new perspective. We as a society (and outside of Mathematics) need to also understand that this AI teleport to the answer thing has less value than we all expected. The value is in the journey.
The open question is if people will still want to take the journey if the teleport machine exists. Or if the existence of the teleport machine halts exploration.
You're overlooking that the teleport machine costs lots of money to use. The AI providers are not going to waste money by teleporting randomly around.
Someone is going to have to find a new destination that is worth spending the money to teleport to (or walk by themselves) and that someone is mathematicians.
You realize that they don't want that, correct?
So how can they possibly solve problems that humans have worked for decades or centuries on without being able to solve? How did they find something new? Is it that they were able to deduce the solution by examining all current data sets? Or simply brute force it until the produced solution passed all necessary tests?
Lots of mathematics entails recognizing that a tool/techniques that works in one area can be adapted to a completely different problem. For a human to do this requires having a robust mental model of the technique and the problem itself, and probably having just the "right" mental model to see how they fit together. And we are talking about extremely hard/sophisticated techniques and extremely hard/sophisticated problems, so the ability to deeply appreciate either can require years of study. And on top of that, even if the mathematician has sufficient understanding to develop the intuition that the technique might work, they will need profound fortitude, skill, and patience to see the solution through.
Unlike human mathematicians, who will develop a deep understanding of only a relative handful of techniques and problems in their lifetime, LLMs have ALL written techniques, problems, solutions, etc stored in their internal representations. So when they ably match a problem to a technique, we can imagine the operation as a supercharged autocomplete, or even an extremely proficient search engine. Their stroke of genius is really a consequence of capacity, moreso than cleverness.
Combining spread out ideas is very powerful and LLMs are great at that.
"Glorified autocomplete" is meant to describe the _underlying mechanism_, but unfortunately it also sounds like one of the things LLMs are actually used for, so it gives the mistaken impression that the claim is all they are good for is a component inside an editor application.