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)
> 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
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.
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)
> 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!
I'd say this is the most optimistic scenario. there are really difficult problems to be solved in terms of access to AI.
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.
> 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.
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.