Mathematics is not like engineering or medicine, where a new discovery can literally save lives. It’s more like philosophy where it moves humanity forward through asking the right questions and learning from the challenges along the way. If AI takes the challenge away, what’s the point?
Another point, OpenAI spent something like $23 million to solve this problem. It’s just a stunt to get to their IPO. The company will get bored of this when it no longer brings them more notoriety. Maybe that would just leave the mathematical community in peace.
but i wouldn't pool all the anti “ techno-luddites “ in the same sac, but yes, indeed, some of them really follow this kind of logic : techno-scientific worship + resentment (softened by some virtuous excuses)
may i propose a simple analogy : if a robot climbs the Himalaya in my name, it would manage it (i won't) but what experience would i have ? nothing
I have liked a lot of what you have written over the years, and I was prepared to like this essay until the last two paragraphs, but particularly the last sentence.
I wish you would expand on this, "That would be a hilarious reversal: the pinnacle of reason preserved as an amateur sport for humans whereas millennium problems become the idle pastimes of the machines."
I can read this as referring to what mathematicians regard as millennial problems, but I can also read this as completely turning all really existential problems (nuclear war, climate change, etc.) over to machines to solve. I realize that is the ideal for some of the people developing AI. I am sure there are many meanings and implications in between, but I am uncertain where you are going with this.
Thanks for the question Guy - I hesitated to end on that note. The "math as sport" thing (or everything as a sport, if you will) is just one possibility of what could end up happening. The "hilarious" part is that it's the upside-down world, not that it is "good." I'm not defending that nor condemning it for what is worth. I'm not picking a side, just trying to so a levelheaded analysis!
Thanks Alberto. It sounded a bit like the idea that we will become the pets of some ASI. I didn't think that was what you meant, but wanted to be sure.
1. The perspective I find missing here is “mathematics as art”
2. The great artworks of the masters are not great simply according to external standards; rather they are setting the standards, only by comparing against them we may decide wether another object deserves the title of a piece of art
3. Likewise, there is no predetermined standards for what count as a meaningful mathematical discovery; rather, it is the historical context shaped by the collective efforts of generations of mathematicians gives meaning to a particular mathematical statement
4. That should be enough reason to regard mathematics without human as complete nonsense
The prize has a clock that no result can skip. The Clay Mathematics Institute's rules, adopted by its board on 26 September 2018, set three conditions before CMI will even consider a proposed solution: publication in a Qualifying Outlet, at least two years elapsed since that publication, and general acceptance in the global mathematics community. CMI also states that it does not accept direct submissions. So whatever is announced this month, a Millennium Prize could not follow a 2026 publication before 2028, and only if the field has agreed by then. The result may well stand. The award procedure is slower than the news cycle.
I don't think "maths is done" just yet. AFAIK, the LLM reasoning agents are using various linguistic representations of maths (primarily Lean) to explore maths as well as natural language. All of that is certainly amenable and doable.
Even so, have you noticed anything about the particular maths problems that AI apparently solves?
AFAIK, all that the AI models have found (by construction) were counterexamples showing some conjectured result to be false, therefore disproving the conjecture.
Admittedly, the agents of course do the heavy lifting of constructing the formalisation in Lean, etc., but this basically reuses and mildly extends/elaborates the known maths frameworks to (a) construct each counterexample and (b) formally prove that it is indeed a counterexample to the conjecture.
In effect, the AIs are being effectively used as formalisation power tools (think: chainsaw) to grind out formally checked and verified maths results - which of course is massively beneficial in itself. But one has to wonder where is the deep creativity or insight one typically expects to see?
As far as I understand it, no new positive mathematical result has (YET) been proposed, formulated and proven by swarms of AI agents operating in complete isolation and without human assistance. The recent formalisation of a proof of Fermat's Last Theorem by an AI (a variant of Claude) doesn't count as it essentially followed an existing proof - no new maths was involved.
Whether already established or discovered for the first time, an insight elicits the same convergence event within the mind that is so very satisfying (and rewarding). Rediscovery from first principles is just as thrilling as discovery.
If the equivalent "millennium problems" of all professions become fodder for machines, then perhaps we really will transition to a life of aristocracy. But at least we will be going into it with eyes open, knowing the value and importance of cognition to the human condition. How... freeing... it might be to live in a world where curiosity and understanding can be pursued for their own sakes and not as complementary means to a survival end.
Hi Juliette, I answered what I think discovery and rediscovery are not the same thing in the article (Newton paragraph) and also as an answer to JLO's comment
The first part of your essay seems to focus on the idea that when AI 'solves' a problem, as in provides a binary answer, much is lost because it takes away the opportunity for a mathematician to 'battle' with the problem and so gain a deeper understanding.
However, can a mathemarician not simply re-explore a problem, even if a binary solution has already been provided by AI, and uncover all the valuable insights along the way anyhow?
And if that is not possible, would then on the individual level any other mathematician solving a problem mean that everyone else loses the chance to uncover those insights? Yiu mentioned one of the millenium problems was already solved in 2002, nearly 25 years ago. Has that destroyed the opportunity for the current crop of mathematicians to explore that problem?
And finally, does solving one problem not lead to new questions, i.e. new opportunities for insights?
1) problems can be re-explored but there are two problems AFAICT, first that the insights that one acquires second-hand can not be acquired first-hand. Meaning, you can't learn by yourself what other people teach you. Whatever you'd have gained by going through a problem yourself is not with you if someone else teaches you (hence my Newton example). And second, mathematics, even if mathematicians insist that is a sort of platonic realm, is embedded in the real world of deadlines, funding, and opportunity costs. Re-exploring sounds like a good idea until you need to explain to someone who's giving you money but doesn't understand math, why the hell are you saving something that's been already solved.
2) (I think explained in 1.a)
3) AI solving a problem is not the same as one solving a problem because, first, it's not you who did and, second, it's not a human who did. So far AI is much better at solving than at explaining or at weighing correctly what is important and what isn't. So yeah, a proof/solution/theorem is valuable but just not as much as the process. That's Tao's point that going through the problems yourself teaches you about the problem but also about how the landscape evolves. AI solving them erodes everything in its way.
I'm confused. You've said many times that you're personal P(doom) is very low but then you are effectively saying that this maths proof is effectively the first step towards an AI dystopia (Not all doom predictions are for the actual physical death of all humans) and nobody seems to have any suggestion for how it might be stopped. Certainly it seems crazily optimistic to suggest that a few top mathematicians getting upset will even move the needle on the political front. Surely, the only thing that can stop this is a Frank Herbert style "Bultlerian Jihad" and I don't think a few people getting upset about data centre water usage is anywhere close to that.
When I talk about p(doom) I always mean "doom" as everyone understands it (that is, the end of our species). It's pretty self-explanatory. If other people use p(doom) in some other way then they should disambiguate.
"Everyone" doesn't understand it like that and I don't think it's a useful distinction. I'm dissapointed that you see it that way. Is death really the worst thing that you can imagine happening to you personally? If not then maybe you should rethink your definition of "doom". At the very least it should be obvious that the common conception of hell, held by millions of people, is not of ceasing to exist but of eternal torment, and if hell isn't doom then I don't know what is.
Good points.
Mathematics is not like engineering or medicine, where a new discovery can literally save lives. It’s more like philosophy where it moves humanity forward through asking the right questions and learning from the challenges along the way. If AI takes the challenge away, what’s the point?
Another point, OpenAI spent something like $23 million to solve this problem. It’s just a stunt to get to their IPO. The company will get bored of this when it no longer brings them more notoriety. Maybe that would just leave the mathematical community in peace.
Totally agreed on both
thanks, very well put
but i wouldn't pool all the anti “ techno-luddites “ in the same sac, but yes, indeed, some of them really follow this kind of logic : techno-scientific worship + resentment (softened by some virtuous excuses)
may i propose a simple analogy : if a robot climbs the Himalaya in my name, it would manage it (i won't) but what experience would i have ? nothing
merci et belle journée
anne
Excellent essay
I have liked a lot of what you have written over the years, and I was prepared to like this essay until the last two paragraphs, but particularly the last sentence.
I wish you would expand on this, "That would be a hilarious reversal: the pinnacle of reason preserved as an amateur sport for humans whereas millennium problems become the idle pastimes of the machines."
I can read this as referring to what mathematicians regard as millennial problems, but I can also read this as completely turning all really existential problems (nuclear war, climate change, etc.) over to machines to solve. I realize that is the ideal for some of the people developing AI. I am sure there are many meanings and implications in between, but I am uncertain where you are going with this.
Thanks for the question Guy - I hesitated to end on that note. The "math as sport" thing (or everything as a sport, if you will) is just one possibility of what could end up happening. The "hilarious" part is that it's the upside-down world, not that it is "good." I'm not defending that nor condemning it for what is worth. I'm not picking a side, just trying to so a levelheaded analysis!
Thanks Alberto. It sounded a bit like the idea that we will become the pets of some ASI. I didn't think that was what you meant, but wanted to be sure.
1. The perspective I find missing here is “mathematics as art”
2. The great artworks of the masters are not great simply according to external standards; rather they are setting the standards, only by comparing against them we may decide wether another object deserves the title of a piece of art
3. Likewise, there is no predetermined standards for what count as a meaningful mathematical discovery; rather, it is the historical context shaped by the collective efforts of generations of mathematicians gives meaning to a particular mathematical statement
4. That should be enough reason to regard mathematics without human as complete nonsense
The prize has a clock that no result can skip. The Clay Mathematics Institute's rules, adopted by its board on 26 September 2018, set three conditions before CMI will even consider a proposed solution: publication in a Qualifying Outlet, at least two years elapsed since that publication, and general acceptance in the global mathematics community. CMI also states that it does not accept direct submissions. So whatever is announced this month, a Millennium Prize could not follow a 2026 publication before 2028, and only if the field has agreed by then. The result may well stand. The award procedure is slower than the news cycle.
I don't think "maths is done" just yet. AFAIK, the LLM reasoning agents are using various linguistic representations of maths (primarily Lean) to explore maths as well as natural language. All of that is certainly amenable and doable.
Even so, have you noticed anything about the particular maths problems that AI apparently solves?
AFAIK, all that the AI models have found (by construction) were counterexamples showing some conjectured result to be false, therefore disproving the conjecture.
Admittedly, the agents of course do the heavy lifting of constructing the formalisation in Lean, etc., but this basically reuses and mildly extends/elaborates the known maths frameworks to (a) construct each counterexample and (b) formally prove that it is indeed a counterexample to the conjecture.
In effect, the AIs are being effectively used as formalisation power tools (think: chainsaw) to grind out formally checked and verified maths results - which of course is massively beneficial in itself. But one has to wonder where is the deep creativity or insight one typically expects to see?
As far as I understand it, no new positive mathematical result has (YET) been proposed, formulated and proven by swarms of AI agents operating in complete isolation and without human assistance. The recent formalisation of a proof of Fermat's Last Theorem by an AI (a variant of Claude) doesn't count as it essentially followed an existing proof - no new maths was involved.
Whether already established or discovered for the first time, an insight elicits the same convergence event within the mind that is so very satisfying (and rewarding). Rediscovery from first principles is just as thrilling as discovery.
If the equivalent "millennium problems" of all professions become fodder for machines, then perhaps we really will transition to a life of aristocracy. But at least we will be going into it with eyes open, knowing the value and importance of cognition to the human condition. How... freeing... it might be to live in a world where curiosity and understanding can be pursued for their own sakes and not as complementary means to a survival end.
Hi Juliette, I answered what I think discovery and rediscovery are not the same thing in the article (Newton paragraph) and also as an answer to JLO's comment
The first part of your essay seems to focus on the idea that when AI 'solves' a problem, as in provides a binary answer, much is lost because it takes away the opportunity for a mathematician to 'battle' with the problem and so gain a deeper understanding.
However, can a mathemarician not simply re-explore a problem, even if a binary solution has already been provided by AI, and uncover all the valuable insights along the way anyhow?
And if that is not possible, would then on the individual level any other mathematician solving a problem mean that everyone else loses the chance to uncover those insights? Yiu mentioned one of the millenium problems was already solved in 2002, nearly 25 years ago. Has that destroyed the opportunity for the current crop of mathematicians to explore that problem?
And finally, does solving one problem not lead to new questions, i.e. new opportunities for insights?
Good questions.
1) problems can be re-explored but there are two problems AFAICT, first that the insights that one acquires second-hand can not be acquired first-hand. Meaning, you can't learn by yourself what other people teach you. Whatever you'd have gained by going through a problem yourself is not with you if someone else teaches you (hence my Newton example). And second, mathematics, even if mathematicians insist that is a sort of platonic realm, is embedded in the real world of deadlines, funding, and opportunity costs. Re-exploring sounds like a good idea until you need to explain to someone who's giving you money but doesn't understand math, why the hell are you saving something that's been already solved.
2) (I think explained in 1.a)
3) AI solving a problem is not the same as one solving a problem because, first, it's not you who did and, second, it's not a human who did. So far AI is much better at solving than at explaining or at weighing correctly what is important and what isn't. So yeah, a proof/solution/theorem is valuable but just not as much as the process. That's Tao's point that going through the problems yourself teaches you about the problem but also about how the landscape evolves. AI solving them erodes everything in its way.
I'm confused. You've said many times that you're personal P(doom) is very low but then you are effectively saying that this maths proof is effectively the first step towards an AI dystopia (Not all doom predictions are for the actual physical death of all humans) and nobody seems to have any suggestion for how it might be stopped. Certainly it seems crazily optimistic to suggest that a few top mathematicians getting upset will even move the needle on the political front. Surely, the only thing that can stop this is a Frank Herbert style "Bultlerian Jihad" and I don't think a few people getting upset about data centre water usage is anywhere close to that.
When I talk about p(doom) I always mean "doom" as everyone understands it (that is, the end of our species). It's pretty self-explanatory. If other people use p(doom) in some other way then they should disambiguate.
"Everyone" doesn't understand it like that and I don't think it's a useful distinction. I'm dissapointed that you see it that way. Is death really the worst thing that you can imagine happening to you personally? If not then maybe you should rethink your definition of "doom". At the very least it should be obvious that the common conception of hell, held by millions of people, is not of ceasing to exist but of eternal torment, and if hell isn't doom then I don't know what is.