I'm reading this paragraph with the precedent of AlphaGo in mind, where none of the Go players would have expected that databases of human games would be valuable training material for machines instead of humans:
> I learned mathematics because others gave their time and knowledge freely, and I wanted to do the same for the next generation. The possibility that a student would go on to do something I could not do was part of the point. But I had imagined this as a continuation of a human activity. [...] I could not imagine treating this knowledge as something to be guarded.
But soon after, AlphaZero was trained purposefully without human games in the training data. The "extractive" training was sufficient to show that a certain machine learning model worked, but unnecessary to produce a machine that performs beyond the level of humans.
For coding agents I imagine it is similar: humans introduce so many subtle bugs, and they're only subtle relative to human cognitive capacity. I'm sure subtle mistakes pass peer review in mathematics too, even if it's generally more rigorous.
What I'm getting at is that labs have demonstrated that a certain model architecture is capable of producing mathematics at a (super)human level, and their extractive methodology was sufficient, but possibly not necessary. You can fight in court whether their methodology is against the law, but it would be naive not to look beyond that, which is my main gripe with the AHM statement the author refers to [1]
Fortunately the author of this article has a more nuanced view:
> I can understand the anger, but I am not sure that withdrawing from AI is the answer
AlphaZero isn’t a precedent for mathematics. Go is a game with a predetermined goal and a closed list of rules. Mathematics has neither. You can’t deduce the axioms of geometry from those of set theory because there’s nothing inherently interesting about geometry. Humans are interested in geometry because we’re interested in building things in the real world.
> Humans are interested in geometry because we’re interested in building things in the real world.
To clarify and be fair, the people who actually invented geometry, or at least worked on the majority, were never interested in building things. Or at least that wasn't their primary concern.
It's more accurate to say that mathematicians were just interested in intellectual exploration, everyone else was interested in building things, and there's an uneasy alliance between the two that is about to be shattered.
Geometry wasn’t invented by mathematicians, it was invented by surveyors, architects, and engineers. These folks were developing and using geometry to build things and to parcel the land for agricultural planning for thousands of years before Euclid came along.
If you look at the "rules of geometry" and look for real world applications. Where do people use those methods?
Use a wire to get a straight line? Used in construction
Use a determined length of wire to compare lengths ? Used in construction
Use a determined length of wire to draw circles? Used in construction
Use 2 lengths of wire to construct parallel lines? Used in construction
Use lengths of wire to construct a perfect 90 degree angle? Used in tool-making (ie. to make a jig that you then use to measure)
The list goes on and on and on. If you've never used CAD packages, you should. The amount of geometry that goes into part design in everything from construction to "tools and construction engineering" as it's called in my country is incredible. Incredible as in they reserve 8 months in university to make sure people who already would have had pretty difficult math in high school are up to date on advanced geometry (plus teaching them how to properly extend what they know into 3d)
There's been updates since Greek times, such as using lasers for drawing straight lines instead of wires, but not as many as you'd think.
This is great because as anyone will point out, doing things this way (with correct equipment) results in accuracy across large distances (up to dozens of kilometers) that you just cannot match with analytical geometry + measurements.
Geometry was obviously invented and fine-tuned for thousands of years, for construction. That's what it was used for 4000 years ago ... and that's close to the only reason to still use (non-analytical) geometry today.
One thing I don't yet understand and would appreciate an opinion from a professional mathematician is: are we okay with mathematicians that outsource the discovery of new theorems to AI while providing verification services themselves?
For instance, imagine a mathematical institute that hires reasonably competent mathematicians who are amazing AI prompters but lack mathematical creativity or "genius". All they produce are mathematical papers with clear proofs and explanations. Very human readable stuff. But the catch is that they are not discovering any of it. All they are doing is "translating" full AI discoveries into a format others can understand.
Former pro mathematician here. There's a big divide amongst mathematicians right now. Some like the idea of going full steam ahead with ideas like this, and others are dead-set against it. There's even "The Association for Human Mathematics" [1] which believes in human mathematics and not using AI at all.
The central problem is that the majority of people go into math because they love discovering mathematics and want to participate in the process. That itself is a huge part of the reward. That creates the community. Simply have huge numbers of people outputting math as if it were an industrial factory kills the joy in the process and so in itself destroys the community.
So in short we're seeing a dichotomy between the "problem miners" who are simply hooked on producing results and mining them and "explorers" for lack of a better term.
There's one thing you've got to keep in mind - mathematics, in reality, is mostly useless in the sense that most discoveries these days will always be useless. What is useful generally is teaching, and applying mathematical abstraction to more realistic problems, a skill that you mainly get from dicovering generally useless stuff. (Math for art's sake.) That's how math works, or at least worked before AI. Once you put AI in the mix, it messes up all the incentives....
Personally, I'm 100% anti-AI because I think there's not much point in simply discovering new things if it's not primarily a human journey. It's revolting to have solutions generated by machines like that.
That's insightful, thank you. The follow up, then, is: will most "explorers" survive in a world where "problem miners" exist? I can see that the very best explorer could do that, but if things like tenure and prestige are decided by how many publications you have and which journals accept you, it will be next to impossible to compete against the miners.
To be clear, I empathize with your sentiment and revolt, despite being more of a "builder" myself in the software world. I just can't see how the explorer mindset can survive the onslaught of AI-augmented output.
At one point, the metrics were fairly aligned with the things mathematicians love, but nobody was ever judged by how much they like their work, right? The theory is that the more you like your work, the better your output will be. But motivations are multifaceted and I could see people who value careers over satisfaction winning if AI helps them.
I'm not saying I like this, just that it seems to be the most likely outcome.
> will most "explorers" survive in a world where "problem miners" exist?
I think that's the key question. To be honest - I have no idea. You said it right when you said "But motivations are multifaceted and I could see people who value careers over satisfaction winning if AI helps them."
It's a complicated question. I can only speak for myself in that I would not want to participate in what's happening today. But it's entirely possible that people will adapt also and find a new niche for themselves within mathematics even if they're not the miner type.
>The central problem is that the majority of people go into math because they love discovering mathematics and want to participate in the process. That itself is a huge part of the reward. That creates the community. Simply have huge numbers of people outputting math as if it were an industrial factory kills the joy in the process and so in itself destroys the community.
Seems like math is split in much the same way software is.
I’m a software engineer and not a mathematician, and, yeah I read this article and immediately recognized the same pattern from my field. Even in this comment thread is somebody making the “AI is a tool, get over it,” argument. I feel sadness for the mathematicians because I already know the misery that’s coming for their field.
Software engineering has always felt somewhat broken and dysfunctional to me even 20 years ago when I started. And LLM economics have turned into something downright hostile to its practitioners. I don’t have a similar intuition about the field of mathematics. Possibly they are in a better state to have a healthy disagreement about the use of this technology.
The question for them is similarly existential - who is going to pay for research grants when the machines do all the work? Why pay for a PhD student to work on some research topic if an llm already proved the theorem?
> So in short we're seeing a dichotomy between the "problem miners" who are simply hooked on producing results and mining them and "explorers" for lack of a better term.
Interesting, thanks for the insight. Very similar to the 'builders' vs 'coders' dichotomy that seems to crop up amongst programmers now.
Going forward, the only humans who will know something (in the relevant domains) are the one who like doing it and want to be someone who knows stuff..
I think they will be the torch bearers and envelope pushers, while the latter will just frantically search the new found expanse for useful derivations..
So do you care more about the way someone solves a problem or about result(the fact that it solves it)? For a student I think the answer is obvious because he has to understand the basics but for a researcher we don’t ask bio researcher to do research with pen and paper alone, do we or just using some deep thinking. AI is a tool. Get over it. Math is not an art.
My dictionary says that art is “the expression or application of human creative skill and imagination”. That seems like a definition that applies to math.
Is it simply a bias in the sampling or are the people in math (the discipline) as shaken as the stuff that shows up on HN makes it look like?
FWIW, it reads very similar to the woes of the SWEs previously, in the sense that it kinda appears like people assigned value to the wrong part of the work within the discipline?
The take-away of it being it being imperative to have open weights models accessible to all, instead of cloud-hosted models owned by few is of course true regardless of those specifics, but it gets a bit muddied by that, which is a shame, as that will be weaponized to dismiss it.
> Is it simply a bias in the sampling or are the people in math (the discipline) as shaken as the stuff that shows up on HN makes it look like?
My local community is in a mixture of denial and despair. Especially for the PhD students and postdocs, and faculty in terms of hiring (how do we hire?).
An author who only posted terse notes until 2025 and then started writing long-form essays about AI, illustrated by AI, all in the same mold (e.g., "How Machines Learn to Write", "How AI Generates: From Noise to Form", "What is Intelligence? From Reflection to Suffering"). Numerous LLMism. And a Pangram flag.
If you read any classic mathematics or computer science textbook, you get the sense that the author wants to impart the idea that this subject leads to an interesting and meaningful life. There are many famous quotes about being the architect or designer of an idea rather than just a technician holding a power tool following instructions someone else wrote. Now all of those adages have become effectively meaningless because so many involved in these careers are cool with the industry dying for some vague notion of progress due to a machine god. It’s a pathetic state of affairs that the elders aren’t speaking out more to protect the next generation of human thinking.
I say this as someone who uses LLMs in the physical sciences (one of the only areas where I would say it is ethical to use them).
Any classic book was arguing against religion which dominated the day
There was an intentional "marketing strategy" among scientists and such to convince young men to go into sciences rather than religious or nation state politics or just be bums fighting and drinking back in the day. They lifted heavily from the tone used by church and military recruiter propaganda
Different tone to the prose merely reflects the zeitgeist of the day. Should be seen as a tool like a wrench.
What exactly to protect? I really don’t get what this fuss is all about.
We still need math research just like we still coding or software or cpu research regardless of how many problems the AI solves.
To me it feels like a bio researcher complaining that his field has no future because AI started to cure diseases so now instead to spend 100 years to run experiments someone can do it in a day with AI(obviously it’s not the case yet).
Are they guys upset that people can “run math” without hiring a math guy?
The main problem is that without safeguards it becomes impossible to adjudicate new work in math or CS, meaning grad students who care about originality in work will have little sense of accomplishment if the AI gives them all the answers and will decide to study something else. Not to mention that AI users in math/CS are often ignoring standards in publishing, including issues with plagiarism and scooping of work. These are pretty serious problems that will end up causing a lot of people to move to other subjects.
But that’s not something specific to mathematics, is it? I am bit against “safeguards”. That reads more like censorship to me from some kind of high priests.
Protecting human thinking. Protecting human endeavors from automation. It's like nobody took Frank Herbert's warnings in Dune novels seriously. You give over thinking to machines, you become enslaved by those machines and the elites controlling them.
So what is stopping you from thinking and using the machines as tools? To me it looks like AI is solving the “low level/grunt work” problems and leaves us the important stuff such as steering it, making products using the discoveries, find new problems to solve/discover and so on…
I use coding as an example, I am glad I can ask AI agents to do all the boilerplate coding so that I can focus on building actual products rather than spending a month to get an API just right. It’s not like I am no longer thinking about the code or the end result. I do but only where it actually matters. Just punishing myself to not use AI would be like writing code with pen and paper. Some people enjoy it but that’s just a personal exercise not an industrial tool. I definitely enjoyed coding without IntelliSense and I think it made me a better programmer but definitely it’s not something I would recommend people to do on their day to day job.
The problem is this: “Astra write me a masters thesis applying new techniques programming language theory to Rust” is not using a tool, it’s hiring someone to do the work and thinking for you.
I have difficulty figuring out how to solve this issue.
You could have done that before AI as well, albeit with a little bit more effort. Plagiarism has been quite common.
That being said, if whatever Astra is spitting is really innovative, the prompt author understands it, proof checks it and can support it in front of a live evaluator I think it’s good enough for me. In practice I think that prompt would be just one of many, perhaps also some “old school” evaluation and improvement before the thesis is done. Again, I see the AI as a tool.
You could ask Astra to generate you an e-commerce website with one prompt. Can you show it to someone and claim it as your project? Can you actually use that in production?
I'm reading this paragraph with the precedent of AlphaGo in mind, where none of the Go players would have expected that databases of human games would be valuable training material for machines instead of humans:
> I learned mathematics because others gave their time and knowledge freely, and I wanted to do the same for the next generation. The possibility that a student would go on to do something I could not do was part of the point. But I had imagined this as a continuation of a human activity. [...] I could not imagine treating this knowledge as something to be guarded.
But soon after, AlphaZero was trained purposefully without human games in the training data. The "extractive" training was sufficient to show that a certain machine learning model worked, but unnecessary to produce a machine that performs beyond the level of humans.
For coding agents I imagine it is similar: humans introduce so many subtle bugs, and they're only subtle relative to human cognitive capacity. I'm sure subtle mistakes pass peer review in mathematics too, even if it's generally more rigorous.
What I'm getting at is that labs have demonstrated that a certain model architecture is capable of producing mathematics at a (super)human level, and their extractive methodology was sufficient, but possibly not necessary. You can fight in court whether their methodology is against the law, but it would be naive not to look beyond that, which is my main gripe with the AHM statement the author refers to [1]
Fortunately the author of this article has a more nuanced view:
> I can understand the anger, but I am not sure that withdrawing from AI is the answer
[1] https://www.ahmath.org/statements
AlphaZero isn’t a precedent for mathematics. Go is a game with a predetermined goal and a closed list of rules. Mathematics has neither. You can’t deduce the axioms of geometry from those of set theory because there’s nothing inherently interesting about geometry. Humans are interested in geometry because we’re interested in building things in the real world.
> Humans are interested in geometry because we’re interested in building things in the real world.
To clarify and be fair, the people who actually invented geometry, or at least worked on the majority, were never interested in building things. Or at least that wasn't their primary concern.
It's more accurate to say that mathematicians were just interested in intellectual exploration, everyone else was interested in building things, and there's an uneasy alliance between the two that is about to be shattered.
Geometry wasn’t invented by mathematicians, it was invented by surveyors, architects, and engineers. These folks were developing and using geometry to build things and to parcel the land for agricultural planning for thousands of years before Euclid came along.
Sure, but you can deduce a proof of e.g. Navier Stokes given its statement and some axioms
If you look at the "rules of geometry" and look for real world applications. Where do people use those methods?
Use a wire to get a straight line? Used in construction
Use a determined length of wire to compare lengths ? Used in construction
Use a determined length of wire to draw circles? Used in construction
Use 2 lengths of wire to construct parallel lines? Used in construction
Use lengths of wire to construct a perfect 90 degree angle? Used in tool-making (ie. to make a jig that you then use to measure)
The list goes on and on and on. If you've never used CAD packages, you should. The amount of geometry that goes into part design in everything from construction to "tools and construction engineering" as it's called in my country is incredible. Incredible as in they reserve 8 months in university to make sure people who already would have had pretty difficult math in high school are up to date on advanced geometry (plus teaching them how to properly extend what they know into 3d)
There's been updates since Greek times, such as using lasers for drawing straight lines instead of wires, but not as many as you'd think.
This is great because as anyone will point out, doing things this way (with correct equipment) results in accuracy across large distances (up to dozens of kilometers) that you just cannot match with analytical geometry + measurements.
Geometry was obviously invented and fine-tuned for thousands of years, for construction. That's what it was used for 4000 years ago ... and that's close to the only reason to still use (non-analytical) geometry today.
One thing I don't yet understand and would appreciate an opinion from a professional mathematician is: are we okay with mathematicians that outsource the discovery of new theorems to AI while providing verification services themselves?
For instance, imagine a mathematical institute that hires reasonably competent mathematicians who are amazing AI prompters but lack mathematical creativity or "genius". All they produce are mathematical papers with clear proofs and explanations. Very human readable stuff. But the catch is that they are not discovering any of it. All they are doing is "translating" full AI discoveries into a format others can understand.
Is that acceptable?
Former pro mathematician here. There's a big divide amongst mathematicians right now. Some like the idea of going full steam ahead with ideas like this, and others are dead-set against it. There's even "The Association for Human Mathematics" [1] which believes in human mathematics and not using AI at all.
The central problem is that the majority of people go into math because they love discovering mathematics and want to participate in the process. That itself is a huge part of the reward. That creates the community. Simply have huge numbers of people outputting math as if it were an industrial factory kills the joy in the process and so in itself destroys the community.
So in short we're seeing a dichotomy between the "problem miners" who are simply hooked on producing results and mining them and "explorers" for lack of a better term.
There's one thing you've got to keep in mind - mathematics, in reality, is mostly useless in the sense that most discoveries these days will always be useless. What is useful generally is teaching, and applying mathematical abstraction to more realistic problems, a skill that you mainly get from dicovering generally useless stuff. (Math for art's sake.) That's how math works, or at least worked before AI. Once you put AI in the mix, it messes up all the incentives....
Personally, I'm 100% anti-AI because I think there's not much point in simply discovering new things if it's not primarily a human journey. It's revolting to have solutions generated by machines like that.
1. https://www.ahmath.org/
That's insightful, thank you. The follow up, then, is: will most "explorers" survive in a world where "problem miners" exist? I can see that the very best explorer could do that, but if things like tenure and prestige are decided by how many publications you have and which journals accept you, it will be next to impossible to compete against the miners.
To be clear, I empathize with your sentiment and revolt, despite being more of a "builder" myself in the software world. I just can't see how the explorer mindset can survive the onslaught of AI-augmented output.
At one point, the metrics were fairly aligned with the things mathematicians love, but nobody was ever judged by how much they like their work, right? The theory is that the more you like your work, the better your output will be. But motivations are multifaceted and I could see people who value careers over satisfaction winning if AI helps them.
I'm not saying I like this, just that it seems to be the most likely outcome.
> will most "explorers" survive in a world where "problem miners" exist?
I think that's the key question. To be honest - I have no idea. You said it right when you said "But motivations are multifaceted and I could see people who value careers over satisfaction winning if AI helps them."
It's a complicated question. I can only speak for myself in that I would not want to participate in what's happening today. But it's entirely possible that people will adapt also and find a new niche for themselves within mathematics even if they're not the miner type.
>The central problem is that the majority of people go into math because they love discovering mathematics and want to participate in the process. That itself is a huge part of the reward. That creates the community. Simply have huge numbers of people outputting math as if it were an industrial factory kills the joy in the process and so in itself destroys the community.
Seems like math is split in much the same way software is.
I’m a software engineer and not a mathematician, and, yeah I read this article and immediately recognized the same pattern from my field. Even in this comment thread is somebody making the “AI is a tool, get over it,” argument. I feel sadness for the mathematicians because I already know the misery that’s coming for their field.
Software engineering has always felt somewhat broken and dysfunctional to me even 20 years ago when I started. And LLM economics have turned into something downright hostile to its practitioners. I don’t have a similar intuition about the field of mathematics. Possibly they are in a better state to have a healthy disagreement about the use of this technology.
The question for them is similarly existential - who is going to pay for research grants when the machines do all the work? Why pay for a PhD student to work on some research topic if an llm already proved the theorem?
> So in short we're seeing a dichotomy between the "problem miners" who are simply hooked on producing results and mining them and "explorers" for lack of a better term.
Interesting, thanks for the insight. Very similar to the 'builders' vs 'coders' dichotomy that seems to crop up amongst programmers now.
Going forward, the only humans who will know something (in the relevant domains) are the one who like doing it and want to be someone who knows stuff..
I think they will be the torch bearers and envelope pushers, while the latter will just frantically search the new found expanse for useful derivations..
So do you care more about the way someone solves a problem or about result(the fact that it solves it)? For a student I think the answer is obvious because he has to understand the basics but for a researcher we don’t ask bio researcher to do research with pen and paper alone, do we or just using some deep thinking. AI is a tool. Get over it. Math is not an art.
> Math is not an art
My dictionary says that art is “the expression or application of human creative skill and imagination”. That seems like a definition that applies to math.
I care that humans solve the problems. That math remains a human endeavor. Not getting over it.
So why do you use computers to communicate and solve problems every day?
Is it simply a bias in the sampling or are the people in math (the discipline) as shaken as the stuff that shows up on HN makes it look like?
FWIW, it reads very similar to the woes of the SWEs previously, in the sense that it kinda appears like people assigned value to the wrong part of the work within the discipline?
The take-away of it being it being imperative to have open weights models accessible to all, instead of cloud-hosted models owned by few is of course true regardless of those specifics, but it gets a bit muddied by that, which is a shame, as that will be weaponized to dismiss it.
> Is it simply a bias in the sampling or are the people in math (the discipline) as shaken as the stuff that shows up on HN makes it look like?
My local community is in a mixture of denial and despair. Especially for the PhD students and postdocs, and faculty in terms of hiring (how do we hire?).
> Is it simply a bias in the sampling or are the people in math (the discipline) as shaken as the stuff that shows up on HN makes it look like?
Given that this appears to be a largely AI-written article, I guess it's AI being shaken by AI.
What gave you the impression?
An author who only posted terse notes until 2025 and then started writing long-form essays about AI, illustrated by AI, all in the same mold (e.g., "How Machines Learn to Write", "How AI Generates: From Noise to Form", "What is Intelligence? From Reflection to Suffering"). Numerous LLMism. And a Pangram flag.
If you read any classic mathematics or computer science textbook, you get the sense that the author wants to impart the idea that this subject leads to an interesting and meaningful life. There are many famous quotes about being the architect or designer of an idea rather than just a technician holding a power tool following instructions someone else wrote. Now all of those adages have become effectively meaningless because so many involved in these careers are cool with the industry dying for some vague notion of progress due to a machine god. It’s a pathetic state of affairs that the elders aren’t speaking out more to protect the next generation of human thinking.
I say this as someone who uses LLMs in the physical sciences (one of the only areas where I would say it is ethical to use them).
Any classic book was arguing against religion which dominated the day
There was an intentional "marketing strategy" among scientists and such to convince young men to go into sciences rather than religious or nation state politics or just be bums fighting and drinking back in the day. They lifted heavily from the tone used by church and military recruiter propaganda
Different tone to the prose merely reflects the zeitgeist of the day. Should be seen as a tool like a wrench.
Some of these academics are still alive. Perhaps it was due to propaganda as you were saying, but I really believe that they meant it.
What exactly to protect? I really don’t get what this fuss is all about.
We still need math research just like we still coding or software or cpu research regardless of how many problems the AI solves.
To me it feels like a bio researcher complaining that his field has no future because AI started to cure diseases so now instead to spend 100 years to run experiments someone can do it in a day with AI(obviously it’s not the case yet).
Are they guys upset that people can “run math” without hiring a math guy?
The main problem is that without safeguards it becomes impossible to adjudicate new work in math or CS, meaning grad students who care about originality in work will have little sense of accomplishment if the AI gives them all the answers and will decide to study something else. Not to mention that AI users in math/CS are often ignoring standards in publishing, including issues with plagiarism and scooping of work. These are pretty serious problems that will end up causing a lot of people to move to other subjects.
But that’s not something specific to mathematics, is it? I am bit against “safeguards”. That reads more like censorship to me from some kind of high priests.
You don't hire someone to "run math" in the context of all these discussions, which are about pure mathematics.
Protecting human thinking. Protecting human endeavors from automation. It's like nobody took Frank Herbert's warnings in Dune novels seriously. You give over thinking to machines, you become enslaved by those machines and the elites controlling them.
So what is stopping you from thinking and using the machines as tools? To me it looks like AI is solving the “low level/grunt work” problems and leaves us the important stuff such as steering it, making products using the discoveries, find new problems to solve/discover and so on…
I use coding as an example, I am glad I can ask AI agents to do all the boilerplate coding so that I can focus on building actual products rather than spending a month to get an API just right. It’s not like I am no longer thinking about the code or the end result. I do but only where it actually matters. Just punishing myself to not use AI would be like writing code with pen and paper. Some people enjoy it but that’s just a personal exercise not an industrial tool. I definitely enjoyed coding without IntelliSense and I think it made me a better programmer but definitely it’s not something I would recommend people to do on their day to day job.
The problem is this: “Astra write me a masters thesis applying new techniques programming language theory to Rust” is not using a tool, it’s hiring someone to do the work and thinking for you.
I have difficulty figuring out how to solve this issue.
You could have done that before AI as well, albeit with a little bit more effort. Plagiarism has been quite common.
That being said, if whatever Astra is spitting is really innovative, the prompt author understands it, proof checks it and can support it in front of a live evaluator I think it’s good enough for me. In practice I think that prompt would be just one of many, perhaps also some “old school” evaluation and improvement before the thesis is done. Again, I see the AI as a tool.
You could ask Astra to generate you an e-commerce website with one prompt. Can you show it to someone and claim it as your project? Can you actually use that in production?
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