As someone who has a degree in math, I still can't help but think mathematicians are getting a little bit of a comeuppance. In a lot of areas of mathematics there had been little effort to make the work understandable and leaves numerous folks who could benefit from the knowledge on the outside looking in. Now AI comes along and do the same to mathematicians. Makes me chuckle a little bit.
I am not a mathematician, but I can’t see how we are going to address the problem which we already see in coding:
Impossibility to independently validate all AI results
And in math it goes even worse. In coding code reviews are typically still the form of action you do within days. In math, historically, the lifecycle of proof is months if not years. Take as an example Millennium problems. They require at least two years of validity after publishing. Two years! In modern times with amount of output AI can produce, it feels like infinity.
We are inches close if not at the moment already when humans can’t reliable validate proofs and mathematics produced by AI. Then next research will be based on this AI-written-no-human-in-the-loop results. And we will end up in just few years in a world where novel and frontier problems will be articulated by AI and proven by AI based on AI results and humans will be incapable of understating the mere nature of the solution.
I don't think that's actually the real problem. Along with the progress in answering mathematical questions, recent progress on AI-powered autoformalisation has been astonishing. All the recent AI discoveries have been accompanied by Lean proofs.
And, yes: that doesn't absolutely guarantee correctness. The Lean kernel has had soundness bugs, and may have some still. But it's pretty strong evidence of correctness nevertheless.
The concern among mathematicians is not mainly that they doubt the correctness of any of these discoveries, but that human understanding may be devalued.
Excellent optimistic post in a sea of negativity, and with actual suggestions, too.
After reading, my mental image is this: think of Olympiads in Ancient Greece.
* A weightlifter was only awarded a laureate if he were able to lift a heavy stone (have no idea what they were lifting, for illustrative purposes only :-)
* Along comes Archimedes who invents what we would call an exoskeleton. Now any regular guy can lift twice as much as last year’s athlete.
* What to do? You can cancel the Olympiads, but they are actually useful as training, motivation, etc So now you have to give the prize on other factors, eg how well he can lift, has he opened a gym in the city, etc
BTW, physics and bio are not exempt, so those researchers better read and try to stay ahead.
I love how this is (without slighting the problems entailed) coming at it from a perspective of infinitude and abundance (we will always have more problems to solve) - which is the correct framing, particularly when dealing with ideas, or fields in the which ideas are the driver/product/output/material, and ideas themselves, the field itself, are infinite.-
> I propose the following reconceptualization of the goal of a mathematics PhD: to become a world expert on some interesting, deep topic, and to be able to convey that interest and understanding to others. Part of operationalizing this might be a thesis, but the degree would be awarded primarily on the basis of a rigorous defense, in which the student explains the topic to their examiners until they are satisfied.
I think this is a refreshingly forward looking idea and I agree with it 100%, especially the the "rigorous defense" part. That is a good measure of how well the topic has been researched and understood by the researcher. This is where the humans can be "in the loop".
> How different would this look from current PhDs? I think students would still meet with an advisor, who might suggest a topic. That topic could be explored with AI assistance, or not, but the student would be responsible for understanding it; it might be much more open-ended and larger than the typical PhD is currently.
Interesting point about "more open-ended" and "...larger than the typical PhD". I think the author has a point. Earlier, the bottleneck was the candidate's/researcher's understanding and knowledge. Now with AI tools, it is so much easier to zero in to relevant knowledge, get your questions answered quickly which might lead to understanding more quickly.
For e.g., before the advent of public libraries and printing press, the knowledge was inaccessible and guarded. So that was the bottleneck.
Then books became ubiquitous and the bottleneck to knowledge and understanding was people's motivation AND knowledge of WHAT books and topics to research.
Then came the internet and free PDFs of books and research articles. Now, the bottleneck was still people's motivation and a mild version of what books and topics to research. I say "mild" because one can lookup articles and newsletters, and book reviews and come up with a list of reading.
Now comes AI and it looks like the only bottleneck is people's motivation.
I believe there was also a silent, yet potent, bottleneck all along which is also removed by AI: personal tutor/coach/teacher/professor etc. Let's say if I am reading a textbook on manifolds or some research paper and I have a question about a specific theorem or even a mathematical operator being used. Before AI my only way to get my questions answered was to read more books (PDFs or print), or ask on math exchange or math overflow and wait for someone to answer, or to ask a professor. This could take up to a week.
Now all of that has been cut down to 1 hour or less with an interactive chatting session.
!!!!!
So....the only bottleneck is people's motivation! QED
Another bottleneck is money required to pay corporations who own the technology to "do mathematics". In that future, there will never be another Ramajunan.
The idea that most any modern "interesting" aspect of mathematics is going to be understood (or often even explained in enough detail to reveal what is interesting) in an hour is pretty rare. Either the student's aptitude, the tutorial, or the mathematics are unique. There is a reason that these are PhDs and not undergraduate HW sets.
I think we often delude ourselves as to how well we understand problems and their solutions. Some instructors even make you feel that you understand better than you do by pointing to a few approximations or simple solution spaces that obscure the larger complexity. Just looking in wonder at the many categories of three-body solutions (currently on hnews) is enough to remind me of this.
> I think so. This machine might produce answers we value, but it would not, in itself, produce human understanding of those answers.
It's nice that the author is optimistic, but won't the AI be best placed to dumb down its increasingly complex proofs into a language us lowly humans can understand? To keep thinking until it can refactor complex proofs into ones from 'the book'?
As they go on to explain, a human understandable proof is different than a human actually understanding the proof. That actual human understanding (like, in a brain of a human) is one of their stated goals.
Producing human understandable proofs is possibly a job best for humans today, but the author appears to agree with you that this is probably fleeting (and argues that even if you disagree, it should probably be treated as if it is fleeting when planning for the future):
> Right now AI systems arguably underperform us at theory-building, asking questions, exposition, … so we could prioritize and reward those skills. I think this is unwise: compare the speed at which the academy adapts to the speed at which model capabilities improve. We need to consider the endgame. If the models remain incapable in some domain, we can adjust later.
The more I see these posts about mathematics institutions reforms and challenges from AI advancements the more it looks like they may need to go through a death. Or to put it another way they may need to start again from first principles.
If math is truly about spreading intuition and understanding then our institutions have dropped the ball decades ago and have not been able to grab hold of it since (if they ever had it to begin with)
As someone who has a degree in math, I still can't help but think mathematicians are getting a little bit of a comeuppance. In a lot of areas of mathematics there had been little effort to make the work understandable and leaves numerous folks who could benefit from the knowledge on the outside looking in. Now AI comes along and do the same to mathematicians. Makes me chuckle a little bit.
I am not a mathematician, but I can’t see how we are going to address the problem which we already see in coding:
Impossibility to independently validate all AI results
And in math it goes even worse. In coding code reviews are typically still the form of action you do within days. In math, historically, the lifecycle of proof is months if not years. Take as an example Millennium problems. They require at least two years of validity after publishing. Two years! In modern times with amount of output AI can produce, it feels like infinity.
We are inches close if not at the moment already when humans can’t reliable validate proofs and mathematics produced by AI. Then next research will be based on this AI-written-no-human-in-the-loop results. And we will end up in just few years in a world where novel and frontier problems will be articulated by AI and proven by AI based on AI results and humans will be incapable of understating the mere nature of the solution.
I don't think that's actually the real problem. Along with the progress in answering mathematical questions, recent progress on AI-powered autoformalisation has been astonishing. All the recent AI discoveries have been accompanied by Lean proofs.
And, yes: that doesn't absolutely guarantee correctness. The Lean kernel has had soundness bugs, and may have some still. But it's pretty strong evidence of correctness nevertheless.
The concern among mathematicians is not mainly that they doubt the correctness of any of these discoveries, but that human understanding may be devalued.
Excellent optimistic post in a sea of negativity, and with actual suggestions, too. After reading, my mental image is this: think of Olympiads in Ancient Greece.
* A weightlifter was only awarded a laureate if he were able to lift a heavy stone (have no idea what they were lifting, for illustrative purposes only :-)
* Along comes Archimedes who invents what we would call an exoskeleton. Now any regular guy can lift twice as much as last year’s athlete.
* What to do? You can cancel the Olympiads, but they are actually useful as training, motivation, etc So now you have to give the prize on other factors, eg how well he can lift, has he opened a gym in the city, etc
BTW, physics and bio are not exempt, so those researchers better read and try to stay ahead.
I love how this is (without slighting the problems entailed) coming at it from a perspective of infinitude and abundance (we will always have more problems to solve) - which is the correct framing, particularly when dealing with ideas, or fields in the which ideas are the driver/product/output/material, and ideas themselves, the field itself, are infinite.-
PS. The validation problem, being one.-
> I propose the following reconceptualization of the goal of a mathematics PhD: to become a world expert on some interesting, deep topic, and to be able to convey that interest and understanding to others. Part of operationalizing this might be a thesis, but the degree would be awarded primarily on the basis of a rigorous defense, in which the student explains the topic to their examiners until they are satisfied.
I think this is a refreshingly forward looking idea and I agree with it 100%, especially the the "rigorous defense" part. That is a good measure of how well the topic has been researched and understood by the researcher. This is where the humans can be "in the loop".
> How different would this look from current PhDs? I think students would still meet with an advisor, who might suggest a topic. That topic could be explored with AI assistance, or not, but the student would be responsible for understanding it; it might be much more open-ended and larger than the typical PhD is currently.
Interesting point about "more open-ended" and "...larger than the typical PhD". I think the author has a point. Earlier, the bottleneck was the candidate's/researcher's understanding and knowledge. Now with AI tools, it is so much easier to zero in to relevant knowledge, get your questions answered quickly which might lead to understanding more quickly.
For e.g., before the advent of public libraries and printing press, the knowledge was inaccessible and guarded. So that was the bottleneck.
Then books became ubiquitous and the bottleneck to knowledge and understanding was people's motivation AND knowledge of WHAT books and topics to research.
Then came the internet and free PDFs of books and research articles. Now, the bottleneck was still people's motivation and a mild version of what books and topics to research. I say "mild" because one can lookup articles and newsletters, and book reviews and come up with a list of reading.
Now comes AI and it looks like the only bottleneck is people's motivation.
I believe there was also a silent, yet potent, bottleneck all along which is also removed by AI: personal tutor/coach/teacher/professor etc. Let's say if I am reading a textbook on manifolds or some research paper and I have a question about a specific theorem or even a mathematical operator being used. Before AI my only way to get my questions answered was to read more books (PDFs or print), or ask on math exchange or math overflow and wait for someone to answer, or to ask a professor. This could take up to a week.
Now all of that has been cut down to 1 hour or less with an interactive chatting session.
!!!!!
So....the only bottleneck is people's motivation! QED
Exciting time!
> a refreshingly forward looking idea
What?
"a rigorous defense, in which the student explains the topic to their examiners until they are satisfied."
is exactly how PhDs were awarded for hundreds of years.
Even my BSc in Applied Physics (1977) had a viva voce that was a substantial fraction of the final exam.
> the "rigorous defense" part
In my country, that's exactly how it is.
Yes, you need to have a thesis to defend, but ultimately it all comes down to the (oral and live) defense/disputation.
I'd say this is the most optimistic scenario. there are really difficult problems to be solved in terms of access to AI.
Another bottleneck is money required to pay corporations who own the technology to "do mathematics". In that future, there will never be another Ramajunan.
The idea that most any modern "interesting" aspect of mathematics is going to be understood (or often even explained in enough detail to reveal what is interesting) in an hour is pretty rare. Either the student's aptitude, the tutorial, or the mathematics are unique. There is a reason that these are PhDs and not undergraduate HW sets.
I think we often delude ourselves as to how well we understand problems and their solutions. Some instructors even make you feel that you understand better than you do by pointing to a few approximations or simple solution spaces that obscure the larger complexity. Just looking in wonder at the many categories of three-body solutions (currently on hnews) is enough to remind me of this.
Why is "Doctor of Philosophy" the correct certificate of "becoming expert in a topic"?
That's a radical departure from "PhD" being a certificate that someone is qualified to produce new research.
What you describe is more like a Masters Degree.
> I think so. This machine might produce answers we value, but it would not, in itself, produce human understanding of those answers.
It's nice that the author is optimistic, but won't the AI be best placed to dumb down its increasingly complex proofs into a language us lowly humans can understand? To keep thinking until it can refactor complex proofs into ones from 'the book'?
As they go on to explain, a human understandable proof is different than a human actually understanding the proof. That actual human understanding (like, in a brain of a human) is one of their stated goals.
Producing human understandable proofs is possibly a job best for humans today, but the author appears to agree with you that this is probably fleeting (and argues that even if you disagree, it should probably be treated as if it is fleeting when planning for the future):
> Right now AI systems arguably underperform us at theory-building, asking questions, exposition, … so we could prioritize and reward those skills. I think this is unwise: compare the speed at which the academy adapts to the speed at which model capabilities improve. We need to consider the endgame. If the models remain incapable in some domain, we can adjust later.
The more I see these posts about mathematics institutions reforms and challenges from AI advancements the more it looks like they may need to go through a death. Or to put it another way they may need to start again from first principles.
If math is truly about spreading intuition and understanding then our institutions have dropped the ball decades ago and have not been able to grab hold of it since (if they ever had it to begin with)