5 ms·

Tao's central point seems to be: "In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving p

Tao's central point seems to be: "In short, the indiscriminate use of powerful solution-extraction tools can achieve the immediate short-term goal of solving problems at hand, but at the cost of sustaining the ecosystem for the next wave of progress, or in understanding the progress already obtained. " I am no mathematician, may have misunderstood his point and would be delighted to receive any corrections.

ramraj07 23d ago
Seems to be it, though Im not particularly concerned about this problem personally. The fact that we all readily accept that modern AI systems can likely solve any math problem that no living genius can, tells me that no task is beyond this system we just need the right harness around it. The exhaustion of meaningful math problems to motivate mathematicians minds seems to be the least of my worries at that point. Inb4 someone suggests that this is not proof that these AIs generalize, I agree thats a popular opinion, but both sides are merely that, with no possible way to prove. I will wallow in my existential dread while you do whatever it is that gives you comfort.
kkotak 23d ago
How is this any different from people in any field that are impacted by AI and lose the utility of their skills and endeavors over the past decades? Are we saying that we're running out of problems to solve because of AI and hence it should be stopped? I am not underestimating the importance of the collective knowledge of the mathematics community and the role of mathematics as the enablers of other sciences, but opposing meaningful progress in that discipline or any for that matter feels counter intuitive. I would rather have the mathematics community start collaborating closely with the this newly evolving and powerful tool to expedite humanity's progress.
- hn_throwaway_99 23d ago
  Do you see an end state in this? When AI is better than humans at everything (and I used to be very sceptical of that claim but I'm getting less and less by the day), I don't see the Wall-E version of humanity as some sort of utopia, and that's the good outcome.
  - largbae 23d ago
    To paraphrase the not-so-great philosopher Ted Kaczynski: Either we will maintain control of the machines or we won't. And if we do, it won't be you or I who control them, but a small group of elites.
- IanCal 23d ago
  His point is twofold: that the process of solving the problems leads to more than just solving the problem in front of you but other interesting things (he has an example of going on a hike to a waterfall and all the other things you might spot over in the distance or nearby on the way, which you’d miss if you were able to jump straight there), and also lots of the simpler open problems are ones early researchers learn on (this is akin to the “if we automate junior engineers how does anyone learn to be a senior?”).
  - chr1 23d ago
    Ability to jump straight to waterfall doesn't prevent you from taking another hike looking specifically for those other things. If solving any problem is possible with ai, the job of mathematician is finding new problems, not trying to do what a machine can do better. As for training, most of it is already done by solving known problems, and no one was suggesting to forget analytical geometry so that solving school geometry problems becomes cutting edge research.
hn_throwaway_99 23d ago
Doesn't this seem to be where all domains are headed? I get kinda freaked out when I feel like all the AI "utopianists" haven't taken the next logical step of thinking about what society looks like when humans are subpar in every domain (and you may argue this won't happen, though I'm becoming more and more a believer that it will, but my point is the utopianists believe that this absolutely will happen, and that it's also a wonderful thing). How motivated do you think folks will be to do the hard cognitive work to focus on things like math problems when there is a good chance AI will do it better?
- ahepp 23d ago
  Doesn't the premise that there's something inferior about these AI solutions, imply that there is something superior about human intelligence and that there will continue to be some kind of useful work for humans to do?
  - palmotea 23d ago
    > Doesn't the premise that there's something inferior about these AI solutions, imply that there is something superior about human intelligence and that there will continue to be some kind of useful work for humans to do? Not necessarily. The "something superior about human intelligence" may have dependencies that "these AI solutions" are able to eliminate, such as the motivation to refine intellectual talent to a high level. Basically, AI could kick the ladder out from under human intelligence but be incapable of actually surpassing it in important ways, enabling a burst of advancement that's also a dead end. Sort of like https://en.wikipedia.org/wiki/The_Road_Not_Taken_(short_story) https://en.wikipedia.org/wiki/The_Road_Not_Taken_(short_stor.... So the AI could be inferior but there's still no useful work for humans, because the environment doesn't allow them to work up to that level anymore. This is kinda feeling a bit like SBF's coin flip bet: https://www.businessinsider.com/sam-bankman-fried-coin-flip-destroy-world-caroline-ellison-trial-2023-10 https://www.businessinsider.com/sam-bankman-fried-coin-flip-.... Achieve human-superior AGI this generation or humanity stagnates.
- znnajdla 23d ago
  I don't think humans necessarily become subpar when AI can do most of the work. Taking my own personal example, I have far more intellectual curiosity and improved my skills in programming far more with Claude Code than for 15 years of programming without AI simply because I was bogged down by boilerplate and grunt work. Now that AI handles most of the boilerplate and grunt work and can handle harder and harder problems, I have the time and space to work on unexplored frontier problems. So, no, my skills have not become subpar, but have only become stronger because of the presence of AI.
oefrha 23d ago
One related problem I see is the pipeline for producing working mathematicians seems to have been completely and irreversibly decimated. What’s the point of doing a long and arduous PhD when all PhD level research problems that used to take months to years can be solved by far less talented people with $100/$1000/$10,000 to spare? How do you even select people into your program (this part is likely hypothetical, classical talent selection probably still works at the moment, but what about in a couple years)? Disclosure: I did a theoretical physics PhD, but got admitted to quite a few top math programs back when I was applying to math and physics programs simultaneously. If you asked me whether I’d do a PhD today I’d say why bother.
- Alien1Being 23d ago
  Here most local STEM PhDs try to get into finance. This is largely due to lack of funding for science and poor opportunities for PhDs. Why be a poorly paid postdoc when Jane Street is offering a million USD signup bonus ? PhDs from poorer overseas do try to get related jobs here, mainly to be able to get a permanent resident visa.
  - oefrha 23d ago
    > Why be a poorly paid postdoc when Jane Street is offering a million USD signup bonus? One reason is despising that line of work. Quant firms were pummeling my @prestigious.edu inbox throughout my PhD and I fucking hated those parasites. Well, jokes on me if AI shatters my current career. Btw a former classmate Caroline Ellison who went to Jane Street eventually landed in prison, lol.
    - irishcoffee 23d ago
      Why put @prestigious.edu and name a famous person? Why not just say you went to Stanford? Lots and lots of people have gone to Stanford.
      - oefrha 23d ago
        prestigious.edu isn’t Stanford; went to a different school for PhD. And the prison bit is just something funny about Jane Street (that sort of vindicates my view of these people) that I edited in later, didn’t think much about it.
Agentlien 23d ago
I definitely see what he is saying. In my work as a graphics programmer I often find that I look at a problem and will immediately see how to solve it, more or less. But the devil is in the details and often nothing works unless you get every detail right. So you spend a lot of time coming up with complex solutions, then boiling them down to simpler versions. In the end you often end up with a fix which is short, simple, and seems obvious. But it gets a lot of subtle details just right and avoids countless potential issues you wouldn't know if you hadn't failed a lot getting there. And that is actually how you learn and master the craft. Now, imagine you describe how you sort of solve it to a machine and it spits out the simple, correct implementation and you nod approvingly, never knowing all the ways it could have gone wrong. If this is how mathematics - or programming - is done from now on, no one will actually master their craft. I definitely see why this would worry someone whose career is built on mastery of the craft and a legacy meant to teach the next generation.
- cadamsdotcom 22d ago
  > no one will actually master their craft. The (new) lack of need to redo solutions to already solved things opens up new craft we've not yet conceived. We shouldn't reimplement solutions and call it self-education; humans have always stood on the shoulders of giants. These tools contain those giants (copyright notwithstanding) and let you receive solutions to solved things; if your mindset is in order you can leverage that to ignore the solved bits and do even more ambitious stuff.
jasonhong 23d ago
There was a recent link on HN about Tao explaining Six Essential Mathematical Concepts https://news.ycombinator.com/item?id=49503521 https://news.ycombinator.com/item?id=49503521 Here's an excerpt of the video where he explains his idea: https://www.youtube.com/watch?v=svl_1upFpQo https://www.youtube.com/watch?v=svl_1upFpQo My summary: Science is like a hike. You have a general sense of what direction you want to go. You explore things along the way. You might make wrong turns but might also discover something new and interesting. While on your journey, you might spot new mountains or waterfalls or other vistas that look like they might be good to explore. In contrast, AI is like taking a helicopter to your destination. Yes you got there quickly, but you missed a lot on the way. Related, Tao later talks about the pipeline. It used to be that seminal proofs were rare, so there was lots of time for the community to review them, process them, share them, and summarize them in textbooks. Nowadays, AI helps a lot with generating more proofs, but not so much with the other parts of the pipeline. Tao also explains the math pipeline issue in his talk at the International Congress of Mathematicians, also shared on HN last month: https://news.ycombinator.com/item?id=49056620 https://news.ycombinator.com/item?id=49056620 My overall take: Tao is very thoughtful and insightful about progress in AI and math, and about what the tradeoffs are. We're seeing similar problems in computer science (my field), where conferences are now getting over 10x the number of submitted papers over just a few years ago (not an exaggeration). It's straining the research community in ability to review, understand, and present the papers.
aaron695 23d ago
[dead]
redwood 23d ago
Indeed it's as if we've harvested all the large oak trees and can no longer make trans-Atlantic wooden ships and now must move on to metal. And doing that actually required stumbling on coal which ultimately was old wood. And then metal enabled us to move past coal. None of this was necessarily visible at the time but is the arc of progress. Of course mathematics being a purely abstract thought medium may not have the benefit of the physical / material world's progression. However it is ironic that here mathematics is progressing specifically because of how far along our material world has come. We're already at the point where a tiny nearly nil percentage of the population can reason at the frontier in mathematics, Terrence Tao obviously included in that small number, and if in the future that number truly reaches nil and only computers can go forward it's not so clear how fundamentally different the world will be to me. The hope is that the innovative manifestations in the real world will be able to be enjoyed by all just as one hopes mathematicians' thought exercises do unlock real blapplications. I think the deeper point Terrence is making is simply that the information revolution which is the culmination of the mathematics revolution has less low-hanging fruit today. That was true ever since Claude Shannon
bwfan123 23d ago
> but at the cost of sustaining the ecosystem John Naur said the same of coding eons ago [1]. That code is only as healthy as the human mental model of its maintainers. Code is not the artifact, it is the intuitions and insight and mental models of its human maintainers. For a project to sustain, it is vital that the human mental models are kept alive. [1] Programming as Theory Building https://pages.cs.wisc.edu/~remzi/Naur.pdf https://pages.cs.wisc.edu/~remzi/Naur.pdf
p0ckets 22d ago
Rephrasing the argument made in the post, math was like a "take a book, leave a book" exchange where you solved a problem, and in the process you discover more problems so the pool of interesting open problems is renewed. What OpenAI and the other labs are doing is akin to - taking all the books in the exchange (which is technically allowed) - destructively scanning them (still technically allowed) - and not leaving any new books in their place (which is frowned upon, but there's no rule that says you have to replace books you take)

ramraj07 23d ago

Seems to be it, though Im not particularly concerned about this problem personally. The fact that we all readily accept that modern AI systems can likely solve any math problem that no living genius can, tells me that no task is beyond this system we just need the right harness around it. The exhaustion of meaningful math problems to motivate mathematicians minds seems to be the least of my worries at that point. Inb4 someone suggests that this is not proof that these AIs generalize, I agree thats a popular opinion, but both sides are merely that, with no possible way to prove. I will wallow in my existential dread while you do whatever it is that gives you comfort.

kkotak 23d ago

How is this any different from people in any field that are impacted by AI and lose the utility of their skills and endeavors over the past decades? Are we saying that we're running out of problems to solve because of AI and hence it should be stopped? I am not underestimating the importance of the collective knowledge of the mathematics community and the role of mathematics as the enablers of other sciences, but opposing meaningful progress in that discipline or any for that matter feels counter intuitive. I would rather have the mathematics community start collaborating closely with the this newly evolving and powerful tool to expedite humanity's progress.

hn_throwaway_99 23d ago
Do you see an end state in this? When AI is better than humans at everything (and I used to be very sceptical of that claim but I'm getting less and less by the day), I don't see the Wall-E version of humanity as some sort of utopia, and that's the good outcome.
- largbae 23d ago
  To paraphrase the not-so-great philosopher Ted Kaczynski: Either we will maintain control of the machines or we won't. And if we do, it won't be you or I who control them, but a small group of elites.
IanCal 23d ago
His point is twofold: that the process of solving the problems leads to more than just solving the problem in front of you but other interesting things (he has an example of going on a hike to a waterfall and all the other things you might spot over in the distance or nearby on the way, which you’d miss if you were able to jump straight there), and also lots of the simpler open problems are ones early researchers learn on (this is akin to the “if we automate junior engineers how does anyone learn to be a senior?”).
- chr1 23d ago
  Ability to jump straight to waterfall doesn't prevent you from taking another hike looking specifically for those other things. If solving any problem is possible with ai, the job of mathematician is finding new problems, not trying to do what a machine can do better. As for training, most of it is already done by solving known problems, and no one was suggesting to forget analytical geometry so that solving school geometry problems becomes cutting edge research.

hn_throwaway_99 23d ago

Doesn't this seem to be where all domains are headed? I get kinda freaked out when I feel like all the AI "utopianists" haven't taken the next logical step of thinking about what society looks like when humans are subpar in every domain (and you may argue this won't happen, though I'm becoming more and more a believer that it will, but my point is the utopianists believe that this absolutely will happen, and that it's also a wonderful thing). How motivated do you think folks will be to do the hard cognitive work to focus on things like math problems when there is a good chance AI will do it better?

ahepp 23d ago
Doesn't the premise that there's something inferior about these AI solutions, imply that there is something superior about human intelligence and that there will continue to be some kind of useful work for humans to do?
- palmotea 23d ago
  > Doesn't the premise that there's something inferior about these AI solutions, imply that there is something superior about human intelligence and that there will continue to be some kind of useful work for humans to do? Not necessarily. The "something superior about human intelligence" may have dependencies that "these AI solutions" are able to eliminate, such as the motivation to refine intellectual talent to a high level. Basically, AI could kick the ladder out from under human intelligence but be incapable of actually surpassing it in important ways, enabling a burst of advancement that's also a dead end. Sort of like https://en.wikipedia.org/wiki/The_Road_Not_Taken_(short_story) https://en.wikipedia.org/wiki/The_Road_Not_Taken_(short_stor.... So the AI could be inferior but there's still no useful work for humans, because the environment doesn't allow them to work up to that level anymore. This is kinda feeling a bit like SBF's coin flip bet: https://www.businessinsider.com/sam-bankman-fried-coin-flip-destroy-world-caroline-ellison-trial-2023-10 https://www.businessinsider.com/sam-bankman-fried-coin-flip-.... Achieve human-superior AGI this generation or humanity stagnates.
znnajdla 23d ago
I don't think humans necessarily become subpar when AI can do most of the work. Taking my own personal example, I have far more intellectual curiosity and improved my skills in programming far more with Claude Code than for 15 years of programming without AI simply because I was bogged down by boilerplate and grunt work. Now that AI handles most of the boilerplate and grunt work and can handle harder and harder problems, I have the time and space to work on unexplored frontier problems. So, no, my skills have not become subpar, but have only become stronger because of the presence of AI.

oefrha 23d ago

One related problem I see is the pipeline for producing working mathematicians seems to have been completely and irreversibly decimated. What’s the point of doing a long and arduous PhD when all PhD level research problems that used to take months to years can be solved by far less talented people with $100/$1000/$10,000 to spare? How do you even select people into your program (this part is likely hypothetical, classical talent selection probably still works at the moment, but what about in a couple years)? Disclosure: I did a theoretical physics PhD, but got admitted to quite a few top math programs back when I was applying to math and physics programs simultaneously. If you asked me whether I’d do a PhD today I’d say why bother.

Alien1Being 23d ago
Here most local STEM PhDs try to get into finance. This is largely due to lack of funding for science and poor opportunities for PhDs. Why be a poorly paid postdoc when Jane Street is offering a million USD signup bonus ? PhDs from poorer overseas do try to get related jobs here, mainly to be able to get a permanent resident visa.
- oefrha 23d ago
  > Why be a poorly paid postdoc when Jane Street is offering a million USD signup bonus? One reason is despising that line of work. Quant firms were pummeling my @prestigious.edu inbox throughout my PhD and I fucking hated those parasites. Well, jokes on me if AI shatters my current career. Btw a former classmate Caroline Ellison who went to Jane Street eventually landed in prison, lol.
  - irishcoffee 23d ago
    Why put @prestigious.edu and name a famous person? Why not just say you went to Stanford? Lots and lots of people have gone to Stanford.
    - oefrha 23d ago
      prestigious.edu isn’t Stanford; went to a different school for PhD. And the prison bit is just something funny about Jane Street (that sort of vindicates my view of these people) that I edited in later, didn’t think much about it.

Agentlien 23d ago

I definitely see what he is saying. In my work as a graphics programmer I often find that I look at a problem and will immediately see how to solve it, more or less. But the devil is in the details and often nothing works unless you get every detail right. So you spend a lot of time coming up with complex solutions, then boiling them down to simpler versions. In the end you often end up with a fix which is short, simple, and seems obvious. But it gets a lot of subtle details just right and avoids countless potential issues you wouldn't know if you hadn't failed a lot getting there. And that is actually how you learn and master the craft. Now, imagine you describe how you sort of solve it to a machine and it spits out the simple, correct implementation and you nod approvingly, never knowing all the ways it could have gone wrong. If this is how mathematics - or programming - is done from now on, no one will actually master their craft. I definitely see why this would worry someone whose career is built on mastery of the craft and a legacy meant to teach the next generation.

cadamsdotcom 22d ago
> no one will actually master their craft. The (new) lack of need to redo solutions to already solved things opens up new craft we've not yet conceived. We shouldn't reimplement solutions and call it self-education; humans have always stood on the shoulders of giants. These tools contain those giants (copyright notwithstanding) and let you receive solutions to solved things; if your mindset is in order you can leverage that to ignore the solved bits and do even more ambitious stuff.

jasonhong 23d ago

There was a recent link on HN about Tao explaining Six Essential Mathematical Concepts https://news.ycombinator.com/item?id=49503521 https://news.ycombinator.com/item?id=49503521 Here's an excerpt of the video where he explains his idea: https://www.youtube.com/watch?v=svl_1upFpQo https://www.youtube.com/watch?v=svl_1upFpQo My summary: Science is like a hike. You have a general sense of what direction you want to go. You explore things along the way. You might make wrong turns but might also discover something new and interesting. While on your journey, you might spot new mountains or waterfalls or other vistas that look like they might be good to explore. In contrast, AI is like taking a helicopter to your destination. Yes you got there quickly, but you missed a lot on the way. Related, Tao later talks about the pipeline. It used to be that seminal proofs were rare, so there was lots of time for the community to review them, process them, share them, and summarize them in textbooks. Nowadays, AI helps a lot with generating more proofs, but not so much with the other parts of the pipeline. Tao also explains the math pipeline issue in his talk at the International Congress of Mathematicians, also shared on HN last month: https://news.ycombinator.com/item?id=49056620 https://news.ycombinator.com/item?id=49056620 My overall take: Tao is very thoughtful and insightful about progress in AI and math, and about what the tradeoffs are. We're seeing similar problems in computer science (my field), where conferences are now getting over 10x the number of submitted papers over just a few years ago (not an exaggeration). It's straining the research community in ability to review, understand, and present the papers.

aaron695 23d ago

[dead]

redwood 23d ago

Indeed it's as if we've harvested all the large oak trees and can no longer make trans-Atlantic wooden ships and now must move on to metal. And doing that actually required stumbling on coal which ultimately was old wood. And then metal enabled us to move past coal. None of this was necessarily visible at the time but is the arc of progress. Of course mathematics being a purely abstract thought medium may not have the benefit of the physical / material world's progression. However it is ironic that here mathematics is progressing specifically because of how far along our material world has come. We're already at the point where a tiny nearly nil percentage of the population can reason at the frontier in mathematics, Terrence Tao obviously included in that small number, and if in the future that number truly reaches nil and only computers can go forward it's not so clear how fundamentally different the world will be to me. The hope is that the innovative manifestations in the real world will be able to be enjoyed by all just as one hopes mathematicians' thought exercises do unlock real blapplications. I think the deeper point Terrence is making is simply that the information revolution which is the culmination of the mathematics revolution has less low-hanging fruit today. That was true ever since Claude Shannon

bwfan123 23d ago

> but at the cost of sustaining the ecosystem John Naur said the same of coding eons ago [1]. That code is only as healthy as the human mental model of its maintainers. Code is not the artifact, it is the intuitions and insight and mental models of its human maintainers. For a project to sustain, it is vital that the human mental models are kept alive. [1] Programming as Theory Building https://pages.cs.wisc.edu/~remzi/Naur.pdf https://pages.cs.wisc.edu/~remzi/Naur.pdf

p0ckets 22d ago

Rephrasing the argument made in the post, math was like a "take a book, leave a book" exchange where you solved a problem, and in the process you discover more problems so the pool of interesting open problems is renewed. What OpenAI and the other labs are doing is akin to - taking all the books in the exchange (which is technically allowed) - destructively scanning them (still technically allowed) - and not leaving any new books in their place (which is frowned upon, but there's no rule that says you have to replace books you take)